UTPP-84

Measurement of the Spin Correlationin the Top Quark Pair Production
using the Dilepton Eventsin 1.96-TeV Proton-Antiproton Collisions

Kenichi TAKEMASA

February 2012

UTPP-84

Measurement of the Spin Correlationin the Top Quark Pair Production
using the Dilepton Eventsin 1.96-TeV Proton-Antiproton Collisions

Kenichi TAKEMASA(Doctoral Program in Physics)

Submitted to the Graduate School ofPure and Applied Sciencesin Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy in
Scienceat theUniversity of Tsukuba

Abstract
One of the most remarkable properties of the top quark is its extremely short lifetime,
which allows us to observe the top quark spin at its production. That means a spin correlation at tt̄ production is possible to be observed. In this thesis, we report on a measurement of the correlation coefficient between top quark spin and antitop quark spin in
the beam basis at top quark pair production. We select tt̄ pair production candidates byrequiring two high transverse momentum leptons, two jets and large missing ET . In orderto reconstruct the angular distribution of top quark decay products, we perform full kinematical reconstruction using predicted distributions of pztt̄ , pTtt̄ and Mtt̄ . We make signal
and background templates from admixture of Monte Carlo simulations and data-based
background modelings. Then, we perform unbinned likelihood fit of angular distributionof data to signal and background templates and obtain result. This analysis is based on
the data of 5.1 fb−1 collected with the Collider Detector Facility (CDF) at the Fermilab Tevatron between March 2002 and June 2009. We observe 334 tt̄ candidate events.
We determine a confidence interval at 68% level for the correlation coefficient κ to be2−0.520 < κ < 0.605 or κ = 0.042+0.563−0.562 on the assumption of Mtop = 172.5 GeV/c .

Acknowledgments
I would like to thank people who have helped me and contribute to my growth in mycollege life.
First of all, I would like to acknowledge a great debt to my advisor, Prof. Shinhong
Kim and Prof. Yuji Takeuchi for giving me the opportunity to work on CDF and suggesting the topic of this analysis. They have gave me a lot of adivices and always guided
me in the right direction. Without their support and encouragement, this thesis would nothave been possible.
I wish to thank staff of the High Energy Physics group of University of Tsukuba,Prof. Fumihiko Ukegawa, Prof. Kazuhiko Hara, Prof. Koji Sato, Dr. Hideki Miyake, Dr.
Masakazu Kurata, Dr. Koichi Nagai for their kindness. They give me a lot of suggestions
to my analysis. I also thank students who belong to the High Energy Physics group of
University of Tsukuba. I shared good times with them. In particular, I would like to thankYuji Sudo who is my peer. He listened to my problem on analysis and supported mylife at Fermilab. Also, I would like to thank academic colleague at Fermilab, Atsunari
Hamaguchi, Daisuke Yamato, Yuki Sakurai, Yujiro Funakoshi, and people who belonged
the High Energy Physics group of University of Tsukuba, Prof. Takasumi Maruyama,
Dr. Naoki Kimura, Dr. Tomonobu Tomura, Dr. Junji Naganoma, Masato Aoki and Dr.
Yoshikazu Nagai. They gave me a lot of advices and provided motivation to analysis.
I wish to express apprecisation to Kazuko Kumashiro, Kyoko Kunori for their constantsupport with their secretarial works.
Conveners of CDF Top Group and Dilepton working group gave me a lot of helps. My
deep thank go to the members of CDF top dilepton working group, Jaroslav Antos, Yen
Chu Chen, Hyunsoo Kim, Chang-Seong Moon, Youngdo Oh, Andy Beretvas, Soo-Bong
Kim, Roman Lysak and Satoru Uozumi. I thank the Fermilab staff and the technical staffs
of the participating insttutions for their vital contributions.
I wish to thank my best friend Junichi Tanaka. Playing with him was a nice change ofpace for me.
Finally, I would like to thank my parents and my sister.

Contents
1 Introduction
1.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Spin Quantum Number . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The Top Quark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Top Quark Production . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Top Quark Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Top Quark Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Top Quark Spin Correlation . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 κmeas , what is measured in this analysis . . . . . . . . . . . . . . . . . .
1.9 Motivation of Spin Correlation Measurement . . . . . . . . . . . . . . .
1.10 Previous Measurement of Top Quark Spin Correlation . . . . . . . . . . .
1.10.1 Dilepton channel . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10.2 Lepton+Jets channel . . . . . . . . . . . . . . . . . . . . . . . .
1.10.3 Combine lepton+Jets events and dilepton events . . . . . . . . . .
1.11 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2
5
5
6
7
9
10
12
13
13
13
14
14
14

2 Experimental Apparatus
2.1 The Fermilab Accelerator Complex . . . . . . . . . . . . . . . . . . . .
2.2 The CDF II Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Tracking System . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Muon Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Luminosity Monitor . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . .

15
15
19
20
23
26
27
28

3 Reconstruction and Event Selection
3.1 Electron Identification . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Central Electrons . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Plug Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Muon Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Jet Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Jet Energy Corrections . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Jet Energy Scale Uncertainties . . . . . . . . . . . . . . . . . . .
3.4 Missing Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31
31
31
35
37
39
40
42
42

i

4 Spin Correlation Measurement
4.1 Outline of the Measurement . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Top Dilepton Event Selection . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Background Composition . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Full Kinematical Reconstruction in Dilepton Channel . . . . . . . . . . .
4.4.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Jet and MET Resolutions . . . . . . . . . . . . . . . . . . . . . .
4.4.4 Performance of the Full Kinematical Reconstruction Method . . .
4.5 Templates of the Signal and the Background . . . . . . . . . . . . . . . .
4.5.1 Template Fit Function . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Signal Template as a Function of κ . . . . . . . . . . . . . . . .
4.5.3 Background Template . . . . . . . . . . . . . . . . . . . . . . .
4.5.4 Crosscheck of the Signal and the Background shape . . . . . . .
4.6 Pseudo-experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Check of the Method for κ Measurement . . . . . . . . . . . . .
4.6.2 Pseudo Experiment Results . . . . . . . . . . . . . . . . . . . .
4.6.3 Correlation between κmeas from leptons and κmeas from b-jets .
4.6.4 Construction of Feldman-Cousins intervals . . . . . . . . . . . .

45
45
45
47
49
49
49
50
51
53
53
53
57
64
65
67
69
71
73

5 Systematic Uncertainties
5.1 Statistical Uncertainty of Signal MC Sample . . . . . . . . . . . . . . . .
5.2 Uncertainties in expected signal, background numbers and backgroundtemplates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Jet Energy Scale Uncertainty . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Initial and Final State Radiation . . . . . . . . . . . . . . . . . . . . . .
5.5 Uncertainty of gg fraction . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Parton Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Systematics from LO versus NLO . . . . . . . . . . . . . . . . . . . . .
5.8 Color Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Total Uncertainty and Feldman-Cousins Intervals . . . . . . . . . . . . .

75
75
75
77
79
80
80
81
82
84

6 Results and Discussions

87

7 Conclusion

91

A Spin States at tt̄ Pair Production
A.1 Production by q q̄ → tt̄ . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Production by gg → tt̄ . . . . . . . . . . . . . . . . . . . . . . . . . . .

93
93
95

B P, CP of tt̄ System

97iiList of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8

Leading-order production diagram for q q̄ → tt̄. . . . . . . . . . . . . . .
Leading-order production diagrams for gg → tt̄. . . . . . . . . . . . . . .
The spin configurations of the gluon fusion and the q q̄ annihilation subprocesses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NLO calculations of σ(pp̄ → tt̄X). . . . . . . . . . . . . . . . . . . . . .
Tree level q q̄ → tt̄ production and decay. . . . . . . . . . . . . . . . . . .
Loop diagrams generating corrections to the theoretical W boson mass. .
Electroweak constraints on the Higgs boson mass. . . . . . . . . . . . . .
The two-dimensional distributions of cosθ+ and cosθ− in beamline basis
for the processes q q̄ → tt̄ and gg → tt̄. . . . . . . . . . . . . . . . . . . .

2.1
2.2
2.3
2.4
2.5
2.6
2.7

6
6
7
8
8
9
11
12

A diagram of the Fermilab accelerator chain. . . . . . . . . . . . . . . .
Isometric view of the CDF II detector. . . . . . . . . . . . . . . . . . . .
The CDF tracking system. . . . . . . . . . . . . . . . . . . . . . . . . .
The CDF silicon detector . . . . . . . . . . . . . . . . . . . . . . . . . .
The CDF calorimeter system. . . . . . . . . . . . . . . . . . . . . . . . .
The single CEM wedge. . . . . . . . . . . . . . . . . . . . . . . . . . .
CDF muon coverage for the CMU, CMP, CMX, and BMU detectors. The
BMU is referred as the IMU. . . . . . . . . . . . . . . . . . . . . . . . .
2.8 A cross section view of a CMU muon chamber. . . . . . . . . . . . . . .
2.9 Total integrated luminosity delivered by the Tevatron and recorded by the
CDF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Data flow in the CDF data acquisition system. . . . . . . . . . . . . . . .
2.11 Block diagram of the Level 1 and Level 2 trigger path. . . . . . . . . . .

16
20
21
22
24
25

3.1

The total uncertainties of JES. . . . . . . . . . . . . . . . . . . . . . . .

43

4.1

Distributions of pztt̄ , pTtt̄ , and Mtt̄ of dilepton candidates in tt̄ PYTHIA
Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mean and σ of the distribution of (ETtrue − ETL5 )/ETL5 as a function of ETL5
for jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
jet1
σ of the distribution of E
6 meas
6 true+ ETjet2 . . . .
x,y − Ex,y as a function of ET
Scatter plots of reconstructed kinematical variables versus the true valueof the corresponding variables. . . . . . . . . . . . . . . . . . . . . . . .
The distribution of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) on
the assumption of κ = +1. . . . . . . . . . . . . . . . . . . . . . . . . .

4.2
4.3
4.4
4.5

iii

26
26
28
29
30

50
51
52
52
54

The distributions of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) on
the assumption of κ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 The distributions of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) on
the assumption of κ = −1. . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 κ dependence of the fit parameters (Ciℓ (κ)) for the (cos θ+ , cos θ− ) distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 κ dependence of the fit parameters (Cib (κ)) for the (cos θb , cos θb̄ ) distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 The distributions of W W /W Z/ZZ background . . . . . . . . . . . . . .
4.11 The distributions of Z → τ τ background . . . . . . . . . . . . . . . . .
4.12 The distributions of Z/γ ∗ → ee, µµ background . . . . . . . . . . . . . .
4.13 The distribution of fake background . . . . . . . . . . . . . . . . . . . .
4.14 The distribution of all backgrounds . . . . . . . . . . . . . . . . . . . . .
4.15 The distribution of cos θ+ , cos θ− and cos θb , cos θb̄ . . . . . . . . . . . .
4.16 An example of pseudo data . . . . . . . . . . . . . . . . . . . . . . . . .
4.17 The distribution of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) in
Herwig Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18 The distribution of κmeas of pseudo-experiments using herwig Monte Carloas signal sample pool, on assumption of Nbkg =0. . . . . . . . . . . . . . .
4.19 The distribution of κmeas of pseudo-experiments using herwig Monte Carloas signal sample pool . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.20 The result of pseudo-experiments . . . . . . . . . . . . . . . . . . . . . .
4.21 The pull distribution of pseudo-experiments . . . . . . . . . . . . . . . .
4.22 σ(κmeas ) as a function of κtrue for leptons only case. . . . . . . . . . . . .
4.23 σ(κmeas ) as a function of κtrue for b-jets only case. . . . . . . . . . . . . .
4.24 Feldman-Cousins confidence interval (statistics only) . . . . . . . . . . .
4.6

5.1
5.2

55
55
56
56
57
59
62
62
63
64
66
67
67
68
69
70
71
71
73

Statistical uncertainty of signal Monte Carlo sample . . . . . . . . . . . .
Uncertainties in expected signal, background numbers and backgroundtemplates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Jet energy scale uncertainty . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Diagram of tt̄ production and decay with ISR and FSR . . . . . . . . . .
5.5 ISR/FSR uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Uncertainty of gg fraction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 PDF uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Feynman diagrams of tt̄ production for LO and NLO . . . . . . . . . . .
5.9 Systematics from LO versus NLO . . . . . . . . . . . . . . . . . . . . .
5.10 Systematics from color reconnection . . . . . . . . . . . . . . . . . . . .
5.11 Total systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . .
5.12 Total uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Feldman-Cousins confidence interval (statistics+systematics) . . . . . . .

78
78
79
79
80
82
83
83
83
84
85
85

The distribution of (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) in data for integrated luminosity of 5.1fb−1 . . . . . . . . . . . . . . . . . . . . . . . .
Feldman-Cousins confidence intervals with measured result . . . . . . . .

87
88

6.1
6.2

iv

76

Mass dependence on κ measurement . . . . . . . . . . . . . . . . . . . .

89

A.1 Differential cross section for tt̄ production as a function of the tt̄ invariant
mass for the Tevatron . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

B.1 Spin configurations of tt̄ system and their P and CP reversal states . . . .

97

6.3

v

List of Tables
1.1
1.2

Fundamental properties of fermions. . . . . . . . . . . . . . . . . . . . .
Fundamental properties of bosons. . . . . . . . . . . . . . . . . . . . . .

3
4

4.1
4.2
4.3
4.4

The number of expected events in data correspond to 5.1 fb −1 with theobserved number of events . . . . . . . . . . . . . . . . . . . . . . . . .
Z → τ τ background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ee channels of Drell-Yan background . . . . . . . . . . . . . . . . . . . .
µµ channels of Drell-Yan background. . . . . . . . . . . . . . . . . . . .

48
58
60
61

5.1

The summary table of systematic uncertainty . . . . . . . . . . . . . . .

84

vii

Chapter 1
Introduction
Particle physics is a branch of physics. Its purpose is to understand the fundamental constituents of matters and the laws governing them. This is done by studying the interactions
between objects, which can range from something as large as whole galaxies to the smallest building blocks of matter such as quark. The fundamental interaction between quarks
and other elementary particles that form everything we see today is described by particle
physics. As of now, the Standard Model (SM), a theory that describes these elementary
particle interactions, has been very successful due to results of high energy experiments
which have thoroughly verified the accuracy of the Standard Model.
Particle physics have been using particle colliders to investigate the fundamental particles and the interactions. The Tevatron is a proton-antiproton collider located at the
Fermi National Accelerator Laboratory (Fermilab). It is a powerful enough to produce
top quarks. At the Tevatron, top quarks are predicted to be produced in pairs via thestrong force, and decay via the electroweak force. The top quark is by far the heaviest
fundamental particle in the Standard Model. Because of its large mass, the top quark has
several unique properties and could provide hints for the original of mass and physics
beyond the Standard Model.
Top quark and tau neutrino are observed in 1995 [1, 2] and in 2000 [3], respectively.
By these two observations, almost all of the Standard Model particles has been observed
directly, except only the Higgs boson. Despite the great success of the Standard Model indescribing the particles and interactions observed up to now, however there is a good theoretical motivation to believe that a new framework must come into play at approximately
the TeV energy scale. Increasingly stringent measurements of the properties of particles
and their interactions at the highest available energies, coupled with direct searches forphenomena not described by the Standard Model, are the main project of modern experimental particle physics. A significant part of the project is the elucidation of the propertiesof the top quark.
This thesis reports measurements of the correlation coefficient between top quark spin
and anti-top quark spin. A measurement of the spin correlation coefficient plays an important role in testing the Standard Model.
11.1

The Standard Model

The Standard Model of elementary particle physics is a quantum field theory that describes the fundamental particles and the interactions between them. It includes thestrong, weak and electromagnetic force. The model itself is a combination of the theory of quantum chromodynamics (QCD) [4, 5] and the Glashow-Salam-Weinberg (GSW)
theory of electroweak interactions [6–8].
The strong force is described by quantum chromodynamics (QCD), a quantum field
theory with an SU (3)C gauge group, and an asymptotically free coupling. Asymptotic
freedom means that the strong force is weak at high energies or small distances, andonly become strong at low energies or large distances. The strong force becomes strong
enough at large distances that when colored quarks and gluons become separated, thebinding energy between them is large enough to create new quarks and gluons. All the freequarks and gluons ultimately become bound into hadrons. This process of hadronization
has a characteristic scale of ΛQCD ∼ 200 MeV, which corresponds to a time scale ofτt ∼ 10−24 s.
The electromagnetic and weak forces are combined together into a single electroweak
force which obeys an SU (2)L × U (1)Y gauge group. The SU (2)L group has three generators and the U (1)Y group has one, corresponding to four gauge bosons that mediate the electroweak force: Wµ1 , Wµ2 , Wµ3 for SU (2)L and Bµ for U (1)Y . The SU (2)L ×
U (1)Y symmetry of the electroweak interaction is spontaneously broken through its coupling to the scalar higgs field. The higgs field has a degenerate ground state at a finite
value of the field and thus spontaneously breaking turn into three massive gauge bosons(W + , W − , Z 0 ) and the massless photon which obeys an unbroken U (1)EM gauge symmetry.
The Standard Model contains the strong interaction which is represented as SU (3)Cand the electroweak interaction which is represented as SU (2)L ⊗ U (1)Y . Therefore, theStandard Model is described as follows, and is locally invariant under transformations of
the group.
G = SU (3)C ⊗ SU (2)L ⊗ U (1)Y .

(1.1)

There is another known force in the nature, graviton, but its interaction is too weak to be
detected in the subatomic experiments. Therefore gravitation is not understood in termsof particle physics, and has not been included the Standard Model.
The fundamental particles are categorized into two categories, spin s = 12 fermions
which are the constituents of normal matter, and spin s = 1 bosons which mediate theforce between fermions.
Spin s = 12 fermions are classified into quarks and leptons. There are six types ofquarks: up (u), down (d), charm (c), strange (s), top (t) and bottom (b). Similarly, there
are six types of leptons: electron (e), muon (µ), tau (τ ), and their respective neutrinos(νe , νµ , ντ ). These fermions are classified into the three generations of left-handed andright-handed quarks and leptons. The left-handed fermions are in weak isospin doublets,
while the right-handed fermions are in weak isospin singlets:
 
 
νe
u
(1)
(1)
(1)
−
1st generation: Le = − , Re = eR , Lq =, Ru = uR , Rd = dR
e L
d′ L
2

2nd generation: Lµ =3rd generation: Lτ =





νµ
µ−





ντ
τ−

,

Rµ = µ −
R,

(2)
Lq =

,

Rτ = τR− ,

(3)
Lq =

L

L

 
c
(2)
(2)
, Ru = c R , Rd = s R
s′ L

 
t
(3)
(3)
, Ru = tR , Rd = bR
b′ L

These fermions are characterized with weak isospin (I) and weak hypercharge (Y ) throughthe relation Q = I3 + 12 Y , where Q is the electric charge. The mass eigenstates of theleft-handed down-type quarks (d′ , s′ , b′ ) are related to flavor eigenstates (d, s, b) throughthe Cabibbo-Kobayashi-Masukawa matrix [9, 10]:  ′ 
d
Vud Vus Vubds′  =  Vcd Vcs Vcb  s
(1.2)
b
Vtd Vts Vtbb′
The fundamental properties of fermions are summarized in Table 1.1.
Particle
uL
uR
dL
dR
cL
cR
Quarks
sL
sR
tL
tR
bL
bR
νe
eL
eR
νµ
Leptons
µL
µR
ντ
τL
τR

Electric charge Q Hypercharge Y
1/3
+2/3
4/3
1/3
−1/3
−2/3
1/3
+2/3
4/3
1/3
−1/3
−2/3
1/3
+2/3
4/3
1/3
−1/3
−2/3
0
−1
−1
−1
−2
0
−1
−1
−1
−2
0
−1
−1
−1
−2

Mass
1.7–3.1 MeV/c2
3.0–4.8 MeV/c221.29 +0.05−0.11 GeV/c
2
100 +30−20 MeV/c

172.9 ± 0.6 ± 0.9 GeV/c2
2
4.19 +0.18−0.06 GeV/c< 2 eV/c2
0.511 MeV/c2< 0.19 MeV/c2
106 MeV/c2< 18.2 MeV/c2
1.78 GeV/c2

Table 1.1: Fundamental properties of fermions.

Quarks have an additional quantum number called color charge, which is three types(r,g and b). Color charge is not seen in nature and therefore the colorless composite
particle can only exist. The colorless composite particle can be made with two ways:
3

bound state of three quarks called baryons such as a proton (ur ug db ), or bound state of
one quark and one antiquark called mesons such as a pion (ur d¯r ).
The interaction between fermions is mediated by spin s = 1 bosons. The photon(γ) carry electromagnetic force, the W ± and Z bosons carry the weak force, and thegluons (g) carry the strong force. The photon is massless, while the W ± and Z bosons
are massive particles. In the Standard Model, the W ± and Z bosons acquire the mass as aresult of the electroweak symmetry breaking through the Higgs mechanism. The gluon is
the massless bi-colored particle and influences only quarks. The properties of bosons are
summarized in Table 1.2.

Particleγ−
W (W + )Z
g

Force
Electromagnetic
Weak(charged)
Weak(neutral)
StrongElectric charge Q Hyper charge Y
00−1 (+1)−1 (+1)
0
0
0
0

Mass
0
80.399 GeV/c2
91.188 GeV/c2
0Table 1.2: Fundamental properties of bosons.

The Standard Model has been successful in describing interactions of the particles
described above, all of which have been discovered experimentally. In addition, manyof the predicted properties of these particles have been confirmed, some to a high degreeof precision. However, in order for the symmetry described in Equation 1.1 to be exact,
the fermions and the W and Z bosons would have to be massless. In order for the Standard Model to be compatible with the large masses of the W and Z bosons and thus thelarge division between the effective weak coupling constant (the Fermi constant) and theelectromagnetic coupling constant (the fine structure constant), spontaneous symmetry
breaking must occur. This symmetry breaking would additionally be responsible for themass hierarchy observed in the fermions. This Electroweak Symmetry Breaking (EWSB)
is accomplished by the introduction of a scalar field known as the Higgs field [11]. The
existence of a massive scalar particle, the Higgs boson, would be associated with the
Higgs field.
The existence of the Higgs boson has yet to be confirmed experimentally, and remains
one of the most important tasks for the field of high energy physics. Direct searches for theStandard Model Higgs boson at the CERN Large Electron Positron (LEP) collider haveset a lower bound on its mass of MH > 114.4 GeV/c2 at the 95% confidence level [12].
Also, direct search at the Tevatron excluded the mass range between 156 and 177 GeV/c2
at 95 % confidence level [13].
In addition, indirect bounds on the mass of the Higgs boson can be set from precision
measurements of the top quark and W boson masses, as these quantities are sensitive toln MH through radiative corrections. Using measurements of the top quark mass and theprecision electroweak measurements made at the LEP, SLD, CDF and DØ, the constraints
on the Standard Model Higgs bosons are [14]42MH = 92+34−26 GeV/c ,MH < 161 GeV/c2 at 95% C.L..

1.2

(1.3)
(1.4)

The Spin Quantum Number

The concept of spin was first introduced in 1924 by Wolfgang Pauli, although at thetime he did not use the term “spin”. It was introduced as an internal degree of freedomof the electron, in order to explain observations in atomic physics, but it soon became
apparent that spin is an important property of all particles. Although the spin of a particle
is an internal property and does not correspond to a physical rotation, it was named spinbecause it does have some properties that are similar to a physical rotation - for example,
it adds to a particle’s total angular momentum, and causes a charged particle to interact
with a magnetic field.
Leptons and quarks have a total value for their spin of 12 . When measuring spin experimentally, a particular axis called the “spin quantization axis” must be chosen. The spin ismeasured along that axis in the rest frame of the particle and can have one of two values.
If a top quark’s spin is oriented in the same direction as the quantization axis, it has spins = + 12 , and if it is oriented opposite the direction of the axis, it has spin s = − 21 . The
choice of the quantization axis is also referred to as a choice of basis, and there are threecommon basis for measuring top quark spin in collider experiments:• Beamline Basis: the spin quantization axis is the direction of the incident colliding
particles in the tt̄ rest frame.
• Helicity Basis: the spin quantization axis is the direction of motion of the top quarkin the tt̄ rest frame.
• Off-Diagonal Basis: a hybrid basis where the spin quantization axis is in between
the direction of the axis in the helicity basis and the direction of the axis in thebeamline basis, with the exact choice of axis dependent on the kinematics of theevent [15].

1.3

The Top Quark

Since the discovery of the bottom quark in 1977, as all other quarks and leptons have asame family partner, the existence of the top quark was anticipated. The existence of the
top quark was inferred for several reasons. For one, the renormalizability of the Standard
Model requires that the sum of electric charges of all left-handed fermions must equal
zero. This condition is only satisfied with the existence of a sixth quark with an electriccharge of +2/3. In addition, the precise measurements involving the isospin of the b-quark
can be made at e+ e− colliders, which can be used to exclude the possibility of the b-quark
being a member of a singlet [16].
5The experimental discovery of the top quark took much longer than originally anticipated because the top quark was not expected to be so heavy. The top (t) quark was
observed at Fermilab in 1995 by the CDF and DØ collaborations [1, 2]. By the end of the
1992-1996 collider run (Run I), combined datasets from both experiments of ∼ 100 pb−1
provided a measurement of the top quark mass of Mt = 178.0 ± 4.3 GeV/c2 [17].

1.4Top Quark Production

Top quark has a large mass. Therefore, the top quarks can currently be produced at two
accelerators in the world: the Tevatron accelerator and the now operational and running
LHC accelerator. At the Tevatron, the top quark is produced predominately in top antitop
pairs via the strong interaction. There are two different mechanisms on the top quark
pair production. They are the quark-antiquark annihilation√ process (q q̄ → tt̄) and thegluon fusion process (gg → tt̄). At a center of mass energy s of 1.96TeV, the processesq q̄ → tt̄ and gg → tt̄ occur approximately 85% ± 5% and 15% ± 5% of the time,
respectively [18]. The leading order diagrams for two processes are shown in Figure 1.1
and 1.2.

q

t
g

−
q

−

t

Figure 1.1: Leading-order production diagram for q q̄ → tt̄.

Figure 1.2: Leading-order production diagrams for gg → tt̄.
The top quark pair production via the q q̄ annihilation has a different spin state fromthat produced via the gluon fusion. When top quark pairs are produced near kinematicthreshold (approximately 345 GeV), the pairs produced via q q̄ annihilation and gluon
fusion are in the following total angular momentum states, respectively (See Appendix A):
q q̄ : J = 1, Jz = ±1,
gg : J = 0, Jz = 0,
where z denotes the initial parton direction.
Therefore, in the case of q q̄ annihilation, the top quark and the antitop quark have thealigned spin on the beam axis, while they have the opposite spin on any axis in the case of6

gluon fusion. Figure 1.3 shows an example of the spin configurations of the gluon fusionand q q̄ annihilation at a threshold production.

Spin
t
q

q
t

Figure 1.3: The spin configurations of the gluon fusion (left) and the q q̄ annihilation(right) subprocesses.

In the actual pp̄ collisions, top quark pairs can be produced at center-of-mass energies
significantly above threshold. Then, for q q̄ → tt̄ process, for top and antitop quark spinstates, (↑↓) and (↓↑) dominate, but a little bit the (↑↑) and (↓↓) spin configuration exist in
the beamline basis. On the other hand, for gg → tt̄ process, the unlike spin configuration
between top and antitop quark is mitigated so that top quark pairs slightly anti-correlated
spins in the beamline basis.
The theoretical prediction of the tt̄ production cross section at next-to-leading order(NLO) is σ N LO (pp̄ → tt̄X) = 6.7+0.7
GeV/c2 [18]. Figure 1.4 shows the−0.9 pb at Mt = 175√
NLO calculation of σ(pp̄ → tt̄X) for pp̄ collisions at s = 1.96 TeV as a function of the
top quark mass.

1.5

Top Quark Decay

The top quarks decay almost completely in to a W boson and a b-quark. Other decay
channels are permitted in the Standard Model, but are heavily suppressed by factors of|Vts |2 /|Vtb |2 ∼ 10−3 and |Vtd |2 /|Vtb |2 ∼ 5×10−4 , where Vij is the element of the CabibboKobayashi-Masukawa (CKM) weak-mixing matrix [10]. The large mass of the top quark
results in a very rapid decay with a mean lifetime of τt ∼ 10−24 s. As this is shorter than
the time scale required for quarks to form bound states (or “hadronize”), the top quark
essentially decays as a “free” quark. Therefore, top quark retain its original polarization
at the production until decay, and due to the decay via parity violating weak interaction,
the information of the parent top polarization is transferred to decay products.
The b-quark resulting from the decay will then proceed to hadronize and manifest
itself in the detector as a jet, or a collimated stream of hadrons. The W boson will decay
rapidly into either a pair of quarks or a pair of charged lepton and a neutrino. Thus, forthe case of a tt̄ pair production and decay, there are six objects: two b-quarks and two
decay products from each of the W boson. Figure 1.5 shows the tree level diagram of tt̄
production and decay in the subprocess q q̄ → tt̄.
7

σ(pp → tt) (pb)

12
10
8
6
4
Cacciari et al. JHEP 0404:068 (2004)
Cacciari et al. ± uncertainty
Kidonakis,Vogt PIM PRD 68 114014 (2003)

2

Kidonakis,Vogt 1PI

0
160

162

164

166

168

170

172 174 176 178 180
2Top Quark Mass (GeV/c )Figure 1.4: NLO calculations of σ(pp̄ → tt̄X) [18] for pp̄ collisions at
a function of the top quark mass.

bνl , q,qtW+–l+ , q
–

b

–
–

t

q

l- , q,W––νl , qFigure 1.5: Tree level q q̄ → tt̄ production and decay.

8√

s=1.96 TeV as

It is the decay mode of the W bosons that defines the decay channels of the tt̄ systemused in its experimental study. These decay channels are classified as:• All-hadronic channel, where both W bosons decay to quark-antiquark pairs, resulting in a final state having an experimental signature of six jets. This decay mode
carries the largest branching ratio of 46 %, but suffers from the largest amount of
irreducible QCD background.
• Lepton+jets channel, where one W decays to a lepton-neutrino pair and the otherto quarks, resulting in an experimental signature of a high momentum lepton, four
jets, and a missing transverse energy associated with the neutrino. Due to the difficulty of identifying τ leptons at a hadron collider, only leptonic states with anelectron or muon in the final state are considered. This channel carries a branchingratio of 30 %.
• Dilepton channel, where both W bosons decay to leptons, resulting in an experimental signature of two high momentum leptons, two jets, and large missing transverse energy associated with two neutrinos. As with the lepton+jets channel, only
leptonic states with an electron or muon in the final state are considered. This channel carries a branching ratio of 4 %. The remaining 20 % of tt̄ decays involve theproduction of a lepton that does not decay to an e or µ. While measurements in thisso-called “τ + X” channel are possible, they do not afford nearly the same precisionthat any of the other three channels does.

1.6

Top Quark Mass

The top quark mass is a fundamental parameter in the Standard Model. Due to its large
value, it has a critical influence on the Standard Model calculations than the other quarks.
It contributes to higher order (radiative) corrections to electroweak processes, as illustrated in Figure 1.6. It also helps to constrain the mass of Higgs boson via the radiative
H
b̄
W+

W+

W+

W+

t

Figure 1.6: Loop diagrams generating corrections to the theoretical W boson mass. On
the left is a fermion loop with the top and b-quarks. On the right is a Higgs boson loop.

correction to the mass of the W boson. Figure 1.7 shows constraints on the Higgs boson mass using the current best measurements of the W boson and the top quark masses,
and a global fit of the Higgs boson mass to several electroweak parameters. As indicated9in Figure 1.7, the most likely value of the Higgs boson mass is ruled out by the direct
searches at LEP. A failure of the Standard Model to properly describe these results may
indicate new physics yet to be discovered.

1.7

Top Quark Spin Correlation

In the quark pair production by the strong interaction, the quark spins are entangled according to the short distance dynamics of quantum chromodynamics (QCD). The spinstate is observable in angular correlations among the quark decay products induced by the
V-A nature of the weak interaction, but is typically destroyed by the depolarizing effectsof hadronization before the decay can proceed. The top quark is an exception to this rule.
Spin-flip time by hadronizing is described as [19]O(mt /Λ2QCD ) ≃ (1.3 MeV)−1(1.5)

On the other hand, the top quark decay width is predicted to be Γt ≃ 1.42 GeV [20].
Therefore we can findO(mt /Λ2QCD ) ≫ 1/Γt

(1.6)

and this means the top quark life time is shorter than the fragmentation timescale, cutting
off the long distance QCD effects and transmitting the tt̄ production configuration to thefinal state.
In top quark decays in the SM the V-A couplings fix the angular distributions of the
decay products according to the polarization of the parent top quark via
1 ± αi cosθi
1 dΓ=
Γ dcosθi2(1.7)

where the positive and negative sign is used for decay products from the top quark whichhas ↑ and ↓ spin state, respectively. θi denotes the angle between the quantization axis
and the flight direction of the decay particle in the top quark rest frame.
In the V-A weak decay, the spin analyzing power coefficient αi is equal to +1.0 for
the charged lepton or down-type quark, -0.41 for the bottom quark, and -0.31 for theneutrino or up-type quark, with the signs reversed for antitop-quark decays [21]. The tt̄
spin correlation connects the daughter flight direction on each side of the decay.
The differential cross-section in these variables is1
1 + καi αj cosθi cosθjd2 σ=
σ d(cosθi )d(cosθj )
4

(1.8)

where i and j refer to top-quark and antitop-quark decay products respectively.
κ is defined asκ=N↑↓ + N↓↑ − N↑↑ − N↓↓N↑↓ + N↓↑ + N↑↑ + N↓↓(1.9)

where Nst st̄ indicate number of t, t̄ in st , st̄ spin state with respect to the quantization axis
for the top quark. κ is a parameter between -1 and 1 that depend on the quantization
10

80.5

July 2011

mW [GeV]LEP2 and Tevatron
LEP1 and SLD
68% CL80.4∆α80.3 mH [GeV]
114
300
155
175

1000
195

mt [GeV]6

mLimit = 161 GeVJuly 2011

Theory uncertainty
∆α(5)
had =

5

0.02750±0.00033
0.02749±0.00010
incl. low Q2 data

∆χ2

4
3
2
1
0

Excluded
30

100

300

mH [GeV]Figure 1.7: Electroweak constraints on the Higgs boson mass using the current best measurements of W boson and top quark mass (top). Global fit of Higgs boson mass to several
electroweak parameters (bottom).

11

axis used and determines the magnitude of the tt̄ spin correlation. κ = +1(−1) give fullcorrelated (anticorrelated) spins and κ =0 corresponds to no spin correlation. Theoretical
calculation of κ which include next-to-leading order effects predicts κ = 0.777 in thebeamline basis, κ = 0.782 in the off-diagonal basis and κ = 0.352 in the helicity basis, at
the Tevatron [15].
tt̄ pair produced via q q̄ annihilation and that produced via gg fusion have different
spin states. Therefore, flight direction of the decay particle also depends on the tt̄ production mechanism. Fig 1.8 shows two dimensional distributions of cosθ+ and cosθ− in thebeamline basis for the processes q q̄ → tt̄ and gg → tt̄ exclusively. These distributions
are made with Herwig event generator [22] on the assumption of CTEQ5L PDF set [23]and Mtop = 172.5 GeV/c2 . As we mentioned in the Section 1.4, κ in the beamline basis
for q q̄ annihilation is expected to be near the unit, while for gg fusion, κ has a negativevalue as expected. By fitting the distributions to a function C0 · (1 + κcosθ+ cosθ− ), we
found κ = 0.958 ± 0.003 for q q̄ → tt̄ and κ = 0.369 ± 0.016 for gg → tt̄.
herwig gg→tt

Entries 579952

EventsEventsherwig qq→tt
10000
8000

500
400

6000

300

4000

200

2000

100

0
1

co 0.5sθ
-Entries 35638

0
1

0
-0.5
-1 -1

-0.5

0

0.5

co 0.5
sθ
-

1

cosθ+

0
-0.5
-1 -1

q q̄ → tt̄

-0.5

0

0.5

1

cosθ+

gg → tt̄

Figure 1.8: The two-dimensional distributions of cosθ+ and cosθ− in beamline basis forthe processes q q̄ → tt̄ and gg → tt̄.

1.8

κmeas, what is measured in this analysis

As described in Section 1.7, the top quark is expected to decay before losing spin polarization at its production and the charged lepton flight direction in the top rest frame ismaximally correlated to the top polarization.
In this measurement, we suppose that tt̄ is once produced with the Standard Modelspin correlations, i.e. κ ∼ 0.8. However top and/or anti-top polarization might be lostwhen they decay because of spin flip by hadronization, such as interaction with gluons.
Suppose κprod is spin-spin correlation coefficient of tt̄ pair at its production, and Ct (Ct̄ )denotes a fraction of the top (anti-top) polarization at production which is kept until it
decays. Then the differential decay rate equivalent to Equation 1.8 is proportional to
1 + Ct Ct̄ κprod cos θ+ cos θ− .
12

Here the standard model predicts top quark decays before losing its polarization, i.e. Ct =Ct̄ = 1.
Substantially, what we measure in this analysis is Ct Ct̄ κprod . If we observe nonzero correlation coefficient κmeas in tt̄ production and decay, that consequently indicatesa direct evidence that top and anti-top are produced with their spins being correlated anddecay as bare quarks before losing their spin polarizations.

1.9

Motivation of Spin Correlation Measurement

Measurement of the spin correlation is important not only as a test of the theoretical prediction, but also as a possible factor in helping to understand new physical interactions.
Known interactions that can produce top quark pairs are well understood, leading to accurate predictions for the tt̄ spin correlation. Any deviation of the measured spin correlationfrom the theoretical predictions would indicate new top quark production processes. An
example of such a process would be tt̄ production via Kaluza-Klein graviton in a RandallSundrum model [24], which would alter the spin correlation because unlike the gluon, thegraviton has a total spin of 2. In particular, resonant production of such a graviton would
produce large changes in the spin correlation as a function of the tt̄ invariant mass compared to current theoretical predictions. Precision studies of the spin correlation and its
behavior as a function of tt̄ mass could provide evidence for or against this model andmany other models of new physical process.
A recent CDF measurement found an asymmetry in the direction that the top quark
from a tt̄ pair travels after it is produced that disagrees with the theoretical prediction
by approximately 3σ [25]. Many models of new physical process have been proposed to
explain this discrepancy, and many of these models also predict that the new interactions
will have an effect on the top quark pair spin correlation [26]. A precise measurementof the spin correlation would differentiate among these models, probing new physicalinteractions.

1.10

Previous Measurement of Top Quark Spin Correlation

Measurements of spin correlation has been performed by the CDF, DØ and ATLAS experiments.

1.10.1 Dilepton channel
The DØ experiment reported on two measurement results in beamline basis using tt̄ dilepton candidates in data of 5.4 fb−1 . One of the two results is measured using a matrix
element approach with 485 tt̄ dilepton candidate events. Its result is κ = 0.57 ± 0.31
(stat+syst) [27]. Another is obtained from differential angular distribution template in
lepton decay angles with 441 tt̄ dilepton candidate events. Its result is κ = 0.10 ± 0.45
(stat+syst) [28].
13ATLAS experiment reported on a result in helicity basis using tt̄ dilepton candidates indata of 0.7 fb−1 . Standard Model predicts κ = 0.32 in helicity basis for tt̄ spin correlationin tt̄ production from proton-proton collisions at a center-of-mass energy of 7 TeV. The
measurement result at ATLAS experiment yielded κ = 0.34 ± 0.07(stat) +0.13−0.09 (syst) [29].
This result is consistent with the Standard Model prediction and it indicates the existenceof a spin correlation in tt̄ events at a level of 3 σ equivalence.

1.10.2 Lepton+Jets channel
The CDF experiment reported on measurement results with 1001 tt̄ lepton plus jet candidates in data of 5.3 fb−1 . Its result is κ = 0.72 ± 0.64 (stat) ±0.26(syst) [30] in beamline
basis.

1.10.3 Combine lepton+Jets events and dilepton events
The DØ experiment reported on a result that obtained by combining a result in lepton
plus jets events with the result in dilepton events, where both results are obtained using
the matrix element approach. Its result is κ = 0.66±0.23 (stat+syst) in data of 5.3 fb−1 [31]in beamline basis. This result indicates the existence of spin correlation in tt̄ events at alevel of 3.1 standard deviations equivalence.

1.11

Outline of This Thesis

In this thesis we measure the strength of the tt̄ spin correlation, κ, from a differential
angular distribution of the two decay leptons in the rest frames of their respective t quark
and the spin quantization axis. To measure κ, we use data of 5.1 fb−1 collected with
the Collider Detector Facility (CDF) at the Fermilab Tevatron between March 2002 andJune 2009. Chapter 1 describes the short summary of the Standard Model and the top
quark physics. Chapter 2 describes the experimental apparatus of Tevatron accelerator
and CDF detector. In Chapter 3, the event reconstruction using information from thedetector responses is discussed. Chapter 4 describes the event selection to select the
dilepton candidate events, the method of the full kinematical reconstruction, preparationof signal/background templates and the construction of Feldman-Cousins confidence belt
from pseudo-experiments. Chapter 5 describes an estimation of systematic uncertainties
in our measurement. Chapter 6 describes the measurement result and discussion, andChapter 7 concludes this analysis.

14

Chapter 2
Experimental Apparatus
This chapter describes the Fermilab accelerator complex including the Tevatron collider,
CDF II detector components, trigger and data acquisition (DAQ) system.

2.1

The Fermilab Accelerator Complex

The Tevatron is a circular particle collider in the United States at the Fermilab. The
Tevatron was the second most powerful proton-antiproton accelerator in the world before
it shut down. The reason why the Tevatron accelerates protons and antiprotons is thefollowing two reasons.
• Synchrotron radiation of protons and antiprotons is much smaller than it of electronsand positrons at high energy in circular accelerators: Although electron-positron
collisions are easier to analyze (because electrons are single point-like particles,
unlike the composite protons), the large synchrotron radiation of electrons prohibits
their use at high energies in circular accelerators. Because synchrotron radiation
increases as the inverse of the fourth power of a particle’s mass, protons, whichhave roughly 2000 times the mass of electrons, radiate much less. Protons andantiprotons are currently the only viable alternative to electrons and positrons.
• A collider’s design is vastly simplified if it collides particles with their antiparticles,
because the particles can travel opposite directions in the same beam-pipe and bebent by the same set of magnets.
The accelerating system consist of a few accelerators(Figure 2.1) where the protons andantiprotons are gradually accelerated up to final energies. The Tevatron is the last in achain of accelerators. The Tevatron’s first physics run, referred to as Run I, occurred from
1992-1996. After a series of upgrades, it began running in 2002 and finished running in
2011.
The detector located at proton-antiproton collision points need to disentangle all theparticles that are produced and need to do this every 396 ns, keeping only those events
that mark “interesting physics” , the anomalies. Not only do we need to understand what
type of particles we expect and how often they should be produced, but we also need toknow how these particles will interact in different environments so we can build detectors
15

in ways that we can distinguish the difference between the types of particle produced.
Until we can be fairly confident in how the detector works and its different idiosyncrasies
that need to be calibrated and understood, we can not make any interesting connections
with what theory tells us.

Figure 2.1: A diagram of the Fermilab accelerator chain.

Proton Source
The protons used in the Tevatron are originally extracted from very pure hydrogen gas.
For ease of insertion into the Booster, the particles accelerated are actually H- ions insteadof protons. Hydrogen gas is moved between two electrodes and a spark ionizes the hydrogen into electrons and H+ ions. The positive ions strike a cathode made of cesium, which
has a low work function and thus releases electrons easily, and occasionally pick up twoelectrons and form H- ions. An electrostatic extractor sends them to the preaccelerator.

Preaccelerator
The preaccelerator is a Cockcroft-Walton-style [32] electrostatic accelerator. Ions fromthe proton source are subjected to a potential of -750 kV, thus producing beams of H- ionswith an energy of 750 keV. The potential is created by the 5-stage diode voltage multiplier,which converts 75 kV AC to the -750 kV DC. The extracted H- ions are accelerated to16

750 keV by passing through the accelerating column. The preaccelerater beam every
66 milliseconds (a 15 Hz repetition rate) whether beam is being requested or not. The Hions are steered and focused by magnets down a transfer line to the Linac.

Linac
The Linear Accelerator is the next Level of acceleration for the negative charged H- ions.
It is abbreviated to Linac. It takes the ions with energy of 750 keV and accelerates themto an energy of 400 MeV. The Linac consists of two main sections, the low energy drift
tube Linac and the high energy side-coupled cavity Linac. The drift tube Linac (DTL)
makes up the first five RF stations. It accelerate the ion beam to 117 meV. Beam in the
DTL is focused by means of quadrupole magnets located inside the drift tubes, which inturn are located inside the RF cavities. The side-coupled cavity Linac (SCL) has sevenside-coupled cavity modules that accelerate the H- ions to 400 meV. The beam travelingthrough the SCL is focused by quadrupoles placed between the accelerating modules(outside of the accelerating cavities).
At the far end of the Linac is a chopper, that electrostatically selects a portion of the
Linac beam to be sent along a transfer line to the Booster. The Linac completes fifteen
acceleration cycles per second.

Booster
The Booster is a proton synchrotron, approximately 150 m in diameter. It takes the
400 MeV H- ions from the Linac and strips the electrons off, which leaves only the proton,
and accelerating the protons to 8 GeV. It has the same duty cycle as the Linac, 15 Hz. The
acceleration is accomplished by 18 ferrite-tuned RF cavities located around the Booster
ring. A 100 kW PA ( Power Amplifier) drives each cavity. 96 conventional magnets witha maximum field of 0.7 T bend the beam into a circular orbit. The Booster is able to holdmultiple batches of particles from the Linac at once to increase beam intensities, often
storing eleven or twelve batches in its ring.
A special set of magnets handles the injection of incoming H- ions from the Linac.
Each magnet bends the beam and focuses the beam either horizontally or vertically. The
H- ions and circulating beam passes through the stripping foil, which removes the electrons from most of the H- ions and yields protons. A similar set of magnets steers thebeam back into the Booster orbit while removing any leftover H- ions.
A set of fast kicker magnets extracts the proton beam from the Booster. The protons
go into a transfer line that leads to the Main Injector.

Main Injector
The Main Injector is a large proton synchrotron with a diameter of about 1 km. It has two
main functions involving the Tevatron: accelerating protons and antiprotons for injection
into the Tevatron, and accelerating protons to be sent to the antiproton source.
Main Injector has 18 accelerating cavities. It uses 344 dipole magnets and 208 focusing quadrupole magnets, all conventional water-cooled electromagnets, to steer the proton17

beam. It can accelerate protons to 120 GeV or 150 GeV. When used to stack antiprotons,
the final energies is 120 GeV. When used to inject into the Tevatron, the final beam energy
is 150 GeV.
As well as accepting protons from Booster, the Main Injector can accept antiprotons
from the Antiproton Source. The Main Injector can accelerate beam as fast as every
2.2 seconds.

Antiproton Source
The antiproton source produces antiprotons for use in Tevatron collisions. The Main
Injector sends 120 GeV protons down a transfer line to a nickel target. (Nickel was chosenbecause it can absorb more heat without melting than other metals.) Antiprotons areamong the products resulting from this collision; they are selected by an electromagnetic
selector and focused down a transfer line to the Debuncher. Studies have shown that
120 GeV is the optimal energy for antiproton production; at this energy, approximately
one antiproton is collected per 105 protons sent to the antiproton source. The resultant
antiprotons have an average energy of about 8 GeV.

Debuncher
The Debuncher is not an accelerator but a triangular storage ring. Its main purpose is to“debunch” the particle beam, removing its RF bunch structure. Magnets in the Debuncher
decrease the momentum spread of the antiprotons by rotating them in phase space, trading momentum spread for time spread. This results in a beam of particles that have no
RF bunch structure but have roughly uniform momentum. Antiprotons remain in the Debuncher until the next batch of protons is sent to the antiproton target, at which point the
antiprotons are sent to the Accumulator.

Accumulator
The Accumulator lies in the same tunnel as the Debuncher. It is a long term antiproton
storage ring, designed to store antiprotons with minimal losses for days. Antiprotons fromthe Debuncher are manipulated by RF systems in the Accumulator to fill a stable regionof phase space, known as the core. The core is kept as small as possible to minimize
the momentum spread of the antiprotons; a smaller beam gives a higher luminosity upon
injection into the Tevatron.
While the antiprotons stay in the Accumulator, they are reduced in transverse momentum through a process called stochastic cooling [33]. This procedure measures themomentum spread of a group of antiprotons and sends a signal across the ring to corrector magnets, which adjust their fields for each group of particles to reduce the momentum
spread of those particles. This results in denser antiproton beams injected into the Tevatron, increasing the resulting luminosity.
Extraction from the Accumulator requires the antiprotons to be collected into bunches
again. Adiabatic activation of RF stations causes a portion of the beam to be collected18into bunches, which are then transferred back to the Main Injector, decelerated to 8 GeV,
and injected into the Recycler.

Recycler
When the Accumulator reaches its maximum optimal capacity, its antiprotons are passed
into the Recycler, a ring of permanent magnets in the same tunnel as the Main Injector.
This storage ring keeps antiprotons at 8 GeV, collecting them until the Tevatron is readyfor injection. In the Recycler, antiprotons are cooled further using a process called electron cooling [34], in which a beam of electrons is accelerated to the same energy as the
antiprotons and runs alongside them. Transverse momentum from the antiproton beam ispassed to the much lighter electrons, causing the antiprotons to lose transverse momentum, making the beam smaller. Antiprotons are injected from the Recycler to the Main
Injector, which accelerates them to 150 GeV for injection into the Tevatron.

Tevatron
The Tevatron is the largest of the Fermilab accelerators, with a circumference of approximately 1 km. It is a circular synchrotron and keeps both protons and antiprotons in thesame beampipe, revolving in opposite directions. The Tevatron accept both protons andantiprotons from Main Injector and accelerate them from 150 GeV to 980 GeV.
Electrostatic separators produce a strong electric field that keeps the two beams fromtouching except at the collision point. The beam is steered by 774 superconducting dipole
magnets and 240 quadrupole magnets with a maximum magnetic field of 4.2 T. They arecooled by liquid helium to 4.2 K, at which point the niobium-titanium alloy in the magnets
becomes superconducting.
The Tevatron holds 36 bunches each of protons and antiprotons. The process of injecting particles into the machine, accelerating them, and initiating collisions, referred to as ashot, starts with injection of protons, one bunch at a time, at 150 GeV from the Main Injector. The antiprotons are injected four bunches at a time from the Recycler through theMain Injector. RF cavities accelerate the beams to 980 GeV, and then some electrostatic
separators switch polarity to cause the beams to collide at two points. Each collision pointlies at the heart of a particle detector: one named D0 and the other named the Collider
Detector at Fermilab (CDF).

2.2

The CDF II Detector

The CDF II detector is a general purpose detector designed to study pp̄ collisions at theTevatron. The detector components are arranged in cylindrical shape. The position of the
sub-detectors are described in cylindrical coordinates (r, φ, z) with fixing the origin to thegeometric center of the detector. The φ = 0 is parallel to the ground and points out of theTevatron ring. The z-axis points along the beam in the direction of the protons.
The outgoing particles are described in spherical coordinates. The z-axis is replacedby the polar angle θ. In the event reconstruction, the transverse momentum of particles
19

pT = |p|sinθ is measured in the transverse plane, and their direction is given by thepseudorapidity η. The pseudo-rapidity is defined as
 
θ
(2.1)
η = −ln tan
2which is a good approximation at high energies (pT ≫ m) to the rapidity
1
E + pzy = ln2
E − pz(2.2)

where E is the particle’s energy and pz is its momentum along the z-axis.
A solid cutaway view of the CDF II detector is shown in Figure 2.2. In the center of thedetector, the charged particle tracking system is enclosed by a superconducting solenoid.
Outside the solenoid is the calorimeter system which is surrounded by the muon detectors.

Figure 2.2: Isometric view of the CDF II detector.

2.2.1 Tracking System
The innermost part of the CDF II detector is the tracking system which is composed ofmulti-layer silicon microstrip detectors, an open-cell wire drift chamber, and a superconducting solenoid magnet. The coverage of these detectors are illustrated in Figure 2.3. It is20

used to reconstruct trajectories of charged particles and precisely measure their momenta.
The reconstructed particle trajectories are called tracks. Good resolution is required todetect displaced secondary vertices, which is a key to detect B hadrons.

CDF Tracking Volumem

2.0

END WALL
HADRONCAL.

n = 1.0

030

END PLUG EM CALORIMETEREND PLUG HADRON CALORIMETER

SOLENOID

1.5

1.0
COT

.5

n = 2.0n = 3.0

3

0

0
0

.5
SVX II
5 LAYERS

1.0

1.5

2.0

2.5

3.0

m

INTERMEDIATESILICON LAYERS

Figure 2.3: The CDF tracking system.

Silicon Detectors
The silicon detector consists of three parts: Layer 00 (L00) [35], the Silicon Vertex Detector (SVXII) [36], and the Intermediate Silicon Layers (ISL) [37], as shown in Figure 2.4.
The detector offers full tracking coverage for |η| < 2.0 as shown in Figure 2.4 (left).
All the three sub-detectors are constructed from wafers of n-type silicon with thin
strips ( ∼ 10 µm) doped with p-type silicon (n-type in addition for SVXII sensors). The
reverse bias voltage extends the depletion region from the p-n junction. When a charged
particle passes through the depleted region, it ionizes the silicon wafer creating electron
and hole pairs. The voltage moves electrons to one side of the sensor, the holes to theother side. Then, collected charge is read out by ASIC chips mounted at the end of thesensors. The spacial resolution is varying depending on each silicon sub-detector, since
pitches are ranging from 25 µm to more than 100 µm.
L00: The L00, the innermost silicon detector, consists of one layer of single-sided silicon21

Figure 2.4: The CDF silicon detector. Left: the coverage of the silicon detector in r-z
plane. Right: Configuration of the silicon detector in r-φ plane.

attached directly to the beampipe, only at ∼1.5 cm radius. Its purpose is to improve
the resolution of the track impact parameter and position of secondary vertices.
SVXII: The SVXII is the main part of the silicon detector, which consists of five concentric layers of double-sided silicon. These layers are placed at radii from 2.4 cm
to 10.7 cm. The hit information of the SVXII provides high resolution tracking information and is especially useful for reconstructing displaced secondary vertices.
ISL: The ISL is the outermost silicon detector, which consists of a single layer at |η| <1.0 (at a radius of 22 cm) and two layers at 1.0 ≤ |η| ≤ 2.0 (at a radius of 20 cm
and 29 cm). This detector helps the connection of tracks between the Central Outer
Tracker and the SVXII. This improves the track resolution and the performance of
forward tracking in |η| ≤ 2.0.
Central Outer Tracker (COT)The COT is a cylindrical drift chamber that sits directly outside of the silicon detectors
in the central region (|η| < 1.0) [38]. The chamber consists of eight cylindrical radial
sections (“superlayers”) of 310 cm long cells at radii between 40 and 132 cm from thedetector center. The eight superlayers are placed in alternating axial and stereo sections:wires in axial superlayers run parallel to the z-axis, while wires in stereo superlayers
are strung at ± 2 degree angles with respect to the z-axis. The number of cells in the
superlayer increases radially outwards, the innermost superlayer consists of 168 cells andthe outermost one consists of 480 cells. Each cell contains twelve sense wires and thirteen
potential wires placed alternately. The chambers are filled with a mixture of argon andethane gasses, which is chosen to have a uniform drift velocity across the cell volume.
22

Along a charged particle passing through the chamber, the gasses in the chamber are
ionized. The electrons are drifted forward the sense wire by the electric field, and thenthey create an avalanche of charges which induce a pulse onto the sense wire. The position
resolution of the COT is about 140 µm per cell, and the transverse momentum resolutionσis pp2T = 0.0015 [GeV/c]−1 .
TTime-of-Flight System (TOF)
The TOF detector [39] is incorporated into the CDF detector in order to identify particles
up to 1.5 GeV/c. By measuring the time it takes for a collision product to reach the TOF,
we can separate particles which have different masses, such as π ± and K ± . This detector
is located between the COT and the superconducting solenoid at a radius of 140 cm witha coverage in |η| ≤ 1.0. In this analysis, we do not use for particles discrimination but
use for the event veto coming from cosmic rays.
Superconducting Solenoid
The superconducting solenoid operated at a current of about 4650 A produces an uniform
magnetic field of 1.4 T parallel to the z-axis. The conductor is made of Al-stabilized NbTi.
This strong magnetic field bends the trajectory of high-pT charged particles, allowing us
to reconstruct their momentum using the tracking system.

2.2.2 Calorimeters
The CDF calorimeters measure the energy of both charged and neutral particles. They
are sampling scintillator calorimeters segmented into towers having a geometry projectedto the detector center. The calorimeter system consists of electromagnetic (EM) andhadronic calorimeters, covering 2π in azimuth over the range |η| < 3.6. The cross sectionof the CDF calorimeter system is shown in Figure 2.5.
The EM calorimeter system consists of two sections: the central EM calorimeter [40](|η| < 1.1) and the plug EM calorimeter [41] (1.1 < |η| < 3.6). Both sections include
the main calorimeter, which mainly measures the energy of particles, and the shower
maximum detector, which helps to improve the position resolution of the calorimeterclusters. The hadronic calorimeter system consists of three sections: the central hadronic
calorimeter (|η| < 0.9), the wall hadronic calorimeter (0.7 < |η| < 1.3), and the plug
hadronic calorimeter (1.3 < |η| < 3.6).
Central Electromagnetic Calorimeter (CEM)
The CEM is segmented into 24 towers in φ and 10 towers in η. The single CEM wedge
is shown in Figure 2.6. It is a lead-scintillator sampling calorimeter having a radiation
length (X0 ) 18 X0 . The energy resolution of the CEM is
σE
13.5%
=p
⊕ 2%
E
E (GeV)

where the notation ⊕ sums the constant and stochastic term in quadrature.
23

(2.3)

Figure 2.5: The CDF calorimeter system.

Central Electromagnetic Shower Maximum Detector (CES)The CES is a proportional chamber with wire and strip readout which improves the position resolution of the calorimeter clusters. The CES is placed at a position at which
the shower for electrons and photons has the maximum number of particles, called theshower maximum. Its position corresponds to a depth of 6 X0 of the EM calorimeter. The
position resolution is 0.2 cm at 50 GeV.
Plug Electromagnetic Calorimeter (PEM)
The PEM is segmented into 12 towers in η and 24 (48) towers in φ for the inner (outer)groups. It is a lead-scintillator sampling calorimeter having a total thickness is 21 X0 .
The energy resolution of the PEM is:
σE
14.4%
=p
⊕ 0.7%
E
E (GeV)(2.4)

Plug Electromagnetic Shower Maximum Detector (PES)The PES [42] measures the shower maximum position similar to the CES. It is located at
6 X0 depth and is made of two layers of 5 mm wide scintillator strips, with each layer
24

Y

Phototubes

Light
GuidesRight

Left

Wave Shifter
Sheets
0

X

1
2
3
4

Lead
Scintillator
Sandwich

5
6
7
8
9

Tow
ers

Strip
ChamberZ

Figure 2.6: The single CEM wedge.

having a 45◦ crossing angle relative to the other.
Central Hadronic Calorimeter (CHA)
The CHA, a iron-scintillator sampling calorimeters, is segmented into 24 towers in φ and8 towers in η [43]. It is located directly outside of the CEM with 32 layers per tower,
which corresponds to 4.7 interaction lengths (λI ) thick. The energy resolution of the
CHA is:
σE
50%
=p
⊕ 3%
(2.5)
E
E (GeV)
Wall Hadronic Calorimeter (WHA)

The WHA extends the CHA coverage to fill the gap between the central and plug regions [43]. It is made of 15 layers of iron (5.0 cm) and scintillator (1.0 cm). The energy
resolution of the WHA is:
σE
75%
=p
⊕ 4%
(2.6)
E
E (GeV)
Plug Hadronic Calorimeter (PHA)

The PHA is made of 23 layers of alternating iron and scintillator. The energy resolutionof the PHA is:
80%
σE
⊕ 5%
(2.7)
=p
E
E (GeV)25

2.2.3 Muon Detectors
The energy of high-pT muons are not measurable with the calorimeter, since they passthrough the detector materials by depositing only minimum ionizing energy. The CDF
muon detectors consist of four systems and are located outside of the calorimeters. The
muon system coverage is shown in Figure 2.7. A cross section view of a muon chamberis shown in Figure 2.8. When a muon passes through the muon system, the drift time ineach layer is registered. We define a muon “stub” which requires a hit in three of the four
layers of drift chambers.
- CMX

- CMP

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00-1
0
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 000- CMU

00 00 00 00 00 00 00 00- IMU
10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000000000000000000000

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000000000000000000000

φη

Figure 2.7: CDF muon coverage for the CMU, CMP, CMX, and BMU detectors. The
BMU is referred as the IMU.

Radial centerline

Muon track
t4

.

.

.

.

.

.

.

55 mm

.

.

. t2

.

.

.

.

.

.

To pp interaction vertex

Figure 2.8: A cross section view of a CMU muon chamber.

26

Central Muon Detector (CMU)
The CMU [44] consists of four layers of planar drift chambers located outside of the CHA.
It covers the central region of |η| < 0.6 and can detect muons with pT > 1.4 GeV/c.
Central Muon Upgrade (CMP)The CMP consists of four layers of planar drift chambers located outermost of the CDF
for walls or behind the magnet return yokes. The CMP covers |η| < 0.6 and can detectmuons with pT > 2.0 GeV/c.
Central Muon Extension (CMX)The CMX consists of four to eight layers of drift chambers depending on the polar angle.
It provides muon detection in the region 0.6 < |η| < 1.0 and can detect muons withpT > 1.4 GeV/c. The scintillator tiles (CSX) is also placed on the inside and outside of
the CMX, used for improved triggering.

Barrel Muon Detector (BMU)
The BMU extends the muon detector coverage to 1.0 < |η| < 1.5. Drift chambers andscintillators are attached surrounding the forward toroid magnets.

2.2.4 Luminosity Monitor
Luminosity
The collision rate of protons and antiprotons is quantified by the instantaneous luminosity.
The luminosity is calculated with the following formula:
fr NB Np Np̄  σl F ∗L=
2π(σp2 + σp̄2 )
β

(2.8)

where fr is the revolution frequency, NB is the number of bunches, Np and Np̄ are thenumber of protons or antiprotons per bunch, σp and σp̄ are the beam sizes at the interaction
point, F is a form factor which corrects for the bunch shape and depends on the ratio of∗∗
the bunch length σl to the beta function√ ∗ β at the interaction point. β is a measure of thebeam spreads, which are given by β ǫ with ǫ being the beam emittance.
There is a continuous effort to maximize the peak luminosity which directly results
in increasing the amount of data delivered by the Tevatron. The
R amount of data collected
through Run II is expressed by the integrated luminosity ( Ldt) which is measured inunits of b−1 , where 1 b−1 is 1024 cm−2 . Figure 2.9 shows the integrated luminosity delivered by the Tevatron and recorded by the CDF.
27

Figure 2.9: Total integrated luminosity delivered by the Tevatron and recorded by the
CDF.

Cherenkov Luminosity Counters (CLC)
The beam luminosity is measured by using the CLC detector [45] located in the forward
region (3.7 < |η| < 4.7) of the CDF detector on both sides. The CLC consists of longconical gaseous Cherenkov counters that monitor the average rate of inelastic pp̄ collisions
per bunch crossing (Rpp̄ ). The instantaneous luminosity (Linst ) is calculated from the next
expression:µ · fBC = σi · Linst · ǫ(2.9)
where µ is the number of interactions per bunch crossing counted by the CLC detector,
fBC is the bunch crossing frequency (2.5 MHz for 36×36 bunch operations), ǫ is the
CLC acceptance for a single pp̄ interaction, and σi is the inelastic pp̄ cross section at theTevatron (60.7 ± 2.4 mb).

2.2.5 Data Acquisition System
The bunch crossing rate of the Tevatron is 2.5 MHz for 36×36 bunch operations whichcorresponds to 396 ns separation. The actual interaction rate is bit lower, about 1.7 MHz.
Since this rate is too high to record every event into disk, we need to discard the mostevents while interesting ones must be identified. The selection of events is performed by
the fast online electronics, called the trigger system. The CDF trigger system has a threelevel architecture (called Level 1, Level 2, and Level 3) and is designed to reduce thedata rate by identifying the physically interesting events. Once an event is accepted by
the Level 3 trigger, then the data are sent to the Consumer Sever/Logger (CSL) that is the
final component in the CDF data acquisition system. The data flow in the CDF trigger
system is shown in Figure 2.10.
28

Figure 2.10: Data flow in the CDF data acquisition system.

Level 1
The Level 1 trigger (L1) discards about 97% of the events, resulting in the acceptance rate
to about 50 kHz. Figure 2.11 shows the trigger path from the Level 1 to the Level 2. The
L1 CAL can make its decision based on clusters of energy in the calorimeters, missing
energy from the energy conservation, or the sum of calorimeter energy. A system called
Extremely Fast Tracker (XFT) [46] reconstructs tracks using the COT information, andthe tracks found by the XFT are used for the L1 trigger decision. The CMU can also
provide L1 trigger for muon candidates.
Level 2
The Level 2 trigger (L2) reduces the event rate to about 300 Hz. The L2 system consistsof several asynchronous subsystems which provide input to programmable L2 processors.
The L2 decisions are made based on the following:• L2 cluster finder (L2CAL): The L2CAL combines adjacent calorimeter towers over
1 GeV threshold starting from a seed tower of minimum 3 GeV.
• CES information: The shower maximum detector information provides the position
resolution for electron and photon showers with better than the cluster location. It
is also used to match with the tracking information.
29

RUN II TRIGGER SYSTEM
Detector Elements
CAL

COT

MUON

XFT

MUON
PRIM.

SVX

CES

XCES

XTRP

L1
CAL

L1
TRACK

L1
MUON

GLOBALLEVEL 1

L2
CAL

SVT

GLOBALLEVEL 2

TSI/CLKPJW 9/23/96

Figure 2.11: Block diagram of the Level 1 and Level 2 trigger path.

• Silicon Vertex Tracker (SVT) [47]: The SVXII information is combined with theL1 XFT track information by the SVT, a system that rapidly analyzes the silicondata to look for a displaced vertex.
• Muon information: The muon trigger combines information from the muon detector
and from the L1 XFT track.
Level 3
The Level 3 trigger (L3) consists of a few hundred of computers. The L3 computer farmreconstructs the event in software and filter the event rate to about 75 Hz. Events that passL3 decision are written to disk.
Consumer Server/Logger
Once an event is accepted by the L3 trigger, it sends to the Consumer Server/Logger (CSL)
system. The CSL is responsible for categorizing events by the trigger path, writing themto the disk, and sending a fraction of events to online processors for real time monitoringof data quality.

30

Chapter 3
Reconstruction and Event Selection′

As outlined in Chapter 1, the tt̄ dilepton events (pp̄ → tt̄X → bW + b̄W − X → bb̄l+ νl − ν̄X)
have a signature that is characterized by two high-pT leptons, electrons or muons, twohigh-ET jets and two neutrinos resulting in a large transverse energy imbalance, missing
transverse energy, E/ T .
This chapter describes in detail the identification variables and criteria applied to identify final state objects in the event by the detector components. We start with the electron
identification in the central and plug regions of the detector, discuss the requirements
for muon candidates, describe the jet reconstruction algorithm and corrections to the jetenergies, and the way we determine the energy of neutrinos escaping the detector.

3.1

Electron Identification

Electrons resulting from the dilepton channel of the tt̄ decay are highly energetic. They
can be identified by a high-pT track in the drift chamber and large energy depositions
in electromagnetic calorimeters. At energies of tens of GeV the dominant energy loss
for electrons is bremsstrahlung. When electrons traverse the lead absorbers in the electromagnetic calorimeter, they interact with the nuclei of the material and emit photons
which produce electron-positron pairs e+ e− . The secondary particles are also very energetic and lead to production of an electromagnetic cascade, called a shower. The shapeand position of the electromagnetic shower is measured by shower maximum detectors,
and the shower characteristics are used in electron identification. A hadronic shower islonger and much broader. Electrons deposit most of their energy in a single electromagnetic calorimeter tower, while a hadronic shower continues into the hadronic section of
the calorimeter and into the adjacent calorimeter towers. In addition to certain shower
properties, for electrons the momentum of the track pointing to the shower should matchthe energy in the calorimeter.

3.1.1 Central Electrons
Central electron candidates traverse the central part of the detector, |η| < 1.1, leaving
the track in the COT and depositing most of their energies in the CEM calorimeter. The
following variables and criteria are used to identify high-pT electrons:31• ET = E sin θ > 20 GeV

The transverse electromagnetic energy ET deposited by the electron candidate in
the CEM cluster. It is given by the total electromagnetic energy of the electron
cluster E multiplied by sinθ of the COT track pointing to the seed tower of thecluster. An electron cluster is formed of a seed EM tower, where are added to theseed tower until the maximum cluster size is reached. The cluster at maximum has
two towers in pseudorapidity and one tower azimuth. The energy E is corrected for
differences due to non-linearities and time-dependent changes.

• pT > 10 GeV

The transverse momentum by the COT track measured by its curvature in the magnetic field. Raw COT resolution is substantially improved by constraining the track
to originate from the beam line. The beam constrained tracking introduces a pT
curvature bias in data, which is removed by correcting the signed curvature Q/pT ,where Q is the charge of the track:Q
Q=− 0.00037 − 0.0011 × sin(φ + 0.28)
corr
pT
pT

(3.1)

Later in the text we refer to the pT of the COT track when beam constrained, pcorr
T .
This correction is applied only to data and not to simulation of events.
• Ehad /Eem < 0.055 + 0.00045 × Etotal

The ratio of the hadronic calorimeter (CHA) energy of the cluster Ehad to its electromagnetic energy Eem (CEM). The value Ehad /Eem is scaled by a factor of 0.00045
multiplied by the total energy of the cluster Etotal to compensate for inefficiency ofthe cut at very high energies, as higher energy electrons have more leakage into thehadronic calorimeter.

• E/P < 2 is applied for electrons with ET < 199 GeV or pT < 50 GeV.

The ratio of the electromagnetic calorimeter energy E to the COT track momentum
P . During the passage through the material inside the COT inner radius the electron might radiate a photon (“external bremsstrahlung”). The photon is collinearwith the electron and it generally deposits its energy in the same calorimeter tower,
thus not much affecting the value of ET , however the momentum pT measured in
the COT after bremsstrahlung will be smaller. This causes a long tail in the E/P
distribution above 1.0.

• Lshr < 0.2,

The lateral shower profile Lshr is a measure of how well the lateral shower development matches that expected from the electromagnetic shower. This variable
compares the energies of CEM calorimeter towers adjacent to the seed tower of the
EM cluster with energies expected from the test beam electrons. It is defined as asum over towers:
32

Lshr = 0.14

Xi

E measured − Eiexpected
q i √(0.14 E)2 + σE2 expected

(3.2)

i

where measured Ei is a measured energy in the CEM tower i; E expected i is anenergy deposit in the i-th tower√expected from the test beam electrons and E is theenergy of the EM cluster. 0.14 E represents the error on the energy measurementand σE2 expected is the uncertainty on the energy estimate. For a typical EM clusteri
Lshr is a two-tower sum. Any extra particles accompanying the one responsible forthe main EM shower will tend to add to the energy in adjacent tower and make Lshr
a large positive number.
• −3.0 cm < Q∆x < 1.5 cm; |∆z| < 3 cm
The distance ∆x / (|∆z|) in the r-φ/(r − z) plane between the COT track extrapolated to the CES and the best matching CES cluster. The cut on ∆x has been
multiplied by the charge Q of the electron and it is asymmetric in r − φ to accountfor energy of the photon emitted in bremsstrahlung radiation. This requirement on
a tight match between the track and shower position removes a large number offake electron due to the coincidence of charged and neutral hadrons in the same
tower, mainly due to π 0 and π ± results in a reconstructed track that matches theelectromagnetic cluster.
• χ2strip < 10The χ2 comparison of the CES shower profile in the r −z view with the shower profile extracted form test beam electrons. The χ2 -fit is performed on the distributionof energy deposited on each of the 11 strips in the CES shower.

• |zvertex | < 60 cm
The interaction position in z, zvertex , is taken from z0 , the z intersection of the trackwith the beam axis in the r − z plane. The longitudinal spread of the event vertex
about the nominal interaction pint z = 0 is a Gaussian with σ = 26 cm. The vertex
position is required to be within 2 σ.
• Track quality cuts
To insure that the track associated with electron is well reconstructed, it must passthrough two axial and three stereo superlayers (SL) of COT with at least 7 hits outof 12 in each SL.
• Fiduciality
This variable insures that the electron is reconstructed in a region of the detector
that is well instrumented. The electron position in the CEM is determined by the
CES shower position and it must satisfy the following requirement.
33

– the electron must lie with 21 cm of the tower center in the r − φ view, so thatthe shower is fully contained in the active region, this corresponds to the cut|xCES | < 21 cm, where the xCES is the local coordinate of the calorimeter
tower.
– the region |xCES | < 9 cm, where the two halves of the central calorimeter
meet is excluded, as well as the region |zCES | > 230 cm, which corresponds
to the outer half of the CEM tower (tower9), as it is more subjected to theleakage into the hadronic part of the calorimeter.
– the region immediately close to the point of penetration of the cryogenic connections to the solenoidal magnet, the chimney, is un-instrumented and therefore excluded. It corresponds to 0.77 < η < 1.0, 75◦ < φ < 90◦ and|zCES | > 193 cm.

– the region 1.05 < |η| < 1.10 is excluded due to the smaller depth of theelectromagnetic calorimeter.
• Not a conversion

Photons produced either directly in the hard scattering or from hadron decays interact with the material in the detector and convert to electron-positron paris. These
photon conversion can be identified by the presence of another track of the opposite
sign near the electron candidate. If the two tracks exhibit small r − φ separationin the pint of conversion |∆XY | < 0.2 cm, and the difference in their polar angle|∆ cot φ| < 0.04, the electron candidate is flagged as a conversion and the whole
event is rejected.

• Isolation = ETiso /ETcluster < 0.1.

whereETiso = ET0.4 − ETcluster is the transverse energy ET0.4 in a cone of radius ∆R =p(∆η)2 + (∆φ)2 ≤ 0.4 around the electron cluster (hadronic + electromagnetic)excluding the electron cluster energy ETcluster . This cut in fact is not an electron
identification requirement. It rejects the electrons that are not isolated from extra
hadronic activity and could be the products of quark semi-leptonic decays, whileelectrons from W and Z decays are expected to be isolated. We therefore consider
both categories of isolated and non-isolated electrons in the event selection.
The isolation is corrected for leakage energy into the neighboring φ wedge outsideof the cone, which increases towards the edges of the φ wedges. The respective correction factor is determined form Monte Carlo data comparison and parametrized
the form:
Eleak = ETcluster · P0 · exp P1 · (|xCES | − 21)(3.3)

where P0 = 0.0511±0.0075 and P1 = 0.33±0.061. If additional interactions occur
in the same bunch crossing the energy in a cone is increased. Therefore isolation is
also corrected for the number of interactions par bunch-crossing, the efficiency ofthe cut then becomes independent of the instantaneous luminosity. This correction
factor is determined similarly to jet multiple interaction energy correction.
34

The corrected isolation energy is thenETcorr = ETiso − Eleak − EMI(3.4)

and corrected isolation is given by
Isocorr = ETcorr /ETcluster(3.5)

Central electron ID efficiency was determined from the Z → e+ e− events were selected as dilepton events of opposite charge falling into Z mass window, 75 GeV < Mee <105 GeV. The number of observed same-sign dielectron event’s is the same Z mass range
served as an estimate of QCD background contamination. At least one electron was required to be tight, i.e. passing all identification cut.

3.1.2 Plug Electrons
Electron candidates deposition their energies in the PEM calorimeter are referred to asplug electrons. We impose the following identification criteria to define plug electron
candidates:• 1.2 < |η| < 2.0

Although plug electron can be identified up to |η| < 2.5, this analysis considers
only those with |η| < 2.0, primarily due to large charge misidentification rate athigh pseudorapidities η. This cut has a small effect on the tt̄ acceptance, as final
products of tt̄ decay events are mostly central and the acceptance falls rapidly atlarge η, while backgrounds considerably increase. The low η-region |η| < 1.2
is excluded due to geometrical reasons since the PES detector does not provide
usable coverage there. The track information from the COT is unavailable in theforward region of the detector, as plug electrons do not traverse the active volumeof the COT. Plug electron candidates identified based only on the presence of an
EM cluster in the PEM calorimeter, i.e. with no track requirements, are called PEM
electrons. PEM electron candidates do not provide charge sign information and are
subject to a large fake rate.
To reduce the misidentification rate the track reconstruction for plug electrons is
performed by utilizing the silicon layer (ISL) residing in the forward part of thedetector and providing usable coverage in 1.2 < |η| < 1.8 rage. The silicon pattern recognition code extrapolates the hits in the outer layers to the inner layers ofsilicon and determines the location of the primary vertex. The tracking efficiency
for tracks pointing to the plug region is considerably lower than for the central partof the detector. It is improved by a special algorithm, called phoenix algorithm.
the Phoenix algorithm makes use of the information of the PES position of an EM
shower. It constraints the track at two endpoints, one is fixed at the interaction
vertex and the other is at the PES position of an EM shower. Adoption the corresponding PEM cluster energy as the momentum of the electron, a helix of the track
can be determined. This defines two possible track trajectories, one is for negative35

and the other is for positive charge. The silicon pattern recognition code further
attempts to reconstruct those tracks by matching hits in the layers of silicon. If oneof these tracks is reconstructed, it is appended to the event record as being associated with the respective electron candidate. If both tracks are reconstructed then thealgorithm performs the χ2 -fit of a possible electron trajectory and adopts the track
best matching the activity in the silicon detector. Such a track is call a Phoenix plug
electron, abbreviated as PHX.
The Phoenix algorithm establishes charge identification for plug electron candidates
beyond the coverage of the forward silicon layers up to |η| < 2.0.
• ET > 20 GeV

Unlike the CEM clusters, the PEM clusters are limited to 2×2-two towers in pseudorapidity by two towers in azimuth.

• Ehad /Eem < 0.05

Similar to the central electrons, the ratio of the hadronic calorimeter (PHA) energyof the cluster Ehad to its electromagnetic energy Eem (PEM).

• U5×9 ≥ 0.65 and V5×9 ≥ 0.6 The quantities U5×9 and V5×9 are essentially isolation
variables for the shower maximum detector independently applied to both the U
and V layers.The PES clustering is performed by ordering PES strips in decreasingenergy with the highest-energy strips used as seeds. Then a fixed-width nine-strip
cluster is formed from each seed. This is done separately for 1D U- and V- layer
clusters. The quantities U5×9 and V5×9 represent the ratios of energy sum in thecentral 5 strips of a PES cluster to the total energy of the PES cluster (in all 9
strips).
• χ23×3 < 10This variable represents the goodness of χ2 − f it measure of the energy distribution in 3×3 towers around the seed tower to energy distributions from test beamelectrons.

• |∆RP ES | < 3 cm

The χ2 − f it also determines the position for the center of the shower. ∆RP ES =p(∆η)2 + (∆φ)2 is the distance between the χ2 )3 × 3 best fit position and theintersection of the centroids in the U- and V-layer PES clusters. ∆R matchingrequirement is also enforced between the PES coordinates and the extrapolated
Phoenix track.

Si• Nhits≥3

Si
Number of Silicon hits Nhits. This requirement is enforced to improve the qualityof the silicon tracks at the cost of some efficiency.

• |z0 | < 60This cut is identical to the CEM electrons.
36• Isolation = ETiso /ETcluster < 0.1

This variable is defined identically to the CEM electrons. The corrections to ETiso
are done differently, however, as the clustering algorithm in the plug region differsfrom the one in the central region. The leakage energy is parametrized dependingon the distance of the tower form the center of the electron shower in θ and φ.

Plug electron ID efficiency was determined from the Z → e+ e− data sample, similar
to how it was done for central electrons. Z → e+ e− events were selected as central-plugdielectron events in the Z mass range. The central electron was required to be a goodelectron, i.e. pass all tight identification criteria, and the other leg was required to be an
EM object fiducial to the plug region. Plug electron ID efficiency was determined as theratio of tight-tight (central-plug) candidates over tight-loose candidates.

3.2

Muon Identification

Muons are minimum-ionizing particles which penetrate matter very easily. Muons resulting from the dilepton channel of the tt̄ decay, as well as electrons, are very energetic. They
are identified by the high-pT track in COT, very little energy deposition in the calorimeters, and matching hits in the muon chambers. A muon candidate is required to havealigned hits in both r − φ and r − z planes on at least 3 separate layers. These hits form amuon stub which is then matched to the COT tracks extrapolated to the muon chambers.
Muons are categorized by the detector region through which they pass. Muons reconstructed in both the CMU and CMP chambers are called CMUP muons. Due to the gaps in
the muon chambers coverage there are also muon that are reconstructed only in the CMUor the CMP muon chambers. These muons are called CMU, CMP respectively. Muons
with stubs in the CMX chamber are called CMX muons. The COT tracks with no muonstubs are also considered as muon candidates. Such muons are required to have minimum
energy depositions in the calorimeter. These muon candidates are called CMIO’s (central
minimum ionizing objects). The following criteria are applied for muon candidates:• pT > 20 GeV

The transverse momentum of the COT track. The track is beam constrained and pT
corrections are applied identically as it is done for electrons.

• |z0 | < 60 cm

As for electrons, the z-position of the muon track is associated with the event vertex
and required to be within 60 cm from the geometrical center of the detector.

• |d0 | < 0.2 cm for tracks with no silicon hits.; |d0 | < 0.02 cm for tracks with silicon
hits.
The impact parameter d0 is the distance between the reconstructed muon track andthe beam axis in the r − φ plane. Unlike others this selection variable is basedon the default muon track, not beam-constrained, and a trigger cut is applied if the
track contains silicon hits. This cut forces the muon to originate from the nominal
37

interaction region and substantially reduces the cosmic muon background. It also
helps to remove muons from kaons and pions that decay in flight.
• Track quality cuts

Track quality cuts are identical to electron track quality cuts. The track is required
to have at least 7 COT hits on at least 2 axial and 3 stereo superlayers.

• Eem ≤ 2 + max(0, 0.0115(P − 100)) GeV

The energy deposited in the electromagnetic calorimeter. High pT muons are notexpected to deposit substantial amount of energy in the electromagnetic calorimeter.
the sliding cut is introduced for muons with P < 100GeV to increase efficiency ofthe cut.

• Ehad ≤ 6 + max(0, 0.0280(P − 100)) GeV

The energy deposited in the hadronic calorimeter is higher, but still quite small incomparison to strongly interacting jets.

• Eem + Ehad > 0.1 GeV for stubless muon only.

Stubless muons are required to have a non-zero energy deposition in the calorimeterto limit backgrounds from electrons escaping the detector through cracks in thecalorimeter.

• |∆x|CMU ≤ 7 cm

The distance in the r−φ plane between the extrapolated COT track and the stub segment in the relevant muon chamber. The muon candidate passing this requirement
falls into CMUP or CMU category.

• |∆x|CMP ≤ 5 cm

Muons in the CMP and CMX detector traverse more material than in the CMU andexperience grater deflections due to multiple scattering. Therefore, the track-to-stubmatching cut are looser. This muon candidate falls into CMUP or CMP category.

• |∆x|CMX ≤ 6 cm

CMX muon category requirement.

• ρCOT > 140 cm

The COT exit radius

ρCOT =

sign(η) · zCOT − z0cot(θ)(3.6)

is based on pseudorapidity η and z0 of the track, where zCOT = 155 cm is the lengthof the COT and θ is the polar angle.
This cut is enforced only for CMX muons to eliminate the data bias due to the XFT
trigger requirement that a track must leave hits in at least four COT superlayers.
38• Isolation = ETiso /pT < 0.1

where ETiso = ETcone − ETtower is the difference between energy in the cone of ∆R =0.4 around the muon track ETcone and the amount of energy in the tower associatedwith the muon track ETtower .

• not-cosmic
Cosmic ray can be identified in the CDF detector as dimuon events leaving a nearlystraight track and therefore mimickintg a very energetic µ+ µ− pair. Since cosmic
rays do not originate from a ppbar collision and enter the detector at random locations, this background is reduced by a cut on the track impact parameter d0 . In
addition, cosmic rays appear randomly in time and can be distinguished by substantial time delay between the hits of the two muons in the hadronic calorimeter,
measured by Time to Digital Converter (TDC), and by using timing informationfrom the Time of Flight detector (TOF). All of this information is analyzed by thesoftware code, named Cosmic Ray Tagger, which makes a decision on flagging an
event as a cosmic. Events with muon identified as cosmics are rejected.
The muon ID efficiencies are measured using Z boson decays Z → µµ similarly toelectrons. Both legs are linked to a CMUP or CMX muon stub, where one leg is required
to pass tight identification criteria and is matched to Level-1 trigger information, while
the second leg is chosen to be fiducial to the tested detector region (e.g. CMUP, CMX), or
non-fiducial to any (for stubless CMIO muons) and examined if it passed muon ID cuts
thus being independent of a trigger requirement.

3.3

Jet Reconstruction

At high energies, as those achieved in Tevatron, jets are the dominant feature of hadronproduction. They result from point-like collisions of a quark or gluon from the protonwith a quark or gluon from the antiproton. A jet is formed from a scattered initiating
parton, which experiences fragmentation leading to the creation of a stream of energeticcolorless particles emitted spatially collimated along the original parton direction.
The jets are observed as cluster of energy located in adjacent detector towers. Typically a jet contains neutral or charged pions to a lesser extent of kaons, and about 10% of light baryons such as protons and neutrons. Pions mostly deposit their energies inelectromagnetic calorimeter, while kaon and baryons leave most of their energies in the
hadronic section of the calorimeter. The energy of the initial parton can be approximatedby summing the tower energies within a cone of specified size. This procedure is calledjetclustering. The cone size is chosen to encompass most of the jet energy without allowing a significantpcontribution from other event activity. It is defined in η − φ spaceby its radius, R = ∆η 2 + ∆φ2 and is centered at the largest calorimeter energy tower
serving as a seed tower of the jet cluster. This analysis is using a cone size of ∆R = 0.4.
After the jet cluster is thus formed, the ET weighted centroid of the cluster is determined, as follows
39

PN

ηcentroid = Pi=1
N

ETi η ii
i=1 ETPN

φcentroid = Pi=1
N

ETi φii
i=1 ET

(3.7)
(3.8)

where the sums are carried out over all calorimeter towers in the cluster. It defined thecentroid tower and a new cone drawn around this position. This process is iterated until
the cluster remains unchanged in two consecutive paths. In some cases two clusters canoverlap and then they are either merged into one, if the sum of the energies in shared
towers exceed 75 % of the energy of the smaller cluster, or left intact.
raw raw
The jet four-momentum (E raw , praw
x , py , pz ) is then determined by the following
sums over the cluster towers:

E raw =
praw
=
x
praw
=
y
praw
=
z

N
X

i=0
N
X

i=0
N
X

i=0
N
X

Ei

(3.9)

Ei sin θi cos φi(3.10)

Ei sin θi sin φi(3.11)

Ei cos φi(3.12)

i=0

These quantities are referred to as raw, since they are affected by mismeasurements
for a variety of reasons due to both to physics and to detector effects and are different
from the true energies of the partons which initiated jets. Therefore proper corrections
need to be applied to reconstruct true momentums of partons.

3.3.1 Jet Energy Corrections
The measured four-vector of jets generally differs from the energies of the initial partons. This is the result from both instrumental and physical effects such as low energy
non-linearities, η crack energy losses, underlying events, and clustering. Some of the corrections are decided by the measurable quantities independent of the theory, while someof them rely on the theory prediction. Thus the row jet energies measured in the calorimeter must be corrected for detector effects at first before they can be compared to physics
predictions/models. The correction strategy is the followings:• Relative Corrections (Level1 Corrections)

The first step in jet energy corrections is to correct the jets for any variation in the
response with detector η. For this correction, dijet event samples are used. Since the40transverse energy of the two jets in a 2 → 2 process should be equal, the energies
of jets in the plug and forward calorimeters are scaled to give the energy of anequivalent jet in the central calorimeter. One well-measured central jet (0.2 < |η| <0.6) is required and a scale factor is derived from the dijet balance to the second jet.
The central calorimeters CEM/CHA are the best understood calorimeters in CDF
and the selected region is far away from the cracks. The gain variation dependingon the time (run range) in the plug calorimeters is also taken into account. The
corrections for the Monte Carlo and data are determined separately since somediscrepancy between data and simulation can be seen due to a lack of the materials
in the detector simulation.
• Multiple Interaction Corrections (Level4 Corrections)
The multiple interaction affects the measured jet energy when the energy from theseminimum bias events falls into the jet clustering cone. The transverse energy in arandom cone is measured in minimum bias data and parameterized as a function of
the number of vertices in the event. This transverse energy is subtracted from eachjet to account for multiple interaction in the same bunch crossing as a function of
the number of vertices in the event. This correction factor is a linear function of thenumber of reconstructed vertices in the event. Only vertices associated with at least
2 COT tracks in minimum bias events are used to decide this correction factor.
• Absolute Corrections (Level5 Corrections)
The jet energy measured by the calorimeters must be corrected for any non-linearityand energy loss in the un-instrumented regions of each calorimeter. The absolute jetcorrections account for the response to particle-level energy in the central calorimeter. This correction depends on the jet fragmentation properties. The calibration
point is derived using a 50 GeV pion from test beam data. For the non-linearity response, the tuned Monte Carlo events are used for the charged and neutral particles.
After fragmentation, the events are processed with a full CDF detector simulation.
Each simulated event is compared to the total pT of all generated particles lying ina cone centered about the measured jet axis. A quadratic spline fit is used to parameterize the mean jet response as a function of ET for the each cone size.
• Underlying Event Corrections (Level6 Corrections)
The underlying event contains all the soft interactions except the hard one. The
underlying event energies must be subtracted from the measured jet energy whenthese particles fall into the clustering cone. The correction procedure is the sameas the multiple interaction correction. Events with only one vertex are used todetermine the underlying event correction.
• Out-of-Cone Corrections (Level7 Corrections)41

The jet clustering may not include all the energy from the initiating parton. Someof the partons generated during fragmentation may fall outside the cone chosen forclustering algorithm. Out-of-cone corrections are applied in order to correct theparticle-level jet energy to the parton energy (as much as theoretically allowed).
These corrections are completely independent of detector/calorimeter performanceand depend on the parton fragmentation functions. The correction factor is parameterized as a function of jet pT . Jet tends to become narrower at large energies, andthe fractional energy deposited outside the cone decreases.
Thus, the jet energy is corrected bypT (R) = [prawT (R) × frel − UEM(R)] × fabs (R) − UE(R) + OC(R),(3.13)

where R denotes the clustering cone size, pT and praw
T are the corrected and row transverse
momenta of jet, frel is the relative jet energy correction, UEM(R) is the multiple interactions correction, fabs (R) is the absolute jet energy correction, UE(R) is the underlying
event correction, and OC(R) is the out-of-cone correction.

3.3.2 Jet Energy Scale Uncertainties
The differences between the data and CDF simulations of the jet responses are treatedas the systematic uncertainties of the jet energy scale (JES) [48]. The uncertainties are
estimated for each type of jet energy corrections. The uncertainties due to the absolute
scale and the jet shape (out-of-cone) are dominant. The main systematic uncertainties
on the absolute scale are obtained by propagating the uncertainties on the single particle
response (E/p) and the fragmentation. Smaller contributions are from the calorimeter
response close to tower boundaries in azimuth, and from the stability of the calorimeter
calibration with time. The uncertainties from the jet shape are estimated by measuring
the energy flow between cones of size 0.4 and 1.3 in both data and MC simulations. The
total JES uncertainties in the central region are shown in Figure 3.1.

3.4

Missing Energy

Neutrinos interact only through weak interactions and therefore cannot be directly detected as they traverse the detector material. Production of neutrinos in an event can be
spotted by the existence of the large imbalance in the calorimeter energy. The longitudinalcomponent of the colliding partons is not known, but the transverse component is subject
to conservation, and the sum of the transverse components of the neutrino momenta canbe measured. This quantity is called missing transverse energy E/~T . The missing transverse energy is two-component vector (E/ Tx E/ Ty ). The raw value of E/~T is defined by thenegative vector sum of the transverse energy of all calorimeter towers:X~/ raw = −E(Ei sin θi )~ni
(3.14)T
tower

42

Fractional systematic uncertainty

0.1
Systematic uncertainties. Cone 0.4

0.08

Quadratic sum of all contributions
Absolute jet energy scale

0.06

Out-of-Cone + Splash-out
Relative - 0.2<|η|<0.6
Underlying Event

0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1

20

40

60

80

100

120

140

160

180

200

Corrected jet PT (GeV)

Figure 3.1: The total uncertainties of JES as a function of corrected jet pT in the central
calorimeter (0.2 < |η| < 0.6).

where Ei is the energy of the i-th tower, ~ni is a transverse unit vector pointing to thecenter of the tower and θi is the polar angle of the line pointing from z0 , z-coordinate of
the event vertex, to the i-th tower. This sum extends to |ηdetector | < 3.6.
~/ raw should be further corrected for escaping muons and jet energy
The value of ETmismeasurements. Muons do not deposit substantial energy in the calorimeter, but may
carry out significant amount of the energy. The sum of transverse momenta of escapingP~/ raw with a negative sign andmuons P~T,µ measured in the COT has to be added to the ETP~
the energy deposited by muons in the calorimeters
ET,µ has to be subtracted from that~/ raw .
sum, as it has been already counted in the ET~/ raw and these values have to be
Only raw values of jet energies contribute to the ET~/ corr is therefore given
replaced in the sum by the corrected ones. The corrected value of E
T
by the following relation:~/ corr −~/ corr = E
E
T
T

X

muons

P~T,µ −

X

~ T,µ
E

muons

corr

!

−

X
jet

corr
~ T,jet
E
−

X
jet

raw
~ T,jetE!

(3.15)~/
Uncertainties in E
are dominated by uncertainties in jet energies. MismeasureT
ments of E/ T result from jets traversing through poorly instrumented regions of detectors,
e.g. cracks, dead zones, and beam halo effects. They may also result from cosmic rays,
muon misidentification and mismeasurements in muon track momenta.
The resolution of the E/ T generally depends on the response of the calorimeter tothe total energy depositedin the event. It is parameterized in terms of the total scalarPtransverse energy
E/ T , which is defined as
43

X

E/ T =

X

Ei sin θi(3.16)

towersThe E/ T resolution in the data is measured with minimum bias events, dominatedby inelastic pp̄ collisions. In minimum bias events the x and y components of E/~T are
distributed as Gaussian around zero with σx = σy = σ:E/ 2Tx ,ydN≃ exp −
dE/ Tx ,y2σ 2!

(3.17)√The E/ T resolution ∆ = hE/ 2T i is then given by ∆ = 2σx,y . It is expected to scaleP
as a squareroot of the total transverse energy in the event,ET , is determined to bepP∆ ≃ 0.64
ET from minimum bias studies.
q44Chapter 4Spin Correlation Measurement
4.1

Outline of the Measurement

In this thesis we measure the strength of the tt̄ spin correlation κ from a differential angular distribution of the two decay leptons and the two decay b-jets in the rest frames of theirrespective top quark and the spin quantization axis. Since the spin correlation is large inbeamline basis and beamline basis is easier to determine experimentally, we chose beamline basis as a spin quantization axis. To measure κ, we use tt̄ dilepton channel, since, as
we discussed in the Section 1.7, the charged leptons from the t → W b → ℓνℓ b decays
are the probes with the highest sensitivity to the direction of the top quark spin. Froman angular distribution of two charged leptons, the information on κ is to be extracted.
In addition to the angular distribution of charged leptons, we use one of b-jets as wellto increase the sensitivity to the κ measurement. In order to obtain the flight directionsof decay products in top and anti-top rest frames, we need to a full kinematical reconstruction of tt̄ system in dilepton decay channel including two neutrinos. The method of
the full kinematical reconstruction will be discussed in Section 4.4. Then we will makethe templates of tt̄ signal as a function of κ as well as the background from each angular distribution(Section 4.5). Using the templates, we fit the angular distribution of data,and once we get measured κ, we set Feldman-Cousins confidence intervals on the true κ.
The construction of Feldman-Cousisns confidence belt from pseudo-experiments will be
discussed in Section 4.6.

4.2

Top Dilepton Event Selection

Dilepton event selection aims at reconstructing tt̄ events with both W’s from top decaying
leptonically. It requires two fully identified electrons or muons with transverse energy
above 20 GeV, 6 ET > 25 GeV and at least two tight jets of ET > 15 GeV. The first,
or trigger lepton can be one of three types: CEM electron, CMUP or CMX muon. The
second, or loose lepton, can be also non-isolated or an isolated PHX electron or one of
the non-trigger muon types: CMU-only, CMP-only and CMIO. Details on the cuts used
to identify each lepton category are contained in Section 3. Non isolated counterpart of
the trigger leptons (i.e. NICEM electrons and NICMUP/NICMX muons) are allowed to45

trigger the event when they come together with a PHX electron to recover most of theacceptance lost by dropping the plug electron dataset.
The jets are corrected up to hadron level (i.e. we do not apply Underlying Eventsand Out-of-Cone Correction, jet is called parton level after these correction). We should
mention here that for the purpose of correcting 6 ET (which is calculated starting fromthe raw transverse energy deposited in each tower of the calorimeter), we used the jetenergy scale calculated ignoring the Multiple Interaction correction in order to avoid overcorrection E
6 T for the presence of energy due to extra interaction.
Extra event topology cuts are imposed to improve the purity of the selection:• Dilepton invariant mass > 5 GeV, where Monte Calro cannot describe properly inthis region.
• COT radius exit cut > 140 cm for the stubless CMIO muon candidate, which isa high pT central track with a minimum energy deposition in the calorimeter but
no matching stub in the muon chambers, in order to remove mismodeling events
around the end of COT in the simulation.
• Cosmic and conversion removal: This cut reject the events which were marked as
the events with the particles coming from cosmic or the events where the electrons
are coming from the conversion of the photon to e+ e− pair within detector. The
cosmic removal is used to remove events where the muons are identified as ancosmic muons. The conversion removal is applied only to central electrons. The
track associated with the electron is checked to all other tracks whether it is close
in all three spatial dimensions to another track and that it has opposite charge. In
such case, the electron is flagged as the conversion electron.
• Z-veto for ee and µµ events. To remove events that contain leptons come from Z
boson, if the event has same flavor lepton pairpwith√ mass in 76 < Mℓℓ <P its invariant
106 GeV, the missing ET significance 6 ET /ET > 4 GeV is required. Thisvariable separates events with real 6 ET due to neutrinos from events where 6 ET isdue to energy measurement fluctuations or energy loss in calorimeter cracks. These
second category of events is expected to have a degraded E
6 T resolution.
• L-cut in the (6 ET , δφmin ) plane to reject Z → τ τ events and events with mismeasured 6 ET from jets pointing to cracks in the calorimeter: this cut requires
the event 6 ET to be above 50 GeV is there is any lepton or jet inside 20◦ of theE
6 T direction. In events with mismeasured jets, the fake E
6 T is pointing close to thedirection of the jet. Moreover, in Z → τ τ events the 6ET vector often points close
to the direction of one of the leptons. This cut remove these backgrounds.
• Scalar sum of ET′ s of leptons, jets, and missing energy: HT > 200 GeV. Becauseof such heavy top quark, there is higher activity in transverse plane in tt̄ events
comparing to background events. Therefore, the variable which sums (scalarly)transverse energies of all particles in the event has higher value for tt̄ events than inbackground events. Such variable we call HT and in our case it is the sum of 6 ET
plus leptons transverse energies plus the ET of all tight jets in the event.
46

• Opposite charge for the two leptons. This cut requires that the charged leptons havethe charges with the opposite sign. In tt̄ events the charged leptons have indeed the
opposite sign of charges because they are coming from opposite charged top quarks(top quark vs. antitop quark). However, some of the backgrounds (Fake events) do
not necessary have opposite sign leptons, so such events are suppressed by this cut.
• |δZℓ | < 4cm is made to make sure that the two leptons are from the same interaction, where |δZℓ | is the absolute value of the Z position difference of the two
leptons. This cut remove events from multiple interaction.
The data we use in this analysis were collected during the period March 2002 - June 2009.
The total integrated luminosity for this period is 5.1fb −1 . Finally the number of observed
events survived after this selection criteria are 334 events.

4.3

Background Composition

We consider four different sources of standard model process that can mimic the signatureof dilepton plus 6 ET plus two or more jets signature: diboson events (W W, W Z, ZZ or
W γ), Drell-Yan production of tau leptons (Z→ τ τ ), Drell-Yan production of electrons ormuons with additional E
6 T (if the event is an actual Drell-Yan event, there is no E
6 T so werefer to this as fake E
6 T ) (DY→ ee/µµ), and QCD production of W boson with multiple
jets in which one jet is misidentified as a lepton (W +jets fakes).
• Diboson
The diboson processes, W W, W Z, ZZ and W γ, can mimic the signature of the tt̄
signal via different mechanisms, with real leptons and E
6 T from W and Z decays andjets produced by boson hadronic decays or initial state radiation for W W events,
the two leptons and the 6 ET are produced when both W ’s decay semi-leptonicallybut the jets require some hadronic radiation external to the diboson system. ForW Z and ZZ events, the two leptons come from the Z boson while the other W
or Z boson provides the jets vis their hadronic decays. As these decays do notcontain any neutrino, some mechanism to produce fake E
6 T is required. Finally forW γ events, one lepton plus E
6 T is generated from the semi-leptonic W decay while
the second lepton is produced from an asymmetric γ conversion in which one of
the two electrons has little energy and is caught spiralling inside the central drift
chamber.
• Drell-Yan to ee/µµZ/γ ∗ → ee/µµ process can mimic the signature of the tt̄ signal when there are atleast 2 jets from initial state radiation. More ever, this process does not have real
physics 6 ET , it can have 6 ET just due to mis-measurement of jets and/or leptons
energies.

• Z → ττ47

Z/γ → τ + τ − events can fake the dilepton + E
6 T plus 2 or more jets signature when−both τ ’s decay semi-leptonically to ℓνℓ ν̄τ ℓ ν̄ℓ ντ and jets from initial state radiation
are present.
• W +jets fakes
Events where one jet is misleadingly reconstructed as a lepton, i.e. jet “fakes” thesignature of a lepton. The main source of such events is production of W bosonwith associated production of at least 3 jets (W + ≥ 3 jets).
Z+jets, and diboson events are generated using ALPGEN, PYTHIA and MADEVENT respectively, where PYTHIA is used to model parton showering and the underlying
event for all generated samples [49–53]. CTEQ6.6 parton distribution functions (PDF) areused in all Monte Carlo simulations [54]. CDFSIM, a GEANT-based simulation, is used
to model the CDF detector response [55, 56].
Events where Z/γ ∗ decays to leptons and Z → τ τ events are generated using ALPGEN, and diboson events are generated with PYTHIA. All other processes pass selection
by mis-identifying one or more leptons in the detector. These fake lepton events are dominated by W boson events with associated jets. Fakes are modeled by W+jets events where
one jet passes selection criteria such that it is a good candidate to fake lepton selection.
The predicted number of events for each background process, along with the numberof expected tt̄ events at the measured cross section, is calculated from the measurement of
the tt̄ cross section in the dilepton decay channel [57]. The number of tt̄ signal and backgrounds expected with the data corresponds to 5.1fb −1 are summarized in the Table 4.1.

Table 4.1: The number of expected events in data correspond to 5.1 fb−1 with the observed
number of events.
CDF II Preliminary(5.1 fb−1 )Process
Number of expected events
WW
11.69 ± 2.35
WZ
3.48 ± 0.55
ZZ
2.25 ± 1.75
Drell-Yan (Z → ee, µµ)22.34 ± 3.23
Z → ττ
12.21 ± 2.17
Fakes
34.27 ± 9.46
Wγ
0.42 ± 0.44
Total background86.24 ± 14.05
tt̄(σ = 7.4 pb)
236.97 ± 11.29
Total SM expectation
323.21 ± 25.12
Data (5.1fb−1 )
334

48

4.4

Full Kinematical Reconstruction in Dilepton Channel

In this section, we describe a full kinematical reconstruction method we use in this analysis. By the event selection, we have two or more jets in events. we use the leading twoenergetic jets as b-jet and b̄-jet candidates.

4.4.1 Principle
In a dilepton event, due to two undetected neutrinos in the final state, we have six unknown
variables (~pν , p~ν̄ ). Also, we have six constrains: W + mass, W − mass, top quark mass,anti-top quark mass, x-axial missing ET and y-axial missing ET . The details of constrains
are as follows.
Mℓ2+ ν
Mℓ2− ν̄Mℓ2+ νbMℓ2− ν̄ b̄
(~pν + p~ν̄ )x(~pν + p~ν̄ )y

=
=
=
=
=
=

2
(|~pℓ+ | + |~pν |)2 − (~pℓ+ + p~ν )2 = MW2(|~pℓ− | + |~pν̄ |)2 − (~pℓ− + p~ν̄ )2 = MW(|~pℓ+ | + |~pν | + |~pb |)2 − (~pℓ+ + p~ν + p~b )2 = Mt2(|~pℓ− | + |~pν̄ | + |~pb̄ |)2 − (~pℓ− + p~ν̄ + p~b̄ )2 = Mt2(6ET )x(6ET )y .

(4.1)

Typically we will have two or four possible solutions of (~pν , p~ν̄ ) as well as two combinations due to b − b̄ ambiguity in the system of the equation above. However without
any additional redundant constraint, no one can tell which solution is the best solution.
The basic idea to select most likely solution is to calculate pztt̄ , pTtt̄ and Mtt̄ for eachsolution and to look how plausible these reconstructed variables are. These variables,pztt̄ , pTtt̄ and Mtt̄ are mostly distributed depending on the initial parton distributions which
we can know to some extent in prior.

4.4.2 Likelihood
In the actual reconstruction method we adopt, we take resolution of b-jet energies andE
6 T into account, i.e. the likelihood are given as a function of assumed p~ν , p~ν̄ , Ebguess , andEb̄guess . We use the following likelihood:   
(4.2)
L p~ν , p~ν̄ , Ebguess , Eb̄guess = P pztt̄ P pTtt̄ P (Mtt̄ ) × meas meas
guess guess − Ejet1− Ejet2
1
1
1 Ejet1
1 Ejet2
×
×
exp −
exp −
σjet1
2
σjet1
σjet2
2
σjet2
 meas meas
6 Ty −E
6 T guess
1 E
6 Tx −E
6 T guess
1
1 E
1
y
x
exp −
× MET exp −
,
σxMET
2
σxMET
σy
2
σyMET
where P (pztt̄ ), P (pTtt̄ ) and P (Mtt̄ ) are probability density function of each variables in a tt̄
candidate, which obtained from dilepton candidates in tt̄ PYTHIA Monte Carlo sample.
4947931

2

P0 exp[-0.5 (x/P 1) ]P0=749.51
P1=190.706

P0[exp(-P1x)+P2exp(-P3x)][1-exp(-P4x-P 5x )]

3

10

M

47931
2

P0=3197.15
P1=0.187744
P2=0.172679

102

10

Entries

PT (t t )Entriestt

Events102

EntriesP z (tt )EventsEvents

3

10

47931

3

10

P 0exp(-P1(x-340))[1-exp(-P2(x-340))]
P 0=1711.91

102

P 1=0.011281
P 2=0.0390347

P3=0.0419876
P4=0.211576

10

P5=0.0696098

10

1
-500

0

500

0

20

40

60

80

100

PT(tt ) (GeV/c)

Pz (tt ) (GeV/c)

400

500

600

700

800
2

Mtt (GeV/c )Figure 4.1: Distributions of pztt̄ , pTtt̄ , and Mtt̄ of dilepton candidates in tt̄ PYTHIA Monte
Carlo.

Figure 4.1 shows the distributions of pztt̄ , pTtt̄ and Mtt̄ of dilepton candidates. Each distributions is fitted to an analytic function, and fit result and parameters are shown in themeasplots. In Equation (4.2), Ejet1,2
denote the measured energy of the jets which are assigned
as b-jets and b̄-jets, respectively. The measured energies of jets are corrected by a dilepton
specific correction which is explained in the following sub section, after the Level5 jet energy correction. E
6 T measx,y are x, y components of the measured missing transverse energy.
METσjet1,2 and σx,ydenote the resolution of measured jet energy and missing transverse enguessergy which are also described in the Sec 4.4.3. Ejet1,2
and E
6 T guessjet1,2 in Equation (4.2) are
guess
guess
the quantities deduced from the assumed p~ν , p~ν̄ , Eb , and Eb̄ , respectively. We takeone representative set of (~pν , p~ν̄ , Ebguess , Eb̄guess ) which gives maximum likelihood in each
event as reconstructed quantities. In both of two cases for b − b̄ assignments, the likelihood is calculated and the assignment which gives the better likelihood ia chosen as thesolution.

4.4.3 Jet and MET Resolutions
As mentioned in the previous subsection, we use corrected measured jet energies after
the Level5 jet energy correction in the likelihood for the full kinematical reconstruction(Equation (4.2)) as well as the resolutions of the jet energies. The correction is performedmeas
so that the mean of Ejet1,2
goes to b-quark or b̄-quark energy.
To obtain the correction factor and the resolution, we use reconstructed jets in dilepton
candidates of tt̄ Monte Carlo sample, which is matched to b-quark or b̄-quark within∆R < 0.4. We divide the matched jets into the following three pseudo-rapidity regions:|η| < 0.7, 0.7 < |η| < 1.3, and 1.3 < |η| < 2.5. Then using the true b(b̄)-quark transverse
energy (ETtrue ) and the observed transverse energy after Level5 correction (ETL5 ), we makethe distribution of (ETtrue − ETL5 )/ETL5 for each ETL5 . The plots in Figure 4.2 show meanand σ of the distribution of (ETtrue − ETL5 )/ETL5 as a function of ETL5 for each pseudorapidity region. Each plot in Figure 4.2 is fitted to the function exp(P0 + P1 x) + P2 , andwe use these parameters for the correction and the resolution in Equation (4.2).
For E
6 T , the resolutions are also used in the likelihood for the full kinematical reconstruction (Equation (4.2)). They are obtained by the σs of the distributions of E
6 T meas
x,y −
jet1
true
meas
true
E
6 T x,y . Figure 4.3 shows the plot of σ of E
6 T x,y − E
6 T x,y as a function of ET + ETjet2 .
50

Figure 4.2: Mean and σ of the distribution of (ETtrue − ETL5 )/ETL5 as a function of ETL5
for jets matched within ∆R < 0.4 in dilepton candidates of Pythia Monte Carlo, for eachregion of |η| < 0.7, 0.7 < |η| < 1.3, and 1.3 < |η| < 2.5.

4.4.4 Performance of the Full Kinematical Reconstruction Method
To check the performance of the full kinematical reconstruction method, we look at the
scatter plots of generated values of kinematical variables versus reconstructed values. The
full HEPG information, including the identity and kinematics of all particles in an event,is available in Monte Carlo samples. We use dilepton candidates where both of the twocharged leptons in the events are matched to HEPG leptons within ∆R < 0.1 and both ofthe leading two jets are matched to b-quark of b̄-quark within ∆R < 0.4 in tt̄ PYTHIA
Monte Carlo sample. The ratio of the lepton and jet matched events to the whole dilepton
candidates is about 76%. Three plots in Figure 4.4 show the scatter plots of cosθbeam ,cosθt , and cosθ∗ , which are cosine of the angle of ℓ+ (ℓ− ) flight direction w.r.t. the beamaxis in the top(anti-top) quark rest frame, cosine of the angle of top flight direction w.r.t.
the proton direction in the tt̄ rest frame, and cosine of the angle of ℓ+ (ℓ− ) flight directionin the W + (W − ) rest frame w.r.t. W + (W − ) flight direction in the top(anti-top) quark rest
frame, respectively.
The events in the diagonal region in the scatter plots indicate that the variables arecorrectly reconstructed, while the events in the off-diagonal region indicate that the variables are incorrectly reconstructed. We can see bands of events in the diagonal regions inthe scatter plots, which indicate the method reasonably successfully reconstructs events
including neutrinos as well as b and b̄ assignment.
The ratio of the events where b and b̄ are correctly assigned to the two jets by thefull kinematical reconstructed method to the lepton and jet matched dilepton candidate isfound to be 75%.

51

jet1
jet2
Figure 4.3: σ of the distribution of E
6 meas
6 truefor dilepton
x,y − Ex,y as a function of ET + ET
candidates of Pythia Monte Carlo.

Entries

72670

1

0.5

36335

1

0.5

0

-0.5
-1
-1

Entriest Prod. Ang.

cos θ * (rec)cos θbeamcos θt (rec)cos θbeam (rec)

1

0

0.5

1

0

cos θbeam (gen)-1-1Entries

72670

0.5

-0.5
-0.5

W hel.

0

-0.5
-0.5

0

0.5

1

cos θt (gen)

-1
-1

-0.5

0

0.5

1

cos θ* (gen)Figure 4.4: Scatter plots of reconstructed kinematical variables versus the true value ofthe corresponding variables. The kinematical variables are cos θbeam (left), cos θt (middle),
and cos θ∗ (right). The areas of the square are proportional to the entries in the bins. For
each plot, we use dilepton candidates which satisfy following two requirements. Both of
the two reconstructed charged leptons obtained from the full kinematical reconstruction
are matched to HEPG charged leptons within ∆R < 0.1, respectively. Both of the tworeconstructed leading jets obtained from the full kinematical reconstruction are matched
to HEPG b−quark or HEPG b̄−quark within ∆R < 0.4, respectively.

52

4.5

Templates of the Signal and the BackgroundIn this section, we discuss about the two dimensional distribution of reconstructed cosθ+ cosθ− and cosθb -cosθb̄ of the tt̄ signal and background. We use these expected distribution
as templates to extract measured κ from observed reconstructed distribution by likelihood
fitting. Therefore, for the tt̄ signal, the templates should be as a function fo any assumed
κ.

4.5.1 Template Fit Function
To obtain the templates as an analytic function, f (x, y), we assume as follows:• Polynomial function of x and y up to O((x, y)4 ).
f (x, y) =X

Cij xi y j

(4.3)

i+j≤4

• Assume P is conserved. On this assumption, physical quantities at cos θ± are equivalent to ones at − cos θ± under P reversal(See Appendix B), ie. we assumef (x, y) = f (−x, −y)(4.4)• Assume CP is conserved. cos θ± change to cos θ∓ under CP reversal(See Appendix B), i.e. we assumef (x, y) = f (y, x)(4.5)
On the assumption above, we obtainC0
x2 + y 2f (x, y) =1 − C1 − C3 − C5 + 3C1
4
2

4
4
x +y
x3 y + xy 3
2 2
+ C2 xy + 5C3+C4+ 9C5 x y
2
2
and

Z 1

−1

dx

Z 1

dyf (x, y) = C0 .

(4.6)

(4.7)

−1

Hereafter we suppose C0 = 1 unless it’s specified.

4.5.2 Signal Template as a Function of κ
We use a tt̄ Monte Carlo sample generated by Pythia with Mt = 172.5 GeV/c2 . Inthis
 sample, there is nospin correlation between generated t and t̄. We put a weight,
hepg
hepg
/ (1 + |κ|), on a dilepton candidate in its Monte Carlo sample,
1 + κcosθ+
cosθ−
hepg
where κ is assumed spin-spin correlation coefficient in the beamline basis and cosθ±53

Signal (cosθ+, cosθ-)κ = 1.0

400
300
200
100
0
1
co 0.5

sθ

-

200

0
-0.5

-1 -1

-0.5

0

0.5

cos0.5
θ 0-0.5

1

cos θ +

κ = 1.0

b

-1 -1

-0.5

0

Sig. Fit (cosθb, cosθb)

0.5

1

cos θ b

κ = 1.0

CDF II Preliminary(5.1fb-1)

Prob.

Prob.

-

300

0
1

Sig. Fit (cosθ+, cosθ- )sθκ = 1.0b100

CDF II Preliminary(5.1fb-1)

0.4
0.3
0.2
0.1
0
1
co 0.5Signal (cosθb , cosθ )CDF II Preliminary(5.1fb-1)

EventsEvents

CDF II Preliminary(5.1fb-1)

0.2
0.1
0
1

0
-0.5

-1 -1

-0.5

0

0.5

cos0.5
θ 0-0.5

1

cos θ +

b

-1 -1

-0.5

0

0.5

1

cos θ bFigure 4.5:
The distributions of reconstructed (cos θ+ , cos θ− ) (left) and(cos θb , cos θb̄ )(right) of weighted dilepton candidates in tt̄ Pythia Monte Carlosample on the assumption of κ = +1, and their fit results.

represent true cosθ± using HEPG information. Then, we obtain the reconstructed distributions of (cosθ+ ,cosθ− ) and (cosθb ,cosθb̄ ) for dilepton candidates from tt̄ signal with anassumed κ.
We made the 10 by 10 bin distributions of reconstructed (cosθ+ ,cosθ− ) and (cosθb ,cosθb̄ )from weighted dilepton candidates in tt̄ Pythia Monte Carlo sample with assumed κ ranging from -1 to 1 with 0.2 step. For each distribution we fit the resultant distribution to thefit function of Equation (4.6), and obtain fit parameters Ciℓ (i=1,· · · ,5) for leptons, andCib (i=1,· · · ,5) for b-jets as a function of κ.
Figure 4.5, 4.6 and 4.5 shows the reconstructed distributions and their fit results incases of κ = +1,κ = 0, and κ = -1, respectively.
Figure 4.8 and 4.9 show κ dependence of the fit parameters Ciℓ and Cib , respectively.
Since the differential cross-section corresponding to each events a linear function of
κ, each parameter except for normalization parameter C0 should be a linear function of κas well.
We fit Ciℓ (κ) and Cib (κ) (i = 1, · · · , 5) to a linear function of κ so than we can describe
signal templates with any κ assumed.

54

Signal (cosθ+, cosθ-)κ = 0.0

800
600
400
200
0
1
co 0.5

sθ

-

400

0
-0.5

-1 -1

-0.5

0

0.5

cos0.5
θ 0-0.5

1

cos θ +

κ = 0.0

b

-1 -1

-0.5

0

Sig. Fit (cosθb, cosθb)

0.5

1

cos θ b

κ = 0.0

CDF II Preliminary(5.1fb-1)

Prob.

Prob.

-

600

0
1

Sig. Fit (cosθ+, cosθ- )sθκ = 0.0b200

CDF II Preliminary(5.1fb-1)

0.4
0.3
0.2
0.1
0
1
co 0.5Signal (cosθb , cosθ )CDF II Preliminary(5.1fb-1)

EventsEvents

CDF II Preliminary(5.1fb-1)

0.2
0.1
0
1

0
-0.5

-1 -1

-0.5

0

0.5

cos0.5
θ 0-0.5

1

cos θ +

b

-1 -1

-0.5

0

0.5

1

cos θ bFigure 4.6:
The distributions of reconstructed (cos θ+ , cos θ− ) (left) and(cos θb , cos θb̄ )(right) of dilepton candidates in tt̄ Pythia Monte Carlo sample on
the assumption of κ = 0, and their fit results.

Signal (cosθ+, cosθ-)κ = -1.0

400
300
200
100
0
1
co 0.5

sθ

-

200

-1 -1

-0.5

0

0.5

cos0.5
θ 0-0.5

1

cos θ +

κ = -1.0

b

-1 -1

-0.5

0

Sig. Fit (cosθb, cosθb)

0.5

1

cos θ bκ = -1.0

CDF II Preliminary(5.1fb-1)

Prob.

Prob.

-

300

0
1

0
-0.5

Sig. Fit (cosθ+, cosθ- )sθκ = -1.0b100

CDF II Preliminary(5.1fb-1)

0.4
0.3
0.2
0.1
0
1
co 0.5Signal (cosθb , cosθ )CDF II Preliminary(5.1fb-1)

EventsEvents

CDF II Preliminary(5.1fb-1)

0.2
0.1
0
1

0
-0.5

-1 -1

-0.5

0

0.5

cos0.5
θ 0-0.5

1

cos θ +

b

-1 -1

-0.5

0

0.5

1

cos θ bFigure 4.7:
The distributions of reconstructed (cos θ+ , cos θ− ) (left) and(cos θb , cos θb̄ )(right) of weighted dilepton candidates in tt̄ Pythia Monte Carlosample on the assumption of κ = −1. The two surfaces shown in lower row are fit resultsof the distributions above.

55

Cl0 (κ )2000

Cl1(κ )

Cl2 (κ )C2l(κ )0C1l(κ )

C0l(κ )

3000

1.5
1

-0.2

0.5
1000

0
-0.5

-1

-0.5

0

0.5

1

-1

κ

Cl3 (κ )

0

0.5

1

-1

κ

-0.1

-0.5

0

0.5

1

0

0.5

1

κ

Cl5 (κ )C5l(κ )0

-0.5

Cl4(κ )

C4l(κ )

C3l(κ )

0

-0.4

0
-0.5

0.1

0.05

-0.2
-1
-0.3
-1

-0.5

0

0.5

1

-1.5

κ

-1

-0.5

0

0.5

1

0

κ

-1

-0.5

κ

Figure 4.8: κ dependence of the fit parameters (Ciℓ (κ)) for the (cos θ+ , cos θ− ) distribution. We assume Ciℓ (κ)(i = 1, · · · , 5) are a linear function of κ.

bb

C0 (κ )b

C2 (κ )b

C2 (κ )bb

C1 (κ )

C1(κ )C0 (κ )

3000

0.3

0.5

2000
0.2
1000

0

0.5

1

0

κ

b
C3 (κ )

-0.5

0

0.5

1

-1

κ

b
C4(κ )0b

C4 (κ )

0

-1

b

-0.5

C5 (κ )-1b

C3 (κ )

0

0

0.1

-0.5

0

0.5

1

0

0.5

1

κ

b
C5 (κ )

0

-0.01

-0.2

-0.1

-0.02
-0.4

-0.2

-0.03
-0.04

-1

-0.5

0

0.5

1

κ

-1

-0.5

0

0.5

1

κ

-1

-0.5

κ

Figure 4.9: κ dependence of the fit parameters (Cib (κ)) for the (cos θb , cos θb̄ ) distribution.
We assume Cib (κ)(i = 1, · · · , 5) are a linear function of κ.

56

4.5.3 Background Template
Next, we describe background templates. We use the table 4.1 as expected number ofbackground. We consider diboson (W W, W Z, and ZZ), Drell-Yan (Z/γ ∗ → ee, µµ),Z →τ τ , and fake (W +QCD jet, where a jet is misidentified as a lepton) processes as background events. We neglect W γ process, since it is found to be very small contribution.
For diboson, Drell-Yan, and Z → τ τ backgrounds, we estimate the background distributions using Monte Carlo basically. For fake backgrounds, we use real data to estimate
the background distributions.

Diboson
We use W W/W Z/ZZ Monte Carlo samples generated by Pythia event generator to estimate diboson background. We made the distribution of reconstructed (cosθ+ ,cosθ− )and (cosθb ,cosθb̄ ) of dilepton candidates in W W, W Z and ZZ simlated samples, respectively. Then, we normalize the distributions for each sample to each expected numberof events and accumulate them. Figure 4.10 shows the distributions of reconstructed(cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) of dilepton candidates in W W /W Z/ZZ samples. The
right column plots in the figure indicate the magnitude of 1σ uncertainty of each bin of
the left column distributions.

Figure 4.10:
The distributions of reconstructed (cos θ+ , cos θ− ) (upper) and(cos θb , cos θb̄ )(lower) of dilepton candidates in W W /W Z/ZZ simulated samples. The
distribution in the right column indicates the magnitude of 1σ uncertainty of each bin of
the left distribution.

57

Z → ττ
We use Monte Carlo samples of Z/γ ∗ + n partons where Z/γ ∗ → τ τ which are generated
by ALPGEN event generator to estimate Z → τ τ background.
Table 4.2 shows processes in Z → τ τ background, number of dilepton candidates
found in Monte Carlo of each process divided by generated number of events and crosssection in pb for each process. We accumulate distributions from each process with appropriate weight considering its acceptance and cross-section, then finally normalize theaccumulated distribution to expected number of event of Z → τ τ background.
Table 4.2: Table of processes in Z → τ τ background. Number of dilepton candidates
found in Monte Carlo of each process divided by generated number of events and crosssection in pb for each process are also shown.
Process
Z peak
Z + 0 parton
Z + 1 parton
Z + ≥2p partonsMZ [20-75]
Z + 0 parton
Z + 1 parton
Z + ≥2 partonsMZ [105-800]
Z + 0 parton
Z + 1 parton
Z + 2 partons
Z + 3 partonsZ + ≥4 partons# of candidate/# of generated X-section[pb]0/7.8M
47/7.8M
1045/3.05M

157.7
21.5
4.14

0/1.45M
1/1.47M
78/2.90M

160.0
8.38
1.82

0/350k30/261k401/346k
825/350k
206/71.7k

4.07
0.706
0.117
0.01850.0033Figure 4.11 shows the distributions of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ )of dilepton candidates in Z → τ τ samples.
Drell-Yan
We use Monte Carlo samples of Z/γ ∗ + n partons where Z/γ ∗ → ee, µµ which are
generated by ALPGEN event generator to estimate Drell-Yan background.
Table 4.3 and 4.4 shows processes in Drell-Yan background, number of dilepton candidates found in Monte Carlo of each process divided by generated number of events andcross-section in pb for each process. We accumulate distributions from each process withappropriate weight considering its acceptance and cross-section, then finally normalize
the accumulated distribution to expected number of events of Drell-Yan background.
Figure 4.12 shows the distributions of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ )of dilepton candidates in Z/γ ∗ → ee, µµ simulated samples.
58

Figure 4.11:
The distributions of reconstructed (cos θ+ , cos θ− ) (upper) and(cos θb , cos θb̄ )(lower) of dilepton candidates in Z → τ τ simulated samples. The distributions in the right column indicates the magnitude of 1σ uncertainty of each bin of theleft distribution.

Fake
For fake events where a jet fakes charged lepton, we hardly estimate this backgroundwith Monte Carlo simulation. Instead, we use data to estimate the fake background, andvalidate the fake background using same sign dilepton events where the contribution fromfake background is dominated. We pick up an event have high-pT lepton (e or µ) and at
least one jet which can fake lepton from 5.1fb−1 data, and forcibly fake the jet to anelectron or muon, then set an event weight for the event with faked lepton by its fakerate. The fake rate is a lepton type dependent probability that a fakeable object, that is anobject which shares some of the jets and some of the high pT lepton characteristics, canbe reconstructed as a good lepton. The fake rate is calculated in [57].
Subsequently we apply the selection to the event and perform the kinematical reconstruction to make distributions of dilepton candidates, taking the weight of each event
into account, and finally normalize the distributions to expected number of events offake background. Figure 4.13 shows the distributions of reconstructed (cos θ+ , cos θ− )and (cos θb , cos θb̄ ) of dilepton candidates obtained from lepton + fakeable jet events in
5.1fb−1 data.
The right column plots in the figure indicate the magnitude of 1σ uncertainty of eachbin of the left distribution.
Peaky bins in these plots come from single events with high fake rate jet (typically
low-pT fakeable jet), therefore have large statistical uncertainties.
59

Table 4.3: Table of processes in Drell-Yan background. Number of dilepton candidates
found in Monte Carlo of each process divided by generated number of events and crosssection in pb for each process are also shown.
Process# of candidate/# of generated X-section[pb]Mee [75-105]
ee + 0 parton
0/3.46M
157.7
ee + 1 parton
25/3.45M
21.6
ee + 2 partons
801/6.07M
3.46
ee + 3 partons
218/678k
0.550
ee + ≥4 partons393/675k
0.0990Mee [8-20]
ee + 0 parton
0/682k1512.4ee + 1 parton
0/682k19.7
ee + ≥2 partons
3/671k
6.97
Mee [20-75]
ee + 0 parton
0/686K
160.0
ee + 1 parton
3/666K
8.38
ee + 2 partons
726/6.06M
1.60
ee + 3 partons344/675k
0.233
ee + ≥4 partons
788/671k
0.0398Mee [105-600]
ee + 0 parton
7/669k4.07
ee + 1 parton
166/675k
0.706
ee + 2 partons
1395/664k
0.117
ee + 3 partons3528/681k
0.0185ee + ≥4p partons
6071/679k
0.00333

All Background
Finally we accumulate distributions from each component of background: W W /W Z/ZZ,
Z → τ τ , Z/γ ∗ → ee, µµ, and fake samples.
Figure 4.14 show the resulting distributions of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ )of accumulated dilepton candidates from all components discussed above. The distribution in the right column indicates the magnitude of 1σ uncertainty of each bin of the upper
distribution.
The distributions indicate no significant P violating component. Hence we adopt thesame fit function form (Equation 4.6) as the signal templates to fit the distributions. The
surfaces in the lower row are fit results. The χ2 /ndf’s of fit are found to 100.761/94
for (cos θ+ , cos θ− ) distribution and 94.9453/94 for (cos θb , cos θb̄ ) distribution, which arecorresponding to 29.8% and 45.3% for χ2 probabilities, respectively.

60

Table 4.4: µµ channels of Drell-Yan background.
Process# of candidate/# of generated X-section[pb]Mµµ [75-105]µµ + 0 parton
1/3.48M
157.7µµ + 1 parton
31/3.48M
21.6µµ + 2 partons
970/6.14M
3.46µµ + 3 partons
268/686k
0.548µµ + ≥4p partons
480/685k
0.0994Mµµ [8-20]µµ + 0 parton
0/682k1512.6µµ + 1 parton
2/677k
19.7µµ + ≥2 partons
13/682k
6.98Mµµ [20-75]µµ + 0 parton
0/676k
160.0µµ + 1 parton
3/681k
8.39µµ + 2 partons
864/6.07M
1.60µµ + 3 partons352/675k
0.233µµ + ≥4 partons
967/681k
0.398Mµµ [105-600]µµ + 0 parton
16/686k
4.07µµ + 1 parton
213/683k
0.706µµ + 2 partons
1997/683k
0.117µµ + 3 partons4071/671k
0.0185µµ + ≥4 partons6822/681k
0.0033261

Figure 4.12:
The distributions of reconstructed (cos θ+ , cos θ− ) (upper) and(cos θb , cos θb̄ )(lower) of dilepton candidates in Z/γ ∗ → ee, µµ simulated samples. The
distribution in the right column indicates the magnitude of 1σ uncertainty of each bin of
the left distribution.

Figure 4.13:
The distributions of reconstructed (cos θ+ , cos θ− ) (upper) and(cos θb , cos θb̄ )(lower) of fake dilepton candidates obtained from lepton + fakeable jet
events in 5.1fb−1 data. The distribution in the right column indicates the magnitude of
1σ uncertainty of each bin of the left distribution.

62

Background (cos θ+, cos θ- )Background (cos θb, cos θ )

2
1.5
1
0.5
1
co 0.5

sθ

0
-0.5

-0.5

0

0.5

sθ

1

cos θ +

b

0
-0.5

-1 -1

0

0.5

1

0

0.5

1

-0.5

0

0.5

1

-0.5

-0.5

Bkg. 1σ Err. (cos θ , cos θ-)

Bkg. 1σ Err. (cos θ , cos θ )CDF II Preliminary(5.1fb-1)

CDF II Preliminary(5.1fb-1)

Events

-1 -1

2.5
2
1.5
1
0.5
1
co 0.5

Events-b

CDF II Preliminary(5.1fb-1)

EventsEvents

CDF II Preliminary(5.1fb-1)

b

+

2
1.5
1
0.5
0
1
c 0.5

os

θ-

b

2
1
0
1

0
-0.5

-1 -1

-0.5

0

0.5

cos0.5
θ 0-0.5

1

cos θ +

b

Bkg. Fit (cos θ+, cos θ- )

-1

Prob.

CDF II Preliminary(5.1fb )

0.3
0.2
0.1
0
1

cos0.5 0
θ-0.5

-1 -1

-1 -1

cos θ b

Bkg. Fit (cos θb, cos θb)-1

CDF II Preliminary(5.1fb )Prob.

cos θ b

-0.5

0

0.5

0.4
0.3
0.2
0.1
0
1

cos0.5
θ 0-0.5

1

cos θ +b

-1 -1

cos θ bFigure 4.14:
The distributions of reconstructed (cos θ+ , cos θ− ) (left) and(cos θb , cos θb̄ )(right) of accumulated dilepton candidates from W W /W Z/ZZ, Z → τ τ ,Z/γ ∗ → ee, µµ, and fake samples. The distributions in the middle row indicate themagnitude of 1σ uncertainty of each bin of the distributions in left column. The surfaces
in the lower row are fit results of each distribution.

63

4.5.4 Crosscheck of the Signal and the Background shape
We check the modeling of background amd kinematical reconstruction method by comparing distributions of observed candidates in 5.1fb −1 with prediction of signal and background.
We make the distribution of one dimentional distribution of reconstructed cosθ ofleptons and b-jets, which are hardly depend on κ. The distributions are therefore suitable
to use as a cross-check, keeping results blinded in this analysis.
Considering that cosθ+ and cosθb are changed to cosθ− ans cosθb̄ under CP reversal,
we compare the distribution of cosθ+ and cosθ− , and the distribution of cosθb and cosθb̄
between data and prediction.
Figure 4.15 shows the resultant distributions. The cross indicates 5.1 fb−1 data withstatistical errors and red band indicates expected number of signal and background total
events with 1σ uncertainty.
We perform the Kolmogrov-Smirnov test to compare cosθ distribution of data to cosθ
distribution of prediction, for leptons and for bottoms, respectively [58]. The KolmogrovSmirnov probability are found to 0.71 for cosθ distribution of leptons and 0.22 for cosθ
distribution of b(b̄). The expected distributions and the distribution of data are in good
agreement within the uncertainties for both of leptons and bottoms.

Figure 4.15: The distribution of cos θ+ , cos θ− (left) and cos θb , cos θb̄ (right). The crossindicates 5.1fb−1 data and red band indicates expected number of signal and backgroundtotal events with 1σ uncertainty.

64

4.6

Pseudo-experiments

In this section, we describe a method to extract measured κ from observed distributionsof (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) first, and in order to verify the method and study
sensitivity to κ measurement, we perform pseudo-experiments.
In the previous section, we obtain signal and background templates, and using thesetemplates we define the following likelihood as a function of assumed κ:Yii
L(κ) =f ℓ (cos θ+, cos θ−; κ)f b (cos θbi , cos θb̄i ; κ) ,(4.8)
i

where i is index of candidates, andf ℓ,b (x, y; κ) ≡

sig
Nexp

f ℓ,b (x, y; κ) +bkg sig

sig
Nexp+ Nexpbkg
Nexp

f ℓ,b (x, y) .
sig
bkg bkg
Nexp + Nexp

(4.9)

sig
bkg
Here Nexp, and Nexp
represent the expected number of events for signal and backℓ,bℓ,bground, respectively. fsig (x, y; κ), and fbkg
(x, y) represent template functions for (cos θ+ , cos θ− )and (cos θb , cos θb̄ ) of signal and background, respectively. We take κmeas as a measured
κ, which gives maximum of the likelihood.
Note that this likelihood doesn’t take correlations between (cos θ+ , cos θ− ) and (cos θb , cos θb̄ )into account. In reality, we observe a correlation, though it is found to be very small. The
likelihood might give a wrong estimate for statistical uncertainty due to this correlation.
So, we do not use the likelihood to estimate statistical uncertainty of κmeas , To exclude
this inexpedience, we construct Feldman-Cousins confidence intervals from the distribution of κmeas itself by perform many pseudo-experiments as mentioned below. We exclude
correlation by suppose prior probability with σ(κmeas ) which contain correlation between(cos θ+ , cos θ− ) and (cos θb , cos θb̄ ).
For one pseudo-experiment, first we assume the following input values:• true κ• Nobs : number of observed candidates
sig• Nexp: number of expected tt̄ signal eventsbkg• Nexp: number of expected background events

Then, we generate the following two random numbers:
sigsig• Nobs: Poisson distributed with expected value of Nexp
bkg
bkg
• Nobs: Poisson distributed with expected value of Nexp

and repeat this generation until the condition
sig
bkg
Nobs
+ Nobs= Nobsis satisfied.
65(4.10)

sig
bkg
sig
Once we obtain Nobs
and Nobs, we pick up Nobs
set of (cos θ+ , cos θ− ) and (cos θb , cos θb̄ )randomly from tt̄ candidate event pool of signal Monte Carlo sample with probability
hepg
hepg
bkg
proportional to 1 + κ cos θ+
cos θ−. We next generate Nobs
random number set of(cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) which are distributed by the background templates
functions.

Then, we make the unbinned likelihood (Equation 4.8) to obtain κmeas .

+ -Entriescos θbeam b-b

334

EventsEventscos θbeam l -l

20
10
0
1

co 0.5 0sθ-0.5-

Entries

334

20
10
0
1

-1 -1

-0.5

0

0.5

co 0.5
sθ 0
-0.5

1

cos θ+

b

-1 -1

-0.5

0

0.5

1

cos θbFigure 4.16: An example of pseudo data on the assumption of κtrue = +1.0, Nobs =sigbkg
334, Nexp= 236.97, and Nexp
= 86.24. Upper left: the distribution of reconstructed(cos θ+ , cos θ− ). Upper right: the distribution of reconstructed (cos θb , cos θb̄ ). Lower:−2∆ log L as a function of assumed κ. Measured κ (κmeas ) is defined as the maximum
point of the likelihood. By fitting −2∆ log L to a quartic polynomial function, we allow
unphysical region outside of [-1,1] as κmeas . Measured uncertainty of κmeas (σ meas ) isdefined as the region of −2∆ log L < 1.

Figure 4.16 illustrates an example of pseudo data on the assumption of κtrue = +1.0,
sigbkg
Nobs = 334, Nexp
= 236.97, and Nexp
= 86.24, and −2∆ log L as a function of assumed
meas
κ. κ
is defined as the maximum point of the likelihood. By fitting −2∆ log L toa quartic polynomial function, we allow unphysical region outside of [-1,1] as κmeas inorder to construct Feldman-Cousins confidence intervals later. We also define σ meas by theregion of −2∆ log L < 1 as a measured uncertainty of κmeas to evaluate pull distribution
later.
66

4.6.1 Check of the Method for κ Measurement
In order to check the method to extract κmeas , signal templates, and procedure of thepseudo-experiment, we use tt̄ Monte Carlo sample generated with Herwig. In this sample,
standard model spin correlation is incorporated, and we found κtrue = 0.882 ± 0.003 by
hepg
hepg
fitting distribution of (cos θ+, cos θ−) in the whole dilepton events to a function of
hepg
hepg
1 + κ cos θ+ cos θ− .

Figure 4.17:
The distribution of reconstructed (cos θ+ , cos θ− ) (left) and(cos θb , cos θb̄ )(right) of dilepton candidates in Monte Carlo sample which is madewith Herwig event generator where tt̄ spin correlations are included.
Figure 4.17 shows distributions of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ )of dilepton candidates in herwig Monte Carlo sample in 10 by 10 bin histograms.
sigbkg
First we assume Nobs = Nexp
= 130, and Nexp
= 0, which corresponds a scenario
with ideal b-tag situation that has 55% event efficiency for tt̄ event and perfect rejection
for background. On the assumption, we perform 10,000 times pseudo-experiment bypicking up Nobs events from dilepton candidate pool in herwig Monte Carlo, and look at
the distribution of κmeas .

Figure 4.18: The distribution of κmeas of 10,000 pseudo-experiments using herwig Montesigbkg
Carlo as signal sample pool on assumption of Nobs = 130, Nexp
= 130, and Nexp
= 0.
meas
The curve is the Gaussian fit of the distribution, and we find hκ
i = 0.879 ± 0.006 andσ(κmeas ) = 0.612 ± 0.004.
67

Figure 4.18 shows the resultant distribution. By the Gaussian fit of the distribution,mean of κmeas is found to be hκmeas i = 0.879 ± 0.006, that is consistent with κtrue , anduncertainty of κmeas is found to be σ(κmeas ) = 0.612 ± 0.004.
sigbkg
Next, we assume numbers of 5.1fb−1 , i.e. Nobs = 334, Nexp
= 236.97, and Nexp=
86.24.
On the assumption, we perform 10,000 times pseudo-experiment using herwig Monte
Carlo signal pool for pseudo signal event and background template for pseudo background
event.

Figure 4.19: The distribution of κmeas of 10,000 pseudo-experiments using herwig Montesigbkg
Carlo as signal sample pool on assumption of Nobs = 334, Nexp
= 236.97, and Nexp=
meas
86.24. The curve is the Gaussian fit of the distribution, and we find hκ
i = 0.872 ±meas
0.005 and σ(κ) = 0.534 ± 0.004.
Figure 4.19 shows the resultant distribution. By the Gaussian fit of the distribution,mean of κmeas is found to be hκmeas i = 0.872 ± 0.005, that is consistent with κtrue as well,and uncertainty of κmeas is found to be σ(κmeas ) = 0.534 ± 0.004, that is nearly equal tothe case with ideal b-tag scenario above.

68

4.6.2 Pseudo Experiment Results
Here in order to obtain hκmeas i and σ(κmeas ) as functions of κtrue , we perform 10,000
pseudo-experiments for each κtrue ranging from -1 to +1 with 0.1 step, using the prescription described in the beginning of this section to cope with Pythia tt̄ sample which doesn’t
have spin correlation.
κmeas

Nexp

κ true

CDF II Preliminary(5.1fb-1)

1
1000

0.5

κ true = -1.0

1000

κ true = 0.0
κ true = 1.0

0
500

500

-0.5
-1
-4

-2

0

2

κ

4

0

0
-4

meas

-2

κ meas sigma

κ meas mean

0.8

P 1= 1.0041 ± 0.0020

0.6

0

0.4
-1
-0.5

0

4
meas

CDF II Preliminary(5.1fb-1)

f = P0 + P1 κ true
P 0= 0.0002 ± 0.0012

-1

2

κ

CDF II Preliminary(5.1fb-1)

1

0

0.5

1

f = P0 + P1κ true + P2(κ true)2
P 0= 0.5499 ± 0.0013
P 1=-0.0048 ± 0.0014
P 2=-0.0101 ± 0.0026-1

κ true

-0.5

0

0.5

1

κ trueFigure 4.20: The result of 10,000 pseudo-experiments on the assumption of κtrue =sig
bkg−1.0 ∼ +1.0, Nobs = 334, Nexp
= 236.97, and Nexp
= 86.24. Upper left: The dismeas
truetribution of κ
(horizontal axis) as a function of κ(vertical axis). Upper right: The
meas
true
true
distribution of κin case of κ
= −1.0, κ= 0.0, and κtrue = +1.0, and Gaussianmeas
fit curve, respectively. Lower left: hκ
i as a function of κtrue , and linear fit. Lowerright: σ(κmeas ) as a function of κtrue , and parabolic fit.

Figure 4.20 shows the result of 10,000 pseudo-experiments on the assumption ofsig
bkg
κtrue = −1.0 ∼ +1.0, Nobs = 334, Nexp
= 236.97, and Nexp
= 86.24. By Gausmeas
true
meas
sian fit of a distribution of κfor each κ , we obtain hκ
i and σ(κmeas ) for eachκtrue . Finally, we obtain hκmeas i as a linear functionhκmeas i = P0 + P1 κtrue
P0 = 0.0002 ± 0.0012
P1 = 1.0041 ± 0.0020
69(4.11)

and σ(κmeas ) as a parabolic function
σ(κmeas )
P0
P1
P2

=
=
=
=

P0 + P1 κtrue + P2 (κtrue )2
0.5499 ± 0.0013−0.0048 ± 0.0014−0.0101 ± 0.0026(4.12)

by fitting of plots, respectively.
A small offset is seen in hκmeas i. This offset will be properly incorporated in the final
result since we construct Feldman-Cousins confidence intervals with this hκmeas i itself
and σ(κmeas ).
Pull

CDF II Preliminary(5.1fb-1)

Nexp

κ true1400

0.5

κ true = -1.0
κ true = 0.0

400κ true = 1.0

300
0
200
-0.5

200

100

-1

-2

0

0

2

-2

0

2

Pull

Pull
CDF II Preliminary(5.1fb-1)

Pull sigma

Pull mean

CDF II Preliminary(5.1fb-1)
f = P0 + P1 κ
P0= 0.0043 ± 0.0023
true

0.2

P1= 0.0184 ± 0.0038

0

f = P0 + P1 κ
P0= 1.0462 ± 0.0018
true

1.4

P1= 0.0010 ± 0.0029

1.2
1

-0.2

0.8
-1

-0.5

0

0.5

1

-1

κ true

-0.5

0

0.5

1

κ trueFigure 4.21: The pull distribution of 10,000 pseudo-experiments on the assumption ofsig
bkg
κtrue = −1.0 ∼ +1.0, Nobs = 334, Nexp
= 236.97, and Nexp
= 86.24. Upper left:
meas
true
meas
true
The pull(≡ (κ− κ )/σ) distribution as a function of κ . Upper right: The
truepull distribution in case of κ
= −1.0, κtrue = 0.0, and κtrue = +1.0, and Gaussian
fit curve, respectively. Lower left: pull mean as a function of κtrue , and linear fit. Lowerright: pull width as a function of κtrue , and linear fit.
Figure 4.21 shows the pull distribution of 10,000 pseudo-experiments for each κtrue .
The pull is defined bypull ≡ (κmeas − κtrue )/σ meas .
(4.13)
By fitting pull distribution for each κtrue to a Gaussian curve, we obtain pull mean andpull sigma as functions of κtrue . We observe pull sigma is a little bit grater than unit. We
found this is mainly caused since σ meas is estimated smaller due to correlation between
the distributions of (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ). We check this by extracting κmeas
with only (cos θ+ , cos θ− ) and with only (cos θb , cos θb̄ ), separately.
70

4.6.3 Correlation between κmeas from leptons and κmeas from b-jets
In order to check correlation between κmeas obtained only from (cos θ+ , cos θ− ) and κmeas
obtained only (cos θb , cos θb̄ ), we performed 10,000 pseudo-experiments on the same condition in 4.6.2, but for likelihood instead of Equation 4.8 we just use the following Lℓand Lb , separately:
Lℓ (κ) =b

L (κ) =Yiif ℓ (cos θ+, cos θ−; κ) ,iY

f ℓ (cos θbi , cos θb̄i ; κ) .

i

Figure 4.22: σ(κmeas ) as a function of κtrue for leptons only case.

Figure 4.23: σ(κmeas ) as a function of κtrue for b-jets only case.

Fig 4.22 and 4.23 show the σ(κmeas ) as a function of κtrue for leptons only case andthat for b-jets only case, separately.
In Section 4.6.2, σ(κmeas ) is about 0.55 at κtrue = 0.
Comparing this, σ(κmeas ) of leptons only case is about 0.60 at κtrue = 0 and σ(κmeas )of b-jets only case is about 1.10 at κtrue = 0. A naive combination of statistical uncertain71

ties of these results suppose they were independent is
q

1

2
1
+
0.60

2
1
1.10

= 0.53 .

This value is close to 0.55 but a little bit smaller. Hence we conclude there is a small but
some correlation between κmeas obtained only from (cos θ+ , cos θ− ) and κmeas obtained
only from (cos θb , cos θb̄ ). And leptons+ b-jets case in Section 4.6.2, 0.55, is improved byabout 8% comparing with leptons only case.

72

4.6.4 Construction of Feldman-Cousins intervals
Using hκmeas i and σ(κmeas ) obtained in Sec 4.6.2, we construction of Feldman-Cousins
confidence interval [59] as a function of κtrue .
We first suppose P (κmeas |κtrue ) as the probability that κmeas is observed at a giventrue
κ . We suppose the probability is given by normal distribution of N (hκmeas i, σ(κmeas )).
Then we set an interval [κmeas, κmeas] so that the integral of the probability from κmeas
1
2
1
to κmeas
isagiven
confidence
level,i.e.
2Z κmeas2
κmeas
1

P (κmeas |κtrue )dκmeas = C.L. ,(4.14)

using Feldman-Cousins’s ordering principle. In the principle, we define κtrue
best whichtrue
gives maximum probability within physically allowed region of κ
when a κmeas is
given, then requireR(κmeas) = R(κmeas),(4.15)1
2
whereR(κmeas ) ≡P (κmeas |κtrue ).
P (κmeas |κtrue
best )(4.16)

Figure 4.24 shows confidence intervals at 68% and 95% level, as a function of κtrue ,respectively.
Feldman-Cousins interval

True κ

1
68% C.L. (Stat only)
95% C.L. (Stat only)

0.5

Best value

0
-0.5
-1

-2

-1

0

1

2

Measured κ

Figure 4.24: Confidence interval at 68% and 95% level based on Feldman-Cousins prescription as a function of κtrue , where the distribution of κmeas is assumed to be Gaussianwith hκmeas i and σ(κmeas ) obtained from pseudo-experiments. Only statistical errors are
considered here.

73

Chapter 5
Systematic Uncertainties
We have performed systematics studies with respect to known sources. They are Statistical uncertainty of signal MC sample, Uncertainties in expected signal backgroundnumbers and background templates, jet energy, ISR/FSR, gg fraction, PDF, NLO andcolor reconnection. To estimate these systematics, we performed pseudo experiment.

5.1

Statistical Uncertainty of Signal MC Sample

In this analysis, the signal templates depend on Monte Carlo tt̄ sample. The number oftt̄ candidates in the sample is finite. Therefore, the signal templates themselves suffer
statistical fluctuation.
To estimate the effect of signal MC statistics, we perform 10,000 pseudo experiments
for each κtrue , where we have 100 signal only events and no background for each experiment. Then we obtain σ(κmeas ) for 100 signal only case.
In the nominal tt̄ signal sample we use, we have 47,898 candidates. Consequently, we
obtain σ(κmeas p) due to the statistical fluctuation of signal templates by re-scaling by thescale factor of 100/47898.
Figure 5.1 shows expected σ(κmeas ) for the case of 47,898 signals.
We add this σ(κmeas ) into systematic uncertainty in quadrature as systematics due tostatistical uncertainty of signal Monte Carlo sample.

5.2

Uncertainties in expected signal, background numbers and background templates

As described in Section 4.6, in a nominal pseudo experiment, we generate pseudo background events as random numbers of (cosθ+ ,cosθ− ) and (cosθb ,cosθb̄ ) which distribute thebackground templates defined in Section 4.5.3.
However, we have to consider uncertainties in the background templates and look at
effects on measured κ. The possible uncertainties in the background templates come from• Uncertainty of expected number of each background component75

-1σ(κ meas)

CDF II Preliminary(5.1fb )

0.035

0.03

f = P0 + P1κ true + P2(κ true)2
P0= 0.0332 ± 0.0001
P1=-0.0007 ± 0.0001
P2= 0.0000 ± 0.0002

-1

-0.5

0

0.5

1

κ trueFigure 5.1: Expected σ(κmeas ) for the case that all entries (47,898)
p in signal MC is used.
measThis is obtained by scaling σ(κ) of 100 signal only case by 100/47898.
• Statistical uncertainties of Monte Carlo and/or data samples which are used to define background templates.
To estimate these effects, we perform an alternative set of pseudo-experiments described below. In these pseudo-experiments, we also consider the uncertainty in expected
number of signal events.
1. We use uncertainties of expected numbers of signal and each background component in table 4.1. Following these uncertainties, we generate following Gaussian
randoms:
′sig
sig
sig
Nexp
= Gaus(Nexp, δNexp
)
′W W
WW
WW
Nexp
= Gaus(Nexp, δNexp
)
...
...
′fake
fake
fake
Nexp = Gaus(Nexp, δNexp),

where N ′ indicates fluctuated expected number.
Then we take sum over all background components:
X
′bkg
′bkgi
Nexp
=
Nexp
,
i = W W, W Z, ..., fake
i

′sig
′bkg
2. Using Nexp
, Nexp, and Nobs , we generate observed numbers as Poisson randoms:
sig
′sig
Nobs
= Poisson(Nexp
)
bkg
′bkg
Nobs
= Poisson(Nexp
),
sig
bkg
where Nobs
+ Nobs= Nobs .

76

sig
3. Next generate pseudo signal events by picking up Nobs
events randomly from thesignal candidate pool.

4. Generate pseudo background events: To estimate systematic uncertainties of background templates, we generate background events from fluctuated background template. There are five parameters in background template. Easy way to make fluctuated background templates is to fluctuate five parameters by normal distribution.
However, five parameters have correlations therebetween. Therefore, we have tofluctuate five parameters considering these correlations. The method is as follows.
~ = (C1 , · · · , C5 )T with consideration of• First, we fluctuate fit parameters C~ ′ = (C1′ , . . . , C5′ )T (fluctuated C)~
correlation between each fit parameters. C~′ = C~ + LG,~ where G~ is vector of independent normalized
is given by C
Gaussian (0,1) and L is a matrix from Cholesky factorization [60], which
satisfies Σ = LLT , where Σ is the covariance matrix of the fit parameters.
• Next, generate background template using Ci′ (i = 1, · · · , 5).

bkg• Then generate Nobs
random number sets of (cos θ+ , cos θ− ) and (cos θb , cos θb̄ )which are distributed by the fluctuated background template functions.

5. Calculate unbinned likelihood for pseudo events of signal+background as a function
sig
bkg
′sig
of κ on the assumption of nominal Nexp
and Nexp, since we never know Nexp
nor
′bkg
meas
Nexp . Then obtain κ.
6. Make the distribution of κmeas by performing many pseudo-experiments (10,000 foreach κtrue ), then obtain hκmeas i′ , and σ(κmeas )′ as a mean and a standard deviationof each distribution.
7. Finally we calculate a shift in the mean of κmeas from the nominal and an increaseof the standard deviation of κmeas , i.e.
∆κmeas = hκmeas i′ − hκmeas ipσ′ =σ(κmeas )′2 − σ(κmeas )2

We suppose the ∆κmeas represents the effect from uncertainty of background templatebkg
sigand σ ′ represents the effect from uncertainties of Nexp
and Nexp i , as well as statisticalfluctuation in background distribution.
Figure 5.2 shows resultant ∆κmeas and σ ′ . We include both of two in quadrature asa systematic uncertainty from uncertainties in the expected signal, background numbers
and background templates.

5.3

Jet Energy Scale Uncertainty

Jet energy uncertainty also affects dilepton acceptance as well as kinematical distributions. To estimate this effect, we perform pseudo experiments (10,000 experiments foreach κ) where jet energy scale (JES) is shifted by 1σ uncertainty as well as shifted by77

-1

CDF II Preliminary

Increase of σ

CDF II Preliminary(5.1fb )∆ κ meas

0.2

0

-0.2

f = P0 + P1κ true + P2(κ true)2
P0= 0.0405 ± 0.0026

0.6
0.4

-0.5

0

0.5

P0= 0.0961 ± 0.0107
P1= 0.0144 ± 0.0093
P2= 0.0638 ± 0.0179

0.2

P1=-0.0125 ± 0.0028
P2=-0.0159 ± 0.0052-1

f = P0 + P1κ true + P2(κ true)20

1

κ true(a) shift of hκmeas i

-1

-0.5

0

0.5

1

κ true(b) increase of σ(κmeas )Figure 5.2: (a) Shift of hκmeas i and (b) increase of σ(κmeas ) from nominal pseudoexperiment case when we take fluctuations of uncertainties of signal, background numbers
and background template function into account in the pseudo-experiment.

sig-1σ. We also take shifts in tt̄ acceptance into account. Nexp
is shifted by +3.4% for JES+1σ shift, and by -3.5% for JES −1σ shift.
Figure 5.3 shows shift of hκmeas i from nominal pseudo-experiment case for jet energy
scale +1σ case and −1σ case.
We look at shifts of hκmeas i for two cases are in opposite direction. Therefore wetake half difference as systematics from JES uncertainty, and add this into systematic
uncertainty in quadrature.

-1-1

CDF II Preliminary(5.1fb )0.2∆ κ meas∆ κ meas

0.2

0

-0.2

CDF II Preliminary(5.1fb )f = P0 + P1κ true + P2(κ true)2
P0=-0.0193 ± 0.0025-0.2

P1= 0.0088 ± 0.0027
P2=-0.0231 ± 0.0051

-1

-0.5

0

0

0.5

1

f = P0 + P1κ true + P2(κ true)2
P0= 0.0397 ± 0.0026
P1=-0.0154 ± 0.0028
P2=-0.0193 ± 0.0052-1

κ true(a) JES +1σ

-0.5

0

0.5

1

κ true(b) JES −1σ

Figure 5.3: Shift of hκmeas i from nominal pseudo-experiment case when jet energy scaleis (a) increased by 1σ and (b) decreased by 1σ.

78

5.4

Initial and Final State Radiation

Extra jets originating from the incoming partons and/or outgoing partons are called initialand final state radiations(ISR and FSR). Figure 5.4 shows tt̄ diagram with ISR and FSR.
Differences due to QCD ISR and FSR are estimated using specially generated Pythia
samples in which the QCD parameters for the parton shower evolution are varied based on
the studies of the CDF Drell-Yan data and recommendations from the authors of PYTHIA.
ISR and FSR systematic uncertainties are estimated by changing the parameters related to ISR and FSR from their default values to half and double.
We look at shifts of hκmeas i for two cases are in opposite direction. Figure 5.5 shows
shift of hκmeas i from nominal pseudo-experiment case for more ISR/FSR case and less
ISR/FSR case. Therefore we take half difference as systematics from ISR/FSR uncertainty, and add this into systematic uncertainty in quadrature.

Figure 5.4: Diagram of tt̄ production and decay with ISR and FSR.

-1

-1

CDF II Preliminary(5.1fb )0

0.2∆ κ meas∆ κ meas

0.2

CDF II Preliminary(5.1fb )f = P0 + P1κ true + P2(κtrue)2
P0=-0.1509 ± 0.0025
P1=-0.0314 ± 0.0027
P2=-0.0072 ± 0.0051

-0.2

0

-0.2
-1

-0.5

0

0.5

1

f = P0 + P1κ true + P2(κ true)2
P0= 0.0208 ± 0.0025
P1=-0.0089 ± 0.0028
P2=-0.0150 ± 0.0051-1κ true(a) ISR/FSR more

-0.5

0

0.5

1

κ true(b) ISR/FSR lessFigure 5.5: Shift of hκmeas i from nominal pseudo-experiment case when (a) ISR/FSR
more sample is used as signal pool in the pseudo-experiment and (b) ISR/FSR less sample
is used.

79

5.5

Uncertainty of gg fraction

The nominal signal sample we use in this analysis is found to have about 6% of tt̄ production via gluon gluon fusion process. However, this fraction is expected to be muchbigger, at the level of around 15%, in case of NLO calculation [61] [18].
In the full kinematical reconstruction method described in Section 4.4, the probability
density functions of pztt̄ , pTtt̄ , and Mtt̄ used in Eqn. 4.2 depend on tt̄ Monte Carlo which
has only 6% gg fusion process. Due to difference of initial partons, we expect differenceof the probability density functions between q q̄ and gg processes.
To estimate how much shift we observe in measured κ if we increase the fractionof gg fusion process, we perform pseudo-experiments (10,000 experiments for each κ),
where 15% is assumed for gg fusion. We realize this by adopting an event by change of(1-0.15)/(1-0.06) if the event originates from q q̄ process when we pick up the event fromthe signal pool.
Figure 5.6 shows shift of hκmeas i from nominal pseudo-experiment case. We add thisshift into systematic uncertainty in quadrature, though it is found to be almost negligibly
small effect.
-1

CDF II Preliminary(5.1fb )∆ κ meas

0.2

0

-0.2

f = P0 + P1κ true + P2(κ true)2
P0=-0.0002 ± 0.0025
P1=-0.0024 ± 0.0028
P2= 0.0051 ± 0.0051

-1

-0.5

0

0.5

1

κ trueFigure 5.6: Shift of hκmeas i from nominal pseudo-experiment case (Fgg ∼ 6%) when
Fgg = 15% is assumed in the pseudo-experiment.

5.6

Parton Distribution Functions

For nominal signal sample, CTEQ5L parton distribution function (PDF) is supposed.
To evaluate the systematics due to PDF uncertainties, we perform pseudo-experiments(10,000 experiments for each κ) on the assumption of different PDF sets, such as MRST72,
MRST75, CTEQ6L, CTEQ6L1 and CTEQ6M, and look how much shifts of measured κ
are caused.
To realize signal events with different PDF sets, we adopt the weighted MC method.In
this method, we reweigh each event of a signal MC sample we already have, instead ofgenerating a different set of MC events with each different PDF set. We calculate the80relative probability of the event on the assumption of a different PDF set, based on theparton momentum fractions x1 , x2 , Q2 and initial partons in the event. Then we pickup an event from the signal event pool according to new probability of an event in thepseudo-experiments.
To look at the effect of difference between MRST72 (ΛQCD = 228MeV) and CTEQ5L(ΛQCD = 300MeV) PDF set, we compare between hκmeas i with MRST72 and one with
CTEQ5L.
And next, to look at the effect of different αs value, we compare between MRST72
and CTEQ75 cases.
Then, to look at the effect of uncertainties on the fitting parameters of CTEQ6M
PDF set, we compare between CTEQ6M and CTEQ6Mn±, where CTEQ6Mn± means
CTEQ6M with nth of 20 orthogonal parameters are changed by ±90%. We sum up positive shifts and negative shifts asymmetrically in quadrature.
Finally, we take the larger effect either of the 20 eigenvectors uncertainty or the difference between MRST72 and CTEQ5L PDF sets, and add effect of different αs in quadrature.
Figure 5.7 show resulting shifts of measured κ. Shift of hκmeas i for MRST72 to
CTEQ5L, difference of hκmeas i between MRST72 and MRST75, and shifts of hκmeas iby uncertainties of 20 eigenvectors in CTEQ6M PDF set are shown in the figure. And
total uncertainties toward positive and negative direction are also shown.
We add them into into systematic uncertainty in quadrature as systematics from PDF
uncertainty.

5.7

Systematics from LO versus NLO

In this analysis, we assume standard model tt̄ production and top (anti-top) decays except
for correlation coefficient between top and anti-top at the decays. This analysis relys
on Monte Carlo with LO calculations for tt̄ production kinematics such as pztt̄ , pTtt̄ , andMtt̄ . These might be changed at NLO calculations. Also there might be QCD interaction
between final state products at NLO.
We examine the effect of NLO using tt̄ Monte Carlo with an event generator at
NLO(MC@NLO [62] [63]).
We perform pseudo experiments (10,000 experiments for each κ), using tt̄ MonteCalro generated from MC@NLO event generator with CTEQ5M and MRST02 for PDF,
respectively.
Note that there is no spin correlation in these samples, therefore we treat κ on thesesamples with the same prescription with nominal Pythia tt̄ sample. Also note that about
11% of events are generated with negative weight in MC@NLO.
To cope with negative weight events, We pick up a positive weight event to cancel out
negative event, if we pick up a negative weight event when generate performing pseudo
experiment.
Figure 5.9 shows shift of hκmeas i from nominal pseudo-experiment case for MC@NLO
with CTEQ5M and MRST02 cases, respectively.
81

-1

-1

CDF II Preliminary(5.1fb )∆ κ meas∆ κ meas

CDF II Preliminary(5.1fb )

0.1

0.1

0

0

-0.1

-0.1
-1

-0.5

0

0.5

1

-1

κ true

-0.5

0.5

1

κ true-1-1

CDF II Preliminary(5.1fb )CDF II Preliminary(5.1fb )0.2∆ κ meas∆ κ meas

0

0.1

0

0
f = P0 + P1κ

-0.2
-0.1
-1

-0.5

0

0.5

1

+ P2( κtrue 2)

P 0=-0.0271 ± 0.0025

P 1=-0.0062 ± 0.0028

P 1=-0.0057 ± 0.0028

P 2 = 0.0068 ± 0.0051

P 2=-0.0338 ± 0.0051-1κ truetrue

P 0 = 0.0379 ± 0.0025

-0.5

0

0.5

1

κ trueFigure 5.7: Shift of hκmeas i for P.E. when MRST72 is assumed for PDF comparing withnominal P.E. with CTEQ5L (Upper left), difference of hκmeas i between P.E. with MRST72
and MRST75 (Upper right), shifts of hκmeas i for CTEQ6M where each parameter of 20
eigenvectors is shifted by ±90% comparing with P.E. with nominal CTEQ6M (Lower
left), and total uncertainties toward positive and negative direction of hκmeas i from threesources of PDF uncertainties above (Lower right).

We take larger shift as systematics from NLO effect, and add this into systematic
uncertainty in quadrature.

5.8

Color Reconnection

It has been suggested that color reconnection effects [64] could cause a bias in κ measurement. To estimate this effect, we perform pseudo-experiments (10,000 experiments
for each κ), using Pythia tune A [65] and Pythia tune ACR [65] as signal Monte Carlosamples, respectively. Pythia tune A is very similar to the tune for CDF nominal, Pythia
tune ACR includes color reconnection effect into the tune A. We compare between hκmeas iwith Pythia tune A and one with Pythia tune ACR .
Figure 5.10 shows difference of hκmeas i between with and without color reconnection
effect. We add this shift into systematic uncertainty in quadrature as systematics due tocolor reconnection effect.

82

(a) Leading order(b) Next to leading orderFigure 5.8: Feynman diagrams of tt̄ production for LO and NLO.

-1

-1

CDF II Preliminary(5.1fb )CDF II Preliminary(5.1fb )0.2∆ κ meas∆ κ meas

0.2

0

-0.2

f = P0 + P1κ true + P2(κ true)2
P0=-0.0397 ± 0.0025-0.5

f = P0 + P1κ true + P2(κ true)2
P0= 0.0902 ± 0.0025-0.2

P1=-0.0964 ± 0.0027
P2= 0.0096 ± 0.0050

-1

0

0

0.5

1

P1=-0.1071 ± 0.0027
P2= 0.0114 ± 0.0050-1κ true

-0.5

(a)

0

0.5

1

κ true(b)

Figure 5.9: (a) Shift of hκmeas i from nominal P.E. case when MC@NLO event generator with CTEQ5M as PDF is used. (b) Shift of hκmeas i from nominal P.E. case when
MC@NLO event generator with MRST02 as PDF is used.

-1

CDF II Preliminary(5.1fb )∆ κ meas

0.2

0

-0.2

f = P0 + P1κ true + P2(κ true)2
P0= 0.1032 ± 0.0025
P1=-0.0122 ± 0.0028
P2=-0.0009 ± 0.0051

-1

-0.5

0

0.5

1

κ trueFigure 5.10: Difference of hκmeas i with tune A and one with tune ACR .

83

5.9

Total Uncertainty and Feldman-Cousins Intervals

We summarize here all possible systematics described above. And by adding this into
statistical uncertainty in quadrature, we obtain total uncertainty including statistic andsystematic effects to construct confidence belt as a function of κ. Table 5.1 shows summary of systematic uncertainty. Figure 5.9 shows total systematic uncertainty in hκmeas ifrom all sources we discussed. We evaluate systematic uncertainties toward positive andnegative direction separately. The figure 5.9 shows quadratic sum of statistical uncertainty
and systematical uncertainty.
SourceStatistical uncertainty of signal MC sample
Expected signal, background numbers and background templates

JES
ISR/FSRgg fraction
PDF
NLO calculation
Color reconnection
Total systematic uncertainty

Systematic uncertainty
0.033 ∼ 0.034
0.104 ∼ 0.175
0.019 ∼ 0.044
0.071 ∼ 0.093
0.000 ∼ 0.007

0.037∼0.051 (toward positive)
0.027∼0.067 (toward negative)

0.068 ∼ 0.209
0.115 ∼ 0.092±0.200 ∼ ±0.294

Table 5.1: The summary table of systematic uncertainty.

-1

σsyst

CDF II Preliminary(5.1fb )

0.5

0

-0.5

f = P0 + P1κ true + P 2 (κ true)2
P0 = 0.2018 ± 0.0052
P1=-0.0232 ± 0.0050
P2 = 0.0686 ± 0.0097

-1

-0.5

0

P0 =-0.1998 ± 0.0052
P1= 0.0208 ± 0.0050
P2 =-0.0736 ± 0.0096

0.5

1

κ

true

Figure 5.11: Total systematic uncertainty on κmeas from all sources we consider. Uncertainties toward positive(blue) and negative(red) direction are shown separately.
Figure 5.13 shows confidence interval at 68% and 95% level based on FeldmanCousins prescription as a function of κtrue , where the distribution of κmeas is assumed
to be Gaussian with convolution of statistical and systematic errors.
Once we obtain a measured κ (κmeas ) from beam data, we can set a confidence intervalon true κ from this plot.
84

σstat ⊕ σsyst-1

CDF II Preliminary(5.1fb )0.7

Statistics Only
Stat. ⊕ Syst. (Positive Side)

0.65

Stat. ⊕ Syst. (Negative Side)

0.6

0.55
0.5
0.45

-1

-0.5

0

0.5

1

κ trueFigure 5.12: Statistical uncertainty (green) and statistical ⊕ systematical uncertainties(blue and red) on κmeas are shown separately.

Feldman-Cousins interval

True κ

1
68% C.L. (Stat only)
68% C.L.

0.5

95% C.L.
Best value

0
-0.5
-1

-2

-1

0

1

2

Measured κ

Figure 5.13: Confidence interval at 68% and 95% level based on Feldman-Cousins prescription as a function of κtrue , where the distribution of κmeas is assumed to be Gaussianwith convolution of statistical error and systematic error. Also Feldman-Cousins confidence interval with only statistical error considered is shown in the same figure.

85

Chapter 6
Results and Discussions
Finally, we look at the distribution of (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ) in data for integrated luminosity of 5.1fb−1 , which has been kept blinded until all studies on all possible
systematic uncertainties are done.
Then we calculate unbinned likelihood by Equation 4.8 to obtain measured κ (κmeas ).

∫ L dt = 5.1 fb

20
15
10
5
0
1
co 0.5

sθ

-

0
-0.5

-1 -1

0

-0.5

0.5

-1

20
15
10
5
0
1
co 0.5sθ 0

1

cos θ+-0.5b

Log Likelihood-2∆ Log(L)∫ L dt = 5.1 fb

Observed (cosθb, cosθb)
CDF II Preliminary-1

EventsEventsObserved (cosθ+, cosθ- )CDF II Preliminary

-1 -1

-0.5

0

0.5

cos θb

CDF II Preliminary∫ L dt = 5.1 fb-14

κ =0.042

2

0
-1

-0.5

0

0.5

1

κ

Figure 6.1: Upper left: the distribution of reconstructed (cos θ+ , cos θ− ) in 5 by 5 bins.
Upper right: the distribution of reconstructed (cos θb , cos θb̄ ) in 5 by 5 bins Lower:−2∆ log L as a function of assumed κ. The function is found to be minimum atκ = 0.042.

871

Figure 6.1 shows the distributions of reconstructed (cos θ+ , cos θ− ) and (cos θb , cos θb̄ ),and −2∆ log L as a function of assumed κ.
From the function, we observe κmeas = 0.042.

True κ

1

Feldman-Cousins interval

CDF II Preliminary

68% C.L. (Stat only)
68% C.L.

0.5

95% C.L.
Best value∫ L dt = 5.1 fb-1

0

Observed κ = 0.042-0.5-0.520 < κ < 0.605 (68%C.L.)+0.563

κ = 0.042-0.562

-1

-2

-1

0

1

2

Measured κ

Figure 6.2: Feldman-Cousins confidence intervals with measured result. Interval at 68%
C.L. on true κ corresponding to κmeas = 0.042 is shown.

Figure 6.2 shows the Feldman-Cousins confidence intervals constructed in Section 4.6.4
and measured result κmeas = 0.042. From this we obtain the following results:−0.52 < κ < +0.61 (68%C.L.)(6.1)

κ = 0.042+0.563−0.562(6.2)

oron the assumption of Mtop = 172.5GeV/c2 . In this analysis, we assume Mtop = 172.5GeV/c2 .
The result will be changed if we assume another top mass. To look at top mass dependence of the result, we repeat the same analyses, but in which we change the assumptionof top mass to each of 167.5, 170.0, 175.0, and 177.5 GeV/c2 .
Figure 6.3 shows 68% C.L. intervals on κ as a function of Mtop . Solid line indicates
center values of κ.
Due to signal sample statistics of each top mass, the center values are fluctuated.
Since systematics except for signal sample statistics described in Section 5.1 are common
among results for each top mass, The fluctuation is caused by only signal sample statistics.
The numbers of dilepton candidates in MC with assumption that top mass = 167.5, 170.0
and 177.5 GeV/c2 is 9K each. Their signal sample statistics is about 0.08. The numbersof dilepton candidates in MC with assumption that top mass = 175.0 GeV/c2 is 30K. Its
signal sample statistics is about 0.04. Considering this uncertainty, we do not see any
conclusive top mass dependence of κ measurement.
88

κ

1

68% Confidence Interval for κ

CDF II Preliminary∫ L dt = 5.1 fb

-1

0.5
0
-0.5

68% C.L.
Best value

-1

168

170

172

174

176

Mtop [GeV/c2]
Figure 6.3: 68% C.L. intervals on κ as a function of Mtop . Solid line indicates center
values of κ.

Comparing our result with previous measurements, our result of κ = 0.042+0.563−0.562 isconsistent with other measurements. Sensitivity of our measurement is comparable to theresult reported by CDF II lepton plus jets channel of 0.72 ± 0.64 (stat) ± 0.26 (syst), andlittle bit inferior to DØ results of 0.66 ± 0.23 (stat + syst).
Currently, statistical uncertainty is dominating in our measurement. As data are accumulated, the sensitivity of the measurement is expected to increase. We will be able toanalyses data of 10 fb−1 in the near future. We consequently can reduce the uncertainty
as integrated luminosity increases.

89Chapter 7
Conclusion
We have measured the spin correlation coefficient κ of top and antitop quark in the beambasis at top quark pair production in proton-antiproton collisions at a center of mass energy of 1.96 TeV. Collected data corresponds to an integrated luminosity of 5.1 fb−1 between March 2002 and June 2009 with the CDF Detector at Fermilab Tevatron.
We selected tt̄ candidate events which contain two high transverse momentum charged
leptons, muon or electron, high missing ET and two jets, originating from top quark pair
production process. We found 334 tt̄ candidate events with an expected background of
86 ± 14 events.
We performed full kinematical reconstruction to obtain angular distribution of charged
leptons and b-jets. Then we made the templates of tt̄ signal as a function of κ as wellas background for each angular distribution using the simulation as well as data-based
background modeling.
Using the templates, we fit the angular distribution of data, and we obtained measuredκ = 0.042 on the assumption of Mtop =172.5 GeV/c2 . And this corresponds to−0.520 < κ < +0.605 (68%C.L.)

(7.1)

or
κ = 0.042+0.563−0.562 .

(7.2)

Also, we can not find any conclusive top mass dependence on the κ measurement.
This result indicates the null correlation rather than the existence of correlation. However, it does not mean the inconsistency with the Standard Model because this result has alarge uncertainty. There is no hint for anomalous production or decay of top quark pairs.
This result is similar to one of DØ result which obtained from differential angular
distribution template in dilepton channel of κ = 0.10 ± 0.45 (stat+syst). Also, it indicatesdifferent tendency from other results (Lepton + jets channel at CDF, Lepton + jets channelat DØ, Dilepton channel at DØ with matrix element method). These results indicate theexistence of spin correlation in tt̄ with 3 or more standard deviations.
Our analysis provides an independent measurement to others above. Besides, ourresult is currently limited by the statistical uncertainty. The total integrated luminosity ofdata collected with the CDF Detector at Fermilab Tevatron reached√about 10 fb−1 . Since
the statistical uncertainty is expected to be reduced by the factor 1/ 2 as the statistics of91

the data sample increasing√ N times, the sensitivity to the spin correlation coefficient will
be improved by about 2 maximally.

92

Appendix A
Spin States at tt̄ Pair Production
A.1 Production by q q̄ → tt̄
In highly relativistic interactions, particle helicity is same as its chirality. For a fermionof energy E ≫ m, we can use the following relations,
11 − γ 5 u = uL21
1 + γ 5 u = uR .
2

(A.1)
(A.2)

where u indicates spinor of fermion, uL and uR indicate spinor of right-handed fermionand spinor of left-handed fermion, respectively, and 21 (1 ± γ 5 ) project out the helicity
λ = ± 12 components of a spinor, respectively.
Also, for antifermion we can use the following relations,
11 − γ 5 v = vR2
1
1 + γ 5 v = vL .
2

(A.3)
(A.4)

where v indicates spinor of antifermion, vR and vL indicate spinor of right-handed antifermion and spinor of left-handed antifermion, respectively, and 21 (1 ± γ 5 ) project out
the helicity λ = ∓ 21 components of a spinor, respectively.
Strong interaction is mediated by gluon and it has vector structure. Therefore, quark
antiquark annihilation is described as the following equation.
v̄ ′ γ µ u = (v̄ ′ L + v̄ ′ R ) γ µ (uL + uR ) = v̄ ′ R γ µ uL + v̄ ′ L γ µ uR

(A.5)

The proof is as follows.
v̄L = vL† γ 0 = v †v̄R = vR† γ 0 = v †11
1 + γ 5 γ 0 = v̄ 1 − γ 5
2
2



1
1
1 − γ 5 γ 0 = v̄ 1 + γ 5
2
2
93

(A.6)
(A.7)

since γ 5 = γ 5† and γ 5 γ 0 = −γ 0 γ 5 . Hence,




1 ′
1
v̄ 1 − γ 5 γ µ 1 − γ 5 u = v̄ ′ γ µ 1 + γ 5 1 − γ 5 u = 0 (A.8)
4
4




1
1 ′
v̄ 1 + γ 5 γ µ 1 + γ 5 u = v̄ ′ γ µ 1 − γ 5 1 + γ 5 u = 0, (A.9)=
44v̄ ′ L γ µ uL =v̄ ′ R γ µ uR2

where we have used γ 5 γ µ = −γ µ γ 5 and (γ 5 ) = 1.
Equation A.5 shows that initial quark and antiquark can only have opposite helicity
states. Therefore, quark and antiquark annihilation process is in the following total angular momentum state,
J = 1, Jz = ±1
Top quark spin and antitop quark spin are 12 each and they move oppositely in the tt̄
rest frame, therefore top quark and antitop quark will have opposite helicity if they havethe same spin directions.
q q̄ annihilation process at Tevatron is high energy interaction but their velocity issmaller than velocity of light. So, small amount of J = 1, Jz = 0 state exists. q q̄ → tt̄
process is in following total angular momentum states,

J = 1, Jz = +1 :|+>|+>
1
J = 1, Jz = 0: √ [ | + > | − > +| − > | + >]2
J = 1, Jz = −1 :|−>|−>
where | + > and | − > indicate “spin up” state and “spin down” state, respectively.
The exact energy dependence of the spin states for beamline basis can be determinedby looking at the matrix elements for the production of like spin and unlike spin top quark
pairs via q q̄ → tt̄. These matrix elements are given in Equations A.10 and A.11 [66].
β 2 (1 − β 2 )sin2 θ∗(1 − βcosθ∗ )2
(1 − βcosθ∗ − β 2 sin2 θ∗ )2
4= 8g 1 +(1 − βcosθ∗ )2X|M(q q̄ → tt̄)|2 = 8g 4

(A.10)X|M(q q̄ → tt̄)|2(A.11)

↑↑,↓↓

↑↓,↓↑

In these equations, g is the QCD coupling strength, θ∗ is the top quark scattering angle
with respect to the proton beam in the tt̄ center of mass system, and β = v/c is top quark
velocity, which depends on the tt̄ pair production energy and is limited to be between 0
and 1.
Notice the factor β 2 (1−β 2 ) in the like-spin pair amplitude (Equation A.10). It supplies
suppression of this component for both small and large value of β. In contrast, the unlikespin pair amplitude (Equation A.11) contains a contribution which is independent of β.
94

A.2 Production by gg → tt̄
Because gluon is a massless particle, gluon spin state is in J = 1, Jz = ±1 state. Therefore, total angular momentum projection of gluon pair can be in Jz = +2, 0, or −2. Also,
total angular momentum of gluon pair can be in J = 2, 1, 0 state.
When tt̄ pairs are produced near kinematic threshold, pair of top and antitop quark do
not have orbital angular momentum. Since tt̄ pair can not be in Jz = ±2 state, gg → tt̄
process can not be in Jz = 2 state.
Therefore, gg → tt̄ process can be in J = 1, Jz = 0 or J = 0, Jz = 0 state.
rr
1
1
J = 1, Jz = 0 :| + 1, −1 > −| − 1, +1 >(A.12)
2
2
r
r
r
1
1
1
J = 0, Jz = 0 :| + 1, −1 > −|0, 0 > +| − 1, +1 > (A.13)
3
3
3
where, | Jz1 , Jz2 > indicate Jz of each gluon. Each gluon can not be in Jz = 0 (longitudinal) state. Therefore J = 0, Jz = 0 state is rewritten as follows.
r1[ | + 1, −1 > + | − 1, +1 > ](A.14)
J = 0, Jz = 0 :2
In addition, because gluon is a boson, when one swaps two gluons, the wavefunctionof the system has to be unchanged. Therefore, J = 1, Jz = 0 state is excluded and
J = 0, Jz = 0 state is only allowed, if the wavefunction of color part in two-gluon system
is a symmetric with exchange of two gluons.
The exact energy dependence can be seen by looking directly at the matrix elements
for gg → tt̄ production, as given in Equations A.15 and A.16 with the angular factor Y
defined in Equation A.17 [66].
X
16 4
g Y(1 − β 2 )|M(gg → tt̄)|2 =
3
↑↑,↓↓


∗
2
2
2 ∗
3
2 ∗ β − cosθ× 1 + β + β cos θ + 2β sin θ(A.15)(1 − βcosθ)2X
16 4 2 2 ∗g Yβ sin θ|M(gg → tt̄)|2 =
3
↑↓,↓↑


(1 − β 2 )2 + (1 − βcosθ∗ − β 2 sin2 θ∗ )2
× 1+(A.16)(1 − βcosθ∗ )2
7 + 9β 2 cos2 θ∗Y =(A.17)(1 − β 2 cos2 θ∗ )2Equation A.15 shows that the like-spin pairs coming from gluon-gluon fusion will be
suppressed for large β, while Equation A.16 tells us that unlike-spin pairs are disfavoredat low β.
The breakdown of the total tt̄ cross section into like- and unlike-spin pairs as a function of the tt̄ invariant mass is given in Figure A.1 for the Tevatron using the beamline
basis [66].
95Figure A.1: Differential cross section for tt̄ production as a function of the tt̄ invariant
mass,Mtt̄ , for the Tevatron with center of mass energy 2.0 TeV, decomposed into ↑↓ + ↓↑and ↑↑ + ↓↓ spins of the tt̄ pair using beamline basis for both q q̄ and gg components. [66]96Appendix B
P, CP of tt̄ System
Figure B.1 illustrates spin configurations of tt̄ system and their P and CP reversal states.
Since helicities are flipped under P reversal,
P (N↑↓ ) = N↓↑ ,P (N↓↓ ) = N↑↑ ,

(B.1)
(B.2)

where Nst s̄t indicates number of t, t̄ in st , s̄t state with respect to the quantization axis forthe top quark.
Since top and antitop are exchanged under C reversal,
CP (N↓↓ ) = C(N↑↑ ) = N↑↑ .

(B.3)

Therefore, if P is conserved at tt̄ production, N↑↓ = N↓↑ , or, vise versa, if P isviolated, there would be an asymmetry between N↑↓ and N↓↑ . If CP is conserved at tt̄
production, N↓↓ = N↑↑ , or if CP is violated, there would be an asymmetry between N↓↓
and N↑↑ .

t−

t

t

t−

t

t− CP−reversal t−

P−reversaltFigure B.1: Spin configurations of tt̄ system and their P and CP reversal states. In thecase above, P (N↑↓ ) = N↓↑ , P (N↓↓ ) = N↑↑ and CP (N↓↓ ) = C(N↑↑ ) = N↑↑ . So if P and
CP are conserved, N↑↓ = N↓↑ and N↓↓ = N↑↑ .
Next, we consider the effect of P reversal and CP reversal on cosθ (θ denotes theangle between the quantization axis and the flight direction of the decay particles fromthe top quark in the top quark rest frame). Under P reversal cosθ± changes to −cosθ±
since helicity states of top quark and antitop quark is flipped under P reversal. Under C97reversal cosθ± changes to −cosθ∓ since top and antitop are exchanged under C reversal.
Therefore, Under CP reversal, cosθ± changes to cosθ∓ .

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