TOP QUARK PAIR IN ASSOCIATION WITH AN EXTRA JET:
PHENOMENOLOGICAL ANALYSIS AT THE TEVATRON
By
Moh’d Ahmad Hussein

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics and Astronomy
2011

ABSTRACT
TOP QUARK PAIR IN ASSOCIATION WITH AN EXTRA JET:
PHENOMENOLOGICAL ANALYSIS AT THE TEVATRON
By
Moh’d Ahmad Hussein

The first measurement of the cross section of the top quark pair in association with an
extra hard jet (t¯+jet) has been performed with 4.1 fb−1 of data collected at CDF. The
t
measurement is an important test of perturbative QCD, as NLO effects play an important
role in the calculation of the theoretical cross section. In addition, it is also important
for future measurements at the LHC, for which almost half of the top quark events will
be produced with extra jets. Therefore, this process will be a substantial background for
many new physics signals. The measurement is performed using SecVtx tagged events in
the lepton plus jet channel. A data-driven approach is used to predict the background
content, and a 2D likelihood is formed to simultaneously measure the t¯+jet and t¯ without
t
t
extra jet cross sections. The measured result is σt¯
t+jet = 1.6±0.2stat ±0.5syst pb which is
+0.16
in agreement with the recent NLO SM prediction σt¯
t+jet = 1.791(1)−0.31pb .

In order to elucidate the kinematic profile of the extra jet, an isolation algorithm has been
developed. The algorithm has extracted correctly the extra jet out from the final state jets
∼ 60% of the time. This allowed for correcting the measured distributions of the extra jet
for purity/efficiency in order to compare them with the MC distributions. The differences in
the kinematics of the extra jet using different SecVtx requirements and different MC models
(PYTHIA & MCFM) have been studied, and found to be in agreement with the SM prediction.

c Copyright by
MOH’D AHMAD HUSSEIN
2011

DEDICATION
To Ahmad Abdullah Hussein; first as a father, then as a mentor, and now as a
pillar of support . . .
It is he; the first, who recited for me Luqmˆn’s advice to his son as it was revealed:
a
O my dear son! Almighty Allah will bring all things to light, be they as small as
a grain of a mustard seed, be they hidden inside a rock, or high above in the
heavens, or deep down in the earth. Almighty Allah knows the finest mysteries
and is well aware of all things [verse 16].
O my dear son! Establish prayers, enjoin good and forbid evil. Endure with patience
whatever befalls you, surely, all that is of the matters requiring determination
[verse 17].
Do not speak to the people with your face turned away with pride, nor walk proudly
on earth; for Almighty Allah does not love any self-conceited boaster [verse 18].
Be moderate in your pace and lower your voice; for the harshest of voices is the
braying of a donkey [verse 19].
The Holy Qura’n: Chapter 31
I know that you weren’t able to pursue your own PhD for a very noble cause.
I hope that you consider this dissertation yours as well.

iv

ACKNOWLEDGMENTS
This acknowledgment is intended for those who have had a direct involvement with my
graduate career.
Aous Abdo: I have to say right off that I can’t pay back your unquestionable help. Awsy
(as his Mom calls him) the then-research associate was not only among the few nearest and
dearest to me but was also my best grad fellow. He was there on August 22, 2004 to pick me
A
up from Metro airport and he was there last Fall to give me his dissertation L TEX template.

In between, Awsy was always there.
Joey Huston: I’ll be unfair if I offer my praise to Joey for being a mere adviser or for only
admitting me into MSU. With his forbearance, Joey went above and beyond the duties of
an adviser. I have learned from him how to multitask and prioritize. I will never forget his
peculiar way of examining my histograms (out of his glasses frame) and then pointing to my
mistakes! I have benefited from his excellent communication skills with a large collaboration
like the one CDF is. The only two things I may have failed to mimic were his astonishing
ability of throwing jokes while we were in formal meetings that discuss a very complicated
QCD issues, and walking while barefoot in the hallway!
Wayne Repko: Over the past six years, Repko was the ray of advice I always followed.
This goes side by side with the conclusive fact that he is among the very few great teachers
to learn from and to emulate. He inspired me so much when I took his rigorous QED class;
when he rarely looked up at his notes while he was still recovering from a wicked stroke that
hit him. The Sunday afternoon coffees I used to share with him once a while are among the
delightful memories I will bear forever.
Thomas Rockwell: I came in to HEP having no idea how to copy files in a Unix directory!
I looked around in the grad office and was told: It is Tom . . . and indeed it was him! When
v

C++ was keeping me down it was he who I would rush into his office. When my makefiles
were (making) me crazy it was Tom who (made) them a real makefiles. I am indebted to
Tom not only for his fruitful help with the bash scripts, ROOT, debugging etc, but for the
neat smile he used always to welcome me with.
Gene Flanagan: Gene was my second adviser who gave me my first piece of code to play
with. Since the first moment I arrived at Fermilab in May 2006 where he was waiting for
me near the Highrise, he since then has worked hard trying to transfer his expertise to me.
I’d bet that he answered more than a thousand emails from a novice new grad student in
the first two years!
Thomas Schwarz: After all, it was TomTrees that I inherited and ran my code over!
Tom is the one who did the majority of the work of the background estimation, which
many people, including me, have used. I’ve benefited from his crystal clear reasoning and
pinpointed critiques. An Arabic proverb says (more or less) that a pot needs three stones
to be stable; Tom was the third of the three advisers that I have; each was an expert in his
field.
Brian Martin: Brian is an irreplaceable colleague and friend. Thank you for your help in
SpartyJet, for the good coffee you used to prepare, for the joy and the humor you were able
to create, and for keep greeting me with my traditional greeting; Salam! I am pretty sure
you will be a pioneer scholar and a great father as well (It’s soon I guess!). I wish you and
Kate the best of luck wherever you end up after Argonne lab.
Brenda Wenzlick: A big thumbs-up to our HEP secretary, for her ability to make things
possible. Thank you Brenda for preparing my trips to Fermilab even though I used to
give you short notice. I still remember when you came down all of the stairs to charge
me up with a fresh cup of coffee while I was setting in the early morning for the qualifier;
and how the faculty-member proctor got jealous and insisted on knowing who you were! A
special acknowledgment goes also to Debbie Barratt, our very professional graduate program
secretary.
vi

Dean Halderson: A truly special praise is owed to Dean, from WMICH, for believing in
me and supporting me throughout my first year in the US. Your kindness kept me rational.
I want to thank JJ Schmidt, Stephen Hahn and William Badgett for their gentleness and
true help while I was on shifts at CDF.
Lastly, but certainly not least, in physics we study a lot of paradoxes but I believe that
the Joey-Repko-Rockwell-Dean paradox is not addressed in any literature yet! The huge contradiction between the quality of these people, their tranquility, their dignity, compared with
the values and creeds of those who are located at 1600 Pennsylvania Avenue is bewildering...

vii

TABLE OF CONTENTS
LIST OF TABLES

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x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Relevant Theory . . . . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Standard Model of Particle Physics . . . . . . . .
1.3 Quantum Gauge Field Theories . . . . . . . . . . . . .
1.3.1 Quantum Electrodynamics (QED) . . . . . . . .

1.4

1.5

1.6

1.3.2 Quantum Chromodynamics (QCD)
1.3.3 The Weak Interaction . . . . . . .
1.3.4 The Electroweak Theory . . . . . .
Top Quark Physics Review . . . . . . . . .
1.4.1 Top Production Mechanism . . . .
1.4.2 Top Decay Mechanism . . . . . . .
1.4.3 Intrinsic Properties . . . . . . . . .
Top quark Cross Section . . . . . . . . . .
1.5.1 Perturbative QCD (pQCD) . . . .

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1.5.2 The Predicted Inclusive Cross Section of t¯ . . .
t
¯. . .
1.5.3 The Measured Inclusive Cross Section of tt
1.5.4 The Predicted Exclusive Cross Section of t¯+jet
t
Measurement of σ(p¯ → t¯+jet) . . . . . . . . . . . . .
p
t

2 Experimental Apparatus . . . .
2.1 Introduction . . . . . . . . . . .
2.2 The Accelerator Complex . . .
2.2.1 Proton Production . . .
2.2.2 Anti-Proton Production
2.2.3 Tevatron . . . . . . . . .

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2.3
2.4

2.5

Coordinates, Conventions & Kinematics
The CDF Detector . . . . . . . . . . . .
2.4.1 Tracking System . . . . . . . . .
2.4.2 Calorimetry . . . . . . . . . . . .
2.4.3 Muon System . . . . . . . . . . .
ˇ
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Data Acqusition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79
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3 Event Selection & Background Modeling
3.1 Introduction . . . . . . . . . . . . . . . . .
3.2 Lepton Identification . . . . . . . . . . . .
3.2.1 CEM, CMUP & CMX Triggers . .
3.2.2 Offline Selection . . . . . . . . . . .
3.3 Jet Reconstruction & Corrections . . . . .
3.4 b-jet Tagging . . . . . . . . . . . . . . . .
3.5 Missing ET & HT Reconstructions . . . .
3.6
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4 Results . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Counting Analysis . . . . . . . . . . . . . . . . .
4.2.1 Calculating The Cross Section . . . . . . . . .
4.2.2 Systematics . . . . . . . . . . . . . . . . . . .
4.2.3 The Measured σt¯
t+jet & σt¯
t+0jet . . . . . .

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4.3

4.4

Kinematic Analysis . . . . . . .
4.3.1 The Truth Extra Jet . .
4.3.2 The Isolation Algorithm
4.3.3 The Corrected Measured
4.3.4 The Extra Jet Profile . .
4.3.5 The MCFM Extra Jet . .
Conclusions . . . . . . . . . . .

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3.8

Selection Cuts & Data Sets . . . . .
MC Models . . . . . . . . . . . . . .
3.7.1 Event Generator . . . . . . .
3.7.2 The Extra Hard Jet . . . . . .
Backgrounds . . . . . . . . . . . . . .
3.8.1 The Potential Backgrounds .
3.8.2 Background Estimate (Method
3.8.3 Full Background Prediction .

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Extra Jet
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121

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
ix

LIST OF TABLES
1.1

Quantum numbers in the electroweak theory. . . . . . . . . . . . . . . . . . .

15

1.2

The LO, NLO and approximate NNLO prediction for the total cross section
√
of t¯ events at LHC ( s = 14 TeV) and Tevatron, from [64]. . . . . . . . . .
t

50

1.3

The best predictions for t¯ [pb] production at the Tevatron for a pole mass of
t
mt = 173 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

1.4

Cross section for t¯+jet [pb] for different values of PT jet cut for µ= mt . As
t
a numerical value, mt =174 GeV. The upper and lower indices are the shifts
towards µ= mt /2 and µ= 2mt , from [79]. . . . . . . . . . . . . . . . . . . . .

60

3.1

ELECTRON-CENTRAL-18 CEM Trigger. . . . . . . . . . . . . . . . . . . . . . .

87

3.2

MUON-CMUP18 and MUON-CMX18 Triggers. . . . . . . . . . . . . . . . . . . . . .

87

3.3

Offline selections for tight electron. . . . . . . . . . . . . . . . . . . . . . . .

89

3.4

Offline selections for tight muon. . . . . . . . . . . . . . . . . . . . . . . . . .

91

3.5

Summary of the data sets used in our analysis. . . . . . . . . . . . . . . . . . 103

3.6

The observed t¯ event counts as a function of jet multiplicity. . . . . . . . . . 104
t

3.7

Summary of the background and signal models used in our analysis. . . . . . 113
x

3.8

Theoretical cross sections for MC based backgrounds. . . . . . . . . . . . . . 116

3.9

The predicted pre-tagged acceptance (A) for t¯+0jet. . . . . . . . . . . . . . 117
t

3.10 The predicted pre-tagged acceptance (A) for t¯+jet. . . . . . . . . . . . . . . 117
t
3.11 The predicted tagging efficiency (ǫ) for t¯+0jet. . . . . . . . . . . . . . . . . 117
t
3.12 The predicted tagging efficiency (ǫ) for t¯+jet. . . . . . . . . . . . . . . . . . 118
t
3.13 Fitted pre-tagged QCD fraction, FQCD . . . . . . . . . . . . . . . . . . . . . 119
3.14 Fitted tagged QCD fraction, FQCD . . . . . . . . . . . . . . . . . . . . . . . 119
3.15 The heavy flavor fraction (fHF ) for W +jet sample. . . . . . . . . . . . . . . 121
3.16 The tagging efficiency (ǫ) for W +jet sample. . . . . . . . . . . . . . . . . . . 122
3.17 Background normalizations for ≥ 1 SecVtx jets using Method II. . . . . . . 123

xi

LIST OF FIGURES
1.1

Artist’s depiction of a proton (uud ) fluctuating into a Λ(uds) baryon and a
K + (¯u) meson, in the background of vacuum fluctuations of gluons. The
s
false color scale denotes the intensity of the gluon field at a given moment,
proceeding through the rainbow from blue to red (most intense), from [1]. For
interpretation of the references to color in this and all other figures, the reader
is referred to the electric version of this dissertation. . . . . . . . . . . . . . .

2

The known fermions and their properties in the SM, from [3]. For more precise
values, see Reference [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Measurement of αs as a function of energy, from [2]. . . . . . . . . . . . . . .

9

1.4

Overview of the CTEQ6M PDFs, from [7]. . . . . . . . . . . . . . . . . . . .

12

1.5

LO Feynman diagrams for top quark pair production: (a) quark–anti-quark
annihilation and (b) gluon-gluon fusion, from [22]. . . . . . . . . . . . . . . .

21

Representative Feynman diagrams for electroweak single top quark pair production: (a) s-channel, (b,c) t-channel and (d,e) associated production. The
diagrams for single anti-top quark can be obtained by interchanging quarks
with anti-quarks, from [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

A tree level Feynman diagram of t¯ production and subsequent lepton + jets
t
decay, from [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

(a): Virtual top quark loops contributing to the W and Z boson masses. (b):
Virtual Higgs boson loops contributing to the W and Z boson masses, from
[22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Summary of the input measurements and resulting Tevatron average mass of
the top quark, from [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

1.2

1.6

1.7
1.8

1.9

xii

1.10 The confidence band in Γfit for a 95% C.L. The fitted value from the data is
t
indicated by an arrow which corresponds to a limit of Γt < 13.1 GeV, from [37]. 34
1.11 The expected and observed Bayesian probability density for the Γt . The most
probable value for Γt is defined by the peak of the probability density function.
The hatched area represent one standard deviation around the most probable
value, from[38]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

1.12 Angular momentum conservation in top quark decay does not allow righthanded W bosons when b quarks are assumed massless, from[22]. . . . . . .

37

1.13 The distribution of the helicity basis angles for the decay product of the t¯,
t
from [45]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

1.14 Dependence of the total cross section on the top quark mass defined in the
pole scheme. The NLL and NLO+NNLL bands indicate the scale uncertainties and PDF uncertainties combined linearly, while the blue band represents
the dependence of the D∅ measurement of the cross section on mt . For comparison, a CDF combined measurement is also shown, from [62]. . . . . . . .

52

1.15 Combination of CDF measurements of the t¯ production versus the theoretical
t
predictions as a function of the center-of-mass energy. For RunII, the central
values of the predictions from [64] are shown at top quark mass of 170, 172.5
and 175 GeV for CTEQ6.6. The uncertainty of the 172.5 GeV prediction is
shown by the filled blue region. For RunI, the predictions are shown at a top
quark mass of 170 and 175 GeV fro CTEQ6M, from [66]. . . . . . . . . . . .

53

1.16 The measured differential cross section for the t¯ invariant mass with 2.7 fb−1
t
of data at CDF, from [72]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

1.17 The measured differential cross section for the t¯ transverse momentum with
t
1 fb−1 of data at D∅, from [74]. . . . . . . . . . . . . . . . . . . . . . . . . .

56

1.18 Representative sets of LO diagrams for gg fusion and q¯ annihilation in t¯+jet
q
t
production, from [79]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

1.19 Representative sets of pentagon diagrams for gg fusion and q¯ annihilation in
q
¯+jet production at NLO QCD, from [79]. . . . . . . . . . . . . . . . . . . .
tt

59

1.20 Scale dependence for the cross section for t¯+jet production, where µR and µF
t
scales are set equal to µ. The individual contributions of the various partonic
channels are shown also separately, from [80]. . . . . . . . . . . . . . . . . .

63

xiii

2.1

An aerial view of Fermilab. The top ring in the photograph is the Tevatron.
Protons circle Tevatron clockwise as anti-protons circle counter-clockwise.
The beams are steered to collide at the top-left of the ring (CDF) and at
the top-right (D∅). The small ring at the bottom is the main injector, from
[82] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

2.2

A cross-sectional view of silicon detectors in the r-φ plane, from [82]. . . . .

73

2.3

Installing the COT, from [82]. . . . . . . . . . . . . . . . . . . . . . . . . . .

75

2.4

Muon coverage in CDF, from [84]. . . . . . . . . . . . . . . . . . . . . . . . .

79

2.5

Data flow through CDF trigger system, from [84]. The diagram depicts 132 ns
bunch crossing times (L1 42 clock cycles deep). However, the current crossing
time is 396 ns (corresponds to L1 14 clock cycles deep). . . . . . . . . . . . .

82

A cartoon depicts the topology of t¯+jets event where extra hard jets are
t
produced by the ISR, FSR and radiation off from the decay products of the
top quark and the hadronic W boson. . . . . . . . . . . . . . . . . . . . . .

85

The three levels of a jet at CDF event. The time flow direction in the cartoon
is upward, from [100]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

3.1

3.2
3.3

Nvtx verses the instantaneous luminosity L . . . . . . . . . . . . . . . . . .

95

3.4

CDF SecVtx algorithm b-tagging efficiency, from [107]. . . . . . . . . . . . .

98

3.5

CDF SecVtx algorithm mistag efficiency, from [107]. . . . . . . . . . . . . .

99

3.6

t¯+jet event’s topology where the extra hard jet is produced by the ISR. . . 102
t

3.7

t¯+jet event’s topology where the extra hard jet is produced by the radiation
t
off of the decay prodcuts of the top quark. . . . . . . . . . . . . . . . . . . . 107

3.8

t¯+jet event’s topology where the extra hard jet is produced by the radiation
t
off of the decay prodcuts of the hadronic W boson. . . . . . . . . . . . . . . 108

3.9

t¯+jet event’s topology where the extra hard jet is produced by the FSR. . . 109
t

¯
3.10 W+bb event’s topology, from [28]. . . . . . . . . . . . . . . . . . . . . . . . . 111
3.11 W+c¯ event’s topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
c

xiv

4.1

Systematic uncertainties for the measured t¯+0jet cross section. The numbers
t
in the (∆ σ pb) column are rounded but the percentile column represents the
exact values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.2

Systematic uncertainties for the measured t¯+jet cross section. The numbers
t
in the (∆ σ pb) column are rounded but the percentile column represents the
exact values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.3

The truth plus jet multiplicity. . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.4

The number of truth plus jets in 4 reconstructed jet events. . . . . . . . . . . 134

4.5

The number of truth plus jets in 5 reconstructed jet events. . . . . . . . . . . 135

4.6

All the possible combinatoric W boson and top quark mass distributions using
the ttop25 MC sample with ≥ 2 SecVtx tagged jets. . . . . . . . . . . . . . 138

4.7

The best reconstructed W boson and top quark mass distributions using the
ttop25 MC sample with ≥ 2 SecVtx tagged jets. . . . . . . . . . . . . . . . . 139

4.8

The best reconstructed W boson mass distribution using the ttop25 MC sample with ≥ 2 SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . . . . . 140

4.9

The best reconstructed top quark mass distribution using the ttop25 MC
sample with ≥ 2 SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . . . 141

4.10 ∆R value between the dijet combination that best reconstructs the W boson
and the truth W boson mass distributions using the ttop25 MC sample with
≥ 2 SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.11 ∆R value between the trijet combination that best reconstructs the top quark
and the truth top quark mass distributions using the ttop25 MC sample with
≥ 2 SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.12 ∆R value between the extra ttop25 jet and the truth plus jet using the ttop25
MC sample with ≥ 2 SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . 144
4.13 ∆R value between the extra ttop25 jet and the truth plus jet within a ± 10
GeV window of the truth W boson mass distribution using the ttop25 MC
sample with ≥ 2 SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . . . 145
4.14 ∆R value between the extra ttop25 jet and the truth plus jet within a box of
± 10 and ± 20 GeV of the truth W boson and the truth top quark masses
respectively, using the ttop25 MC sample with ≥ 2 SecVtx tagged jets. . . . 146
4.15 The best measured W boson and top quark mass distributions with ≥ 2
SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

xv

4.16 The best hadronic W boson and top quark mass distributions using the ttop25
MC sample with ≥ 1 SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . 150
4.17 ∆R value between the extra ttop25 jet and the truth plus jet using the ttop25
MC sample with ≥ 1 SecVtx tagged jets. . . . . . . . . . . . . . . . . . . . . 151
4.18 ∆R value between the extra ttop25 jet and the truth plus jet within a box
of ± 10 and ± 20 GeV of the truth W boson and the truth Top quark
mass distributions respectively, using the ttop25 MC sample with ≥ 1 SecVtx
tagged jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.19 The ET distribution of the extra jet as found in data, ttop25 and the truth
samples for the double tag event samples. . . . . . . . . . . . . . . . . . . . . 154
4.20 The corrected observed ET distribution of the extra jet compared with the
truth distribution for the double tag event samples. . . . . . . . . . . . . . . 155
4.21 The corrected observed ET distribution of the extra jet compared with the
truth distribution for the single tag event samples. . . . . . . . . . . . . . . . 156
4.22 The ET distribution of the extra jet as found in data, ttop25, truth and
Method II background samples for the single tag event samples. . . . . . . . 157
4.23 ∆R value between the ttop25 jets and the truth plus jet for the double tag
event samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.24 ∆R value between the ttop25 jets and the truth plus jet for the single tag
event samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.25 The best W boson mass distribution for LO t¯+jet MCFM sample. . . . . . . . 163
t
4.26 A combinatoric W boson mass distribution for LO t¯+jet MCFM sample. . . . 164
t
4.27 The likelihood for the extra ISR gluon to be the lowest PT jet for LO t¯+jet
t
MCFM sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

xvi

Chapter 1
Relevant Theory
1.1

Introduction

High Energy Physics (HEP) endeavors to enumerate the elementary particles which constitute known matter and to characterize the interactions which govern their behavior. The
best theory to date is what has come to be called the Standard Model (SM) of particle
physics, which has proved to be a very successful theory at the energy scales attainable by
accelerators all the way up to the present-day Tevatron at the Fermi National Accelerator
Laboratory (Fermilab) and the Large Hadron Collider (LHC) at the European Organization
for Nuclear Research (CERN). Hundreds of measurements varying in analysis technique and
detection apparatus have all confirmed the predictions of the SM at a remarkable level of
precision.
With the observation of the top quark in 1995, nearly the full catalog of SM particles has
been directly observed, with only the Higgs boson evading detection. With the approximate
mass of a gold nucleus, the top quark is the heaviest known elementary particle. Due to
this feature and others, there is a very active program at the Tevatron to determine the
properties of the top quark.
This thesis concerns a first measurement of the top quark pair production in association
1

Figure 1.1: Artist’s depiction of a proton (uud ) fluctuating into a Λ(uds) baryon and a
K + (¯u) meson, in the background of vacuum fluctuations of gluons. The false color scale
s
denotes the intensity of the gluon field at a given moment, proceeding through the rainbow
from blue to red (most intense), from [1]. For interpretation of the references to color in this
and all other figures, the reader is referred to the electric version of this dissertation.

2

with an extra hard jet and the kinematics of the extra jet, utilizing 4.1 fb−1 of data collected
at Fermilab.
I’ll begin the thesis by covering the relevant physics of interest underlying the top quark
within the SM framework. I’ll proceed with a description of the experimental apparatus,
both the Collider Detector at Fermilab (CDF) and the series of Fermilab accelerators which
culminate in the Tevatron to accelerate protons and anti-protons to a center-of-mass energy
of 1.96 TeV. A discussion of how the various decay objects from the top are reconstructed
in the detector will follow, along with a discussion of the selection cuts used to obtain the
data sample for the measurement, and the Monte Carlo events used in the construction and
evaluation of the analysis. Chapter 4 describes both the counting analysis and the kinematic
analysis at length, followed by the results and a brief conclusion.

1.2

The Standard Model of Particle Physics

The SM of particle physics is a quantum field gauge theory describing the dynamics of
particles that interact via the strong, electromagnetic and weak interactions based on the
SU(3) gauge symmetry of Quantum Chromodynamics (QCD) and the SU(2)×U(1) gauge
symmetry of Quantum Electrodynamics (QED). Gravity is not included in the SM, but it’s
extremely weak compared to the other forces, and not important at the relevant energy scale
probed by particle physics.
These SM particles are half-integer spin fermions, which interact via the three forces
mediated by integer spin bosons. Each force has at least one force carrier boson. The
massless, neutral photon (γ) mediates the electromagnetic force, which is a long-range force
and the one most familiar from every day experience. The strong force is mediated by a set of
eight bosons collectively referred to as gluons. The gluon (g) is also massless and neutral but
itself carries the corresponding charge (color). Strong interactions bind quarks together into
nucleons, and the nucleons into atomic nuclei. The weak interaction is mediated by three
3

massive bosons, the W + ,W − (MW = 80.4 GeV/c2 ) and neutral Z ◦ (MZ = 91.19 GeV/c2 )
[2]. Outside of HEP, the weak force shows up primarily through the decay of radioactive
nuclei. However, the electromagnetic and weak interactions are unified at high energies (see
section 1.3.4).
The fundamental fermions are further categorized into quarks and leptons. Quarks carry
color as well as fractional electric charge and participate in all interactions. All of the
fermions participate in the weak interactions, and all of the charged fermions interact in the
electromagnetic interaction. It is only the weak interactions that can change the “flavor”
or type of the quarks, but other interactions leave the flavor intact. Quarks of electric
1
charge + 2 are grouped with quarks of electric charge − 3 into SU(2) doublets. Leptons are
3

colorless, and charged leptons are paired with neutral leptons, called neutrinos, into SU(2)
doublets. Each pair is called a “generation” or a “family” of particles. The SM has three
of them. The first generation is composed of the up and down quarks, and the electron
and electron-type neutrino. These four particles are the constituents of ordinary matter.
Their observed stability is a result of their being the lightest particles of their kind and the
apparent conservation of lepton and baryon number.
The other two generations are successively more massive and are unstable; they exist only
as the product of collisions by cosmic rays and at man-made accelerators. There is a very
vast discrepancy between the different masses of the fermions. The hierarchy of the three
generations and their basic properties are shown in Figure 1.2. Each of these particles has
an antiparticle, identical to the original (mass and spin) but with all the opposite quantum
numbers.
It’s still an open question why the top quark is so heavy compared to the rest of the particles. Therefore, understanding the properties of the top quark is essential to fundamentally
understand the mechanism that gives rise to mass.

4

Figure 1.2: The known fermions and their properties in the SM, from [3]. For more precise
values, see Reference [2].

5

1.3

Quantum Gauge Field Theories

The three forces of the SM are described by so called quantum gauge field theories which
obey local symmetries and are mediated by field quanta. The quanta of these fields carry
spin 1 and are therefore called gauge bosons. I will describe these theories in some detail
in the next few pages. There are several texts thoroughly describing the SM. I have relied
heavily on [4, 5].

1.3.1

Quantum Electrodynamics (QED)

QED is the gauge theory that describes the electromagnetic force. It’s the simplest and the
most accurate theory in the SM. The Lagrangian L for the massless electromagnetic field
Aµ interacting with a spin- 1 field ψ of bare mass m is
2

1
¯
L = − Fµν F µν + ψ(iγ µDµ − m)ψ
4

(1.1)

Here, Fµν is the electromagnetic field tensor

Fµν = ∂µ Aν − ∂ν Aµ

(1.2)

and Dµ is the covariant derivative

Dµ = ∂µ + ieAµ Q

(1.3)

where e is the unit of electric charge and Q is the charge operator (i.e. Qψ = eψ) and A
is the gauge field, the quantum of which is the photon. This Lagrangian is invariant under

6

local U(1) gauge transformations:

ψ −→ eiα(x) ψ

(1.4)

and

Aµ −→ Aµ −

1
∂µ α(x)
ge

The coupling constant ge is the elementary unit of the charge e =

(1.5)
√

4πα, where the current

1
measured vlaue for α is 137.035999679(94) [2]. Since it is small, we compute all matrix elements

as a perturbation series, using α as the expansion parameter (see section 1.5). The U(1)
symmetry of QED leads to conservation of electric charge.

1.3.2

Quantum Chromodynamics (QCD)

QCD is the gauge theory that describes the strong force. Unlike QED, the phenomenology
of this theory is considerably richer and more complicated; it is different in many respects.
The Lagrangian L for three colored quarks of each quark flavor and eight gluons is

1 µν
µ
¯
L = − Fa Faµν + ψj (iγµDjk − Mj δjk )ψk
4

(1.6)

where the indicies a, j and k refer to color and assume the value a = 1, . . . , 8 and j,k = 1,2,3.
The covariant derivative D acting on a quark field is

µ
Djk = δ jk ∂ µ + igs (Ta)jk Gµ
a

7

(1.7)

µ

where Ga are the gluon fields, Ta are the SU(3) generators and gs is the strong coupling
√
constant 4παs analogous to that of QED;

Ta =

1
λa
2

(1.8)

The λ matrices are Gell-Mann matrices, which are to SU(3) what the Pauli spin matrices
are to SU(2). The field tensor is more complicated than the corresponding QED case:

µν
Fa = ∂ µ Gν − ∂ ν Gµ − gs fabc Gµ Gν
c
a
a
b

(1.9)

The commutators of the λ matrices define the structure constants fabc :

[λa, λb] = 2ifabc λc

(1.10)

As a result, QCD predicts that gluons can interact with one another and hence themselves
carry color charge. The consequence of this is that the interaction gets stronger at long
distances, and weaker at short distances, a trend opposite to QED. This is referred to as
asymptotic freedom. Therefore, whereas the QED coupling variation is extremely small over
the accessible energy range, the variation in the QCD coupling is substantial and runs as :

12π

αs (q 2) =

(11n − 2f )ln

q2
Λ2
QCD

(1.11)

where ΛQCD is the QCD energy scale, n and f are the number of the colors and the flavors
(3 and 6 respectively). Here it can be seen that for high energy scales (q 2 ≫ Λ2
QCD ),
αs is small enough to allow perturbative calculations that are similar to QED , but as the
8

0.5

July 2009

αs(Q)

Deep Inelastic Scattering
e+e– Annihilation
Heavy Quarkonia

0.4

0.3

0.2

0.1
QCD
1

α s (Μ Z) = 0.1184 ± 0.0007
10

Q [GeV]

100

Figure 1.3: Measurement of αs as a function of energy, from [2].

interaction energy decreases, the coupling becomes strong and diverges at ΛQCD ≈ 0.2 GeV
making perturbative calculations impossible. The variation of αs with energy is shown in
Figure 1.3 .
The strong coupling at low energy is responsible for the phenomenon of quark confinement; in nature, quarks are not observed in an isolated state. Instead, they are confined in
colorless bound states of combinations of two or three quarks and anti-quarks, the hadrons.
Hadrons come in one of two types: baryons (which are fermionic bound state of three quarks)
and mesons (which are bosonic bound states of a quark and an anti-quark). For example,
9

the negative charged pion (π − ) is composed of d and u quarks. On the other hand, the
¯
proton is comprised of two u and one d quarks as depicted in Figure 1.1.
Hadronization
These (uud ) constituent quarks of the proton are the “valence” quarks. Other virtual quark–
anti-quark pairs appear inside the proton due to gluon interactions (they are created and
annihilated in the vacuum). These quarks are called the “sea” quarks. Thus, when a parton
receives a large amount of energy in a collision, as it moves away from other particles,
the binding energy between them increases. When the energy is sufficiently large, a new
quark–anti-quark pair is created from vacuum. Experimentally, this process repeats until
the parent parton’s energy is expended, resulting in not a single parton but collimated sprays
of partons moving approximately in the same direction. This process is called parton shower.
The interaction of these showers with the detector result in the measured jets. The jets play
a very key role in HEP; by examining their signatures in the detector, we can reconstruct the
event. Hence, jet reconstruction is a source of systematic uncertainty for our measurement
(see section 3.3). An exception to this is the top quark; because of its exceptionally large
mass, it decays before hadronization takes place (see section 1.4.3).
Parton Distribution Functions
Hadron colliders produce inclusive collisions; the partons of the hadrons are what interact
and hence the initial state of the actual hard scatter interaction is complicated. On the
other hand, and thanks to asymptotic freedom, if the momentum of the incoming particles
is high enough (≫ ΛQCD ), it is possible to make the approximation using perturbative
QCD (pQCD), that the interaction occurs between two and only two partons, one in each
incoming particle. However, in the collisions, the component of the initial momenta parallel
to the beamline (the z momenta in CDF) of the partons are unknown. Each parton carries
a fraction (x ) of the total momentum of the hadron according to a statistical distribution
10

that depends on its type (the gluon or quark flavor) and on the momentum scale (Q); this
distribution is known as a Parton Distribution Function (PDF). These functions can not at
present be calculated from first principles but must be determined by data. Many data sets
from a wide variety of experiments are used for an improved determination of PDFs.
There are several collaborations dedicated to this endeavor, among them the Coordinated
Theoretical Experimental Project on QCD (CTEQ) collaboration. The MSU HEP faculty
members are pioneer players in this collaboration . Figure 1.4 shows the CTEQ6M PDFs
at Q = 2 and 100 GeV. The manner in which PDFs are involved in calculating the cross
section will be discussed in detail in section 1.5.1. The systematic uncertainties in the PDFs
are evaluated in section 4.2.2 .
Initial & Final State Radiation
At the Tevatron, p¯ pairs collide occasionally with a large momentum transfer that is sufp
ficient to produce a t¯ pair. In addition to the t¯ pair, gluons can be radiated off of the
t
t
incoming or outgoing partons. Given sufficient momentum, these gluons can hadronize and
form additional jets. This process is labeled Initial or Final State Radiation (ISR/FSR). It
is not yet well-modeled and hence is a source of systematic uncertainty (section 4.2.2). A
very interesting aspect of our cross section measurement is that in a way it is a measure of
ISR (see section 3.7.2).

1.3.3

The Weak Interaction

Enrico Fermi proposed the first theory of β-decay in 1934 [8]. He proposed that a neutrino
is emitted along with an electron in every beta decay. Of course, there was no W boson in
Fermi’s theory. The weak interaction vertices were given by direct 4-fermion couplings, but
from the modern perspective we understand this to be an approximation: that is, Fermi’s
theory is reproduced whenever the W -mass is much larger than any other energy in the

11

(a) Q = 2 GeV

(b) Q = 100 GeV

Figure 1.4: Overview of the CTEQ6M PDFs, from [7].
12

problem (this anyway applies to all the particle except, interestingly, the top quark).
The value of the Fermi constant is set by an accurate measurements of the lifetime of the
muon. The lifetime for the muon is given by

192π 3
τµ = 2 5
G F Mµ
where GF is defined by

√
2
GF =
8

The coupling constant gW is the weak charge

gW
MW
√

(1.12)

2

(1.13)

4παw , analogous to ge and gs in QED

and QCD respectively. Experimentally, τ is measured to be equal to (2.197019 ± 0.000021)
×10−6 sec [2]. This then implies:

GF = 1.16637(1) × 10−5GeV−2

(1.14)

which was the early result from Fermi’s theory!
Currently, it is possible to independently measure the mass of the W boson, and so we
can deduce the strength of the weak coupling constant gW . Equation (1.13) finds gW =
1
0.653 ⇒ αw = 0.034 ≃ 29 which is a surprise - the weak coupling strength is ∼ five times
1
larger than the electromagnetic strength α = 137 !

Weak interactions are indeed weak - but not because the coupling is small. Instead, they are
feeble because the gauge bosons that mediate the interactions are very massive compared
to all the particles (expect for the top). At low energies, much less than the mass of the
weak bosons, the electromagnetic force dominates. For example, the π ◦ can decay electromagnetically into two photons; a π ± decays into lepton and its corresponding anti-neutrino.
The longer lifetimes that are observed for the π ± with respect to the π ◦ indicate that the
13

force responsible for the decay is weaker than the electromagnetic force. On the other hand,
in the Large Electron-Positron Collider (LEP) experiments for instance - which operated at
energies comparable to the mass of the Z Boson - weak interactions were observed to be
stronger than the electromagnetic ones.
Today, Fermi’s theory has been superseded by the modern-day Electroweak Theory.
This model unifies weak and electromagnetic interactions into a more unified whole. This
SU(2)×U(1) model undergoes a spontaneous symmetry breaking that gives mass to the
particles. This is covered next.

1.3.4

The Electroweak Theory

It was observed experimentally that the weak force violates parity i.e. one could tell whether
or not one was observing weak interactions in a mirror. This means that there are two types
for each particle that we present in Figure 1.2: a left-handed and a right-handed particle, and
the weak interaction only couples to the left-handed particles. Additionally, it was observed
that aside from coupling quarks of different flavors (up with down for instance), the weak
force also couples quarks of different families or generations (up with strange for example)
(see the CKM section below). The simplest gauge group that fulfills such observations is
SU(2), where the interaction is only between the left-handed component of the particle and
the right-handed component of the antiparticle.
As James Maxwell showed successfully that electricity and magnetism could be unified
into a single theory involving a vector field (the electromagnetic field); Glashow, Weinberg
and Salam described how it is possible to unify the weak and the electromagnetic forces
through the SU(2)×U(1) symmetry. In 1961 Glashow [9] suggested that the two interactions
were unified, and in 1967-8 Glashow’s model was modified by Weinberg [10] and Salam [11]
to make the weak bosons massive. In 1971 ’t Hooft and Veltman [12] showed that this theory
is renormalizable and in 1983 the W and Z bosons were discovered [13].

14

Fermions

1st

2 nd

3 rd

Q

I3

Y

Leptons

νe
e L

νµ
µ L

ντ
τ L

µR

τR

0
−1

+1

+1/2
−1/2

-1
+2

u
d L

c
s L

t
b L

uR
dR

cR
sR

tR
bR

+2/3
−1/3

+1/2
−1/2

+1/3
-4/3
+2/3

eR

Quarks

-2/3
+1/3

0

0
0

Table 1.1: Quantum numbers in the electroweak theory.
In their theory, they assumed an SU(2)L × U(1)Y gauge symmetry which would generate
four vector bosons. U(1)Y is a symmetry analogous to that of the QED and generates one
vector boson B ◦ with a new quantum number, the weak hypercharge Y. On the other hand,
SU(2)L generates three vector bosons: W 1 , W 2 and W 3 with a new quantum number, the
weak isospin I3 . The up-type fermions are assigned I3 = + 1 and the down-type are assigned
2
I3 = - 1 . With electric charge Q as the quantum number of QED, Y = 2(Q - I3 ). Table
2
1.1 summarizes these quantum numbers. In their work, neutrinos are all left-handed and
massless. Recent experimental evidence indicates that neutrinos do have a mass (so righthanded neutrinos should exist as well) [14]. This means that SM needs to be extended in
this regard.
The electroweak lagrangian L is:

1
1
¯
L = − W µν Wµν − B µν Bµν + iψγ µ Dµψ
4
4

15

(1.15)

where the covariant derivative is

Dµ = ∂µ + igW Wµ · T + igY Y Bµ

(1.16)

+
−
3
= ∂µ + igW (Wµ T + + Wµ T − ) + igW Wµ · T3 + igY Y Bµ

While gY Y Bµ is just a number, W · T is a 2×2 matrix and Ti are the SU(2) generators (Ti
1
= 2 σi ) which satisfy

[Ti, Tj ] = iǫijk Tk

(1.17)

The two bosons W 3 and Bµ mix together to form the electromagnetic field Aµ of equation
(1.2) and the neutral Z, while the remaining W bosons will combine into the charged W ±
bosons:
1
1
2
±
Wµ ≡ √ (Wµ ± iWµ )
2
3
Wµ = cosθW Zµ − sinθW Aµ

(1.18)

Bµ = sinθW Zµ + cosθW Aµ
where θW is a free parameter of the theory, called the weak mixing angle or Weinberg angle.
Up to this point, the particles are massless; the Higgs mechanism (next paragraph) gives
mass to them. After solving the gauge equations of motion, we obtain the solution:

gW sinθW = gY cosθW = ge = e

(1.19)

This angle clearly relates both the weak and the electromagnetic coupling constants men-

16

tioned in the previous sections. Experimentally [2];

expt
sin2θW = 0.23116 ± .00013 ⇒ θW = 28.74◦

(1.20)

The Higgs Mechanism
The lagrangian L in Equation (1.15) describes only massless bosons and fermions. The

problem is not that we don’t know how to give a particle mass - terms such as 1 M 2 Bµ B µ
2
1 M 2 B B µ for example would destroy
¯
or mψψ are indeed mass terms. However, adding
2

µ

the U(1) gauge invariance of the theory. This in turn renders the theory unrenormalizable.
The resolution to this dilemma was found by Higgs in 1963 [15], and then applied to the
weak interactions by Salam and Weinberg as described in the previous section. It involves
introducing a scalar field/particle that couples to the electroweak bosons in a particular way
that the photon remains massless but the Z does not.
The Higgs field is a weak isospin doublet under SU(2) of complex fields, which I will call
Φ = (φ+ , φ◦ ). The term LH is added to the lagrangian L

LH = |DµΦ|2 − V (Φ+Φ)

(1.21)

where the scalar potential takes the form:

V (Φ) = µ2Φ+Φ + λ(Φ+Φ)2

(1.22)

where λ and µ are real constants. The symmetry is broken because the minimum of the
potential is not at Φ = 0 (the trivial ground state), but rather at Φ+ Φ = υ = µ . This allows
λ
us to write Φ = (0, υ + h(x)) i.e. only one of the Higgs field , the h(x) field, has acquired
a mass (the Higgs particle), while the other three fields remain massless. By inserting this
17

form for Φ and solving for the equations, the W ± , Z and the Higgs particle acquire the
following mass terms:

MW

gW υ
,
=
2

gW υ
MZ =
,
2 cos θW

mH =

√

2µ

(1.23)

yielding the relationship

MW = MZ cos θW

(1.24)

Having the mass of the W and the value of the gW , it’s not hard to show that υ ∼ 246
=
GeV. However, the SM makes no prediction as to what mH is - the parameter µ must be
input into the model besides the other free parameters. Saying this, it is possible to obtain
bounds on mH indirectly from the SM using different precision measurements, such as the
W boson mass and the top quark mass. The Higgs boson itself is the subject of extremely
intense searches at the Tevatron and the LHC. How the top quark can be used to constrain
the Higgs mass is what we shall discuss in section 1.4.3.
Yukawa Coupling
The Higgs mechanism allows the bosons to acquire mass. After all, this was the aim of the
whole procedure. On the other hand, fermions acquire their masses directly (we add the
mass-term into the equations), but why not make use of the Higgs mechanism to generate
masses for them as well? This would provide a unified picture of the origin of mass, in
which all particles attain their internal masses through their interactions with the Higgs
particle. This is done by adding, for each fermion, a term to the lagrangian L of the form,
¯
¯
LYukawa = −Yf fL φfR where Yf is the Yukawa coupling of the fermion and fL and fR
are the left-handed and right-handed components of the fermion field f respectively. After
symmetry breaking, each fermion would be coupled to the Higgs; a large coupling of the

18

Higgs to the top becomes the large top quark mass, and a small coupling of the Higgs to the
electron becomes the small electron mass.
For the top quark, the Yukawa coupling turns to be very large:

Ytop =

√

2
mtop ⇒ Ytop ∼ 0.99
=
υ

(1.25)

This is a theoretically interesting value, leading to speculation that important new physics
may be accessed via top quark by probing the Higgs-top interaction; another reason that
triggers us to study its phenomenology!
CKM Matrix
As mentioned before, besides coupling quarks of different flavors, the weak force also couples, with lower probability, quarks of different generations. The amplitudes involved in
calculating this probability is given by a 3×3 unitary matrix known as the CKM matrix (for
Cabbibo-Kobayashi-Maskawa)[16]:





 Vud Vus Vub 




 Vcd Vcs Vcb 




Vtd Vts Vtb

19

(1.26)

The current measured values of these entries is [2]:





+0.00016
 0.97428 ± 0.00015 0.2253 ± 0.0007 0.00347-0.00012 




 0.22520 ± 0.0007 0.97345+0.00015 0.0410+0.0011

-0.00016
-0.0007




+0.00026
+0.0011
+0.000030
0.00862-0.00020
0.0403-0.0007
0.999152-0.000045

(1.27)

We see empirically that the third generation (t,b) hardly mixes at all with the other
two generations. The SM doesn’t explain the origin of the values of the entries in the
CKM matrix. The best we can currently do is to measure experimentally these values for
consistency using the constraints imposed by the SM assumption of the unitary of the CKM
matrix. Having Vtb much much larger than Vtd and Vts, we will look only for top decays
into bottom quarks as we will explain this in detail later.

1.4

Top Quark Physics Review

Though predicted to exist almost immediately after the discovery of the bottom quark at
Fermilab in 1977 [17], the top quark wasn’t discovered for 17 more years. The CDF and
D∅ collaborations jointly announced discovery of the top quark in 1995 [18], but it wasn’t
until the recent high luminosity and large data sets of Run II at the Tevatron that precision
measurements of the top quark were possible.
In the subsequent subsections I will discuss the top production mechanism, the decay
mechanism and the intrinsic properties (such as it’s mass, charge, width/Vtb and spin correlations). There are many review articles summarizing the latest measurements and studies
of top quark. I have particularly consulted these [19, 20, 21, 22].

20

Figure 1.5: LO Feynman diagrams for top quark pair production: (a) quark–anti-quark
annihilation and (b) gluon-gluon fusion, from [22].

1.4.1

Top Production Mechanism

The first thing we may ask ourselves about is how top quarks are produced and how they
afterward do decay. The production of top quarks is only possible at highest center-of-mass
√
energies ( s), set by the scale of top quark mass (mt ). The energies needed for production
of top quarks in the SM are currently only accessible at hadron colliders. The Tevatron is
√
√
operating at s = 1.96 TeV while the LHC currently provides proton-proton collisions at s
= 7 TeV.
In the framework of the SM, top quarks can be produced in pairs, t¯, predominantly via
t
the strong interaction, and singly via the electroweak interaction. The Leading Order (LO)
Feynmann diagrams for t¯ production are shown in 1.5 .
t
To produce a t¯, the energy needed at the tree level is x1 x2 ≥ 4m2 /s. Assuming
t
t
x1 ≈ x2 = x yields as threshold for t¯ production:
t



 0.176 (Tevatron RunII), √s = 1.96 TeV
2mt
x = √ ≈
(1.28)
√

s
 0.025 (LHC), s = 14 TeV

Since large momentum fractions are required for t¯ production at the Tevatron, and by
t
21

examining Figure 1.4, the process is dominated by quark–anti-quark annihilation (Figure
1.5(a)) of the valence quarks. This subprocess accounts for ∼ 85% of all t¯ events, with gg
t
being responsible for 15% [23]. At the LHC, gluon-gluon fusion dominates with a contribution
of about 90% because a small momentum fraction suffices for t¯ production [24]. This can
t
be verified by recalling that the probability for colliding a quark and its anti-quark is small
because the anti-quark can’t be a valence quark and thus the annihilation mechanism is
suppressed.
Single top quark production can only take place via the weak interaction involving the
Wtb vertex (Wts and Wtd vertices are strongly CKM suppressed - as discussed before). The
LO Feynman diagrams are shown in Figure 1.6. The t-channel and s-channel modes are the
dominant at the Tevatron (∼ 65% and ∼ 30% respectively); associated W production is not
expected to be observable at the Tevatron with the amount of data collected to date.
Electroweak single top production was finally observed, simultaneously, by the CDF
collaboration using 3.2 fb−1 of data [25], and by the D∅ collaboration using 2.3 fb−1 of data
[26]. The Tevatron has combined these two measurements and reported the first inclusive
single top cross section in both the s- and t-channels of 2.76+0.58 pb for a top quark of
−0.47

mass 170 GeV [27].

1.4.2

Top Decay Mechanism

After production, top quarks decay very rapidly through the weak interaction into t → Wq
, with q being one of the down-type quarks. The contribution of each quark flavor to the
total decay width is proportional to the square of the respective CKM matrix element Vtq .
Consequently, the decay t → Wb is absolutely dominant. The b quark will convert into a B
hadron, which will in turn decay into a jet of particles, tagged as a b-jet (see section 3.4).
W bosons decay into two fermions, either leptons (a charged lepton-neutrino pair
lνl , l = e, µ, τ , with equal probability per lepton flavor at Born level), or into quark–anti-

22

Figure 1.6: Representative Feynman diagrams for electroweak single top quark pair production: (a) s-channel, (b,c) t-channel and (d,e) associated production. The diagrams for single
anti-top quark can be obtained by interchanging quarks with anti-quarks, from [22].

23

¯
quark pairs q¯ with q = u,c and ¯ = d,¯ (W → tb is not kinematically allowed). At LO level,
q
q ¯s
the hadronic decay widths are enhanced over the leptonic modes by a color factor of three
(taking the three possible quark color into account), and scaled by the appropriate squared
CKM matrix element |Vqq |2 . In summary, W bosons decay leptonically with a Branching
¯
Ratio (BR) of ≈ 1/9 per lepton flavor and ≈ 1/3 for each of the hadronic decays (u,d and
c,¯). For t¯, there are two W bosons in each event. Thus, there are 81 (9×9) possible
s
t
combinatorics of decays. At higher orders, the QCD corrections to the partial widths (see
section 1.4.3 page 32) to quarks enhance the BRs to quarks, causing the BRs to leptons to
decrease. In agreement with this prediction, the measured value of the BR for W → lνl is
(10.80±0.09)% which is less than 1/9, [2].
The four t¯ decay channels are:
t
• Dilepton channels: Both W bosons decay leptonically (lνl , l = e,µ), resulting in a final
/
state comprised of two isolated high ET leptons, missing energy (ET ) corresponding to
the two neutrinos, and two b-jets. This channel has a BR of approximately 5%. While
these channels give samples of the highest t¯ signal purity, they suffer from limited
t
statistics due to the small BR.
• All-hadronic channels: Both W bosons decay to q¯ pairs, resulting in a six-jet final
q
state. With a BR of ≈ 45%, this channel yields the highest statistics of t¯ events but
t
also suffers from large background (will be defined later, in section 3.8) from multijet
production.
• Lepton + jets channels: One W boson decays leptonically and the other one hadroni/
cally, yielding a final state topology of one isolated high ET lepton, ET and four jets.
These channels which exhibit a BR of approximately 30%, offer (for many analyses)
the best trade off between purity of sample and available statistics. For our analysis,
we use a part of this channel, using only electrons and muons. This BR should then be

24

encountered in the cross section measurement as will be explained in (Equation 1.54).
Figure 1.7 shows a Feynman diagram for this channel.
• Hadronic τ channels: Final states where at least one W boson yields a charged τ lepton
that in turn decays into hadrons ( and a neutrino) are called hadronic τ channels, and
together comprise a BR of ≈13%. The identification of τ → hadrons makes the final
states challenging to reconstruct.
In all of the above final states, two of the jets are b-jets.
From an experimental viewpoint, the issue becomes finding an appropriate set of selection
cuts to apply to the decay products in order to reconstruct the t¯ event (identify the jet’s
t
/
ET , the lepton, ET , and the b-tagging). These along with other selection cuts are discussed
in detail in chapter 3. The τ τ final state remains the only channel that has not yet been
explicitly analyzed. The properties of top quarks have been extracted mainly from the first
three of the above channels, and especially from the lepton + jet channel.

1.4.3

Intrinsic Properties

To establish that the top quark discovered at the Tevatron is indeed the SM top quark,
it is important to confirm its fundamental quantum numbers and to set limits on possible
deviations from the SM. In this section, measurements of the top quark’s mass, width, spin
and electric charge performed thus far at the Tevatron are described. Where appropriate, I
comment also on some production and decay properties of the top quark in both the strong
and electroweak interactions.

Mass
For a free particle, the physical mass is usually taken to correspond to the pole of their
propagator, i.e. the value of the four-momentum squared, p2 = E 2 − p 2 . Because of
25

Figure 1.7: A tree level Feynman diagram of t¯ production and subsequent lepton + jets
t
decay, from [28].

26

confinement, quarks can’t exist as free particles and this definition becomes ambiguous.
The definition of the top quark pole mass is still possible on an order by order basis in
the perturbation theory, but is considered to be intrinsically ambiguous on the order of
the confinement scale O(ΛQCD ). To determine the top quark mass mt , one has to find
observable measurements that can be compared to theory predictions which in turn depend
on the top quark mass. In HEP, there are two fundamentally different approaches:
(1) Direct reconstruction of top quark decay products: The momenta of the decay products
are related to mt according to

2



mt(i)2 = pt(i)2 = 

j

pj (i)

(1.29)

where the sum is over all decay products j of the top quark t in a specific event i. Since the
squared sum of the four-momenta as given in Equation (1.29) enters in the denominator

p2 − m2 + imtΓt
t
t

(1.30)

of the propagator term, the measured mt at the Tevatron and LHC taken to be the pole
mass.
(2) The indirect constraints are obtained from fits of the SM prediction as a function of
the top quark mass to precision measurements of electroweak observables (i.e. non-colored
particles). This was the technique used at the LEP.
In perturbation theory, predictions for observables receive contributions (corrections)
from loop diagrams (see section 1.5.1). The size of these corrections depends on the values of
the mass of the particles in the loops. Of particular importance for SM fits is the dependence
of the W boson mass on the top quark and Higgs boson masses. The W boson mass of

27

Equation (1.24) can now be expressed as:

MW
MZ

2

= 1 − sin2 θW (1 + ∆r)

(1.31)

where ∆r contains all the one-loop corrections (is 0 at tree level). Contributions to ∆r
originate from the top quark by the one-loop diagrams shown in Figure 1.8 (a), which
contribute to the W and Z masses via:

3 GF
m2
(∆r)top ≃ − √
t
8 2π 2 tan2 θW

(1.32)

Also the Higgs boson contributes to ∆r via the one-loop diagrams, shown in Figure 1.8 (b):

(∆r)Higgs

2
3 G F MW
≃ √
8 2π 2

5
m2
ln H −
m2
6
Z

(1.33)

While the leading mt dependence is quadratic, i.e. very strong, the leading mH dependence is only logarithmic, i.e. rather weak. Therefore the inferred constraints on mH are
much weaker than those on mt . This was used to successfully predict the top quark mass
from the electroweak precision data before it was discovered. This provided confidence in
the precision and predictive power of radiative corrections in the SM. Therefore, the SM fit
to the electroweak precision data including the direct measurements of the top quark and
the W boson masses is used to infer on the mass of the SM Higgs boson. This will also be a
service to the LHC experiments which optimize their analysis techniques and strategies for
the search for the yet elusive SM Higgs boson in the lower mass range, preferred by the SM
electroweak fit.
With up to 5.9 fb−1 of data analyzed at CDF, and up to 6.7 fb−1 at D∅, the Tevatron has
28

Figure 1.8: (a): Virtual top quark loops contributing to the W and Z boson masses. (b):
Virtual Higgs boson loops contributing to the W and Z boson masses, from [22].

recently excluded, at the 95% confidence level, a Higgs boson with a mass between (158-175)
GeV and between (100-109) GeV and showed that the Tevatron experiments are beginning
to be sensitive to a low-mass Higgs boson [29].
Two main techniques are used in top quark mass analyses at the Tevatron: the Template
Method and Matrix Element Method. The first method begins by choosing an observable
correlated with the top quark mass. The distribution of this observable is reconstructed
in the dataset under consideration. The resulting distribution is then compared in a fit
with template contributions from signal (with varying top quark masses) and background
simulations. The observable most correlated with mt is the reconstructed invariant mass of
the t¯ decay products. In the Matrix Element Method and based on the LO matrix elements
t
of contributing signal and background processes, the four-vectors of the reconstructed objects
in each event define a probability density as a function of the top quark mass. The total
likelihood for the event sample is given as the product of the individual event likelihoods.
This method is also referred to as the Dynamical Likelihood Method [31].
The Tevatron updated average mass of the top quark is mt = 173.32 ± 0.56(stat) ±
29

0.89(syst) GeV. Adding the statistical and systematic uncertainties in quadratic yields a
total uncertainty of 1.06 GeV, corresponding to a relative precision of 0.61% on the top
quark mass. Rounding off to two significant digits in the uncertainty, the combination
provides mt = 173.3±1.1 GeV [32]. The input measurements and the resulting Tevatron
average mass of the top quark, using up to 5.6 fb−1 , are summarized in Figure 1.9. The
combination includes Run I & II measurements using the first three decay channels presented
in section (1.4.2).
The overall conclusion is that the top quark mass measurement is in good agreement
with indirect constraints valid within the SM and thus there is no significant evidence for
phenomena beyond the SM.

Mass Difference
There are three discrete symmetries of interests in HEP. Charge conjugation (C): whenever a
particle could undergo certain interactions, so could its anti-particle. Parity (P): if a system
is described by ψ(x), then the transformed wavefunction Pψ(x) has all coordinates inverted
through the origin. Time reversal (T): which transforms the time coordinate (t) into (-t).
It is known that the weak interactions were found to violate C and P maximally. While
the product CP seemed to be conserved, evidence for CP-violation was later found in Kaon
decays. On the other hand, the triplet product CPT has never been seen to be violated in any
experiment, and in fact, there are several tests that constrain the magnitude of CPT violation
in various sectors of the SM [34]. This requires that the particle and its corresponding antiparticle masses be identical. Thus top quark provides a unique opportunity to measure a
(q − q ) mass difference free from the QCD effects and to probe CPT-violation to a greater
¯
degree than the other sectors of the SM.
CDF presented a preliminary measurement in the lepton plus jet channel using a data
sample of 5.6 fb−1 and found [35]: ∆mtop = mt − m¯ = −3.3 ± 1.4 stat ± 1.0 syst GeV =
t
30

Mass of the Top Quark
14

July 2010

(* preliminary)

CDF-I dilepton

167.4 ±11.4 (±10.3 ± 4.9)

DØ-I dilepton

168.4 ±12.8 (±12.3 ± 3.6)

CDF-II dilepton *

170.6 ± 3.8

(± 2.2 ± 3.1)

DØ-II dilepton *

174.7 ± 3.8

(± 2.9 ± 2.4)

CDF-I lepton+jets

176.1 ± 7.4

(± 5.1 ± 5.3)

DØ-I lepton+jets

180.1 ± 5.3

(± 3.9 ± 3.6)

CDF-II lepton+jets *

173.0 ± 1.2

(± 0.7 ± 1.1)

DØ-II lepton+jets *

173.7 ± 1.8

(± 0.8 ± 1.6)

CDF-I alljets

186.0 ±11.5 (±10.0 ± 5.7)

CDF-II alljets

174.8 ± 2.5

(± 1.7 ± 1.9)

CDF-II track

175.3 ± 6.9

(± 6.2 ± 3.0)

Tevatron combination *

173.3 ± 1.1

(± 0.6 ± 0.9)
( ± stat ± syst)

χ2/dof 2.7 (± (81%)
12.4 ± = 6.1/10 1.5 ± 2.2)

CDF March’07

0
150

160

170
180
mtop (GeV/c2)

190

200

Figure 1.9: Summary of the input measurements and resulting Tevatron average mass of the
top quark, from [33].
31

−3.3 ± 1.7 GeV, consistent with CPT invariance.
D∅ performed the measurement in the same channel using ∼ 1.0 fb−1 and found [36]:
∆mtop = mt − m¯ = +3.8 ± 3.4 stat ± 1.22 syst GeV = +3.8 ± 3.8 GeV, also consistent with
t
CPT invariance. These measurements are statistically limited though.

Width
The total width, Γt , (also referred to as decay rate) of the top quark is determined in the
SM once its mass is specified. It is defined as

Γt =

# of decays per unit time
# of particles

(1.34)

giving it the units of sec−1 as expected. Because the top quark mass is large, so is Γt , and
consequently its lifetime τt , ∼ 1/Γt , is extremely short. The partial width is defined for only
a specific decay channel, for example t −→ Wb.
The Γt at the LO is given by:

Γo
t

GF m3
√t
=
8π 2

(1.35)

The Γt in the SM, including first-order QCD corrections, can be expressed as follows [22]:

Γt =

Γo|Vtb|2
t

m2
1− W
m2
t

2

m2
1+2 W
m2
t

2αs
1−
3π

2π 2 5
−
3
2

,
(1.36)

where the above formula assumes m2 /m2 → 0, m2 ≫ m2 and ignore corrections of order
t
t
b
W
2 ). The other SM decays, t −→ Wd and t −→ Ws, contribute negligibly to the total
O(αs
decay width Γt =

2
2
q Γtq because of proportionality to |Vtd | and |Vts | . Γt is 1.02, 1.26,

1.54 GeV for the top quark masses of 160, 170, 180 GeV respectively.

32

The resulting lifetime of the top quark τt = Γ−1 ≈ (1.3 GeV)−1 is approximately 5.10−25
t
s, and significantly shorter than the hadronization time-scale τhad ≈ Λ−1 ≈ (0.2 GeV)−1
QCD
≈ 3.10−24 s. This means that, unlike lighter quarks, the top quark does not hadronize
before decaying but rather decay as a free particle. In particular, there can be no t¯ bound
t
states, toponium, (Γtt ∼ 2 Γt ). Consequently, this means that the intrinsic properties of the
¯
top quark (spin, charge, etc) can be measured directly through reconstruction of its decay
products before color confinement occurs.
Nevertheless, although the top quark can generally be considered as a free quark, residual
non-perturbative effects associated with hadronization should still be present in the top quark
events, and the fragmentation and hadronization processes will be influenced by the color
structure of the hard interaction.
CDF used 955 pb−1 of data in the lepton+jet channel and employed a kinematic fit to
obtain the mrecon distribution. The shape of the mrecon distribution is sensitive to the Γt
t
t
and a template method of a nominal mt of 175 GeV was employed to measure for the first
time the Γ . Figure 1.10 shows 95% C.L in Γfit ; the fitted value from data. CDF allowed
t

t

negative Γfit values that represent mrecon narrower than the nominal due to statistical
t
t
fluctuations. The Γfit = -4.8 GeV corresponds to a limit of the Γt < 13.1 GeV at 95% C.L.
t
which corresponded to τt > 5.2 10−26 s [37]. These limits would improve if the nominal mt
used in the template was closer to the current Tevatron average recorded in Figure 1.9.
Since the SM Γt is of the order mentioned above, the production and decay vertices
of the top are separated by O(10−16 )m, which is orders of magnitude below the spatialexperimental resolutions at both the Tevatron and the LHC. Consequently, a direct measurement technique of the Γt as the one described above will be limited by detector resolution.
However, it is possible to extract Γt using the measured cross section for the electroweak
production of the single top quarks, as addressed before in [25, 26] . By using the 2.3 fb−1
+0.69
of data in the t-channel [38], D∅ measured Γt = 1.99−0.55 GeV which is equivalent to a

+1.3
lifetime of τt = 3.3−0.9 10−25 s in agreement with the SM prediction. This is displayed in

33

Figure 1.10: The confidence band in Γfit for a 95% C.L. The fitted value from the data is
t
indicated by an arrow which corresponds to a limit of Γt < 13.1 GeV, from [37].

34

Figure 1.11

Helicity of the W boson
The top quark decays, before forming a bound state, via the left-handed charged current
weak interaction t −→ Wb. This is reflected in the observed helicity states of the W boson,
which can be exploited to directly probe the couplings at the tWb interaction vetrex.
The emitted b quark can be regarded as massless compared to the top quark, and hence
expected to be predominantly of negative helicity (left-handed). The emitted W boson,
being a massive spin-1 particle, can assume any of the three helicities: one longitudinal
(W0 ) and two transverse states (W− , left-handed and W+ , right-handed). To conserve
angular momentum in the t −→ W+ b decay, the spin projection of the W boson onto its
momentum must vanish if the b quark’s spin points along the spin of the top quark, while
a left-handed W boson is needed if the b quark’s spin points opposite to the spin of the top
quark. In the limit of a massless b quark, a right-handed W boson can not contribute to
the decay, as illustrated in Figure 1.12. For the decay of an anti-top quark, a left-handed W
boson is forbidden.
At LO, the expected fractions of decays with different W boson helicities, taking the
finite b quark mass into account, are given by:

1
Γ0
≈
Γt
1 + 2x2

(1.37)

2x2
Γ−
≈
f− =
Γt
1 + 2x2

(1.38)

Γ+
2x2
2
f+ =
≈y
Γt
(1 − x2)2(1 + 2x2)

(1.39)

f0 =

where the scaled masses x = mW /mt , y = mb /mt were used. Inserting mt = 175 GeV,
35

Figure 1.11: The expected and observed Bayesian probability density for the Γt . The most
probable value for Γt is defined by the peak of the probability density function. The hatched
area represent one standard deviation around the most probable value, from[38].

36

b

b

t

t

W

W

longitudinal (W0)

left-handed (W−)

b

?

t
W

right-handed (W+)
(forbidden if mb = 0)

Figure 1.12: Angular momentum conservation in top quark decay does not allow righthanded W bosons when b quarks are assumed massless, from[22].

mW = 80.4 GeV, and mb = 4.8 GeV, the partial helicity rates are found to be [22]: f0 =
0.703, f− = 0.297, f+ = 0.00036. NLO corrections lower the f0 by 1.1% and increase the f−
by 2.2% while the f+ remains unchanged at the per mill level. Consequently, any observation
of f+ at the percent level would signal the presence of physics beyond the SM.
¯
Thus far, four analysis techniques have been deployed at the Tevatron in tt events to
extract the W boson helicity fractions. This is explained in detail in [22]. All the measurements are compatible with the SM expectation. The results of a couple of analyses combined
with 1.9 fb−1 of data at CDF found [39]:
f0 = 0.66 ± 0.16(stat) ± 0.05(syst), f+ = −0.03 ± 0.06(stat) ± 0.03(syst).
D∅ has investigated the datasets of RunII with a total luminosity of 2.2–2.7 fb−1 and found
[40]:
f0 = 0.490 ± 0.106(stat) ± 0.085(syst), f+ = −0.110 ± 0.059(stat) ± 0.052(syst).

37

t¯ Spin Correlations
t
The SM predicts that the top quark decays before its spin flips. This is in contrast with the
lighter quarks, which are depolarized by QCD interactions long before they decay. The spin
of a heavy quark is flipped by its chromomagnetic moment, which is inversely proportional
to its mass, mQ . The spin-flip time is therefore proportional to mQ /Λ2
QCD . The spin-flip
time of the top quark is estimated [41] to be ≈ (1.3 MeV)−1 . This is much longer than the
anticipated top quark lifetime given by Equation (1.36) ≈ (1.3 GeV)−1 . Because of this, the
top quark assures that its spin information is passed on to its decays products, and is reflected
in their corresponding angular distributions. This provides experimental confirmation that
the top can indeed be considered as a free quark, thereby setting an upper bound on its width,
which is proportional to the combination of the CKM elements |Vtd |2 + |Vts |2 + |Vtb |2 .
Although Tevatron and LHC use unpolarized beams and therefore produced unpolarized
t¯ pairs, their spins are expected to have a strong correlation and point along the same axis
t
in the t¯ rest frame on an event-by-event basis. This is what called the Helicity Basis [42]. In
t
this basis, when a top quark’s spin is oriented in the same direction as the quark’s direction
of motion in the t¯ rest frame, it is said to be a right-handed quark (tR ), while when the spin
t
is oriented against the quark’s direction of motion, it is called a left-handed top quark (tL ).
CDF has recently performed , in the helicity basis, the first measurement of the t¯ spin
t
correlation [43]. I’ll briefly discuss the theory and summarize the results in turn.
The spin correlation is dependent on the initial particles in the collision which created the
t¯ pair and on the energy of the t¯ pair. Because the initial particles will have a particular ant
t
gular momentum, which is conserved, there will be a correlation between the helicities of the
¯
¯
¯
¯
t¯ pair in the event. There are four possible helicity states for t¯ pair: tR tR , tL tL , tL tR , tR tL .
t
t
Two of these states have t¯ pairs with the same helicity and two have t¯ pairs with opposite
t
t

38

helicites. In [43], F OH is defined to be the fraction of opposite helicity t¯ pairs:
t

FOH

¯
¯
No
σ(tLtR ) + σ(tRtL )
=
=
(1.40)
¯
¯
¯
¯
σ(tR tR ) + σ(tL tL) + σ(tLtR ) + σ(tR tL) No + Ns

where No and Ns are the numbers of t¯ with opposite and same helicity respectively. The
t
same helicity fraction F SH can be defined similarly, and it is apparent that F OH + F SH =
1. The t¯ spin correlation coefficient κ is defined as:
t

κ=

No − Ns
No + Ns

(1.41)

The coefficient κ can vary between +1, a fully negative correlation, and κ = -1 for a fully
positive correlation. If there were no t¯ spin correlation, F OH = 0.5, then κ = 0.
t
As described in Section (1.4.1), t¯ production is dominated by q¯ annihilation, where the
t
q
interaction is mediated by a gluon with spin 1 (J =1), so that the total angular momentum
of the t¯ pair must be 1. Close to the energy threshold for t¯ production, the t¯ pair does not
t
t
t
have an orbital angular momentum, then conservation of momentum requires that t¯ must
t
spin in the same direction. Thus, t¯ pair will be in a 3 S1 spin state with three eigenstates:
t
1
| + + , √ [| + − + | − + ], | − − .
2

In the helicity basis, since the two quarks move oppositely in the t¯ rest frame, they will have
t
opposite helicities. Thus, at the threshold, two of the three possible states for the t¯ pair will
t
have opposite helicities (67%). The t¯ pair produced by the gg fusion tends to have the same
t
helicity, since in this case the total angular momentum of the t¯ is (J =0, Jz =0). Hence, and
t
1
at the threshold, the t¯ will be in a 1 S0 spin state with only one eigenstate: √ [|+− −|−+ ].
t
2

Since t¯ production at the Tevatron proceeds mainly via q¯ annihilation, as opposed to the
t
q
LHC where main contribution comes from gg fusion, the observable correlation will have
opposite signs at the two colliders.

39

As a spin- 1 fermion, and at LO calculation, the SM predicts an opposite helicitiy fraction
2
F OH = 0.70 , corresponding to a t¯ spin correlation coefficient κ = 0.40. NLO calculations
t
predict κ = 0.352. Any deviation of the measured κ from the theoretical predictions would
indicate physics beyond the SM.
Utilizing 5.3 fb−1 of CDF data, a measurement of the t¯ spin correlation coefficient κ has
t
been performed in the lepton plus jet channel by studying the distributions of the helicity
angle for the decay product of the t¯. This is shown in Figure 1.13. The opposite helicity
t
fraction was measured to be FOH = 0.74±0.24stat ±0.11syst which yielded a value for the t¯
t
spin correlation coefficient κ = 0.48±0.48stat ±0.22syst , [44]. The experimental difficulties
of distinguishing the down-type jets from other light jets can be avoided by focusing on the
dilepton channel, where the charged down-type leptons are clearly identified. In the offdiagonal basis , and using 2.8 fm−1 of data, CDF found κ = 0.320 ± 0.545stat ± 0.775syst
using the dilepton channel [46]. These results are consistent with the SM predictions though
they are statistically limited.
As opposed to the unpolarized t¯ pair produced via the strong interaction, single top
t
quarks produced via the weak interaction are expected to be highly left-handed polarized.
This polarization will be reflected in the kinematic distributions of the top quark decay
products. Since the W boson couples solely to left-handed fermions, single top quarks are
≈ 100% polarized along the direction of the down-type quark i.e. the spin of it is always
correlated with the direction of the down-type quark in the event [22]. Unfortunately, there
has been no spin polarization measurement as yet at the Tevatron, although it would greatly
benefit from the increase in data already available.

Gluon Fusion Fraction in t¯ Production
t
As mentioned before in Section (1.4.1), t¯ pairs at the Tevatron are produced dominantly
t
through q¯ annihilation, while about 15% of t¯ pairs are predicted to be produced via gg
q
t

40

Helicity Basis Bilinear Cos(θl)*Cos(θb)
240
220

-1

CDF Run II preliminary L=5.3 fb

Opposite Helicity

200

Same Helicity

180

Backgrounds

Events

160

Data

140

κ: 0.48 +/- 0.48 +/- 0.22

120
100
80
60
40
20
0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
cos(θl)*cos(θb)

Figure 1.13: The distribution of the helicity basis angles for the decay product of the t¯,
t
from [45].

41

fusion. Because of uncertainties in the gluon PDF, the prediction of this fraction has a
large ambiguity and will change by up to a factor of 2 (from 10% to 20%) [47]. Hence,
measurement of the gg fraction can assist in reducing the uncertainties in the gluon’s PDF
within protons as well as tests the QCD prediction.
There are two ways allow us to separate the two different production mechanism and to
measure their relative contributions.
As mentioned in the previous section, the t¯ pair of the gg fusion will be in a 1 S0 spin state,
t
while the q¯ annihilation yields 3 S1 spin state. This difference manifests itself efficiently as an
q
azimuthal correlation of charged leptons in the t¯ dilepton channel. Utilizing the difference
t
in these azimuthal angles between the two opposite charged leptons in the dilepton channel
would enable us to distinguish t¯ pair produced via gluon-fusion from q¯ annihilation. With
t
q
a total integrated luminosity of 2.0 fb−1 of dilepton events [48], CDF found the fraction of
+0.36
gg fusion Fgg = 0.53−0.38. This is statistically limited and can not be used to restrict the

theoretical uncertainties mentioned above.
Alternatively one can discriminate between the similar final state signatures of gg →
¯
¯
tt and q q → tt by taking advantage of the characteristic differences between soft gluons
¯
radiated by quarks and gluons. QCD predicts that the average multiplicity of any type of
object in a gluon jet should become 9 times greater than that in a quark jet. As a consequence
4
of the higher multiplicity, the fragmentation function of a gluon jet would be expected to
be softer, i.e. more concentrated at low values of the momentum fraction x, than that of a
quark jet. Another characteristic difference is that gluon jets should be broader than quark
jets [49]. In [50], CDF used the low momentum track analysis and found a gg fusion fraction
in lepton plus jet channel of Fgg = 0.07 ± 0.14(stat) ± 0.07(syst), corresponding to an upper
limit of 0.33 at 95% C.L., in 0.955 fb−1 of data. This is in agreement with the SM prediction
though it is still statistically limited.
A third analysis done by CDF had used the kinematics of the production and decay of
the t¯ to differentiate the two production mechanisms. The kinematic variables are then used
t
42

to train a neural network (NN) to increase the sensitivity of the method. This analysis relies
on Monte Carlo (MC). A full description of the NN method can be found here [51]. The
low-momentum tracks and the NN methods are very complementary. Firstly, the physics
is complementary in that the number of tracks is an indirect measurement versus the NN
analysis which use very precise kinematics of the different processes. Secondly, the experimental techniques complement each other in that one is data driven and one is MC driven
(the NN). CDF combined the two measurement to increase the sensitivity of both. Using a
+0.15
total integrated luminosity of 955 pb−1 CDF found Fgg = 0.07−0.07 [52].

D∅ did not measure the fraction up to date.

The Electric Charge
The electric charge of the top quark predicted in the SM to be + 2 in units of e. An exper3
imental determination of the top quark electric charge serves as a check of SM consistency.
¯
So far the assumption is made that t → W + b and t → W −¯ However, it is possible that
b.
what is being observed at the Tevatron is an exotic top-like quark that decays to t → W − b
¯
and t → W +¯ Such a possibility has been put forward in [53]. In the SM, t is the SU(2)
b.
partner of the left-handed b quark and the right-handed b is a singlet. In [53] such a particle
can be found in a four generation model, Exotic Model (EM), and the b is allowed to mix
4
with a heavy quark (Q1 ) of charge - 1 whose doublet partner (Q4 ) has charge of - 3 . In the
3

EM, (Q4 ) has the reported mass at the Tevatron while the left-handed real top quark has a
mass around 274 GeV and would thus far have escaped detection. It is worth noting that
the precision electroweak data can be satisfactorily described by this model.
In order to determine the charge of the top quark, one can either measure the charge of
its decay products, in particular of the b-jet via jet charge techniques, or investigate photon
¯
radiation in t¯ events. The later method is based on measuring the cross section of p¯ → ttγ,
t
p
which is proportional to the square of the top charge. However, studies show that Tevatron

43

¯
needs around 20 fb−1 of data to measure the top charge at 95% C.L. using ttγ events [54].
For the former technique, the three ingredients needed to establish the top charge are the
charge of the W, the flavor of the b-jet and the most important the Wb pairing fraction:
W + b (corresponds to SM) or W +¯ (corresponds to EM).
b
Likelihood methods were used at the Tevatron to test the consistency of the data with
the SM and the EM scenario. Using 2.7 fb−1 [55], the CDF data strongly favored the SM
4
over the EM by excluding the exotic - 3 charged quark interpretation at 95 % C.L. . Using

370 pb−1 [56], D∅ data confirmed the SM and and excluded the EM at the 92 % C.L. .

Forward-Backward Charge Asymmetry
At LO in QCD, t¯ production is symmetric under charge conjugation, implying it does not
t
discriminate between top and anti-top quarks. Considering that the initial p¯ state at the
p
Tevatron is not an eigenstate of charge conjugation, this symmetry is a coincidence. However,
at NLO, radiative corrections involving either virtual or real gluon emissions lead to a small
charge asymmetry; an excess of top over anti-top quarks. This asymmetry has its origin in
q¯ mechanism from interference between amplitudes that are symmetric and antisymmetric
q
under the exchange of top and anti-top quarks while the gg mechanism remains charge
symmetric. The real emission corrections tend to push the top quark backward, opposite to
the proton beam direction, while the virtual corrections push it forward, in the direction of
the proton beam [19]. Due to this, the charge asymmetry can be interpreted as a top quark
forward-backward charge asymmetry; Af b .
Af b is frame dependent and can be defined either in the laboratory frame p¯ or in the
p
parton rest frame t¯. The latter frame is more dificult to experimentally reconstruct but the
t
p¯
p

¯

t
asymmetry in the former is predicted to be less. At NLO, Af b and Aftb are expected to be

0.051(6) and 0.078(9) respectively [57].
In the p¯ frame, we look at the angle between the top quark and the proton direction and
p

44

measure the rapidity y t . However, both CDF and D∅ have measured Af b using the lepton
plus jet channel. This makes it problematic to use the y t distribution since acceptances are
different for y had and y lep . We know on the other hand which quark (top or anti-top)
decayed hadronically by the charge of the lepton in the event. This means that y had is
either y t or −y¯. Because top quark and anti-top quark differs by their charge, a top quark
t
moving forward is equivalent to an anti-top quark moving backward. This allows us to write
yt = −q.yhad , where q is the lepton charge. With N as the number of events with a given
rapidity, Af b can hence be written as [58]:

p¯
Afp =
b

N (−q.yhad > 0) − N (−q.yhad < 0)
N (−q.yhad > 0) + N (−q.yhad < 0)

(1.42)

In the t¯ rest frame [58]:
t
¯
lab
rest
ytop = ytop − y tt
1
= yt − (yt + y¯)
t
2
1
(y − y¯)
=
t
2 t
1
∆y
=
2 top
1
=
q∆y
2

t¯
Aftb =

N (q∆y > 0) − N (q∆y < 0)
N (q∆y > 0) + N (q∆y < 0)

(1.43)

(1.44)

CDF has measured recently the asymmetry in both frames using 5.3 fb−1 and found [59]
p¯
p

¯

t
Af b = 0.150 ± 0.050stat ± 0.024syst and Aftb = 0.158 ± 0.072stat ± 0.017syst . This result

is discrepant at the level of three standard deviations (3 σ) for a Gaussian distribution; it
neither confirms nor rejects the SM, nor beyond SM, either. In 4.3 fb−1 of data collected by
t¯
D∅, the asymmetry in the t¯ frame was found to be [60] Aftb = 0.08 ± 0.04stat ± 0.01syst .
t

45

In contrast to the CDF results, this number presents the raw measured value without being
corrected for acceptance and resolution effects. In the dilepton channel, CDF aimed very
recently to measure the equivalent quantity in the t¯ frame. Using 5.1 fb−1 , CDF found the
t
¯

t
asymmetry to be [61] Aftb = 0.417 ± 0.148stat ± 0.053syst .
p¯
p

It should be noted that Af b vanishes at the LHC due to the symmetric initial state, in
contrast to the Tevatron. The dominance of t¯ production via the charge symmetric gg fusion
t
reduces the observable charge asymmetry at the LHC.

1.5

Top quark Cross Section

What quantity should the experimentalist measure and the theorist calculate in a scattering
process? Is it the exclusive scattering cross section or the inclusive (total) scattering cross
section or the differential scattering cross section? Well, the cross section σ for the former
is a mere number (the number of events scattered given a certain amount of data), while
for a theorist, it is a mere number multiplied by a phase factor; this number is related to
the invariant amplitude of a given initial state scattering to a given final state (exclusively).
The amplitude can be imagined as one element in a matrix, usually called the scattering
matrix (S -matrix), which includes the amplitudes of all possible initial and final states
(inclusively), and so is usually called the matrix element M. The matrix element contains
only the dynamical information; we calculate it by evaluating the relevant Feynman diagrams
of the process. The phase space factor contains the kinematical information; it depends on
the masses, energies and momenta of the particles.

1.5.1

Perturbative QCD (pQCD)

Current QCD is unable to calculate the cross section exactly; instead, a perturbative expansion is made in powers of the running αs . These perturbative terms are conveniently
represented by Feynman diagrams. To make optimal use of the experimental measurements
46

requires theoretical predictions of similar precision. The diagrams with the fewest possible
2
number of vertices for a certain process represent the LO term of the expansion (O(αs )). At
hadron colliders, these are often insufficient to describe the process, and including higherorder corrections in QCD is mandatory.
To date, theoretical predictions for t¯ production are based on NLO calculations of the
t
total cross section (inclusive), differential cross section and the forward-backward asymmetry
(see previous section). However, the NLO computations suffer from theoretical uncertainties
larger than 10% [62], both for Tevatron and LHC. These uncertainties are partly due to our
imperfect knowledge of the PDFs and partly to the truncation of the perturbative series in
αs , which introduces a dependence on the unphysical renormalization (µR ) and factorization
(µF ) scales into the physical predictions. The later uncertainty is typically reduced by
including more terms in the perturbative series, and for this reason the calculation of the
cross section to next-to-next-to-leading order (NNLO) has been an area of active research.
However, due to the complexity of the calculations, complete results for the cross section at
NNLO are not yet available. I discuss the systematics of these scales briefly.
One of the unpleasant feature of the NLO diagrams is that they contain closed loops.
The calculations require an integral to be performed over the momenta of the particles in the
loop, and the integral often diverges. This divergence can be dealt with by a process called
renormalization, in which a renormalization scale (µR ) is introduced ad hoc to truncate
the integral before it diverges. This gives a finite result to the calculation that agrees well
with experiment for many processes. The running αs introduces a problem for practical
calculations because the energy at which the αs is calculated must be chosen. This causes
the uncertainty in the cross section calculations and the generation of simulated events as
well.
Another kind of divergence is called an infrared divergence. In QCD, and at low momentum, the αs becomes larger than one. In this case, each successive term in the perturbation
is larger than the one before it, making perturbation diverges without giving a valid result.
47

Such divergence is dealt with by introduction of a factorization scale (µF ) which truncates
the integral before it reaches the nonperturbative region. Thus, particles with momentum
scale above µF are treated perturbatively and below which are treated as a part of modified
PDF.
In practice, an appropriate choice of these scales should minimize the dependence. For
the t¯ cross section, the µR and µF scales are set to be the same scale at which αs is evaluated.
t
Fortunately, the energies of the t¯ are well above the cutoff for valid perturbation (ΛQCD );
t
αs is set to mt and varied from mt /2 to 2mt to determine the uncertainty associated with
this choice of scale.
We consider the process

¯
N1 (P1) + N2(P2) → t(p3) + t(p4) + X(pX )

(1.45)

where X is an inclusive hadronic final state. At Born level this proceeds only through the
q¯ annihilation and gg fusion channels as shown in Figure 1.5
q
¯
q(p1 ) + q (p2 ) → t(p3 ) + t(p4 ),
¯
¯
g(p1 ) + g(p2 ) → t(p3 ) + t(p4 ),
where p1 = x1 P1 and p2 = x2 P2 . We define the kinematic invariants
S = (P1 + P2 )2 , s = (p1 + p2 )2

S and s denote the hadronic center-of-mass energy squared and the partonic center-of-mass
energy squared. The total hadronic cross section for t¯ production is obtained from the
t
convolution of the factorized partonic cross section σij with the parton luminosities Lij [63]
ˆ

48

:
S

σp¯→ttX (S, mt) =
p ¯

4m2
t

i,j=q,¯,g
q

ds Lij (s, S, µF ) σij (s, mt, αs(µR ), µF ),
ˆ
(1.46)

S

1
Lij (s, S, µF ) =
S

s

dˆ
s
s
ˆ
s
fi/p( , µF ) fj/¯( , µF )
p
s
ˆ
S
s
ˆ

(1.47)

Here f i/p(x, µF ) is the PDF describing the density of partons of flavor i in the proton
carrying a fraction x of the initial proton momentum, at factorrization scale µF . The QCD
coupling constant αs (µR ) is evaluated at the renormalization scale µR . The top quark mass
mt appearing in Equation (1.46) is the mass renormalized in the on-shell (pole-mass) scheme
while the sum runs over all the light flavors u,d,c,s,b which considered massless.
In perturbative QCD the partonic cross section σij (s, mt , αs (µR ), µF ) can be expanded
ˆ
in the QCD coupling constant up to any order. Up to NNLO with mt = µR = µF we read :
(0)

(1)

(2)

σij = a2 σij (s, mt) + a3 σij (s, mt , µR , µF ) + a4 σij (s, mt , µR , µF ) + O(a5 ),
ˆ
sˆ
sˆ
sˆ
s
(1.48)
with as = αs /π. In LO only the parton channels q¯ and gg contribute and the respective
q
Born cross sections are given by:

(0)
σq q
ˆ ¯

4π 3 1
β(3 − β 2),
=
27 s
(0)

(1.49)

σgq = σgq = 0,
ˆ ¯
ˆ (0)
(0)
σgg
ˆ

(1.50)

π3 1
1+β
=
(33 − 18β 2 + β 4 )ln
48 s
1−β

− 59β + 31β 3 , (1.51)

49

Cross Section [pb]

Tevatron

LHC

σLO
σNLO
σNNLO (approx)

5.820
7.229
7.814

583.7
877.4
923.0

Table 1.2: The LO, NLO and approximate NNLO prediction for the total cross section of t¯
t
√
events at LHC ( s = 14 TeV) and Tevatron, from [64].
with β =

√

1 − ρ and ρ = 4m2 /s. At higher orders in αs the partonic cross section receives
t

corrections from virtual loop diagrams and real gluon emissions in the q¯ and gg channels,
q
as well as from the other partonic channels g¯ and gq.
q

1.5.2

The Predicted Inclusive Cross Section of t¯
t

Table 1.2 compares the LO, NLO and approximate NNLO prediction for the total cross
√
section at LHC ( s = 14 TeV) and Tevatron using mt = 171 GeV, the PDF set CTEQ6.6
and µR = µF = mt .
Recently, different calculations using different PDFs have been reported for the t¯ cross
t
section. Table 1.3 summarizes the results for some of them at the Tevatron using mt = 173
GeV. The results are presented in the form

+∆σ

+∆σ

σ = σ(central) −∆σµ+ −∆σPDF+
µ−

PDF-

(1.52)

where σ(central) is the best prediction, and ∆σµ± and ∆σPDF± quantify the uncertainties
due to the scale and PDF choices respectively. The explicit dependence of the cross section
on the top quark mass and the hadronic center-of-mass energy as shown in Equation (1.46)
is depicted in Figures 1.14 and 1.15 respectively. We can see that the small changes in

50

Reference

CTEQ

MRST

[64]

+0.24 +0.41
7.34−0.38 −0.41

+0.24 +0.14
7.04−0.36 −0.14

[65]

+0.35 +0.43
6.79−0.74 −0.34

+0.45 +0.21
7.13−0.84 −0.17

Table 1.3: The best predictions for t¯ [pb] production at the Tevatron for a pole mass of mt
t
= 173 GeV.
the center-of-mass energy (by upgrading the Tevatron from RunI to RunII) results in large
changes in the cross section. On the other hand, Figure 1.14 shows that the cross section
changes by ±1 pb for each ±5 GeV change in the top quark mass (a heavier top is harder
to produce). This is actually a manifestation of the fact that t¯ are produced predominantly
t
near the kinematic threshold at the Tevatron. This situation is different at the LHC, where
the center-of-mass energy is far above the threshold.

1.5.3

The Measured Inclusive Cross Section of t¯
t

It is particularly important to measure the t¯ inclusive production in the different top quark
t
SM decay channels since new physics contributions can affect the various t¯ final states,
t
differently. Within uncertainties, all the measured t¯ cross sections in almost all possible
t
final states conducted by CDF and D∅ agree with each other. A combination of all these
results yields an improved precision on the t¯ cross section.
t
+0.98
Combining three measurements using about 1 fb−1 , D∅ finds σtt = 8.18−0.87 pb for
¯

mt =170 GeV [67]. CDF performs a combination of four measurements, using up to 4.6
−1

of data. For mt =172.5 GeV [68], the combination yields σtt = 7.50 ± 0.48 pb. These
¯

measurements are in good agreement with the predictions and have uncertainties comparable
to those associated with the theoretical expectations. Unfortunately, no combined cross
section measurement from both experiments exists to date, unlike for the single top cross
51

Figure 1.14: Dependence of the total cross section on the top quark mass defined in the
pole scheme. The NLL and NLO+NNLL bands indicate the scale uncertainties and PDF
uncertainties combined linearly, while the blue band represents the dependence of the D∅
measurement of the cross section on mt . For comparison, a CDF combined measurement is
also shown, from [62].

52

10
9

CDF Run I
mt =175 GeV/c 2 with 0.11 fb-1

CDF Run II Preliminary
mt =172.5 GeV/c2 with 4.6 fb-1

σ(pp → tt) (pb)

8
7
6
5
4
mt =170 GeV/c2 CTEQ6.6

2

3
2

mt=170 GeV/c CTEQ6M

mt =172.5 GeV/c2 CTEQ6.6
mt=175 GeV/c2 CTEQ6M

mt =175 GeV/c2 CTEQ6.6

1
Cacciari et al. JHEP 0404 (2004) 068

0

Langenfeld, Moch & Uwer, arXiv:0906.5273 (2009)

1800

1960

s (GeV)
Figure 1.15: Combination of CDF measurements of the t¯ production versus the theoretical
t
predictions as a function of the center-of-mass energy. For RunII, the central values of the
predictions from [64] are shown at top quark mass of 170, 172.5 and 175 GeV for CTEQ6.6.
The uncertainty of the 172.5 GeV prediction is shown by the filled blue region. For RunI,
the predictions are shown at a top quark mass of 170 and 175 GeV fro CTEQ6M, from [66].

53

section and the top quark mass as shown previously.
In addition to the total inclusive cross section, kinematic distributions are also of interest.
The differential cross section for the t¯ invariant mass (Mtt ) can be used as a complementary
t
¯
method for measuring mt . The theoretical interpretation can be drawn from Equation
(1.46): all of phase space is integrated over but for a region of the invariant mass (dMtt ).
¯
The presence of bumps in the smoothly decreasing Mtt distribution would be a clear signal
¯
of a heavy resonance, which is predicted in many new physics scenarios . If there was a
massive particle x that strongly coupled to the top quark it would be seen as a resonance
(distortion) in dσ/dMtt in the region of mx . It is possible to imagine a scenario where such
¯
a process would not change the inclusive cross section to an appreciable degree. In this way
the Mtt distribution provides the window for the new physics models.
¯
Such searches have been pursued at the Tevatron, and results for the Mtt distributions
¯
were recently obtained from 2.7 fb−1 of data collected by CDF [71]. The measured dσ/dMtt
¯
is shown in Figure 1.16 which concludes that there is no evidence for non-SM physics in
the Mtt distribution. Limits on KK gravitons of mass 600 GeV which decay to t¯ are set.
t
¯
D∅ has measured the differential cross section as a function of the transverse momentum
t
(PT ) of the top quark, dσ/dPT using ≈ 1 fb−1 of data. Results [73] from higher order QCD
corrections agree with the measured results as shown in Figure 1.17 .
The ATLAS and CMS experiments at the LHC have both recently reported their first
√
measurement of the total t¯ cross section at center-of-mass energy of s = 7 TeV in a data
t
sample of 2.9 and 3.1±0.3 pb−1 respectively. The cross section is measured to be [75, 76]
σtt = 145 ± 31(stat) +42 (syst) and σtt = 194 ± 72(stat) ± 24(syst) ± 21(lumi) for ATLAS
¯
¯
−27
and CMS respectively. These are in agreement with each other, as well as with NLO QCD
√
prediction (σtt increases at s = 14 TeV ≈ 100-fold relative to the Tevatron as displayed in
¯
Table 1.2).

54

CDF II Preliminary

dσ/dMt t [fb/GeV/c2]

SM Expectation
SM Uncertainties

10

CDF II Data,

∫ L ≈ 2.7 fb

-1

1

10-1

0

200

400

600

800

1000

1200

1400

2

Unfolded Mt t [GeV/c ]
Figure 1.16: The measured differential cross section for the t¯ invariant mass with 2.7 fb−1
t
of data at CDF, from [72].

55

dσ / dpT [pb/GeVc-1]

0.16

DØ

0.14
0.12
0.1

data, 1 fb-1
NLO pQCD
Approx. NNLO pQCD
MC@NLO
PYTHIA
ALPGEN

0.08

0.06
0.04
0.02
0

50

100 150 200 250 300 350 400

top quark p [GeV/c]
T

Figure 1.17: The measured differential cross section for the t¯ transverse momentum with 1
t
−1 of data at D∅, from [74].
fb

56

1.5.4

The Predicted Exclusive Cross Section of t¯+jet
t

As mentioned in Section 1.5.1, the scale dependence of the NLO calculations render the
theoretical results quite uncertain which calls for the NNLO calculations of the top quark
production. In general, there are four classes of contributions that need to be calculated for
the NNLO top quark production [78]. The first class involves the two-loop contributions,
which has to be folded with the LO Born term. Second, there are the squared one-loop
diagram contributions. Further, the third class of diagrams consists of the one-loop gluon
emission contributions that are folded with the one-gluon emission graphs. The later is the
topic of present section. Finally, there are the squared two-gluon emission contributions that
are of tree-type. Bits and pieces of the NNLO calculation are now being assembled. Some
recent studies can be found in [62].
An important prediction of the NLO calculation is the possibility of additional hard
partons produced in association with the t¯ pair. If one is produced with sufficiently high
t
PT and large angular separation from other objects in the event, it is detected as a jet.
In the work of [79, 80] the full, exact NLO calculations to t¯+jet were presented for the
t
first time. When integrating over the full phase space of the jet, this calculation can be
turned into a NNLO calculation of the third class as aforementioned above. The predicted
cross section calculation for 14 TeV is also important as a preview of the LHC, for which
[79, 80] has found a cross section of 376 pb for t¯+jet with a PT above 50 GeV (see Table
t
1.4). This is almost half of the total inclusive t¯ cross section evaluated at NLO in [64] (Table
t
1.2). This additional jet activity could signal new physics such as top-quark compositeness.
It also provides a sensitive tool to search for anomalous top-gluon couplings. As mentioned
in section 1.4.3 on page 44, the charge asymmetry for inclusive t¯ appears first at one-loop
t
level, and thus a NLO calculation for the inclusive production provides only a LO calculation
of the asymmetry. On the other hand, an asymmetry is present at LO for t¯+jet production,
t
thus the NLO calculation for this cross section is truly a NLO calculation for the t¯+jet
t

57

asymmetry. Apart from its significance as a signal process, it turns out that t¯+jet production
t
is also an important background to various new physics searches, where the number of jets
in the process is often taken as a discriminator against the background [81]. A prominent
example is Higgs production via vector boson fusion. This reaction represents an important
discovery channel for a SM Higgs boson with a mass of up to several hundred GeV. The
major background to this reaction is due to t¯+jet, again underlying the need for precise
t
theoretical predictions for this process. I will briefly describe the calculation and present the
results.
¯
At LO, t¯+jet production receives contributions from the partonic processes q q → ttg, qg →
t
¯
¯ ¯
¯¯
¯
ttq, qg → ttq , and gg → ttg. The first three channels are related by crossing symmetry.
¯ ¯
¯
Therefore, the required generic matrix elements are 0 → ttq qg (first three), 0 → ttggg, where
for convenience all particles are listed as outgoing. Representative sets of Born diagrams
for the gg and q q channels are depicted in Figure 1.18 . In total, there are 16 LO diagrams
¯
¯
¯¯
for 0 → ttggg and 5 for 0 → ttq q g. Evaluating 2→3 particle processes at NLO level, is not
trivial, both in the analytical and numerical parts of the calculation. There is significant
complexity due to the presence of one-loop pentagon diagrams and the involved tensor integrals which render computer codes quite lengthy and CPU time consuming. Prototypes of
the pentagon graphs are shown in Figure 1.19 . The total number of diagrams (self-energy,
vertex, box-type and pentagon-type) is about ∼400. The challenging step in this context is
the numerically fast and stable reduction of the tensor integrals to box integrals which in
turn are reduced to scalar integrals.
Figure 1.21 shows the dependence of the LO and NLO cross section for t¯+jet on the µR
t
and µF scales, which are set equal to µ. The dependence for the LO case is rather large,
illustrating the well-known fact that the LO predictions can only provide a rough estimate.
At the Tevatron (Figure 1.21 (a)) the q¯ channel dominates by about 85%, followed by the
q
gg channel with about 7%. This is rather similar to the inclusive t¯ pair cross section as
t
discussed in Section 1.4.1. In contrast to the inclusive case, the gq and g¯ subprocesses
q
58

Figure 1.18: Representative sets of LO diagrams for gg fusion and q¯ annihilation in t¯+jet
q
t
production, from [79].

Figure 1.19: Representative sets of pentagon diagrams for gg fusion and q¯ annihilation in
q
t¯+jet production at NLO QCD, from [79].
t

are not suppressed. At the LHC, the gg channel comprises about 70%, followed by gq with
about 22%. The scale dependence of the NLO cross section is shown in the right plots.
For comparison, the LO results are included as well. As expected, the NLO corrections
significantly reduce the scale dependence.
Table 1.4 provides the LO and NLO predictions for the cross section with PT cuts of
20 and 50 GeV on the hard jet. The values presented are for the central scale µ = µR =
59

√

s

NLO [pb]

20

+0.96
1.583(2)−0.55

+0.16
1.791(1)−0.31

+0.29
0.4670(6)−0.17

+0.049
0.5244(4)−0.096

20

710.8(8)+358
−221

692.3(3)−40
−62

50

LHC

LO [pb]

50

Tevatron

PT [GeV]

326.6(4)+168
−103

376.2(6)+17
−48

Table 1.4: Cross section for t¯+jet [pb] for different values of PT jet cut for µ= mt . As a
t
numerical value, mt =174 GeV. The upper and lower indices are the shifts towards µ= mt /2
and µ= 2mt , from [79].
µF = mt . The uncertainty due to the numerical integration is quoted in parentheses. The
scale dependence is indicated by the upper and lower indices. The upper (lower) index
represents the change when the scale is shifted towards µ = mt /2 (µ = 2mt ). Rescaling the
common scale µ = µR = µF from the default value mt up (down) by a factor 2 changes the
cross section at Tevatron in LO and NLO by about 60% (35%) and 9% (18%), respectively.
This means that the scale uncertainty is reduced considerably through the inclusion of the
NLO corrections. Comparing the LO and NLO it is found that the large scale dependence
at LHC of about 100% in the LO cross section is considerably reduced after including the
NLO corrections. The ratio of the NLO t¯+jet cross section at LHC to the total inclusive t¯
t
t
cross section is about 47%, 22% and 7% for a PT cut of 50 GeV, 100 GeV and 200 GeV,
respectively. At Tevatron, the ratio for the small PT cut of 20 GeV is almost 30%. This
fraction is reduced to about 8% when 50 GeV is chosen for the cut.
Having presented the results of [79, 80], and for completeness, it is worthy to note that
t¯ pairs were treated as stable particles and their decays, even at LO, were not included.
t
This is not entirely realistic since all cuts designed apply to t¯ decay products. Therefore, it
t

60

is important to account for decays of t¯ in the calculations of NLO corrections, to have full
t
confidence in the results.
How the results in [79, 80] should be compared to our measured cross section is what we
will discuss in section 4.2.3 .

1.6

Measurement of σ(p¯ → t¯+jet)
p
t

The Tevatron experiment is a scattering experiment; (p¯) beams collide and then scatter
p
elastically to (p¯) and inelastically producing new particles as (t¯+jet). The total inclusive
p
t
cross section is the sum of all the possible elastic and inelastic scattering processes. On the
other hand, the exclusive cross section for p¯ → t¯+jet can be thought of as
p
t
σ=

σ=

transition rate
incident flux

# of times (t¯ + jet) produces per unit time
t
# of (p¯) beams per unit area per unit time
p

leading σ to have unit of area as expected, to give a number which is independent of the beam
flux. Because the length scale accessible to a given probe particle is inversely proportional
to its energy, the customary unit of σ is barn. 1 barn (b) is 10−28m2 . The barn was
originally intended to describe the nuclear physics processes, and hence for HEP, the scale
for the inclusive σ is of the order of microbarn, for electroweak and top processes is picobarn
(pb) and it is of the order of femptobarn (fb) for Higgs boson production. So, for example,
production of a Higgs boson with mass of 150 GeV is about an order of magnitude less likely
than t¯ production, while W boson production is three orders of magnitude more likely than
t
the t¯ production.
t
The incident flux, expressed as the instantaneous luminosity (L), measures the amount

61

of data collected. The cross section for t¯+jet then can read:
t

σ=
Note that the integrated luminosity

Ntt+jet
¯
Ldt

(1.53)

Ldt has the unit of inverse barn, so that it can readily

be converted to σ given the number of the events, and vice versa. In our measurement, we
use 4.1 fb−1 of integrated luminosity. On the other hand, counting the number of t¯+jet
t
events is not simple due to backgrounds. Backgrounds are other processes which mimic the
signature of the t¯+jet events in the detector and hence pass the event selection. Therefore,
t
the backgrounds events (Nbkg ) are estimated using MC simulation and data and subtracted
from the t¯+jet observed events. However, in practice, detectors do not offer ideal coverage
t
and efficiency and hence this should be taken into account. Having the factor ǫ to encompass
for the acceptance selections, lepton identification, b-jet tagging and BR into lepton plus jet
channel, Equation (1.53) is modified to:

σ=

Ntt+jet − Nbkg
¯
ǫ Ldt

(1.54)

The two subsequent chapters are presented to pave the way for the measurement and
to describe all the aspects revealed from this equation. Chapter 4 will conclude with the
ultimate results.

62

10

σLO [pb]

p¯ → t¯+jet+X
p
t

CTEQ6L1
√
s = 1.96 TeV
pT,jet > 20GeV

1

0.1

0.01
0.1

total
qq
¯
gg
qg
gq
¯
1

10

µ/mt
(a) LO Scale dependence, Tevatron

6

p¯ → t¯+jet+X
p
t
√
s = 1.96 TeV
pT,jet > 20GeV

σ[pb]
5
4

NLO (CTEQ6M)
LO (CTEQ6L1)

3
2
1
0

0.1

1

10

µ/mt
(b) NLO Scale dependence, Tevatron

Figure 1.20: Scale dependence for the cross section for t¯+jet production, where µR and µF
t
scales are set equal to µ. The individual contributions of the various partonic channels are
shown also separately, from [80].
63

Chapter 2
Experimental Apparatus
2.1

Introduction

The Tevatron is the last stage of an entire accelerator chain at Fermilab. Collisions occur at
two points at the Tevatron (the ring), where the two multipurpose detectors, CDF and D∅,
are placed. In this chapter, I will discuss the accelerator complex and CDF, focusing on the
aspects of CDF most used in the analysis.
The Tevatron began operations in 1985 with a center-of-mass energy of

√

s = 1.8 TeV.

RunI occurred from 1992-1996, during which the top quark was discovered. The current
physics run, RunII, began operation in 2001, after substantial upgrades to the accelerator
√
and both detectors, and is smoothly running with s = 1.96 TeV.
However, the U.S. Department of Energy has decided to shut down the Tevatron by the end
of September of this year for financial deficit. What a great loss!

2.2

The Accelerator Complex

One would wonder, as electrons are lighter and so much easier to accelerate, why not build
electron synchrotrons? This can be done, but there is a significant cost because all charged

64

Figure 2.1: An aerial view of Fermilab. The top ring in the photograph is the Tevatron.
Protons circle Tevatron clockwise as anti-protons circle counter-clockwise. The beams are
steered to collide at the top-left of the ring (CDF) and at the top-right (D∅). The small
ring at the bottom is the main injector, from [82]

65

particles radiate energy as they accelerate. Circularly moving particles at higher speeds
accelerate more and more, losing energy (synchrotron radiation) that varies inversely with
the fourth power of their mass. This makes protons radiate much less.
On the other hand, designing a p¯ collider is more easy than pp collider because for the
p
former case, the particles can travel opposite directions in the same beam pipe and be bent by
the same set of magnets without being disturbed by each other. However, a p¯ machine poses
p
unique problems in that obtaining a beam of anti-protons is much harder than obtaining a
positron beam; anti-protons must be stored once they are produced. In addition, a single
accelerator can not bring particles from rest to TeV scale of energy because no magnets
have the dynamic range necessary. Consideration of these requirements and others led to
the design of a chain of accelerators at Fermilab. I will consider each element in turn. For a
complete updated review, consult Reference [83].

2.2.1

Proton Production

Pure Hydrogen gas (H) is moved between electrodes and a spark ionizes the atoms into electrons and H+ ions. The positive ions strike a Cesium cathode which results in the formation
of H− ions and electrons. These H− ions are sent to the Cockcroft-Walton pre-accelerator
(preacc) which will subject the negative ions to a -750 kV potential, thus producing H−
beams with energy of +750 kV.
The linear accelerator (linac) accelerates the H− ions through drift tubes to 400 MeV.
The tubes use an alternating-current electric field (usually referred to as RF, for Radio
Frequency). RF cavities are designed so that the H− are shielded from the field when it
points opposite their direction of motion. Thus, H− only feel a force from the field when it
will increase their speed. This method of acceleration causes the H− to group together into
bunches which in turn are sent along a transfer line to the Booster.
The Booster is the first synchrotron in the accelerator complex, composed of 96 magnets

66

arranged in a circle approximately 150 meters in diameter. With 18 RF cavities, the booster
receives the H− bunches and strips the electrons off, leaving bare proton bunches (H+ ) with
an energy of 8 GeV. At this point, the bunches go into a transfer line that leads to the main
injector, and then on to the Tevatron.
The main injector is a larger proton synchrotron with a diameter of about one km. It
uses 344 dipole magnets and 208 focusing quadrupole magnets to both accelerate the 8 GeV
protons from the booster to 150 GeV for injection into the Tevatron in two seconds and to
steer the bunches as well. It has another function involving the Tevatron; accelerating the
8 GeV protons to 120 GeV, which are then used to strike the anti-proton source and create
anti-proton bunches.

2.2.2

Anti-Proton Production

The 120 GeV protons sent from the main injector strike a Nickel target, creating a spray
of particles. A magnetic spectroscopy procedure is used to select the anti-protons from
the bending radii information. The resultant anti-protons have a wide range of energy,
averaging 8 GeV. Studies have shown that the 120 GeV is the optimal energy for anti-protons
production; at this energy, approximately one anti-proton is produced per 105 protons sent
to the nickle target.
The Debuncher is a triangular storage ring which accepts the 8 GeV anti-protons bunches.
Its main purpose is to “debunch” the anti-proton beam (which have a large energy spread)
by removing its RF bunch structure. This results in an anti-proton beam that have no
RF bunch structure but have a similar energy. This is referred to as the stochastic cooling
process. The cooled anti-protons are then injected into the accumulator at 8 GeV with a
uniform energy.
The accumulator lies in the same tunnel as the Debuncher. It is a long-term storage ring,
designed to store the anti-protons beams with minimal losses for days. They are manipulated

67

by RF systems to fill a stable region of phase space, known as the core; a smaller core results
in a higher luminosity. They are accumulated and cooled here until the storage reaches its
maximum capacity. At that point they are passed into the recycler.
The recycler is located in the same ring as the main injector. It is a ring of permanent
magnets designed to store anti-protons at 8 GeV. However, the extracted anti-protons from
the accumulator should be collected into bunches again. RF activation causes them to be
grouped into bunches, which are then transferred back to the main injector, decelerated to
8 GeV, and injected into the recycler.
When the Tevatron is ready for a store, anti-protons are injected from the recycler to the
main injector, which accelerates them to 150 GeV and passed them to Tevatron.

2.2.3

Tevatron

The Tevatron, which we colloquially use it to represent the entire chain, is actually the main
large synchrotron at Fermilab. It is 1 km in radius, accelerates p¯ beams from 150 to 980
p
GeV and steers them using 774 superconducting dipole and 240 quadrupole magnets with
a maximum magnetic field of 4.2 T. The magnets are cooled by liquid Helium to 4.2 K, at
which the Niobium-Titanium (NbTi) alloy becomes superconducting.
The process of injecting the beams into the Tevatron, accelerating them, and initiating
collisions is referred to as a shot. It starts with loading the protons, one bunch at a time.
This is repeated 36 times to obtain the 36 bunches needed for the 36×36 store. Afterword,
the anti-protons are injected four bunches at a time, repeated nine times. Since they are
oppositely charged, p¯ beams circle oppositely in helical orbits in the magnetic field, and
p
hence the Tevatron houses both of them.
The Tevatron is not a perfect circle. There are six sectors (A–F), each sector has five
service buildings (0–4). The “0” parts have large straight sections. B0 contains CDF and
the D∅ detector is aptly named for its place along the Tevatron. When all the p¯ beams
p

68

are loaded, RF cavities accelerate them to 980 GeV within around 85 seconds. At B0 and
D0, they are then focused (squeezed) using the quadrupole magnets into very narrow beam
of width of order 32 µm. The beams are then crossed to induce collisions at the center of
each detector, at a collision rate of approximately 2.5 MHz (396 ns). This is done by some
electrostatic separators which switch polarity to cause the beams to collide. In reality, the
collisions don’t take place at the center, but are distributed as a Gaussian around the centers
of the detectors. This is interpreted by the concept of the cross section.
The store can go on and both detectors can accumulate more integrated luminosity until
the anti-protons are used or lost. At that point, the store is ended and the remaining
anti-protons, if any, are sent to the recycler for the next shot.

2.3

Coordinates, Conventions & Kinematics

In performing pQCD calculations for top quarks cross section like in Equation (1.46), the
partonic center-of-mass frame was not the same as the hadronic center-of-mass frame (the
laboratory frame). Since the former frame for the two colliding partons is a priori undetermined with respect to the later frame, the scattering polar angle from the beamline θ is not
a good observable to describe the theory and the experiment as well. Therefore, it would be
more desirable to describe the kinematics of interactions in terms of variables transverse to
the beam axis i.e. invariant under longitudinal boosts.
CDF is cylindrically symmetric about the beamline and extends through the B0 straight
section. Protons enter the detector from the west side, and anti-protons enter from the
east side. The z axis lies along the beamline, with +z direction defined as the direction of
protons traveling from west to east. Longitudinal refers to the component along the z axis,
and transverse refers to components perpendicular to the z axis. θ is the polar angle from the
beamline and r is the radial dimension transverse to the beamline; tan θ = r /z . Commonly,
x and y are not used; +x points to north, while +y points up. The azimuthal angle φ, is
69

defined clockwise when looking from west to east; tan φ = y/x.
Instead of using θ to describe the kinematics, we use the concept of the rapidity, defined
as

y=

1 E + pz
ln
2 E − pz

(2.1)

for a particle with energy E and z-momentum p z . At the TeV scale, the particle mass is
generally negligible compared to its momentum, making the rapidity y be approximated by
the pseudorapidity η defined as

η = −ln tan

θ
2

(2.2)

which has one-to-one correspondence with the polar angle π ≥ θ ≥ 0 for − ∞ < η < ∞. η
is invariant under boost in the beam direction. Another consequence of this is that instead
of E and p, the transverse momentum pT = p sin θ and transverse energy ET = E sin θ
are used. Transverse energy is somewhat ambiguous concept because energy is not a vector,
but it is convenient to treat only the fraction of energy which is attributable to the motion
transverse to the beam. The sum of the transverse energy of all particles in the event should
sum to zero in the absence of any particles escaping detection. Therefore, if a particle has
escaped the detector without interactions, the x and y components of its transverse energy
can be inferred from the transverse energy apparently missing from the event. In practice,
/
this is the case for neutrinos, so this missing energy, (ET ), is the distinctive signature of
neutrinos in our t¯+jet samples (see section 3.5).
t
This concludes that we need to use the kinematical variables (η, φ) to describe events
in CDF and hadronic collisions in general. Therefore, the separation between objects in the
detector is invariant under longitudinal boosts and defined as

∆R =

∆η 2 + ∆φ2
70

(2.3)

A very important consequence for the introduction of separation is that it provides a practical
definition of a hadronic jet, and ∆R specifies the cone size of a jet formed by the multiple
hadrons within it (see section 3.3).

2.4

The CDF Detector

Broadly speaking, the CDF detector contains three main components arranged coaxially
around the beamline. The innermost one is the tracking system, which contains a set of silicon strips and a drift chamber inside a superconducting solenoid to measure the momenta of
charged particles. Outside of the solenoid are the calorimeters, which measure the energies of
electrons, photons and hadrons. Finally, outside of the calorimeters are the muon chambers,
which are used to identify muons.
We can see that θ = 90◦ corresponds to η = 0, and a value of η = 1 corresponds to θ ∼
40◦ . As a quantitative illustration, for two objects back-to-back in the central region of the
detector, ∆η < ∆φ and hence ∆R ≈ ∆φ ∼ π.

2.4.1

Tracking System

The tracking system is used to reconstruct the trajectories and momenta of the charged
particles. Hence, it is placed inside the solenoid closest to the interaction point. It consists
of two major subsystems, the silicon microstrip detectors and the Central Outer Tracker
(COT).
The best position resolution comes from the silicon detectors which are constructed of thin
strips of p-type silicon (100-µm) doped with n-type silicon. When a charged particle strikes
the strip, it ionizes the silicon, creating electron-hole pairs. The voltage applied across the
silicon draws electrons to one end of the strip, the holes to the other end. The resulting charge
is collected by readout chips mounted at the end of the sensors. With double-sided silicon
strips, perpendicular to each other, CDF can measure simultaneously the two-dimensional
71

particle’s position. Multiple layers of the silicon makes three-dimensional reconstruction of
particle tracks possible. However, due to its cost, CDF can not use silicon through out the
entire tracking volume. In total, the silicon detectors form a cylinder 1.9 m long with a
radius of 28 cm and is, in turn, composed of three separate components. An end view of the
three components are shown in Figure 2.2.
Layer 00 (L00) [85] is the innermost silicon component and lies only 1.6 cm from the
beamline. It consists of single-sided silicon strips with improved resistance to radiation
damage and it improves the overall tracking resolution. The main component of the Silicon
detector is the Silicon Vertex Detector (SVX II) which extends from r = 2.1 cm to r =
17.3 cm around the beamline and covers η < 2.0. It consists of five layers of doubled-sided
silicon strips; each layer consists of twelve ladders arranged radially. Strips on one side are
parallel to the z axis (and thus provide r – φ information and are called axial strips) and
those on the other side are 1.2◦ offset from the z axis (and thus provide r-z information
and are called stereo strips). The SVX detector is particularly useful for resolving displaced
secondary verticies; the common feature of b-jets (see section 3.4). Lying outside of the
SVX II is the Intermediate Silicon Layer (ISL). In the central region (η < 1.0), there is one
layer of silicon at r = 22 cm, and two layers at r = 20 and r = 28 cm for 1.0 < η < 2.0,
using double-sided strips. Being further away from the beamline, the ISL is less affected by
radiation damage. It serves to improve the linking of tracks between the silicon detector and
the COT.
The COT is a 3 m cylindrical drift chamber with a total of 2520 cells organized into
eight radial sections (superlayers), lying outside the silicon detector. It extends to a radius
of r = 137 cm, covers the range η < 1.0. The chamber is filled with a nearly equal mixture
of Argon and Ethane gases, with a small ∼ 1% component of alcohol. The superlayers are
arranged in alternating axial and stereo sections. Each cell contains both sense wires and
potential wires. When voltage is applied to the potential wires, a charged particle traveling
through the chamber ionizes the gas, and the electrons are attracted to the sense wires.
72

Figure 2.2: A cross-sectional view of silicon detectors in the r-φ plane, from [82].

73

These signals on the sense wires are processed by COT electronics that provide hit time
and charge deposition information from each wire [86]. Figure 2.3 shows CDF during the
installation of the COT into the center of the tracking system, surrounded by the calorimeter
(blue and red), with the plug pulled out (to the left of the photograph).
To determine the time at which the charged particle passes through the COT, a Time
Of Flight detector (TOF) made out of scintillating counters is used. TOF sits above the
COT just inside the solenoid at η < 1.0. and has a timing resolution of 100 ps. Charged
particles passing through the TOF create bursts of light which are collected by photomultiplier tubes (PMTs) attached at each end of the scintillators [87]. The TOF system measures
the difference in time between the time of a particle interaction and the arrival of a charged
particle. It is used -together with the other information- for particle identification, especially
for particles within b-jets.
The whole tracking system is contained within a superconducting solenoidal magnet that
provides the magnetic field necessary for measuring charged particle momenta in the central
tracking system. It is built from Aluminum-stabilized NbTi conductor and operates with a
current of about 4650 A and provides up to 1.5 T magnetic field in the z direction. The
solenoid (5 m in length with radius of 1.5 m) is kept inside a cryostat where it is cooled with
liquid Helium [88].

2.4.2

Calorimetry

As the tracking system serves to determine the momenta of the charged particles, the
calorimeter system determines the energies of the particles.
The calorimeter uses plastic scintillators to collect energy from the charged particles. A
higher energy particle passing through the scintillator excites the atoms in it, causing light to
be emitted. The light is collected by the PMTs. At the same time, the particle loses its energy
and creates showers of less energetic particles, which in turn, scatter off to create more and

74

Figure 2.3: Installing the COT, from [82].

75

more showers of less energetic particles. Each charged particle passing a scintillator produces
light. The measurement of the full shower profile allows a more precise determination of the
particle’s energy; thus, most calorimeters have multiple layers. However, CDF reduces the
large amount of the scintillator needed by alternating layers of scintillator with layers of a
dense absorber. This creates a sampling calorimeter, which measures a fraction of the total
energy and estimate the true energy based on calibrations with test beams.
Different particles interact differently with the scintillators. Electrons rapidly lose their
energy primarily through the radiation of photons (bremsstrahlung) and form electromagnetic showers. Photons also radiate their energy much faster than the massive particles.
Charged hadrons are not deflected by the electric fields produced by the scintillators (losing
far less energy to ionization). However, when they pass sufficiently close to the nuclei, their
loss energy mediated by the strong force result in showers of less energetic particles. For
muons, which are massive version of electrons, the electric field is not large enough to make
muons ionizing their energy. On the other hand, they do not interact by the strong force.
Hence they generally pass straight through the calorimeters depositing very little of their
energy.
This motivates the construction of two distinct calorimeters; an electromagnetic calorimeter (EM) and a hadronic calorimeter (HA). Muons on the other hand must be measured
separately by muon chambers.
Lead and steel are used as the absorbing material for the EM and HA calorimeters
respectively. The reason behind using a much denser material for the EM calorimeter is to
ensure that the electrons are all absorbed in the EM layer of the calorimetry system. On the
other hand, and because of the slower rate of radiation of hadrons, the HA calorimeter must
be much larger than the EM calorimeter. The energy resolution of the EM calorimeter is
better than that for the HA. The total energy of the initial particle is related to the number
of layers it interacts with in the calorimetry system before losing all of its energy, as well as
to the amount of light captured by the PMTs. This measured energy must be corrected for
76

different aspects. This will be explained in detail in section 3.3 .
Both the EM and the HA calorimeters are composed of subsystems to fully envelop the
rest of the detector. The Central Electromagnetic calorimeter (CEM) [89] and the Central
Hadronic calorimeter (CHA) [90] are located in the central region of the detector (η < 1.1),
immediately outside the solenoid, with CEM being closer to the beam. The CEM has 31
layers of lead and scintillator. It is organized into towers in a projective geometry in η and
φ. The CHA has 32 alternating layers of steel and scintillator with similar geometry of the
CEM.
In the CEM, and at the point where the shower has the greatest number of particles
(called shower maximum), a special proportional wire chamber called the Central Electromagnetic Shower Maximum (CES) is embedded in the CEM. This CES with finer position
resolution allows for better precision than the surrounding layers. On the other hand, the
CHA calorimeter can not be extended as far forward as the CEM one. The gap that is left
due to this limitation is filled by the Wall Hadronic calorimeter (WHA), which has 15 layers
of steel and scintillator. It extends the hadronic calorimetry coverage in η to the endwall
region (0.8 < η < 1.2).
The plug calorimeters, so named because they fit into the end of the detector like a
plug, have analogous components to the central calorimetry. The Plug Electromagnetic
calorimeter (PEM) followed by the Plug Hadronic calorimeter (PHA) cover the region from
1.1 < η < 3.6 . Both PEM (PHA) have 23 layers of alternating scintillator and lead (steel)
respectively [91]. The Plug Electromagnetic Shower Maximum calorimeter (PES) serves the
same as the CES.

2.4.3

Muon System

CDF has additional drift chambers and scintillators located at the outermost part of the
detector to identify where and when muons passed through. These chambers are mounted

77

behind thick steel slab which serves to decrease the non-muonic particles that may make it
through the calorimetry.
The muon system composed of wire drift chambers (CMU, CMP, CMX and MBU) and
scintillators (CSP, CSX and BSU) as explained in the next paragraph. Muons that pass
through the wires will be reconstructed as a stub. Because muons stubs can also be generated
by cosmic ray muons, timing information helps identify muons that come from collisions.
Scintillators paired with the drift wires provide timing information that is used to reduce
the cosmic background. The reconstructed tracks in the tracking system (COT and silicon)
are then extrapolated to these stubs to form muon candidates. A track which is linked to
the CMX is called a CMX muon, while a track linked to both the CMU and CMP is called a
CMUP muon. Both of these muons are used in our measurement. The muon identification
algorithm is discussed further in section 3.2 .
Being located at the outermost part of the CDF detector, the muons detectors are constrained by the size and geometry of the collision hall (B0), sometimes resulting in unusual
configurations. The Central Muon Detector (CMU), directly outside the CHA calorimeter,
sits in the very central region (η < 0.6). However, this region lacks of steel shielding. Hence,
the Central Muon Upgrade (CMP) is constructed with large piece of steel (the return yoke
for the solenoid magnet) to reduce the background [92]. This is why CMP forms a box
rather than a cylinder. The Central Muon Extension (CMX), extends the chamber coverage
to cover the rest of the COT fiducial region (0.6 < η < 1.0), using several pieces (arches,
keystone and miniskirt) which fit in (on the sides, on top and beneath) the detector respectively. Additional forward coverage (1.0 < η < 1.5), is produced by the Barrel Muon
Detector (BMU).
The Central Muon Scintillator Upgrade (CSP) and the Central Muon Extension scintillators (CSX) are used for the timing information and are paired with the CMP and CMX
respectively. Therefore, they provide the same coverage as the chambers. Similarly, the Barrel Muon Scintillator (BSU) are attached to the BMU. The coverage of the drift chambers
78

Figure 2.4: Muon coverage in CDF, from [84].

CMU, CMP and CMX which are used in our measurement is shown in Figure 2.4

2.4.4

ˇ
Cerenkov Luminosity Counters

CDF has luminosity counters that sit near the beamline, in the very forward region (3.7<
ˇ
η <4.7). The Cerenkov Luminosity Counters (CLC), filled with gaseous Isobutane, are
ˇ
designed to detect the Cerenkov radiation that results from charged particles traversing it.
The light is collected and sent to PMTs to be read out. The CLC has 3 layers with each
layer having 16 counters on each side [93].
79

This detection rate is converted to a measure of the instantaneous luminosity via:

L=

f nNp Np
¯
cm-2s-1
4πσxσy

(2.4)

where n is the number of the bunches, f is the frequency of bunch revolution, Np and Np
are the total number of protons and anti-protons per bunch, and 4πσx σy represents the
transverse area of the beam, assuming a Gaussian distribution of particles with widths
σx and σy in the x and y directions. During the store, the instantaneous luminosity falls off
exponentially as expected. On the same token, the integral of the instantaneous luminosity
over a given time period (the integrated luminosity) increases by the time.
The performance of the Tevatron has improved dramatically since the beginning of RunII.
Currently, more than 10 fb−1 of luminosity has been delivered at CDF with more than 8 fb−1
stored to tape. By comparison, the total integrated luminosity for RunI was 126 pb−1 . The
current estimation on the uncertainty of the CDF luminosity measured by CLC is around
5.8% which is derived from the CLC accuracy and the uncertainty on the theoretical cross
section for inelastic p¯ collisions. The systematic will be addressed in section 4.2.2 .
p

2.5

Data Acqusition System

In Section 2.2.3, it was stated that the bunch crossings occur at a rate of 2.5 MHz and
several events result from each bunch crossing. With an event size of ∼ 1/4 MB, saving the
full output would be impractical. On the other hand, a small fraction of the events contain
interesting physics. The actual CDF output to tape is about 20 MB/sec. This means that
CDF rejects ∼ 99.995% of events!
Therefore, the trigger and acquisition system should be designed to keep -with high
efficiency- events that are useful for physics analysis and reject the unwanted ones. It should
also be dead-timeless; meaning the system should make a decision before the next crossing
80

occurs [96]. Three tiered-decision levels, called trigger levels, are used to pick the desired
events. Each level makes a decision using more detailed information, and thus with more
time allowed for the decision than the previous level. An outline of the data flow through
the trigger system is shown in Figure 2.5. Each level is described briefly below.
The Level 1 Trigger system (L1) is hardware-based, which reduces the event rate to
approximately 25 kHz. The XFT, eXtremely Fast Tracker, reads out only some information
from the COT, calorimetry system and the muon chambers and then performs very basic
event reconstruction [97]. These reconstructed events are then passed to the Extrapolation
Unit (XTRP) to make a decision to keep or reject the event. These accepted events are then
buffered for the Level 2 Trigger system (L2), while the rejected events are simply discarded.
The L2 system is hardware and software-based and reduces the event rate further to ∼
300-400 Hz. One of the hardware components, Silicon Vertex Trigger (SVT), [98], reads
Silicon information from the SVX and looks for displaced secondary vertices which are important for our analysis (see section 3.4). L2 reads also the clustering information from
the CES. This information is sent to a single computer that analyzes the data and decides
whether to accept the event. Like the L1 system, events that are accepted by L2 are buffered
for the Level 3 Trigger system (L3), while those that are not are discarded.
The highest level of trigger at CDF, L3, is solely a software-based system, and is implemented as a large computer farm . The events that pass L2 are sent first to the Event
Builder (EVB), which is a small farm of CPUs. The L1 and L2 triggers only use a small
subset of the event data while making a decision, in order to reduce the time needed, while
the rest of the event data is stored in several buffers. The EVB reads out the disparate information and assembles it into a data format readable by the L3 farm. L3 fully reconstructs
the event and analyzes its topology before making the final decision to accept or reject the
event. The accepted events are then sent to the Consumer Server Logger (CSL), where they
stored temporarily before being written to tape for permanent storage at a rate of about 200
Hz. Details about the data set used in this analysis is addressed in section 3.6 .
81

Dataflow of CDF "Deadtimeless"
Trigger and DAQ
7.6 MHz Crossing rate
132 ns clock cycle

Detector

L1 Storage
Pipeline:

Level1:
L1 trigger

42 Clock
Cycles Deep

7.6 MHz Synchronous pipeline
5544ns latency
<50 kHz Accept rate

L1 Accept

Level 2:
L2 Buffers:

L2 trigger

4 Events

Asynchronous 2 stage pipeline
~20µs latency
300 Hz Accept Rate

L2 Accept
L1+L2 rejection: 20,000:1

DAQ Buffers

L3 Farm

Mass
Storage

PJW 10/28/96

Figure 2.5: Data flow through CDF trigger system, from [84]. The diagram depicts 132 ns
bunch crossing times (L1 42 clock cycles deep). However, the current crossing time is 396
ns (corresponds to L1 14 clock cycles deep).
82

Within the context of the CDF three-level trigger system, each level has several different
trigger paths. Each trigger path has specific acceptance requirements that differ than the
other paths. These requirements mimic the selection cuts used in the different analyses. An
event then can be accepted by more than one path in the same trigger level and hence passed
on to the next trigger level. The t¯+jet triggers used are explained in section 3.2 .
t

83

Chapter 3
Event Selection & Background
Modeling
3.1

Introduction

In order to measure the cross section for t¯+jet as described in Equation (1.54), one needs
t
to quantify the signal and the background events and to model the efficiency for detecting
the signal. This means that we need to associate: the tracks in the tracking detectors, the
clusters in the calorimetry and the stubs in the muon chamber (as discussed in detail in
chapter 2) with the physical objects: electrons, muons, jets and neutrinos. This chapter
details how we: trigger leptons and select them afterword, cluster jets and tag those that
/
come from b quarks, calculate the ET and the total transverse energy (HT ), cluster the
extra hard jet, and finally simulate the different backgrounds of our sample and estimate
their contributions. The main goal we keep in mind while we are performing all these tasks
is to maximize the number of signal events (Ntt+jet ) and to minimize the background ones
¯
(Nbkg ).

84

Figure 3.1: A cartoon depicts the topology of t¯+jets event where extra hard jets are produced
t
by the ISR, FSR and radiation off from the decay products of the top quark and the hadronic
W boson.

85

3.2

Lepton Identification

The identification of electrons, muons and jets at CDF begins during data-taking with the
online CDF trigger system, which was described in section 2.5. Each trigger path applies only
basic selection requirements. The events that pass these basic selections will be processed
offline after being written to tape. This allows more comprehensive and tighter requirements
to be applied, which in turn allows for a reconstruction of the objects with a higher quality.
A complete information can be found in [99].

3.2.1

CEM, CMUP & CMX Triggers

We use three lepton path triggers for t¯+jet events: one for electrons detected in the CEM
t
detector and two for muons detected in the CMUP (CMU & CMP) and CMX detectors. The
trigger path dataset for electron is ELECTRON-CENTRAL-18 which is a composite of three CDF
trigger level (L1-CEM8-PT8, L2-CEM16-PT8 and L3-ELECTRON-CENTRAL-18). The trigger
path for CMUP muons is MUON-CMUP18, with three corresponding triggers (L1-CMUP6-PT4,
L2-CMUP6-PT8 and L3-MUON-CMUP-18) while the CMX muon dataset is MUON-CMX18 with
its triggers (L1-CMX6-PT8 and L3-MUON-CMX-18). Each level applies stricter cuts than the
previous one. The CEM trigger requirements are given in Table 3.1 while the CMUP and
CMX triggers requirements are summarized in Table 3.2.
An electron is identified as an isolated track in the COT matched to an EM calorimeter
cluster. At level 1 (Table 3.1), its calorimeter transverse energy (ET ) is required to be
approximately the same as its COT track momentum PT . The ratio of energy deposited in
the hadronic calorimeter to that in the electromagnetic, EHAD /EEM , should be small to
reject showers from hadrons that might imitate electromagnetic showers. The calorimeter
energy restriction is tightened more at level 2 while the track requirement and the ratio
remain the same. Finally, at level 3, the calorimeter energy has to be at least 18 GeV
and both the track requirement and the ratio are also enforced. In order to remove more
86

Trigger Level

Property

Requirement

Level 1

ET
PT
EHAD /EEM

≥ 8 GeV
≥ 8 GeV
≤ 0.125

Level 2

ET
PT
EHAD /EEM

≥ 16 GeV
≥ 8 GeV
≤ 0.125

Level 3

ET
PT
EHAD /EEM
Lshr

≥ 18 GeV
≥ 8 GeV
≤ 0.125
≤ 0.4

Table 3.1: ELECTRON-CENTRAL-18 CEM Trigger.

Trigger Level

Property

CMUP Requirement

CMX Requirement

Level 1

PT
COTPT
# hits

(CMU) ≥ 6 GeV
≥ 4 GeV
≥ 1 CMP stub

≥ 6 GeV
≥ 8 GeV
≥ 4 COT hits

Level 2

COTPT

≥ 8 GeV

no trigger

Level 3

COTPT

≥ 18 GeV

≥ 18 GeV

Table 3.2: MUON-CMUP18 and MUON-CMX18 Triggers.

87

background from hadronic showers, a lateral sharing profile requirement for the EM shower
(Lshr ) is applied. This variable, which quantifies the distribution of EM energy in adjacent
EM calorimeter cells, must be consistent with the expected profile for electrons derived from
test beam data.
Likewise, the muon is identified as an isolated track in the COT matched to a stub in
the muon chambers. At level 1 (Table 3.2), a ≥ 6 GeV stub in the CMU chamber should
be extrapolated to a ≥ 4 GeV track in the COT. In order to label this stub as a CMUP
muon, there must be also at least one track stub in the CMP. Tracks of ≥ 18 GeV should be
extrapolated to both CMU and CMP at level 3. The distance between the tracks and the
stubs should be ≤ 10 cm and 20 cm for CMU and CMP respectively. In order to reduce fake
muons that result from particles that penetrate all the way through to the muon chambers,
an ionizing energy threshold is applied at level 2.
The forward CMX trigger operates very similarly. However, it’s efficiency is not as great
as the CMUP because the coverage is not as extensive. A ≥ 6 GeV stub in the CMX
chamber should be extrapolated to a ≥ 8 GeV track in the COT at level 1. While no
additional requirements are applied at level 2, tracks of ≥ 18 GeV should be extrapolated
to CMX chamber (within ≤ 10 cm) at level 3.

3.2.2

Offline Selection

The offline lepton selection imposes tighter cuts than those used in the triggers in order to
purify the sample by removing fake leptons. Tables 3.3 and 3.4 respectively describe the
standard cuts used for electron and muon selection at CDF. Leptons that pass these cuts
are referred to as isolated or (tight) leptons; otherwise, they are non-isolated or (loose).
Table 3.3 shows the tighter requirements for each of the electron variavles ET , PT ,
EHAD /EEM and Lshr required at the the offline level.
A true electron deposits almost all of its energy in a single calorimeter cell i.e. its energy

88

Property

Requirement

ET
|η|
PT
EHAD /EEM
Lshr
ET /PT
Isolation
COT Quality
Photon Conversion
Q · ∆x (cm)
χ2
strip

≥ 20 GeV
≤ 1.0
≥ 10 GeV
≤ 0.055+(0.00045)E
≤ 0.2
≤ 2 GeV (if ET < 100 GeV)
< 0.1
True
False
−3.0 < Q · ∆x < 1.5 and |∆z| < 3.0
≥ 10

Table 3.3: Offline selections for tight electron.
is well collimated. Isolation is defined as:
∆R<0.4
Electron
ET
− ET
Isolation =
Electron
ET

(3.1)

∆R<0.4 is the the energy deposited in EM calorimeter cells surrounding the cell
where ET

containing the electron within a cone of radius (∆R < 0.4) (see Equation (2.3)). The
isolation variable should be less than 0.1 in order to discriminate between the true electrons
and some jets that fake electrons. The COT track quality cut demands that the electron’s
track must have hits in at least 3 axial and 2 stereo superlayers (see section 2.4.1). Such an
electron is accepted for physics analysis. Photons passing through matter can convert into
electron-positron pairs. An electron resulting from photon conversion is rejected by searching
for its corresponding positron track; an opposite charged track separated from the electron
by less than 2mm in the r − φ plane at the point of closest approach. If such tracks are
found, the electron is rejected because it is more likely to come from photons and not from

89

a hard collision of interest. The COT tracks should be extrapolated to the CES calorimeter
(see section 2.4.2). The difference between the extrapolation position and the actual hits
in the CES in the (r − φ) plane (∆x) is required to be small as shown in the Table, where
Q is the charge of the track. On the other hand, a similar difference in the (r − z) plane

(|∆z|) should be less than 3.0 cm. Lastly, χ2
strip is comparing the shape of the shower profile
measured in the CES to the expected shower profile of the electron.
Table 3.4 summarizes the standard cuts that are applied to all muons. However, additional cuts are required for the different muon chambers. An important cut is the minimum
ionization energy that the muon should leave in the calorimetry system, as discussed in the
muon trigger. This will remove fake muons that punch through the calorimeters, like pions
(in the EM calorimeter), and energetic hadrons (in the HAD calorimeters). As for electrons,
Isolation and COT quality cuts are enforced for muons . A ∆x cut for the stubs in the
different subdetectors is also applied. Cosmic ray vetos and impact parameter cuts (d0 ) are
applied to reject muons from cosmic rays. Cosmic rays passing through the detector would
leave two tracks in the muon chamber separated in φ by 180◦ (back-to-back) and separated
in time as well. Using a specialized algorithm, such cosmic muons are vetoed (see section
2.4.3) . On the other hand, the distance between the position of the primary vertex and the
z axis, d0 , must be small. If there is no match with the Silicon detector, then the impact
parameter cut is made looser. Besides removing cosmic ray muons, the d0 cut also rejects
the decays-in-flight of pions (π → µν) and kaons (K → µν) that do not originate from
the primary vertex.

3.3

Jet Reconstruction & Corrections

The concept of hadronization and forming jets was introduced earlier in section 1.3.2. When
the colored partons receive large amount of energy in a collision, they hadronize into color
neutral hadrons. Although the state of the event before hadronization is not a physical
90

Property

Requirement

ET
|η|
EEM
EHAD
Isolation
COT Quality
∆x (cm)
Cosmic Rays Veto
d0 (cm)

≥ 20 GeV
≤ 1.0
< Maximum(2.0 , 2.0 + 0.0115(P − 100.0))
< Maximum(6.0 , 6.0 + 0.0280(P − 100.0))
< 0.1
True
< 3.0 (CMU), < 5.0 (CMP), < 6.0 (CMX)
True
< 0.02 (with Silicon) and < 0.2 (with no Silicon)

Table 3.4: Offline selections for tight muon.
observable, it is useful when discussing the phenomenology of hadron physics and is referred
to as the parton level. This is used as the basis for the QCD MC event generators as will be
addressed in section 3.7 . These partons, while interacting with each other, radiate quarks
and gluons. The latter, in turn, can also radiate more quarks and gluons and this chain of
radiation is called a parton shower. After all the partons in the event hadronize, the colorless
content of the event (such as pions and kaons) is referred to as the hadron or particle level.
The particles at this level are observables, and they are the states which interact with the
calorimetry system. After the particles interact with the calorimetry system, the resulting
description of the event is referred to as the detector or calorimeter level [101]. In other words,
what we observe/measure in the calorimeter is the jet that is collimated in the direction of
the parton. However, the MC generators can predict all the three levels. The three levels of
the event are depicted in Figure 3.2.
In principle, the sum of the total energy at the calorimeter level within a cone around
the direction of the parton level should be strongly correlated with the parton’s energy. The
task left then is to reconstruct/cluster the calorimeter jet in order to measure its energy.
Jets at the calorimeter level are first reconstructed offline using a cone-based clustering

91

Figure 3.2: The three levels of a jet at CDF event. The time flow direction in the cartoon is
upward, from [100].

92

algorithm called JETCLU [102]. Briefly, an ET ordered list of seed towers with ET > 1.0
GeV is created. Preclusters are formed from the adjacent towers, provided that the towers
are within a cone of ∆R ≤ 0.4 of the seed . The centroid of the jet is calculated by summing
the η − φ positions of the towers weighted by their ET . The preclusters in the cone centered
on this new centroid are then summed (the new centroid is iterated until the jet is stable,
i.e. the set of towers contributing to the jet is stable and a single tower does not contribute
to more than one jet). Overlapping jets are merged if they share ≥ 75% of the smaller jet’s
energy. Otherwise, the towers are assigned to the closest cluster.
However, the jet energy measured is raw jet energy and has to be corrected for various
detector and physics effects before the experimental measurement can be compared with
theoretical and MC predictions. The jets can be corrected to the hadron level and/or the
parton level. The precision to which this correction can be achieved determines the precision
of many measurements. For example, a 1% uncertainty on the overall jet energy correction
(referred to as Jet Energy Scale –JES) results in a 1 GeV uncertainty on the top quark mass
[103]. The JES systematic uncertainties for our measurement will be addressed in section
4.2.2 .
parton
is related to the calorimeter transverse moT
calorimeter and the particle transverse momentum P particle through a series of
mentum PT
T
The parton transverse momentum P

corrections [104]:

P

parton
particle
calorimeter ×C −C )×C
= (PT
−CUE +COOC
η
PU
Abs −CUE +COOC = PT
T
(3.2)

This equation manifests a total of five levels of corrections, numbered 1 through 7 (levels 2
and 3 correspond to corrections which are no longer used at CDF).
The corrections are:
- Level 1: Relative Correction (Cη ).
The CDF calorimeter response to jets is not flat in η. The non-uniformity in η arises
93

from the separation of calorimeter components (cracks) at η = 0 where the two halves
of the central calorimeter join and at |η| ≈ 1.1, where the plug and central calorimeters
join. The response varies across η also due to the different amounts of material that
both the plug and central calorimeters have. This makes the central region (0.2 <
|η| < 0.6) better understood and instrumented than the plug region. Using a sample
of pure dijet events, the relative correction is derived by balancing the ET of a jet in
the central region against a random (probe) jet in the other different regions. Since the
relative jet energy response is different between data and MC, corrections are derived
separately for data and MC.
- Level 4: Pileup Correction (CPU ).
At high instantaneous luminosity, more than one p¯ interaction can occur in the same
p
bunch crossing at the Tevatron due to the large p¯ cross section. The average number
p
of p¯ interactions per bunch crossing (pileups) depends linearly on the instantaneous
p
luminosity. The number of the reconstructed primary z -vertices, Nvtx, is the best
estimate of the pileups. This is shown in Figure 3.3 . Each extra interaction is found
to add ∼ 0.3 GeV of energy on average [100]. This extra energy should be subtracted
from the jet energy.
- Level 5: Absolute Correction (CAbs ).
By applying the absolute correction, the calorimeter jet can be fairly compared with
the particle jet. As particles pass through the CDF calorimeters, some of their energy
is not collected due to calorimeter cracks and the overall nonlinear detector response.
The absolute correction is derived entirely using MC by comparing particle level and
calorimeter level jets. After this correction, the energy scale of a jet is independent of
the CDF detector. All of the jets used in our measurement are corrected up to level 5
i.e. they have the absolute correction applied.
- Levels 6 & 7: Underlying Event Correction (CUE ) & Out-Of-Cone Correction (COOC ).
94

Nvtx Vs Luminosity
6
5

p0
p1

Nvtx

4

1.13
0.01605

3
2
1
0

0

50

100

150

200

250

300

Luminosity

Figure 3.3: Nvtx verses the instantaneous luminosity L .
Levels 6 and 7 corrections are not applied in correcting the jets in our measurement,
but are only used for the systematic uncertainty calculations. In order to correct the
calorimeter jet further back to the parton level, both of these corrections should be
involved.
Most of the time when p and p beams collide, they pass through each other and
¯
undergoes a soft collision. The UE is everything in a collision except the production
of the hard scattering (the UE is also referred to as the beam remnants). It is an
unavoidable background to many collider observables and can not be separated out on
an event by event basis. By studying distributions which are sensitive to the UE, the
MC can be tuned to fit the effects observed in data. The correction thus derived then
removes the UE energy associated with the beam remnant. [105].
On the other hand, the OOC correction accounts for the energy lost due to the finite
size of the jet cone (0.4 in our case). Part of the parton jet energy can lie outside the

95

cone, either because of fragmentation effects or FSR. For example, this OOC correction
adds ∼ 2.2 GeV of energy to a 20 GeV particle jet [100]. On average, the UE correction
tends to partially remove the OOC correction.
For a high PT jet, the uncertainty shifts the jet energy by ∼ 3% while at low PT , the
uncertainty can shift the jet energy by ∼ 10%. Converting the JES uncertainty into a
systematic uncertainty on our own measurement, the resulting systematic uncertainty
is by far the dominant source of uncertainty in our result. This will be further described
in section 4.2.2 .

3.4

b-jet Tagging

In section 1.4.2, I discussed the importance of identifying the jets that come from b quarks.
This tagging process, takes advantage of the relatively long lifetime of the B hadrons. This
will give the B hadrons a chance to travel, transversely away from the primary vertex, a few
millimeters (∼ 5mm) before they decay. The tracks that result from the displaced decay
(referred to as the secondary vertex) play the key role in tagging the b-jets. With the
secondary vertex tagging algorithm (SecVtx) [106], using the excellent position resolution
the Silicon detector provides (as discussed in section 2.4.1), CDF can efficiently tag b-jets.
The SecVtx algorithm uses a large number of selection requirements that can be found
in Ref. [106]. Very briefly, the algorithm starts by using the precise knowledge of the
primary vertex itself. It looks then for a large value for the impact parameter, d0 , (defined
in Table 3.4) and for its significance (d0 devided by its uncertainty). The distance in the
transverse plane that the secondary vertex is displaced from the primary vertex (Lxy ) plays
a very key role in the SecVtx algorithm. It determines whether the vertex is sufficiently far
from the primary vertex. The algorithm looks for Lxy significance as well. However, Lxy is
signed relative to the jet direction. It is defined to be positive when the secondary vertex is
displaced in the same direction as the jet (a signature of a b-jet). A negative Lxy (i.e. the
96

vertex displaced on the opposite side) indicates a tagging of jets that do not contain true b
quarks (i.e. the light jets u, d, s). This case is called a mistag. If the predicted secondary
vertex passes all the selection requirements, the jet is marked as a tight SecVtx jet. With
less stringent selections, the SecVtx algorithm identifies a loose SecVtx jet as well. We use
the tight SecVtx tag jet in our measurement.
Due to the complexity of the SecVtx algorithm, the MC samples can not simulate the
tagging process properly. As a matter of fact, MC events have more tagged b-jets than CDF
data has. However, this overestimated efficiency can be compensated with the use of a scale
factor. The latter is the ratio of the data over MC efficiencies. Applying this factor to the
MC tagging rate gives a corrected estimate of the efficiency of b-tagging in the MC. The
scale factor is calculated to be 0.95 ± 0.05 [107]. Unlike the b-tagging, MC samples tend to
underestimate the mistag rate compared to the data. The mistag rate is badly modeled in
MC as has a strong dependence on the kinematic distributions of the tagged events. The
mistag rate can be parametrized by five jet variables ( jet ET , HT , η, φ and number of good
SVX tracks). Using this mistag matrix gives the mistag rate [107].
Figure 3.4 shows the CDF efficiency to tag b-jets in top quark MC samples as a function
of the jet ET . The bands represent the systematic error on the scale factor. While the
efficiency for finding a tight b-jet is only about 40%, the efficiency for a mistag is less than
1%, as shown in Figure 3.5. This is a clear manifestation of how b-tagging increases our
sample purity. One more feature that we can read from both plots is that while the loose
b-jet requirement increases the acceptance, and hence the statistics of our sample, it does
decrease, on the other hand, the sample’s purity. The decrease in efficiency at high jet ET
is due to the declining yield of good Silicon tracks passing the quality cuts.
However, the SecVtx algorithm can not distinguish a b-jet from a charm jet. This is why
SecVtx algorithm is sometimes called the heavy flavor tagging algorithm. The charm jet
tagging efficiency is about 8%. Because it’s scale factor is not measured, the same b jet scale
factor is applied to jets matched to charm, with the uncertainty doubled to be conservative.
97

SecVtx Tag Efficiency for Top b-Jets

0.7
0.6
0.5
0.4
0.3
0.2
Loose SecVtx

0.1
0

Tight SecVtx

20

40

60

80 100 120 140 160 180 200

Jet Et (GeV)
Figure 3.4: CDF SecVtx algorithm b-tagging efficiency, from [107].

98

SecVtx Mistag Rate

0.06
0.05
0.04
0.03
0.02
Loose SecVtx

0.01

Tight SecVtx

0

20

40

60

80 100 120 140 160 180 200

Jet Et (GeV)
Figure 3.5: CDF SecVtx algorithm mistag efficiency, from [107].

The systematic uncertainties for both the b-tagging and the mistag will be addressed in
section 4.2.2 .

3.5

Missing ET & HT Reconstructions

/
I mentioned in section 2.3 that the ET is the distinctive signature of neutrinos in our t¯+jet
t
/
sample. In general, the ET could be caused by poor energy measurement from particles
/
passing through the cracks in CDF and due to detector noise. Another source for the ET is
/
from cosmic ray events. However, a large portion of the ET is due to neutrinos. In lepton
plus jet channel, many of the backgrounds in our sample, which have no final state neutrinos,
/
can be removed by applying a ET cut.

99

−
→
/
The missing transverse energy vector (ET ) is defined as follows:

−
→
/
ET = −
where

iˆ
ETni

(3.3)

i

ni is a unit vector perpendicular to the beam axis and pointing from the primary
ˆ

vertex to the ith calorimeter tower. The sum in this equation is over all the towers in the
/
calorimeter. The energy ET is the raw energy measured in the towers. The ET is simply
−
→
/
the absolute magnitude of the ET after correcting the ET up to level 5 and correcting the
primary vertex position at the trigger level. This definition applies exactly for events with
electrons. However, if muons are found in the event (which leave minimal energy in the
/
calorimeters), the ET is corrected by first subtracting the muon ET ’s, and adding instead,
the muon PT ’s to the vector sum in the equation. In our measurement, of t¯+jet events, we
t
/
require ET > 20 GeV.
The total sum of the transverse energy of the event in the detector, HT , is simply the
scalar sum of the transverse energy of all the physical objects the event contains. Therefore,
/
HT is the sum of the electron ET or the muon PT , the ET of all jets and the ET . For an
event such as a t¯+jet event, the HT is expected to have a larger value than the backgrounds.
t
The baseline HT cut used by the Top Physics Group at CDF is 200 GeV. However, to further
reduce the backgrounds, the HT in our measurement is required to be at least 220 GeV.

3.6

Selection Cuts & Data Sets

The cartoon in Figure 3.6 depicts a t¯+jet event topology in the lepton plus jet channel. We
t
/
thus would expect to see 5 jets, of which two are b tagged, one lepton and large ET in the
final state. Consequently, each event is required to have in particular the following cuts:

100

- One tight electron (muon) with ET (PT ) ≥ 20 GeV in the region |η| ≤ 2 (CEM, CMUP
and CMX) as described in section 3.2 .
- At least three tight L5 corrected jets with ET ≥ 20 GeV and |η| ≤ 2 (see section 3.3).
- At least one of the jets above must be a tight SecVtx tagged jet (see section 3.4).
/
- ET ≥ 20 GeV & HT ≥ 220 GeV (see section 3.5).
- Dilepton veto: in order to separate the t¯+jet lepton plus jet channel from the dilepton
t
channel, any event with a second tight lepton is removed.
- Z boson veto: the events are removed if a tight lepton and a second object form an
invariant mass consistent with a Z boson (76 < Mll < 106 GeV). If the tight lepton
is an electron, the second object must be an isolated EM object, a reclustered jet
with 95% of energy deposited in the EM calorimeter, or an opposite-signed isolated
track. If the tight lepton is a muon, the second object must be an isolated muon or an
opposite-signed isolated track.
- Primary vertex reconstruction: The z-position of the primary vertex is used to cluster
jets and to ensure that leptons and jets belong to the same interaction. To ensure the
lepton originates from the primary vertex, events are rejected if the lepton z-intercept
is not within 5 cm of the primary vertex z-position. This reduces the number of events
in the sample where jets and the lepton are part of different interactions [108].
Data at CDF is divided into run periods to take into account the detector conditions that
may vary over time. In our measurement, we have used data collected through the periods
0 to 22 (from February 4, 2002 to February 10, 2009), corresponding to a total of 4.1 ± 0.2
fb−1 of integrated luminosity [109]. All the data sets with the triggers used are listed in
Table 3.5. These runs are contained in the Good Run List v28, as outlined in detail in [109].

101

Figure 3.6: t¯+jet event’s topology where the extra hard jet is produced by the ISR.
t

102

Data set ID

Trigger

Run Range

bhel0d
bhmu0d

CEM
CMUP&CMX

138425-186598
138425-186598

bhel0h
bhmu0h

CEM
CMUP&CMX

190697-203799
190697-203799

bhel0i
bhmu0i

CEM
CMUP&CMX

203819-228596
203819-228596

bhel0j
bhmu0j

CEM
CMUP&CMX

228664-241664
228664-241664

bhel0k
bhmu0k

CEM
CMUP&CMX

241665-261005
241665-261005

bhel0m
bhmu0m

CEM
CMUP&CMX

261119-272214
261119-272214

Table 3.5: Summary of the data sets used in our analysis.
After the selection cuts been applied over the data collected, the number of events that
pass through, as a function of jet multiplicity, is shown in Table 3.6 . The signal region
requires ≥ 3 jets, leading to a total of 2326 events. This represents the number (Ntt+jet ) in
¯
Equation (1.54). Next I will estimate the other terms in Equation (1.54).

103

3jets 4jets ≥5jets
1115

882

329

Table 3.6: The observed t¯ event counts as a function of jet multiplicity.
t

3.7
3.7.1

MC Models
Event Generator

In the MC world, the role of the collider machine, namely to produce events, is taken over by
event generators. As the name indicates, the output of an event generator should be in the
form of events, with the same average behavior and the same fluctuations as in real data. An
event generator can be used for multiple purposes. It tells us what kind of events one may
expect to find, and at what rate. It can be used as a tool for devising the analysis strategies
that should be used on data, so that signal-to-background conditions are optimized. It also
helps in estimating detector acceptance corrections that have to be applied to raw data, in
order to extract the true physical signal, as we will address in section 3.8.2 [110].
For these purposes, an event generator should contain a simulation of several physics
aspects. If we try to follow the evolution of an event in some semblance of time order, one
may arrange these aspects as follows: the partonic distribution for the two incoming particles,
the ISR shower (spacelike) developed from the partons from each incoming particle, the hard
process involving the two incoming partons, the FSR shower (timelike) from the outgoing
partons from the hard process. In addition, further semi-hard scatterings may occur between
the other partons in the two incoming particles, and a beam remnant is left behind when
the shower initiator is taken out of the beam particle. At relatively long time scale, the
hadronization of the outgoing partons takes place due to QCD confinement mechanism, and
finally the unstable hadrons decay into relatively stable particles [110].
104

This means that the components/routines of an event generator are: PDFs (for the
incoming partonic distributions), Parton Shower Algorithms (for ISR, FSR), Matrix Element Calculator (for the hard process) and Color String Framework (for hadronization).
PYTHIA [111] and HERWIG [112] are the most convenient generators currently used. We used
PYTHIA 6.216 to simulate our signal and then cross-checked the results using HERWIG 6.510
(for systematic studies). For the background processes in our analysis, ALPGEN [113] and
MadGraph/MadEvent [114] models are used. However, both ALPGEN and MadGraph/MadEvent
do not contain parton showering or color string routines. Thus, both are used to calculate
the matrix elements which then are passed to PYTHIA or HERWIG for parton showering and
decay. In our analysis, we used ALPGEN 2.10 and MadGraph/MadEvent 4 and passed the
information to PYTHIA.
The response of the detectors (how the final particles produced by the event generator
traverse the detector, bend in the magnetic fields, shower in the calorimeters, or escape
through cracks) is simulated using the CDF detector simulation programs, which use the
GEANT and GFLASH routines [115, 116] . Ideally, the output of this simulation has exactly the
same format as the real data recorded by CDF, and can therefore be put through the same
event reconstruction and physics analysis chain, except that here we know what the results
should be, and so can cross-check the reconstruction chain [110].
Each process that makes up our own signal and the different backgrounds are modeled
using the MC samples presented here, and then are passed through the same selection cuts.
In section 3.8, I will present the signature of these backgrounds to understand how they are
able to mimic our signal and then show in detail how to estimate their contributions to the
overall observed events.

105

3.7.2

The Extra Hard Jet

In order to model (define) the theoretical extra hard jet, the PlusJetFinder tool [125],
was built to operate on PYTHIA. PlusJetFinder identifies the objects at the parton/particle
levels in PYTHIA that arise from ISR, FSR and from the internal decay of the top system.
It only counts how often a PYTHIA event produces an ISR jet (as depicted in Figure 3.6).
It doesn’t contain jets produced by UE or jets from the decays products of the top quark
or the W boson (Figure 3.7 and Figure 3.8) as these objects are not accounted for in the
NLO calculation in [79] (the top quark is stable). However, it includes the small number of
objects that are radiated off by the top quark before it decays i.e. the FSR (Figure 3.9).
To identify the extra jet, we used the SpartyJet framework [117] with ∆R ≤ 0.4 to
cluster the objects that are fed by the PlusJetFinder. If the jet passes our PT threshold
cut (20 GeV to match [79]), then it is counted as an extra hard jet in the t¯ event. In the
t
lepton plus jet channel, t¯ events may manifest with 3, 4, 5 or even 6 jets in the final state,
t
depending on the kinematic and acceptance cuts applied, and the presence of the additional
ISR/FSR and decay products. Hence, we separated our sample into two: the t¯+jet sample
t
allows the extra jet to be produced from ISR or FSR but from no other sources and the
t¯+0jet sample allows a possible extra jet to be produced from any other source; either from
t
the internal decay of the top system whether or not it passes the threshold cut or from
ISR/FSR not passing the threshold cut. It also contains the t¯ pair events. This separation
t
has allowed us to relate the t¯+jet sample to the NLO calculation of [79]. This will let us
t
also know the probability for the extra jet to be radiated from each of the two cases. When
the cross section values of both of the samples are combined, the inclusive cross section of t¯
t
events can be determined and hence compared to the results listed in Table 1.3 .

106

Figure 3.7: t¯+jet event’s topology where the extra hard jet is produced by the radiation off
t
of the decay prodcuts of the top quark.

107

Figure 3.8: t¯+jet event’s topology where the extra hard jet is produced by the radiation off
t
of the decay prodcuts of the hadronic W boson.

108

Figure 3.9: t¯+jet event’s topology where the extra hard jet is produced by the FSR.
t

109

3.8

Backgrounds

3.8.1

The Potential Backgrounds

The following processes mimic the topology of a t¯+jet event as depicted in Figure 3.6 and
t
thus are considered as backgrounds:
- W + heavy flavor: The largest background to our signal comes from events where a
¯ c
W boson is produced in association with heavy flavor jets (bb, c¯ or c). As depicted
¯
in Figure 3.10, the W +bb events contain a real W and two b jets as well as a real
lepton and neutrino. The SecVtx requirement fails to help in this case because the two
b-jets, produced from a radiated gluon, are real b-jets. This is in particular the largest
background in our analysis. Figure 3.11 shows the other two processes where additional
jets come from higher order QCD radiation. However, the background created by the
¯
later two processes and by the Z + jet production is less than that from W +bb events.

- W + light flavor (Mistag): A substantial contribution also comes from W + jets events
with lighter quarks in the final states; called mistags as addressed in section 3.4 . Such
events enter our sample when u, d, s or gluon jet is mistagged as a b jet.
- Electroweak: Diboson electroweak processes such as WW,WZ and ZZ can produce a
real lepton and neutrino, as well as a number of jets. Therefore, they form a possible
background. Z → ll +jets is another background when one of the leptons is missed or
misidentified as a jet. However, their contribution is small.
- Single top: As shown in Figure 1.6 , single top events contain a real W boson and a
b jet and might enter our sample. Similar to the electroweak backgrounds, single top
contributions (s- and t- channels) are small.
- Non-W (QCD): Many pure QCD multijet events, that contain no W bosons, can
110

¯
Figure 3.10: W+bb event’s topology, from [28].

111

Figure 3.11: W+c¯ event’s topology.
c

112

Process

Data set ID

Type

t¯
t
t¯
t
¯
tt
QCD
QCD
¯
Wbb
Wc¯
c
Wc
Wlf
WW/WZ/ZZ
Single Top (s-channel)
Single Top (t-channel)
¯
Zbb
Zc¯
c
Zlf

ttop25
dtop0s
dtop1s/dtop2s
bhelX
gjt1X
btopXw, dtopXw, btopxx
ctopXw, etopXw, btopxx
stopXw
ptopXw, utopXw, utopxx
itopww, itopwz, itopzz
stop00
stop0m
ztopbX, btopzX
ztopcX, btopzX
ztoppX, btopzX

PYTHIA
HERWIG
PYTHIA IFSR
/
Fake Electrons for ET fit
Jet Electrons for Kinematics
ALPGEN
ALPGEN
ALPGEN
ALPGEN
PYTHIA
MadGraph/MadEvent
MadGraph/MadEvent
ALPGEN
ALPGEN
ALPGEN

Table 3.7: Summary of the background and signal models used in our analysis.
mimic our signal and fake every feature of its topology. The very large cross section
of such processes compared to the electroweak ones makes it a significant background
to our sample. QCD background mainly result from gluon multi-jet events: a dijet
production (where a jet escapes through a crack the calorimeter, and another fakes
an electron, leaving a track in the COT associated to an EM energy deposit) plus
additional jets.
Table 3.7 lists all the MC samples used to simulate these backgrounds, as well as the
signal. The ttop25 PYTHIA sample uses a top mass of 172.5 GeV. This new sample improved
our signal modeling compared to the ttop75 (175 GeV) sample that CDF used in the past.
It is worthwhile to mention at this point that because of the extreme complexity of modeling
the QCD background, a pure data-based approach is used instead. Specifically, we use an
anti-electron data set to model the QCD as described in [118, 119].

113

The modeling is based on the idea that for a QCD event to pass the selection cuts, it
/
has to have low ET because the strong interaction does not produce neutrinos. The electron
has also to pass all the offline selection cuts for electron as listed in Table 3.3 , but to fail
the Isolation cut. The rational behind this is that low Isolation (i.e. > 0.1) means more
tracks thus fake lepton contained in a jet. This is why these fake electrons are misnamed
anti-electrons.

3.8.2

Background Estimate (Method II)

Estimating the background content of processes in the SecVtx tagged lepton plus jet data
samples has been implemented at CDF in a program, colloquially referred to as, Method
II For You. This method is documented in detail in [120, 121]. The method begins by
assuming the composition of the sample; meaning we assume we know of all possible processes
contributing to the data. From this assumption we calculate the normalization of processes
sequentially, basing the normalization on each type of process on the previous one. Because
the MC is not a perfect model of the content of our sample, Method II relies heavily on
data to fill in the gaps, so to speak, for the parts of the background not well understood.
Following the assumption of Method II, the number of events observed in data (Nt¯
t+jet )
should equal the sum of the predicted signal itself and all the backgrounds:
signal
Nt¯
+ Nt¯
t+jet = Nt¯
t+0jet + NEW + NZ+jets + Nsingle top + NQCD + NW+jets
t+jet
(3.4)
The first step of Method II is to calculate the MC-based estimates of the processes. This
includes the electroweak (EW in the equation), Z +jet, single top, t¯+0jet, as well as the
t
predicted signal t¯+jet cross section itself. This is done for both pre-tagged (after applying
t
all the selection cuts except for the SecVtx one) and tagged (with the SecVtx cut) samples.
The QCD fraction is the next step in Method II. Because it is by far the least understood and
/
the most poorly predicted, Method II fits the ET spectrum of the QCD template to data
114

(as mentioned in the previous section) to estimate the fraction of our sample that is entirely
due to QCD processes. Finally, what is not QCD, EW, Z +jet, or single top is considered
W +jet. In the following we will go step by step through the procedure. The results presented
here are currently preliminary and documented in detail here [122, 123, 124].
MC Based Background Estimate
The calculated event numbers for each process (EW, Z +jet, single top and the predicted
signal) in our sample is given by:

Np¯ →X = σp¯ →X · A
p
p

dt · L

Np¯ →X = σp¯ →X · A · ǫ
p
p

dt · L

where σpp →X is the theoretical cross section,
¯

(pre-tagged sample)

(3.5)

(tagged sample)

(3.6)

dt · L is the total luminosity, A is the pre-

tagged selection acceptance (or efficiency) derived from PYTHIA and ǫ is the tagged selection
efficiency. The theoretical cross sections used in Method II along with their uncertainties
are shown in Table 3.8 . The acceptance A is simply the number of events that pass our
selection cuts divided by the total number of events generated in MC:

A=

Nselected
Ngenerated

(3.7)

However, this raw acceptance need to be corrected for the lepton scale factor and the trigger
efficiency as described in [99].
The tagging selection efficiency is more complicated. Because MC simulations do not
model tagging correctly, a tagging scale factor and mistag matrix must be integrated into
the tagging efficiency calculation as described in section 3.4. The expected tag yield is
115

Process

Cross Section [pb]

t¯+0jet
t
t¯+jet
t
Single Top (t-channel)
Single Top (s-channel)
WW
WZ
ZZ
Z →jet

5.5±1.0
1.5±1.0
1.98±0.08
0.88±0.05
12.4±0.25
3.96±0.06
1.58±0.02
787.4±50

Table 3.8: Theoretical cross sections for MC based backgrounds.
calculated by weighting MC events by the probability to be tagged. The probability that an
event is tagged is:
jets
tag
Pevent = 1 −

i

(1 − ptag)
i

(3.8)

tag
For jets matched to heavy flavor, pjet is the tagging scale factor if tagged and zero if not
tag
tagged. If the jet is matched to light flavor, pjet is the mistag probability. The tagging
efficiency is then:

ǫ=

events tag
Pj
j
Ntpre-tagged
¯+jet
t

(3.9)

Tables 3.9 through 3.12 show the corrected acceptance and tagging efficiency for the predicted
events (t¯+jet) and for t¯+0jet events as well.
t
t
QCD Based Background Estimate
To determine the QCD fraction in both the pre-tagged and tagged samples (FQCD ), we fit
/
the ET distribution of a QCD template and a MC signal template to data. The pre-tagged
fraction is essential to the rest of Method II since it provides the starting point for the heavy
116

Trigger

3 Jets

4 Jets

≥5 Jets

CEM
CMUP
CMX
LOOSE

1.9%
1.2%
0.5%
1.4%

1.9%
1.2%
0.5%
1.4%

0.4%
0.2%
0.1%
0.3%

TOTAL

5.0%

5.0%

1.0%

Table 3.9: The predicted pre-tagged acceptance (A) for t¯+0jet.
t

Trigger

3 Jets

4 Jets

≥5 Jets

CEM
CMUP
CMX
LOOSE

1.1%
0.7%
0.3%
0.8%

2.0%
1.3%
0.5%
1.5%

2.1%
1.3%
0.6%
1.6%

TOTAL

2.9%

5.3%

5.6%

Table 3.10: The predicted pre-tagged acceptance (A) for t¯+jet.
t

Trigger

3 Jets

4 Jets

≥5 Jets

CEM
CMUP
CMX
LOOSE

57%
57%
57%
56%

61%
61%
62%
62%

62%
63%
64%
62%

TOTAL

57%

61%

62%

Table 3.11: The predicted tagging efficiency (ǫ) for t¯+0jet.
t
117

Trigger

3 Jets

4 Jets

≥5 Jets

CEM
CMUP
CMX
LOOSE

53%
53%
54%
54%

58%
58%
59%
57%

61%
61%
62%
62%

TOTAL

53%

58%

61%

Table 3.12: The predicted tagging efficiency (ǫ) for t¯+jet.
t
flavor fraction and mistag estimate. To perform the fit in the pre-tagged sample, we take the
/
ET distribution of W +jets MC events as the template for signal and for the QCD template
we use the anti-electron sample as described previously in [118, 119]. Both templates are
/
fitted to the ET distribution of isolated pretag data events using a binned likelihood fitter
to calculate the QCD fraction. Once the fraction is calculated the normalization is:

pre-tagged
= FQCD · Ntpre-tagged
NQCD
¯+jet
t

(3.10)

For the tagged QCD fraction we again use anti-electron sample but this time weighted
by the tagging rate (the scale factor and the mistag rate). After calculating the fraction
from the fit, the normalization is:

tagged
NQCD = FQCD · Nttagged
¯+jet
t

(3.11)

tagged
Notice that N ¯
is 2326, as given in Table 3.6. Tables 3.13 and 3.14 show the QCD
tt+jet
fraction results for the pre-tagged and tagged samples respectively.

118

Trigger

3 Jets

4 Jets

CEM
25.5% 17.8%
CMUP 3.3% 1.2%
CMX
3.2% 0.0%
LOOSE 7.2% 3.5%

≥5 Jets
22.7%
4.6%
1.0%
5.0%

Table 3.13: Fitted pre-tagged QCD fraction, FQCD .

Trigger

3 Jets

4 Jets

≥5 Jets

CEM
11.8%
CMUP 3.0%
CMX
7.0%
LOOSE 3.6%

5.5%
0.8%
0.0%
1.6%

6.4%
1.5%
0.0%
0.0%

Table 3.14: Fitted tagged QCD fraction, FQCD .

119

W +jet Background Estimate
In the pre-tagged sample, the number of W +jets events is calculated by subtracting the MC
based processes and the QCD backgrounds from data:

pre-tagged
NW+jet
= Ntpre-tagged − NQCD − NEW − NZ+jet − Nsingle top
¯+jet
t

(3.12)
pre-tagged

NW+jet

pre-tagged

= Nt¯
t+jet

pre-tagged

· (1 − FQCD

) − NEW − NZ+jet − Nsingle top
(3.13)

For the tagged estimate, the W +jet sample is broken into two categories: heavy and light
flavor. Each of these processes produces a tagged jet very differently and therefore requires
different treatment in calculating the normalization.
The contribution of the heavy flavor background to our signal region is calculated by:

tagged
pre-tagged
NW+HF = NW+jet · fHF · K · ǫ

(3.14)

This equation reads that the estimated number of W with heavy flavor is a fraction of
the W +jet sample. The fraction of the sample with jets matched to heavy flavor, fHF , is
calculated by ALPGEN . Therefore, fHF has to be corrected by another scale factor K. The
calculation of the K factor is described here [123]. We used a K factor of (1.5 ± 0.3 stat+sys ).
ǫ is the tagging efficiency as described in (Equation 3.9). Both fHF and ǫ are calculated
¯
for Wbb, Wc¯ and Wc separately, which defines the rates for each of these processes. The
c
uncorrected fHF and ǫ for our sample are shown in Tables 3.15 and Table 3.16 , where 1 or
2 jets are matched to a b or c quark.
As discussed in section 3.4, the mistag matrix is parametrized by five jet variables. This
matrix then gives the probability that a jet with given values of the tag parametrization
120

f HF

1 Jet

2 Jets

3 Jets

4 Jets

≥5 Jets

f 1b
HF

0.76% 1.60%

2.54%

3.40%

3.80%

f 2b
HF

0.00% 0.90%

1.82%

2.99%

4.16%

f 1c
HF

5.66% 9.22% 11.72% 13.01%

12.79%

f 2c
HF

0.00% 1.53%

7.69%

3.31%

5.53%

Table 3.15: The heavy flavor fraction (fHF ) for W +jet sample.
variables will be negatively tagged. This technique is applied to estimate the number of
events in our sample due to mistag. The predicted number of background events from W
with light flavor jets is:

tagged
pre-tagged
pre-tagged
NW+LF = (NW+jet − NW+HF ) · ǫMistag

(3.15)

where the pre-tagged predicted amount of all the background events described so far is
subtracted from the total pre-tagged observed events leaving an estimate for the W plus
light flavor fraction. This estimate is multiplied by the mistag matrix ǫMistag .

3.8.3

Full Background Prediction

Table 3.17 shows the full background estimate used in our measurement as a function of jet
multiplicity. The bins with ≤ 4 jets are exclusive bins, while the ≥ 5 bin is obviously an
inclusive sample. The t¯+jet signal clearly emerges from the background in the ≥ 5 jets bin:
t
∼ 55% of the events in this bin are attributed to t¯+jet, with ∼ 40% due to t¯+0jet and the
t
t
rest to backgrounds. It is ∼ 1.5 times more likely for the extra hard jet to be radiated off
121

ǫ

1 Jet

2 Jets

3 Jets

4 Jets

≥5 Jets

ǫ1b

28.7% 30.7% 33.3% 33.7%

38.2%

ǫ2b

0.0%

52.8% 52.5% 53.0%

55.8%

ǫ1c

6.4%

7.6%

9.8%

12.6%

ǫ2c

0.0%

12.7% 14.5% 16.0%

18.0%

8.9%

Table 3.16: The tagging efficiency (ǫ) for W +jet sample.
of the ISR/FSR than to be radiated off from the decay products of the top quark and the
W boson as discussed before in section 3.7.2 . The requirement of having at least three jets
in the events to minimize the background can be justified by looking at the first two bins;
where the background dominates the data sample.

122

Process

1 Jet

2 Jets

3 Jets

4 Jets

≥5 Jets

Pre-tagged

7445

10947

6380

2724

782

Wbb
Wcc
Wc
Mistags
Non-W
WW
WZ
ZZ
Z+jets
Top -S
Top -T
¯
tt+0jet
¯
tt+jet

50.2±15.5
24.4±7.7
32.6±10.3
111.4±11.2
41.6±12.5
2.9±0.3
1.0±0.1
0.1±0.0
3.8±0.4
1.2±0.1
0.4±0.0
8.6±1.7
0.5±0.3

176.2±54.3
76.9±24.3
75.2±23.7
181.7±26.8
116.4±34.9
19.0±2.5
7.1±0.8
0.9±0.1
16.3±1.9
32.6±3.2
32.9±2.9
179.3±35.0
16.4±10.3

128.4±39.8
65.8±20.8
41.6±13.2
101.2±18.2
71.7±21.5
14.8±2.0
5.0±0.6
1.2±0.2
16.7±2.1
16.5±1.6
18.7±1.6
534.4±104.2
86.7±54.5

Predicted
Observed

50.9±16.9
10.2±6.9
27.3±9.2
6.0±4.0
13.1±4.4
2.4±1.6
33.2±9.4
6.2±7.4
25.5±20.4
9.3±7.5
6.1±0.8
2.0±0.2
1.9±0.2
0.5±0.1
0.5±0.1
0.2±0.0
6.6±0.8
1.8±0.2
4.1±0.4
0.8±0.1
4.9±0.4
0.9±0.1
555.1±108.1 105.7±20.6
163.1±102.6 182.1±114.5

278.6±37.2 930.9±117.3 1102.8±144.6 892.3±157.0 328.2±118.1
304
917
1115
882
329

Table 3.17: Background normalizations for ≥ 1 SecVtx jets using Method II.

123

Chapter 4
Results
There are two possible outcomes: if the result confirms the hypothesis, then you’ve
made a measurement. If the result is contrary to the hypothesis, then you’ve made
a discovery.
Enrico Fermi (1901-1954)

4.1

Introduction

With the background estimate in hand, and after validating the efficiency of Method II, we
now perform the measurement of the t¯+jet cross section (section 4.2) and study the different
t
kinematics of the extra hard jet (section 4.3). The thesis will conclude by a brief conclusion
in section 4.4 .

4.2
4.2.1

The Counting Analysis
Calculating The Cross Section

We saw how Method II is dependent on the measured cross section itself, and hence extracting the latter value is not simple (i.e. we can’t use Equation (1.54) directly). Instead,
124

we construct a Poisson likelihood where we take into account the background dependence. In
order to extract simultaneously the measured values of the t¯+jet and t¯+0jet cross section,
t
t
we construct a 2D likelihood from the data and the prediction for events with three, four,
or five jets:

k

λi i · e−λi
Pi =
ki!

(4.1)

where k is the number of events in data with (i ) jets, and λ is the predicted number of events
with (i ) jets. More specifically:

¯

¯

¯

¯

tt
tt
tt
tt
λ = A0j · ǫ0j · L · σ0j + A+j · ǫ+j · L · σ+j + Background (σ0j , σ+j)

(4.2)
where A is the acceptance, ǫ is the tagging efficiency, and L is the luminosity. The likelihood
is then [126]:

L = − ln (P3 · P4 · P5)

(4.3)

The likelihood is calculated for several values of the cross section and the resulting points
are fit to a 2D second order polynomial. The minimum point on the fitted parabolic is taken
as the measured cross section values while the uncertainty is extracted from the tangents to
the ellipse at the point where the likelihood is 1.15 units (68% CL) and 3.0 units (95% CL)
above the minimum point. The measured values with the 68% CL statistical uncertainties
are:

σt¯+jet = 1.6 ± 0.2stat
t

pb

(4.4)

σt¯+0jet = 5.5 ± 0.4stat
t

pb

(4.5)

125

4.2.2

Systematics

The systematic uncertainties in our measured results are calculated by varying a given parameter within its uncertainty and redoing the entire measurement. Each systematic is
described below along with any additional relevant quantities. The individual systematic
uncertainties are shown in Figures (Tables) 4.1 and 4.2 at the end of the section.
- JES : We study the effect of the JES (as described in section 3.3) on the measurement by varying it for our signal MC and background models and then re-performing
the measurement. The effect of the JES on this measurement is mainly through the
acceptance of signal and background. For the t¯+jet sample, the JES is by far the
t
largest systematic uncertainty, on the order of the statistical error. It is not hard to
understand why, given how drastically its variation changes the acceptance for the
t¯+jet events. The ET spectrum for the fifth jet is quite soft, very close to the 20 GeV
t
cut. Since the majority of the t¯+jet events sit in the fifth jet bin , any change in the
t
JES shifts the number of the events away from or into the fifth jet bin. A lower HT
cut (such as the baseline cut used by the CDF Top Physics Group) will decrease the
dependence on the JES, but to avoid too great dependence on the QCD and W +jet
modeling (mostly for the three jet bin), we raise the HT cut to 220 GeV.
- Tagging : The scale factor applied to each b or c jet is 0.95±0.05 as mentioned in
section 3.3 . The uncertainty associated with the scale leads to a systematic on the
measurement. The effect on the measured value is calculated by fluctuating the scale
factor within its uncertainty, applying it to each jet and repeating the entire measurement. The uncertainty for both the b and c scales propagated to the t¯+jet cross
t
section is ∼ 5% (as shown in Figure 4.2), much less than the uncertainty of the JES.
- Mistag : The uncertainty on the probability for a jet to be mis-tagged, as calculated by
the mistag matrix, is on the order of 20% (see section 3.3). This is taken as a systematic
126

uncertainty on the measurement. The mistag rate on any jet is then fluctuated up and
down by 20% and the entire measurement is repeated to quantify the effect.
- K factor: For modeling the W +jet sample, the heavy flavor fraction was corrected
by the K factor as described in section 3.8.2 (page 120). We used a K factor of
(1.5 ± 0.3 stat+sys ) for this measurement.
- Luminosity : The current estimation on the uncertainty of the luminosity measured
by CLC is around 5.8% as discussed in section 2.4.4 . The luminosity used in the
measurement is fluctuated within this uncertainty and the measurement redone. This
uncertainty is the second largest systematic in our measurement.
- QCD : As addressed in detail in section 3.8.1, a pure data-based set is used to estimate
the QCD fraction instead of a MC model. In section 3.8.2 we used the anti-electron data
set for the QCD background estimate. To estimate the uncertainty on this background,
fits are performed with a different data set; the jet electrons as listed in Table 3.7 .
The fits are based on the idea that for a QCD event to pass the selection cuts, a jet
has to fake an electron. We require the event to pass all the offline selection cuts
for electron as listed in Table 3.3 , but to fail two of the five non-kinematic cuts :
∆x, ∆z, Lshr , EHAD /EEM and χ2
strip . The events that fail these cuts should be
QCD-enriched, since the non-kinematic cuts serve primarily to filter out non-W events
(i.e. to reject fake electrons), but still have most of the kinematic properties of real W
events, including isolation. This is why these fake electrons are named jet electrons.
The resulting difference in the fits is 30% which is taken as a systematic uncertainty
in the measurement.
- ISR/FSR : An interesting aspect of this measurement is that the result of Equation
(4.4) is a measure of the ISR (see section 1.3.2 page 11 and section 3.7.2). Because
of this, we do not include this effect as a systematic, rather we cross check that our
127

result is invariant to using a MC sample where ISR/FSR have been increased (PYTHIA
IFSR in Table 3.7). Unfortunately, the FSR effect, which would be a true systematic
is present in this cross check, but we assume the effect is quite small based on previous
cross section measurements.
- MC Generators : Differences in MC models for parton showering (section 3.7.1) are
studied simply by replacing our PYTHIA t¯ sample with the HERWIG sample as listed
t
in Table 3.7 . HERWIG is separated into t¯+0jet and t¯+jet events exactly as PYTHIA is
t
t
and the measurement was repeated.
- Lepton ID : Detector specific corrections are applied to the MC to more correctly
model the relative trigger efficiencies between CEM, CMUP, and CMX events, as
discussed in detail in section 3.2 . The corrections are data-derived from a pure sample
of Z bosons and have a small uncertainty associated with them. There are two types
of corrections, lepton ID and trigger efficiencies. Each are fluctuated within their
uncertainty, separately, and the resulting errors are added in quadrature.
- PDF : Uncertainty in the PDFs (section 1.3.2 page 10) are evaluated by a re-weighting
scheme. PDFs are re-weighted in our signal MC to simulate 46 different PDF parametrizations. The measurement is performed for each different parametrization. A prescription
for evaluating the uncertainty derived from the result of this is documented in detail
here [127].

4.2.3

The Measured σtt+jet & σtt+0jet
¯
¯

The first measured cross section of t¯ events in association with an extra jet with ET ≥ 20
t
GeV and |η| ≤ 2 is:

σt¯+jet = 1.6 ± 0.2stat ± 0.5syst
t
128

pb

(4.6)

Figure 4.1: Systematic uncertainties for the measured t¯+0jet cross section. The numbers
t
in the (∆ σ pb) column are rounded but the percentile column represents the exact values.

129

Figure 4.2: Systematic uncertainties for the measured t¯+jet cross section. The numbers in
t
the (∆ σ pb) column are rounded but the percentile column represents the exact values.

130

while the measured cross section for t¯ events without an extra jet is:
t

σt¯+0jet = 5.5 ± 0.4stat ± 0.7syst
t

pb

(4.7)

which when measured inclusively gives a t¯ cross section:
t

σt¯ = 7.1 ± 0.3stat
t

pb

(4.8)

The result of Equation 4.6 is in agreement with the NLO prediction [79, 80]
+0.16
σt¯
t+jet = 1.791(1)−0.31 as listed in Table 1.4 . The value of the measured inclusive cross

section (Equation 4.8) is also in agreement with the predicted SM value as listed in Tables
1.2 and 1.3 .

4.3

Kinematic Analysis

So far we have studied the total production rate for the extra hard jet associated with the t¯
t
events. However, this result does not provide us with complete kinematic information about
the extra jet. Therefore, for the completeness of our analysis, we examine the kinematic
profile of the extra jet to see if it agrees with expectations from the theoretical predictions.
In order to do so, we develop an algorithm (a simple handle) that isolates correctly the extra
jet from the other final state jets. In the following subsections, we will introduce the analysis
by studying the extra jet as defined in section 3.7.2 using the PYTHIA ttop25 MC sample.
With the selection cuts we used in the cross section analysis (as listed in section 3.6), we
will proceed by developing the isolation algorithm and verifying its efficiency, along with
correcting the measured extra jet ET distribution for the purity/efficiency. The question of
whether the extra jet is the softest (least energetic) jet among the jets in the final state will

131

be described afterword, followed by cross checking our results using a different MC model
(MCFM).

4.3.1

The Truth Extra Jet

In order to be consistent with the counting analysis, we will apply same selection cuts on
the MC sample. We will first study the case where we have ≥ 2 SecVtx tagged jets in the
events (We will, from now on, refer to these as double tag events) and compare to the case
where we have only ≥ 1 SecVtx tagged jet (single tag events). For the double tag events,
Figure 4.3 shows the extra hard jet multiplicity in the final state as defined and clustered by
the tools described in section 3.7.2. This is why we will term the hard extra jet the “truth
plus” jet to distinguish it from the reconstructed jets of the MC sample, the ttop25 sample
. The histogram demonstrates that having more than one truth jet is highly suppressed, as
expected in QCD. The MC predictions indicate also a good agreement with the measured
cross section of t¯+jet; the ratio of the t¯+jets events to the total inclusive t¯ sample (∼
t
t
t
11k/62k) is close to the ratio of the exclusive cross section (Equation 4.6) to the inclusive
cross section (Equation 4.7) (∼ 1.6 pb/7.1 pb). This reflects that the MC sample that we
use provides a reasonable description of the data.
As we are interested in t¯ events with at least three jets after the t¯ pair decays, we
t
t
investigated how often t¯ events with 4 reconstructed jets would have an extra jet which is
t
a truth plus jet (Figure 4.4), and how often t¯ events with 5 reconstructed jets would have
t
an extra jet which is a truth plus jet (Figure 4.5). These plots verify that most of the time,
events with only 4 jets would not have a truth plus jet, but more than half of the time
events with 5 jets would have a truth plus jet. This, however, doesn’t necessarily mean that
the fifth reconstructed jet is actually the truth plus jet. This will be addressed in the next
section.

132

Events

50000

Sum of bin content: 61898
0 truth plus jet: 50936
1 truth plus jet: 8920
2 truth plus jets: 1717
>2 truth plus jets: 325

40000

30000

20000

10000

0

0

1

2

3

4

5

# of Truth plus jet
Figure 4.3: The truth plus jet multiplicity.

133

Events

ttop25 (≥ 2 SecVtx jets)
Entries 27192

20000

Mean

0.1455

RMS

0.3526

15000

0 truth plus jet:23236

1 truth plus jet:3956
10000

14% of the time, events with 4
jets have a truth plus jet
5000

0

0

1

2

3

4
5
# of Truth plus jet

Figure 4.4: The number of truth plus jets in 4 reconstructed jet events.

134

Events

5000

ttop25 (≥ 2 SecVtx jets)
Entries 9478

4000

Mean 0.5266

RMS

0.4993

0 truth plus jet:4487

3000

1 truth plus jet:4991

2000

53% of the time, events with ≥5
jets have a truth plus jet

1000

0

0

1

2

3

4
5
# of Truth plus jet

Figure 4.5: The number of truth plus jets in 5 reconstructed jet events.

135

4.3.2

The Isolation Algorithm

Double Tag Events
Based on the results from the previous plots, we use the events which have 5 reconstructed
jets where the truth plus jet is found, i.e. the 4991 events of Figure 4.5 . These events have
exactly five jets in the final state as depicted, for example, in Figure 3.6 . One jet is for sure
the extra jet. The main difficulty is how to isolate or extract this extra jet from among the
5 jets.
Requiring two jets to be SecVtx jets would leave us with three jets, one of which should
be the extra jet. Out of these three jets, two jets should reconstruct the hadronic W boson.
There are three possible dijet combinations to reconstruct the mass of the hadronic W boson,
among which we will choose the combination that is closest to the W mass. The other two
combinations are considered as a background to the signal. The jet that remains should in
turn represent the extra jet. We can further reconstruct the hadronic top quark by forming
the 3-jet mass using the best dijet combination along with one of the SecVtx jets.
Figure 4.6 shows all the possible combinatoric mass distributions to reconstruct the
hadronic W boson and the hadronic top quark masses , while Figure 4.7 shows only the
best mass distribution. It is clear how the former distribution fills a much larger phase space
than the best distribution. Figures 4.8 and 4.9 show separately the best reconstructed mass
distributions of the hadronic W boson and the hadronic top quark respectively. Figures 4.10
and 4.11 show ∆R separation (see Equation 2.3) between the dijet that best reconstructs
the hadronic W boson and the truth W boson, and ∆R separation between the trijet that
best reconstructs the hadronic top quark and the truth top quark respectively. Figure 4.12
indicates that by using this algorithm, and after calculating the ∆R separation between the
candidate jet that is not part of the W boson and the truth plus jet, we are able to isolate
the correct extra jet ∼ 52% of the time (i.e. ∆R is found to be ≤ 0.4 in 52% of the events in
the histogram). By requiring the dijet combination that best reconstructs the mass of the
136

hadronic W boson to be within ± 10 GeV of the truth W boson mass, our efficiency of correctly isolating the extra jet increases to ∼ 57% without a significant reduction in statistics.
This is displayed in Figure 4.13 . With this ± 10 GeV constraint on the W boson mass,
and by applying a ± 20 GeV constraint on the hadronic reconstructed top quark mass, the
efficiency increases to ∼ 61%. This is presented in Figure 4.14 .
In data, we of course don’t know which jet the extra jet is. However, we can still verify
the ability of the algorithm to reconstruct the measured hadronic W boson and the measured
hadronic top quark masses in the 5 jets bin events. Figure 4.15 shows that the mean values of
the measured distribution are close to those of the reconstructed MC distribution of Figure
4.7 .

137

Combinatoric W Mass (GeV/c 2)

ttop25 (≥ 2 SecVtx jets)

29946

Mean x

177.4

Mean y

92.63

RMS x

68.85

RMS y

600

Entries

54.57

400

200

0
0

200

400

600

Combinatoric Top Mass (GeV/c2)

Figure 4.6: All the possible combinatoric W boson and top quark mass distributions using
the ttop25 MC sample with ≥ 2 SecVtx tagged jets.

138

Best W Mass (GeV/c 2)

500

ttop25 (≥ 2 SecVtx jets)

450

Entries

4991

Mean x 168.9
400

Mean y 78.68
350

RMS x
RMS y

300

18.07
16.33

250

200

150

100

50

0
0

50

100

150

200

250

300

350

400

450

500

Best Top Mass (GeV/c2)
Figure 4.7: The best reconstructed W boson and top quark mass distributions using the
ttop25 MC sample with ≥ 2 SecVtx tagged jets.

139

Events

1600

ttop25 (≥ 2 SecVtx jets)

1400

Entries

4991

Mean

1000

78.68

RMS

1200

16.33

800

600

400

200

0
0

50

100

150

200

250

300

350

400

450

500

Best W Mass (GeV/c2)
Figure 4.8: The best reconstructed W boson mass distribution using the ttop25 MC sample
with ≥ 2 SecVtx tagged jets.

140

Events

ttop25 (≥ 2 SecVtx jets)

1800

1600

Entries

4991

1400

Mean

168.9

1200

RMS

18.07

1000

800

600

400

200

0
0

50

100

150

200

250

300

350

400

450

500

Best Top Mass (GeV/c2)
Figure 4.9: The best reconstructed top quark mass distribution using the ttop25 MC sample
with ≥ 2 SecVtx tagged jets.

141

Events

2000

ttop25 (≥ 2 SecVtx jets)

Entries

4991

1800

Mean 0.9472
1600

RMS

1.108

4

5

1400

1200

1000

800

600

400

200

0

0

1

2

3

6

∆R
Figure 4.10: ∆R value between the dijet combination that best reconstructs the W boson
and the truth W boson mass distributions using the ttop25 MC sample with ≥ 2 SecVtx
tagged jets.

142

Events

1400

ttop25 (≥ 2 SecVtx jets)

1200

Entries

4991

Mean
RMS

1000

1.138
1.124

4

5

800

600

400

200

0

0

1

2

3

6

∆R
Figure 4.11: ∆R value between the trijet combination that best reconstructs the top quark
and the truth top quark mass distributions using the ttop25 MC sample with ≥ 2 SecVtx
tagged jets.

143

Events

2500

ttop25 (≥ 2 SecVtx jets)

Entries

4991

Mean

1.05

RMS

1.222

4

5

2000

1500

1000

500

0

0

1

2

3

6

∆R
Figure 4.12: ∆R value between the extra ttop25 jet and the truth plus jet using the ttop25
MC sample with ≥ 2 SecVtx tagged jets.

144

Events

1800

ttop25 (≥ 2 SecVtx jets)

Entries

3118

1600

Mean 0.9175
1400

RMS

1.18

1200

1000

800

600

400

200

0

0

1

2

3

4

5

6

∆R
Figure 4.13: ∆R value between the extra ttop25 jet and the truth plus jet within a ± 10
GeV window of the truth W boson mass distribution using the ttop25 MC sample with ≥
2 SecVtx tagged jets.

145

Events

1600

ttop25 (≥ 2 SecVtx jets)

1400

Entries

2698

Mean 0.8645
RMS

1200

1.167

4

5

1000

800

600

400

200

0

0

1

2

3

6

∆R
Figure 4.14: ∆R value between the extra ttop25 jet and the truth plus jet within a box of ±
10 and ± 20 GeV of the truth W boson and the truth top quark masses respectively, using
the ttop25 MC sample with ≥ 2 SecVtx tagged jets.

146

Best W Mass (GeV/c 2)

500

CDF Run II Preliminary L = 4.1 fb-1

450

Entries

400

115

Mean x 164.4
350

Mean y
RMS x

21.37

RMS y

300

76.3
18.88

250

200

150

100

50

0
0

50

100

150

200

250

300

350

400

450

500

Best Top Mass (GeV/c2)
Figure 4.15: The best measured W boson and top quark mass distributions with ≥ 2 SecVtx
tagged jets.

147

Single Tag Events
For the double tag case, in order to isolate the extra jet, we looked at three possible combinations to reconstruct the hadronic W boson. We want to study the impact of requiring
only one SecVtx tagged jet in our sample since the majority of our sample has only one
SecVtx tagged jet. The number of possible combinations will increase and this will have
an impact on determining the extra jet. In single tag events, there are four non b-tagged
jets, two of which should reconstruct the hadronic W boson. Therefore, we have six possible combinations for the reconstruction of the hadronic W boson; one combination is the
signal and the other 5 are considered backgrounds. By choosing the combination that best
reconstructs the W boson mass, two jets remain where, one of them should be the extra
jet. We then calculate the mass of this best W boson combination with each of these two
remaining jets and choose the trijet combination that best reconstructs the correct mass of
the top quark. The jet selected in the best trijet combination should represent the other
SecVtx jet that has to be present, by definition, in the final state, while the other jet should
in turn represent the extra jet.
Figure 4.16 shows the dijet combination that best reconstructs the hadronic W boson
mass and the trijet combination that best reconstructs the top quark mass. It is clear that
the latter is much broader (more background) than that shown in Figure 4.7 for the double
tag events. This is expected, because in the double tag case, one can always reconstruct the
hadronic top quark because one of the SecVtx jets would definitely come from the hadronic
top decay, while in the single tag events, the SecVtx jet could equally likely come from the
leptonic top decay as well as from the hadronic decay, and the former can not be directly
reconstructed. Figure 4.17 verifies that by using this algorithm and after calculating ∆R
separation between the jet that is left over and the truth plus jet, we are able to isolate the
correct extra jet ∼ 50% of the time. With a box of ± 10 and ± 20 GeV around the truth
W boson and the truth top quark masses respectively, the efficiency (Figure 4.18) increases

148

to ∼ 59%. This is slightly less than the analogous value for the double tag case as shown in
Figure 4.14 .

4.3.3

The Corrected Measured Extra Jet

As we are able to isolate the correct extra jet ∼ 60% of the time, we can rely on our isolation
algorithm to correct our observed extra jet distribution back to the truth plus jet distribution.
Therefore, we use all the events in the 5 jets bin for data distribution, while for the ttop25
MC distribution, we use only the events in the 5 jets bin where we find the extra jet (not all
the 5 jets bin MC events).
For the double tag events, we can assume that there are no non-top backgrounds (i.e.
Method II backgrounds) and we have to care only about the combinatoric backgrounds as
the ones in Figure 4.6. Figure 4.19 shows the ET distribution of the extra jet as found in
data, in the signal (ttop25 MC) distribution and in the truth distribution (section 4.3.1)
using the mass box constraint. By determining bin by bin the efficiency for keeping the
correct extra jet, we correct the observed ET distribution of the extra jet for the efficiency
and compare it to the truth plus jet ET distribution. This is shown in Figure 4.20 .
On the other hand, and for the single tag events, we first subtract the Method II backgrounds (as listed in Table 3.17) from the data, leaving only the signal and the combinatoric
backgrounds. Then we correct the observed ET distribution for the efficiency (within the
mass box constraint) and compare it to the truth plus jet distribution. This is what Figure
4.21 displays. Figure 4.22 shows the extra jet ET distribution as found in data (before being
corrected), in the signal (ttop25 MC) distribution, in the truth distribution and in the Method
II background distributions. The Method II background contributions are relatively small,
so we ignore this kind of background for the more pure double tag event samples as shown
in Figure 4.19 .
However, the corrected measured ET distributions (Figure 4.20 and Figure 4.21) are

149

Best W Mass (GeV/c 2)

500

ttop25 (≥ 1 SecVtx jets)

Entries

16838

Mean x

171.9

Mean y

450

79.83

RMS x

20.37

RMS y

9.27

400

350

300

250

200

150

100

50

0
0

50

100

150

200

250

300

350

400

450

500

Best Top Mass (GeV/c2)
Figure 4.16: The best hadronic W boson and top quark mass distributions using the ttop25
MC sample with ≥ 1 SecVtx tagged jets.

150

Events

8000

ttop25 (≥ 1 SecVtx jets)

Entries 16838
Mean

6000

1.082

RMS

7000

1.24

5000

4000

3000

2000

1000

0

0

1

2

3

4

5

6

∆R
Figure 4.17: ∆R value between the extra ttop25 jet and the truth plus jet using the ttop25
MC sample with ≥ 1 SecVtx tagged jets.

151

Events

7000

ttop25 (≥ 1 SecVtx jets)

Entries 11621

6000

Mean 0.9016
RMS

5000

1.204

4

5

4000

3000

2000

1000

0

0

1

2

3

6

∆R
Figure 4.18: ∆R value between the extra ttop25 jet and the truth plus jet within a box
of ± 10 and ± 20 GeV of the truth W boson and the truth Top quark mass distributions
respectively, using the ttop25 MC sample with ≥ 1 SecVtx tagged jets.

152

slightly steeper (peak more at low ET value) than the truth ET distributions, especially for
the double tag event samples. On the other hand, the theoretical predictions (the ttop25
MC distribution and the truth distribution) behave similarly as shown in Figure 4.19 and
Figure 4.22 .

153

Events (Norm 1)

0.3

CDF Run II Preliminary L = 4.1 fb-1
Data

0.25

ttop25 (≥ 2 SecVtx jets)
Truth

0.2

0.15

0.1

0.05

0
0

10

20

30

40

50

60

70

80

90

100

ET (GeV)
Figure 4.19: The ET distribution of the extra jet as found in data, ttop25 and the truth
samples for the double tag event samples.

154

Events (Norm 1)

0.3

CDF Run II Preliminary L = 4.1 fb-1
Data corrected for efficiency

Truth
0.2

0.1

0
0

20

40

60

80

100

ET(GeV)
Figure 4.20: The corrected observed ET distribution of the extra jet compared with the
truth distribution for the double tag event samples.

155

Events (Norm 1)

0.3

CDF Run II Preliminary L = 4.1 fb-1
(Data - Method II) corrected for efficiency
Truth

0.2

0.1

0
0

20

40

60

80

100

ET (GeV)
Figure 4.21: The corrected observed ET distribution of the extra jet compared with the
truth distribution for the single tag event samples.

156

Events (Norm 1)

0.4

CDF Run II Preliminary L = 4.1 fb-1
0.35

Data
ttop25 (≥ 1 SecVtx jets)

0.3

Truth
Method II Background

0.25

0.2

0.15

0.1

0.05

0
0

10

20

30

40

50

60

70

80

90

100

ET (GeV)
Figure 4.22: The ET distribution of the extra jet as found in data, ttop25, truth and Method
II background samples for the single tag event samples.

157

4.3.4

The Extra Jet Profile

Full Machinery ∆R Calculations
So far, we discussed how to correctly determine the extra jet and it is time now to determine
which jet, among the five jets of the final state, is the extra jet. The two b jets tend to be
the most energetic jets in the event and we have verified this using MC events. The two
light jets radiated from the hadronic W tend also to be more energetic than the extra jet,
especially if the latter belongs to the t¯+jet sample (i.e. ISR) not to the t¯+0jet. This is
t
t
why the extra jet is expected to be the fifth jet in the final state i.e. to be the softest (least
energetic) jet.
To investigate the above statement, and for the double tag event samples, we calculated
the ∆R separation between the truth plus jet and each of the five MC jets. Figure 4.23 shows
that the extra jet could be equally likely the fifth jet (24.0% of the time) and the fourth jet
(24.1% of the time). The first (lead) and the second jets have a much small likelihood to
be the extra jet. Figure 4.24 shows the results for the single tag event samples. It provides
similar results that show it is equally likely for the fifth jet (23.0% of the time) and the
fourth jet (23.9% of the time) to be the extra jet.
ET and η Discriminants
Using the straightforward ∆R calculations on the signal MC sample does not solve the
dilemma of whether the extra jet is the fifth or the fourth jet. Hence, we tried to tackle this
issue using the following procedure: There are ten possible combinatorics for two out of the
five jets in the final state to be tagged as SecVtx jets. We do not have to look at all of these
combinatorics because the above ∆R calculations have shown that the likelihood for the
first, the second and the third jets to be the extra jet are small and that it is equally likely
for the fifth and the fourth jets. Hence, we looked at only the following three combinations:
- If the first and the second jets are tagged as SecVtx jets, the third, fourth and fifth
158

Events (Norm 1)

0.25

ttop25 (≥ 2 SecVtx)
∆R(extra jet, truth plus jet)

0.2

th

∆R(5 ,truth): 24.0%
th

∆R(4 ,truth): 24.1%
0.15

rd

∆R(3 ,truth): 18.0%

0.1

0.05

0
0

1

2

3

4

5

6

∆R

Figure 4.23: ∆R value between the ttop25 jets and the truth plus jet for the double tag
event samples.

159

Events (Norm 1)

0.25

ttop25 (≥ 1 SecVtx)
∆R(extra jet, truth plus jet)

0.2

∆R(5 ,truth): 23.0%
th

∆R(4 ,truth): 23.9%
th

∆R(3 ,truth): 19.5%
rd

0.15

0.1

0.05

0
0

1

2

3

4

5

6

∆R

Figure 4.24: ∆R value between the ttop25 jets and the truth plus jet for the single tag event
samples.

160

jets could be the extra jet candidate:
- When the third and the fourth jets best reconstructed the W boson, we considered
the fifth jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
- When the third and the fifth jets best reconstructed the W boson, we considered
the fourth jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
- When the fourth and the fifth jets best reconstructed the W boson, we considered
the third jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
- If the first and the third jets are tagged as SecVtx jets, the second, fourth and fifth
jets could be the extra jet candidate:
- When the second and the fourth jets best reconstructed the W boson, we considered the fifth jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
- When the second and the fifth jets best reconstructed the W boson, we considered
the fourth jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
- When the fourth and the fifth jets best reconstructed the W boson, we considered
the second jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
- If the second and the third jets are tagged as SecVtx jets, the first, fourth and fifth
jets could be the extra jet candidate:
- When the first and the fourth jets best reconstructed the W boson, we considered
161

the fifth jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
- When the first and the fifth jets best reconstructed the W boson, we considered
the fourth jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
- When the fourth and the fifth jets best reconstructed the W boson, we considered
the first jet to be the extra jet and hence compared its ET and η distributions
with that of the truth plus jet.
We have seen that the first, second and third jets have the least likelihood to be the extra
jet. However, both the ET and η discriminants are unable to favor the fifth jet or the fourth
jet over the other. The results show that they are still equally likely to be the extra jet.
By definition, the fifth jet ET distribution should drop off more rapidly than the fourth jet.
This is why the η distribution for the former jet is broader than that for the latter jet.

4.3.5

The MCFM Extra Jet

MCFM (Monte Carlo for FeMtobarn) is a parton level MC program which gives LO and NLO
predictions for a range of processes at hadron colliders [128]. Unfortunately, our t¯+jet
t
process is not included in MCFM yet at the NLO level. However, we can still infer some
knowledge from LO MCFM (process 156) in which an extra gluon is radiated as ISR. We
can use MCFM to cross check our analysis as well. In this regard, and using our isolation
algorithm, Figures 4.25 and 4.26 show the best reconstructed mass and one of the possible
combinatoric mass distributions of the W boson respectively. Finally Figure 4.27 shows how
often the extra ISR gluon could be the lowest PT jet among the five jets in the final state.
The distribution (the bold black) falls off as ∼ (1/P2 ) as expected in QCD [49].
T

162

Events

2.2

MCFM (LO- Process 156)

2
Entries 6.588726e+07

1.8
Mean

1.4

80.03

RMS

1.6

9.256

1.2
1
0.8
0.6
0.4
0.2
0
0

50

100

150

200

250

300

350

400

450

500

Best W Mass (GeV/c2)
Figure 4.25: The best W boson mass distribution for LO t¯+jet MCFM sample.
t

163

Events

0.35

MCFM (LO- Process 156)
0.3

Entries 6.588726e+07
Mean

72.73

RMS

0.25

37.64

0.2

0.15

0.1

0.05

0
0

50

100

150

200

250

300

350

400

450

500

Combinatoric W Mass (GeV/c2)
Figure 4.26: A combinatoric W boson mass distribution for LO t¯+jet MCFM sample.
t

164

Events (Norm 1)

0.8

MCFM (LO - Process 156)

0.7

Extra ISR gluon
Extra ISR gluon is the Lowest (42.76%)
Extra ISR gluon is the Next Lowest (35.52%)
Extra ISR gluon is the Lowest OR the NL(78.28%)
Extra ISR gluon is the Highest (5.64%)

0.6

0.5

0.4

0.3

0.2

0.1

0

0

20

40

60

80

100

Extra ISR gluon P (GeV/c)
T

Figure 4.27: The likelihood for the extra ISR gluon to be the lowest PT jet for LO t¯+jet
t
MCFM sample.

165

4.4

Conclusions

The first measurement of the cross section of the top quark pair in association with an
extra hard jet (t¯+jet) has been performed with 4.1 fb−1 of data collected at CDF. The
t
measurement is an important test of perturbative QCD, as NLO effects play an important
role in the calculation of the theoretical cross section. In addition, it is also important
for future measurements at the LHC, for which almost half of the top quark events will
be produced with extra jets. Therefore, this process will be a substantial background for
many new physics signals. The measurement is performed using SecVtx tagged events in
the lepton plus jet channel. A data-driven approach is used to predict the background
content, and a 2D likelihood is formed to simultaneously measure the t¯+jet and t¯ without
t
t
extra jet (t¯+0jet) cross sections. The measured result is σt¯
t
t+jet = 1.6±0.2stat ±0.5syst pb
+0.16
which is in agreement with the recent NLO SM prediction σt¯
t+jet = 1.791(1)−0.31 pb .
The measured cross section for t¯+0jet is σt¯
t
t+0jet = 5.5±0.4stat ±0.7syst pb which when
measured inclusively gives a t¯ cross section σt¯= 7.1±0.3stat pb, in agreement with the
t
t
SM prediction.
In order to elucidate the kinematic profile of the extra jet, an isolation algorithm has been
developed. The algorithm has extracted correctly the extra jet out from the final state jets
∼ 60% of the time. This allowed for correcting the measured distributions of the extra jet
for purity/efficiency in order to compare them with the MC distributions. The differences in
the kinematics of the extra jet using different SecVtx requirements and different MC models
(PYTHIA & MCFM) have been studied, and found to be in agreement with the SM prediction.
The calculations have shown that the likelihood for the first, the second and the third highest
ET jet in the final state of t¯+jet sample to be the extra jet is quite small. On the other
t
hand, the fourth and the fifth highest ET jet in the final state of t¯+jet sample are found to
t
be equally likely the extra jet.

166

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