学位論文

Measurement of the
Forward-Backward Asymmetries forb and c Quarks in e+e− Collisions at√
s between 130 GeV and 209 GeV
重心系エネルギー 130GeV から 209GeV の電子陽電子
衝突に於ける b 、c クォーク前後方非対称性の測定

平成 17 年 1 月博士（ 理学）申請
東京大学大学院理学系研究科
物理学専攻
南條 創

ii

Abstract
A measurement of forward-backward asymmetries for bb and cc production (AbFB and
AcFB ) in e+ e− annihilation is presented. The data were collected by the OPAL detector√
at the LEP e+ e− collider, at center-of-mass energies ( s) ranging from 130 GeV to 209
GeV. Two methods are used to evaluate the production angle of the primary quark andto identify the bb or cc events. One utilizes the hemisphere charge with a powerfulb-tagging algorithm based on lifetime, lepton and event-shape information. The otheruses the information of the leptons from semileptonic decays of heavy hadrons both toevaluate the primary-quark direction and to enhance events with primary b or c quarks.
Finally, the two independent methods are combined with a single likelihood function withwhich the AbFB and AcFB are measured simultaneously by fitting the distribution of theevaluated primary-quark direction. Results are presented at 11 different center-of-massenergies between 130 and 209 GeV, which are in good agreements with the standard√
model predictions. The same analysis is performed for the combined data with s from√
183 GeV to 209 GeV where the Ab,cs. The results of
FB is stable against
AbFB = 0.52 ± 0.09(stat.) ± 0.09(syst.)
AcFB = 0.59 ± 0.12(stat.) ± 0.09(syst.)
are obtained, where the luminosity weighted center-of-mass energy is 197 GeV. These
values are also in good agreements with the standard model predictions. For the data√with s above 189 GeV, this is the first measurement of the AbFB and AcFB . With theresults, new limits on the contact interaction and leptoquarks are obtained.

iii

Contents
1 Introduction

1

2 Theoretical Framework5

2.1

Electroweak Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Calculation of the AFB . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.1

Massless Tree-Level Calculation . . . . . . . . . . . . . . . . . . . .

8

2.2.2

Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Fermions’ Mass Corrections . . . . . . . . . . . . . . . . . . . . . . 10
One-Loop Electroweak Corrections . . . . . . . . . . . . . . . . . . 10
Photonic Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Final State QCD Corrections . . . . . . . . . . . . . . . . . . . . . 13

2.3

Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1

Perturbative QCD Phase . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2

Hadronization Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 15
String Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4

Neutral B meson mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Experimental Apparatus
3.1

3.2

19

The LEP Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1

Injector chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.2

LEP Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

The OPAL detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1

Central Tracking System . . . . . . . . . . . . . . . . . . . . . . . . 26
Silicon Microvertex Detector(SI) . . . . . . . . . . . . . . . . . . . . 26
Vertex Chamber (CV) . . . . . . . . . . . . . . . . . . . . . . . . . 27
Jet Chamber (CJ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Z chambers (CZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Solenoid Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2

Time-of-Flight System (TOF) . . . . . . . . . . . . . . . . . . . . . 30
Barrel Time-of-Flight detector (TB) . . . . . . . . . . . . . . . . . . 30

iv

CONTENTS
Tile Endcap Detector (TE) . . . . . . . . . . . . . . . . . . . . . . 30
3.2.3

Electromagnetic Calorimeters . . . . . . . . . . . . . . . . . . . . . 30
Barrel and Endcap Electromagnetic Presamplers (PB & PE) . . . . 31
Barrel and Endcap Electromagnetic Calorimeters (EB & EE) . . . . 31
Gamma Catcher (FE) . . . . . . . . . . . . . . . . . . . . . . . . . 31
Minimum ionizing particle (MIP) plug . . . . . . . . . . . . . . . . 32

3.2.4

Hadron Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Barrel and Endcap Hadron Calorimeters (HB & HE) . . . . . . . . 33
Pole-tip Hadron Calorimeters (HP) . . . . . . . . . . . . . . . . . . 33

3.2.5

Muon Detectors (MU) . . . . . . . . . . . . . . . . . . . . . . . . . 33
Barrel Muon Detector (MB) . . . . . . . . . . . . . . . . . . . . . . 33
Endcap Muon Detector (ME) . . . . . . . . . . . . . . . . . . . . . 34

3.2.6

Luminometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Forward Detectors (FD) . . . . . . . . . . . . . . . . . . . . . . . . 34
Silicon Tungsten Detector (SW) . . . . . . . . . . . . . . . . . . . . 34

3.2.7

Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Measurements of the Ab,c
FB

37

4.1

Overview of the Quark-Antiquark Pair Production at LEP2 . . . . . . . . 37

4.2

Definition of the AFB at LEP2 . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3

Procedure of the AFB Measurement . . . . . . . . . . . . . . . . . . . . . . 42

4.4

Data and Monte Carlo Samples . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5

4.4.1

Luminosity Measurement . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4.2

Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4.3

Monte Carlo Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Event Preselections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5.1

LEP2 Multi-Hadronic Event Selection . . . . . . . . . . . . . . . . . 47

4.5.2

s0 Determination and Non-Radiative Event Selection . . . . . . . . 50

4.5.3

4-fermion event rejection . . . . . . . . . . . . . . . . . . . . . . . . 52
W+ W− → qq`ν ` selection . . . . . . . . . . . . . . . . . . . . . . . 52
W+ W− → qqqq selection . . . . . . . . . . . . . . . . . . . . . . . . 53
Preselection . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Likelihood based selection . . . . . . . . . . . . . . . . . . . 54
4-fermion event rejection with W+ W− → qq`ν ` and W+ W− →qqqq selection . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.6

4.5.4

Acceptance Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5.5

Effect of the Event Preselection . . . . . . . . . . . . . . . . . . . . 57

Inclusive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

CONTENTS

v

4.6.1

Determination of the Quark-Hemisphere . . . . . . . . . . . . . . . 57

4.6.2

Inclusive B and C Tagging . . . . . . . . . . . . . . . . . . . . . . . 62
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Jet Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Secondary Vertex Reconstruction . . . . . . . . . . . . . . . . . . . 63
Lifetime Tag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Lepton Tag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Kinematics Tag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Combined Likelihood Tag . . . . . . . . . . . . . . . . . . . . . . . 67
Event Classification with b-tagging variables . . . . . . . . . . . . . 71

4.6.3

Unbinned Likelihood Fit . . . . . . . . . . . . . . . . . . . . . . . . 73
Probability Density Function . . . . . . . . . . . . . . . . . . . . . 75
Probability Density Function with Subsamples . . . . . . . . . . . . 76
Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.6.4
4.7

Corrections Applied to the Fitted Values of AFB . . . . . . . . . . . 79

Lepton Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.7.1

Prompt Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.7.2

Lepton Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Electron Identification . . . . . . . . . . . . . . . . . . . . . . . . . 81
Muon Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7.3

b and c Tagging and Flavor Separation . . . . . . . . . . . . . . . . 86
Construction of Artificial Neural Networks . . . . . . . . . . . . . . 86

4.7.4

Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.7.5

Unbinned Likelihood Fit . . . . . . . . . . . . . . . . . . . . . . . . 91
Making Subsamples . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Probability Density Function . . . . . . . . . . . . . . . . . . . . . 92

4.8

Combined Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.9

Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.9.1

Bottom and Charm Physics Modeling . . . . . . . . . . . . . . . . . 98

4.9.2

Effect of the ISR-FSR Interference . . . . . . . . . . . . . . . . . . 100

4.9.3

Final State QCD Corrections . . . . . . . . . . . . . . . . . . . . . 100

4.9.4

The Rates of Gluon Splitting to bb or cc . . . . . . . . . . . . . . . 101

4.9.5

Total K0 and Λ Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.9.6

Four-fermion Background . . . . . . . . . . . . . . . . . . . . . . . 101

4.9.7

Track Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.9.8

Event Pre-Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.9.9

Lepton Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.9.10 Modeling of Artificial Neural Network Inputs . . . . . . . . . . . . . 102

vi

CONTENTS
4.9.11 Monte Carlo Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.9.12 Background Asymmetries . . . . . . . . . . . . . . . . . . . . . . . 102
4.9.13 Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.9.14 Cross-Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.10 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Conclusion

123

A The Standard Model

125

A.1 Elementary Particles in the Standard Model . . . . . . . . . . . . . . . . . 125
A.1.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.1.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.1.3 Chirality and Quark Mixing . . . . . . . . . . . . . . . . . . . . . . 126
A.2 Interactions (Gauge Theory) . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2.1 Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2.2 Quantum Electrodynamics(QED) . . . . . . . . . . . . . . . . . . . 129
A.2.3 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.3 Electroweak Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.3.1 Massless Electroweak Theory . . . . . . . . . . . . . . . . . . . . . 132
A.3.2 Gauge Boson Masses and Higgs Mechanism . . . . . . . . . . . . . 136
A.3.3 Fermion Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.4 The Standard Model Parameters . . . . . . . . . . . . . . . . . . . . . . . 139
B Massless Tree-Level AFB Calculation

141

C Jet Clustering

145

C.1 Jet-Clustering Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.1.1 JADE algorithm with the E0 recombination scheme . . . . . . . . . 145
C.1.2 Durham algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.1.3 Cone algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.2 Evaluation of Primary-Quark Direction . . . . . . . . . . . . . . . . . . . . 147
C.3 Evaluation of Primary-Quark Charge . . . . . . . . . . . . . . . . . . . . . 148
C.4 b-Tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
D Artificial Neural Network (ANN)155

Acknowledgment

169

1

Chapter 1
Introduction
At the beginning of the 20th century, only the electron and photon had been discovered
among the elementary particles known today and the development of the quantum dynamics had just been started. Since then, great progress has been made in particle physics
with the discoveries of elementary particles and constructions of theories to describe theirinteractions.
In the former part of the century, the relativistic quantum theory to describe theelectromagnetic interaction, the quantum electrodynamics (QED), was constructed successfully. Then the progress in understanding the weak interaction followed. In the middleof the century, parity non-conservation in the weak interaction was discovered in beta decay [1], which is one of the most significant characteristics of the weak interaction. This
made large amount of contribution to determine the structure of the weak interaction as
V − A type which means the Lagrangian contains only the vector coupling (v) and axial
vector coupling (a) as
ψe γ µ (v − aγ 5 )ψνe .

(1.1)

Then the possibility of the intermediating W± bosons in the charged current interaction
was considered [2]. Based on the insight about the local gauge invariance [3], the electroweak theory to unify the weak and electromagnetic interactions was constructed [4–6].
The electroweak theory suggested the existence of the neutral current interaction with theintermediating neutral Z boson. Then many experiments to test the electroweak theory
were performed, finding the neutral current interactions, and the W± and Z bosons .
The neutral current interaction was indicated in the elastic νµ e− scattering [7] and foundin the muon-less neutrino-induced inelastic interactions [8]. At the SPS in proton andanti-proton collisions, the W± [9, 10] and Z [11, 12] bosons were discovered at the masses
expected in the framework of the electroweak theory with the experimental results atlower energies. The studies of the neutral current interaction with e+ e− colliders have√been performed at PEP at the center-of-mass energy ( s) of 29 GeV, PETRA at the√√√s from 14 to 46 GeV, TRISTAN at the s from 50 to 64 GeV, SLC at the s of 91.3

2

CHAPTER 1. INTRODUCTION

GeV and LEP at the√s from 89 GeV to 209 GeV. With the great success in the agree-

ment between these experimental results and the theoretical predictions, the electroweak
theory has been established. On the other hand, the strong interaction is described by
the quantum chromodynamics (QCD). The electroweak theory and the QCD compose
the standard model, which describes phenomena in particle physics successfully. Further
efforts have been and will be made to construct a theory to unify the electroweak theory
and QCD and to include the gravitational interaction.
In the above-mentioned e+ e− experiments, the reaction from the electron-positronannihilation to a fermion and anti-fermion pair with intermediating γ or Z have been
analyzed. The Feynman diagram of the reaction1 is shown in Fig. 1.1.





Figure 1.1: The tree-level Feynman diagram for e+ e− → ff.
.
The forward-backward asymmetry (AFB ) is one of the most important observables ofthe fermion-pair processes, since it is a sensitive probe of the electroweak theory, which
is mainly related to the parity-violating nature of the weak interaction and the γ − Zinterference at the propagator. The AFB represents the difference between the crosssection for the fermion emitted to the electron direction and that for the fermion emitted
to the positron direction defined as
R 1 ¡ dσ ¢R 0 ¡ dσ ¢
d
cos
θ
−
d cos θθ−1 d cos θ
AFB = R01 ¡ d cos,¢
R 0 ¡ dσ ¢
dσ
d
cos
θ
+
d
cos
θ
0 d cos θ−1 d cos θ(1.2)

where θ is angle of the fermion direction measured with respect to the electron beam
direction. The AFB for each lepton-pair final state can be measured. As for the quarkpair final state, the AFB for b or c quark is especially important, since bb or cc events canbe identified efficiently.
1

The final stats of e+ e− and νe νe have another t-channel diagram.

3
The AFB for bottom quarks (AbFB ) and for charm quarks (AcFB ) have been measured
at PEP [13–19], PETRA [20–31], TRISTAN [32–34], SLC [35] and LEP [35] as shown in
Fig. 1.2. In Fig. 1.2, the standard model predictions are given by lines2 .

c
AFB

b
AFB

2

2

1.5

1.5

1

1

0.5

0.5

0

SM prediction

0

SM prediction

PEP combined

PEP combined

PETRA combined

PETRA combined

-0.5

-0.5

-1

-1
TRISTAN combined-1.5

TRISTAN combined-1.5
LEP I+SLD combined-2

LEP I+SLD combined-2

20 40 60 80 100 120 140 160 180 200 220

LEP II (OPAL) 20 40 60 80 100 120 140 160 180 200 220

s[GeV]

s[GeV]LEP II (OPAL)

Figure 1.2: The results of the AbFB and AcFB measurements in e+ e− collisions at the PEP,PETRA, TRISTAN, SLC and LEP are shown, where the standard model predictions are
given by lines.
√s below the Z resonance, the pure Z cross-section increases as the s increases.
√
It becomes comparable to the pure γ cross-section at the s of approximately 60 GeV√
for bottom quarks and at the s of approximately 70 GeV for charm quarks, where theAt√

AFB is minimum due to the large γ–Z interference.
At the Z resonance, the fermion-pair production is dominated by the pure Z decays,
where the AFB is sensitive to the effective electroweak mixing angle3 for both the initiale
fstate electron (θeff) and the final-state fermion (θeff). Since the dependence of the AFB onbce
the θeffor θeff
is small compared to the θeff
, the measurement of the Ab,c
FB has given one ofethe most precise values of θeff
[35].

Above the Z peak, the pure Z cross-section decreases and becomes comparable tothe pure γ cross-section. Hence the contribution of the γ–Z interference becomes larger.
The AFB is sensitive to new physics beyond the standard model. Possible new physics
beyond the standard model, for example, existence of the additional heavy neutral bosons(Z0 ), leptoquarks or R-parity violating squarks which would mediate in addition to γ/Z,b,ccould alter the standard model value of the Ab,c
FB [36–39]. The AFB measurement above

Z resonance is important for both testing the standard model and searching new physics√
The rapid change in the standard model prediction of AFB around s of 110 GeV is due to the event
selection described in Section 4.5.2.
3
The effective electroweak mixing angle (θeff ) is related to the electroweak mixing angle (θW ) throughthe higher order corrections as described in Section 2.2.2. The θW is one of the parameters in theelectroweak theory defined in Eq. A.33.
2

4

CHAPTER 1. INTRODUCTION

beyond the standard model.

√
In this thesis, a measurement of AbFB and AcFB at s above the Z resonance is presented.
The data were collected by the OPAL detector at the LEP e+ e− collider, at center-ofmass energies ranging from 130 GeV to 209 GeV, which is the highest energy for thee+ e− collisions. In this analysis, two methods are used to evaluate the production angleof the primary quark and to identify the bb or cc events. One utilizes the hemispherecharge4 to evaluate the emitting direction of the primary quark together with the thrust
axis5 , where a powerful b-tagging algorithm based on lifetime, lepton and event-shapeinformation is used to enhance events with primary b and c quarks. The other utilizes
the information of the leptons from semileptonic decays of heavy hadrons together withthe thrust axis, both to determine the primary-quark direction and to enrich events withprimary b and c quarks. Finally, two independent methods are combined with a single
likelihood function with which the AbFB and AcFB are measured simultaneously by fitting
the distribution of the evaluated primary-quark direction. In order to compensate for the√small statistics at s above the Z resonance, powerful b or c tagging algorithms are used
for the both analyses, which are more effective than these used in the Ab,c
FB measurements
at the Z resonance [40, 41].
While the basic techniques are similar to those adopted in the previous OPAL mea√
surement [42] at s from 130 GeV to 189 GeV, more powerful b-tagging algorithm andmore sophisticated fitting method are employed in this analysis. The measurement of the√Ab,cs beyond 189 GeV reported here is the first measurement.
FB at theThis thesis is organized as follows. In Chapter 2, the theoretical background concerning
this analysis is described. In Chapter 3, descriptions of the LEP accelerator and the
OPAL detector are given. The AFB measurement is discussed in Chapter 4, where adescription about systematic errors is given and a discussion about the result is followed.
The conclusion is given in Chapter 5.

4
5

A description about the hemisphere charge is given in Section 4.3
A description about the thrust axis is given in Section 4.3

5Chapter 2
Theoretical Framework
The fermion and anti-fermion pair (ff) production in e+ e− annihilation is mainly describedby the neutral current part of the electroweak theory, which is briefly described at firstin this section. Then the calculation of the Ab,c
FB with the electroweak theory is given,
where some higher order effects and QCD effects are also described. Whereas the quark
and anti-quark (qq) production is described by the electroweak theory, the quarks are
only detected at the detector as colorless hadrons. The process to connect the quark andparticles at the detector is termed fragmentation, which is also described in this chapter.
Finally, the neutral B meson mixing (B0 − B0 mixing) is described. Since the charge ofthe primary quark is evaluated with the final state particles, the B0 − B0 mixing dilutes
the charge identification in the AFB measurement.

2.1

Electroweak Theory

The electroweak theory is briefly described here. More detailed description is given inchapter A. The fermions (leptons and quarks) are classified as shown in Table. 2.1 with 3
independent quantum numbers, which are different between their chiral components (Lor R). The quantum numbers are the electric charge1 (Q), the third component of the
weak isospin (T3 ), and the color charge (C).

1

The electric charge is always measured in the unit of the electron charge magnitude(e = 1.602176462(63) × 10−19 C).

6CHAPTER 2. THEORETICAL FRAMEWORKTable 2.1: Quantum numbers for fermions where Q is the electric charge, T3 denotes thethird component of the weak isospin, and C is the color charge.
Fermions

Quantum Numbers
Q
T3
C

Leptons
µ

νe
e−

¶

µ
L

eR

¶

νµ
µ−

µ
L

µR

¶

ντ
τ−

L

τR

0
−1

+ 21
− 21

0

−1

0

0

+ 23
− 31

+ 12
− 21

(R, G, B)

+ 32
− 13

0
0

(R,G,B)(R,G,B)

Quarks
µ

¶

u
d0

µ
L

uR
dR

c
s0

¶

µ
L

cR
sR

¶

t
b0

L

tR
bR

The interactions between the fermions are described as exchanging of a boson between
them, where such bosons are called as gauge bosons. The color charge is the source of thestrong interactions where the gluons act as the gauge boson, which are described by the
QCD. The electric charge and the third component of the weak isospin are the sources ofthe electroweak force where the γ and, W and Z bosons act as the gauge bosons, which
is described by the electroweak theory. In the electroweak theory, the neutral current
interaction where the γ or Z boson are exchanged is described with two Lagrangians as,
LEM

X

= e

f =d,u,s,c,b,t,e,νe ,µ,νµ ,τ,ντ

LZNC

=

e

£

¤
f γ µ Qf f Aµ ,X£

2 sin θW cos θW f =d,u,s,c,b,t,e,ν ,µ,ν ,τ,ν
e

µ

©
ª
¤
f γ µ vf − af γ 5 f Zµ .

(2.1)
(2.2)

τ

In Eq. 2.1, 2.2, the LEM corresponds to the γ exchange and the LZNC corresponds to the Zboson exchange where f means a fermion field, Aµ and Zµ mean the γ and Z boson fields
respectively. The θW is electroweak mixing angle (“Weinberg angle”) which is one of theparameters in the theory as defined in Eq. A.33. The vf is called vector coupling and af

2.1. ELECTROWEAK THEORY

7

is called axial vector coupling which are defined as,vf= T3f (1 − 4 |Qf | sin2 θW ),

(2.3)

af

= T3f ,

(2.4)

and their relations with sin2 θW are shown in Table. 2.2.
Table 2.2: The fermion’s vector and axial vector couplings (v and a respectively).
Fermionsv

aneutrinos
leptons(νe , νµ , ντ )(e, µ, τ )

+ 12
− 12

+2 sin2 θW

+ 12
− 12

up-type quarks
down-type quarks(u, c, t)(d, s, b)

+ 12
− 12

− 43 sin2 θW+ 23 sin2 θW

+ 12
− 12

8

CHAPTER 2. THEORETICAL FRAMEWORK

2.2

Calculation of the AFB

The theoretical calculation of the AFB with the electroweak theory is given in this section.
At first, the tree-level calculation under massless approximation of the fermions is given.
Then the description of some corrections to improve the accuracy is followed.

2.2.1

Massless Tree-Level Calculation

A schematic view of the e+ e− → ff reaction is shown in Fig. 2.1, where the angle θ is theangle of the fermion (f) direction measured from the e− direction.

f
θ

-

e

+

e
f

Figure 2.1: A schematic view of the e+ e− → ff reaction.

The differential cross section in the massless limit for the process e+ e− → ff except
for the final stats of e+ e− and νe νe can be calculated with the electroweak theory [43] as¤dσ
Nc α 2 £=G(s)(1 + cos2 θ) + H(s) cos θ ,
dΩ
4s

(2.5)

2

e
where α is defined as α = 4π
. The G(s) and H(s) are defined as

G(s)

≡ Q2f

+ |χ(s)|2 (ve2 + a2e )(vf2 + a2f ) − 2 Qf ve vf Re{χ(s)},

(2.6)

H(s)

≡ |{z}
0+8|χ(s)|2 ve ae vf af|{z
Z exchange

(2.7)

γ

−4Qf ae af Re{χ(s)},} |{z}γ − Z interference

where χ(s) is defined as
χ(s) ≡

³s´
e2

g2
1
.
2
2
4 cos θW s − mZ + imZ ΓZ(2.8)

For both Eq. 2.6 and Eq. 2.7, the first term corresponds to the γ exchange, the second
corresponds to the Z exchange and the third corresponds to the γ − Z interference. The
total cross section σ(s) can be written as
σ(s) =

4Nc πα2
G(s),
3s

(2.9)

2.2. CALCULATION OF THE AFB

9and the forward-backward asymmetry (AFB ) can be written as,
AFB =3 H(s),
8 G(s)(2.10)and the AFB is related to the angular distribution as,dσ
8∝ 1 + cos2 θ + AFB cos θ.
(2.11)
d cos θ3
In Eq. 2.6, 2.7, contributions of γ exchange, Z exchange and γ − Z interference are
separately written. Those for G(s) are represented with Gγ (s), GZ (s) and Gγ−Z (s) and√those for H(s) are represented with Hγ (s), HZ (s) and Hγ−Z (s), respectively. The s

300

G(s) for cG(s) for b

dependence of these functions and AFB for b and c quarks are shown in Fig. 2.2.

Gtotal
Gγ
GZ
GZ-γ

250
200
150
100
50

250

Gtotal
Gγ
GZ
GZ-γ

200
150
100
50
0

0

0.8

0.6

0.6

0.4

0.4
0.2
0.2
0

0

-0.2
-0.4

-0.2
0

20

40

60

80

100

120

140

160

180

200

-0.4

220

0

20

40

60

80

100

120

140

160

180

100

Htotal
Hγ
HZ
HZ-γ

80
60
40

Htotal
Hγ
HZ
HZ-γ

60
50
40
30
20
10
0

0

-10

15
10
5
0
-5
-10
0

20

40

60

80

100

120

140

160

180

200

-15

220

0

20

40

60

80

100

120

140

160

180

s[GeV]

0.8
0.6

220

0.8
0.6

0.4

0.4

0.2

0.2

-0

200

s[GeV]

c
AFB

b
AFB

220

70

20

8
6
4
2
0
-2
-4
-6
-8
-10

200

s[GeV]

H(s) for c

H(s) for bs[GeV]-0-0.2

AFB(Z)-0.2

AFB(Z)-0.4

AFB(Z+γ )-0.4

AFB(Z+γ )AFB(total)-0.6

AFB(total)

-0.6

-0.8
0 20 40 60 80 100 120 140 160 180 200 220-0.8
0 20 40 60 80 100 120 140 160 180 200 220

s[GeV]

s[GeV]

Figure 2.2: The√s dependence of G(s) and H(s) are shown for b and c quarks.

10

CHAPTER 2. THEORETICAL FRAMEWORK
Large effect of the γ − Z interference is appeared in the AFB around√s where Gγ (s)equals GZ (s). Based on the fact the cross section is proportional to G(s)/s as seen in√
Eq. 2.9, the γ − Z interference is naturally understood as follows. For s below 40 GeV, γ
makes almost all the contribution to the propagator and the interference is small. When√s increases, the contribution of Z increases and becomes comparable to the contribution√of γ at s around 60 GeV for the down-type quarks and around 70 GeV for the up-type√
quarks, where the interference becomes maximum. For s at mZ , the contribution of Z√
is maximum due to the resonance and the interference term vanishes. As the s increasebeyond mZ , the contribution of Z becomes smaller and the interference appears again.
As for the down-type quarks, the contribution of Z becomes close to that of γ but remain√
larger than that of γ. Therefore the AFB increases as the s increases. As to the up-type√
quarks, the contribution of Z becomes smaller than that of γ for s beyond approximately√
140 GeV. Therefore, AcFB increases while s is smaller than approximately 140 GeV. For√√s beyond 140 GeV, it decreases as s increases.

2.2.2

Corrections

There are some corrections to the massless tree level calculation to improve the accuracy. Such corrections can be broken down to four categories, fermions’ mass corrections,electroweak corrections, photonic corrections and QCD corrections.

Fermions’ Mass Corrections
The fermions’ mass can be included in the theoretical calculation as in [43]. The contri√
bution to the AbFB is below 0.5% at s more than 80 GeV and that to the AcFB is below
0.2%, which are negligible in this analysis.

One-Loop Electroweak Corrections
The one-loop electroweak corrections can be broken down into 3 types. One is the casein which a virtual particle loop is inserted in the the propagator which corresponds tothe bosonic self-energy as shown in Fig. 2.3. The second is vertex correction as shown in
Fig. 2.4. The third is the contribution of the box diagram of W or Z bosons as shown in
Fig. 2.5. These corrections can be reflected to the tree-level cross section by introducing
the running α (α(s)), replacing the weak mixing angle by the effective weak mixing angle
f(sin θeff) and re-defining the vector and axial vector couplings. These corrections areproperly treated in some programs, such as Zfitter [44].

2.2. CALCULATION OF THE AFB

11

 
















Figure 2.3: Examples of the diagrams contributing to the bosonic self-energy type electroweak corrections.


















 



 



Figure 2.4: Examples of the diagrams contributing to the vertex-type electroweak corrections.


























˜






























Figure 2.5: Examples of diagrams contributing to the box-type electroweak corrections.

12

CHAPTER 2. THEORETICAL FRAMEWORK

Photonic Corrections
When the final-state fermion is a charged particle, there are both vertex-type and box-type
photonic corrections as shown in Fig. 2.6.


















Figure 2.6: Examples of diagrams contributing to the photonic corrections.

There are also other types of photonic corrections, initial state radiation (ISR) andfinal state radiation (FSR), as shown in Fig. 2.7.












Figure 2.7: Examples of initial state radiation and final state radiation.

The effective center-of-mass energy,√s0 , is defined as the invariant mass of the propa-

gator. The ISR reduce the s0 from s and boost the γ/Z against the photons corresponding
to the ISR, which changes the kinematics and makes the |AFB | smaller.
There is another effect due to the ISR and FSR as follows. Both the ISR and FSR
processes can be written as e+ e− → ff + nγ and are in principle indistinguishable in the
final state. Hence the two processes interfere. Since this ISR-FSR interference (IFI) makes
the definition of the s0 invalid, the IFI should be neglected. The effect in neglecting the
ISR-FSR interference can be evaluated with some programs such as Zfitter or KK2f [45]p√
and is found to be less than 2% with a cut of s0 /s > 0.85 for the s above the Z
resonance.

2.3. FRAGMENTATION

13

Final State QCD Corrections
Hard gluon emissions off final state quarks as shown in Fig. 2.8 distort the kinematics.


























Figure 2.8: Examples of diagrams contributing to the QCD corrections.

b,cSuch QCD effect to the AFB
have been calculated to second order of the strong coupling
constant (αs ) in [46]. As to the first-order correction of αs , corrected asymmetry (AFB QCD )can be written with uncorrected asymmetry (AFB noQCD ) as,
AFB

QCD

= AFB

noQCD

³

αs ´
1−
,
π

(2.12)

where the b and c quark mass is neglected. According to [47], the second-order correction
in [46] and the effect of the non-zero quark mass are considered and the decreases of the
AFB due to the QCD correction are evaluated as 3.30±0.37% and 4.18±0.69% for b and cquarks respectively. The Zfitter can also treat the QCD correction, which gives similar√√
results at the Z resonance and small s dependence of the corrections at s above the Z
resonance. Such QCD corrections of about 3% for b quarks and approximately 4% for cquarks are used in this thesis.

2.3

Fragmentation

The process of quark-antiquark pair production, e+ e− → qq, is described by the electroweak theory as seen in the previous section. A process which transforms quarkantiquark pair into colorless hadrons have to be followed due to the color–confinement
nature of the QCD. This process is termed “fragmentation”. In the AFB measurement,
the direction of primary quarks are evaluated with the final state particles in the detector after the fragmentation process. Hence the fragmentation process is important. The
fragmentation process can be broken down into 3 phases, a perturbative QCD phase, ahadronization phase and a decay phase, as shown in Fig. 2.9.

14

CHAPTER 2. THEORETICAL FRAMEWORK

fragmentation
Non-perturbative QCD(hadronization)
Electroweak

Perturbative QCD

Decay

+

e

q

γ

p

Λ0

π−
0

K

+

D

g

+

e
ν

Z/γ

q

+

B

0

D+
ρ

-

e

cut-off energy scale (typically around 1GeV)

Figure 2.9: The process of quark-antiquark pair production, e+ e− → qq and the subsequent fragmentation process.

1. Perturbative QCD phase
The radiation of a gluon form a quark and subsequent conversion of gluon to further
quark-antiquark pair or gluon pair occur. This multiplicative process is called as“parton shower”, which can be calculable with the perturbative QCD while theenergy scale of such partons are high enough.
2. Hadronization phase
During the later stage of the cascade, the energy scale of the partons becomes
sufficiently small (and αs becomes sufficiently large) and the perturbative QCD
calculation becomes invalid. The cut-off energy scale is typically around 1 GeV. The
remaining shower evolution and subsequent formation of colorless hadrons have to
be simulated with some phenomenological models. This non-perturbative process
is termed “hadronization”. At LEP, due to the high center-of-mass energy, many

2.3. FRAGMENTATION

15

hadrons are produced along an orientation of the momentum vector of the initialquark, which is called a “jet”.
3. Decay phase
Unstable hadrons decay into stabler hadrons and they can be observed by detectors.

2.3.1

Perturbative QCD Phase

There are two methods to simulate the perturbative process. One is the matrix-element
method, in which Feynman diagrams are calculated, order by order. Although this is a correct approach in principle, the applicability is limited by a large difficulty in higher-order
calculations which are important to keep the accuracy. The second is the parton-shower
method, where multi-parton processes are treated as series of a branching of one parton
into two, such as q → qg, g → gg and g → qq. The calculations of the probabilities ofthese branchings are usually performed in the leading logarithm approximation (LLA),
where only the leading terms in the perturbative expansions are kept. This method is employed in the event generator, Pythia [48], which was used to simulate the fragmentation
process of Monte-Carlo events used in this analysis.

2.3.2

Hadronization PhaseAt small energy scale where perturbative calculations are not valid, phenomenologicalmodels are used to simulate the hadronization process. Among some models, the string
fragmentation model is used in the Pythia.
String Fragmentation
The string fragmentation [49] is based on the idea that oppositely colored quark andantiquark are confined by a potential energy due to the color field between them. The
potential energy linearly rises as a distance between the pair increases. When such quarks,
q and q, move apart, the color field between them can be viewed as a narrow flux tube of
uniform energy density, which is stretched like a string between the partons. As the quarks
move further apart, the potential energy of the string rises linearly with the distance until
the string finally breaks up in a new q0 q0 pair. Thus, two new colorless systems qq0 andq0 q are formed. As long as the invariant mass of the produced strings is large enough,further splittings may occur and generate further q00 q00 pairs. Finally, each colorless quarkantiquark pair forms a hadron.
The mechanism of such new q0 q0 generation can be interpreted as quantum mechanicaltunneling. Such view implies a suppression of heavy-quark production. In the Pythia,
the ratio of the new q0 q0 generation rate for u, d, s and c quarks are set as 1 : 1 : 0.3 : 10−11 .
The tunneling mechanism can also be used to explain the production of baryons.

16

CHAPTER 2. THEORETICAL FRAMEWORKFurthermore, a flavor-independent Gaussian spectrum for the transverse momentum(pt ) of q0 q0 pair is naturally introduced with the view. Since the string is assumed to haveno transverse excitations, this pt is locally compensated between the q0 and q0 .
On the other hand, the sharing of the longitudinal momentum is given by some arbitrary probability distribution f (z), where z is the fraction of the sum of energy (E) andlongitudinal momentum (pk ) taken by the newly generated hadron (qq0 ) from original
quark (q). For the adjacent quarks, q00 and q00 , the same procedure is repeated in termsof the longitudinal momentum of q0 . Then the z can be written as(E + pk )qq0 = z · (E + pk )q .

(2.13)Such function, f (z), is called as “fragmentation function”. Among several fragmentation
functions proposed, “Lund symmetric function” and “Peterson function” are frequently
used. Further descriptions are given about the two functions, which was used in the
Pythia to simulate the fragmentation process of the Monte-Carlo events in this thesis.
• Lund symmetric function
If the equivalence in the choice of the starting the iterative procedure from the quarkend (q) or antiquark end (q) is assumed, the shape of the fragmentation function isessentially fixed as,
1
f (z) ∝ z aα
z

µ

1−z
z

¶ aβ

µ

bm2
exp − tz¶,

(2.14)

where m2t is a transverse mass (m2 + p2t ) of the quark whose mass is m. There are
two kinds of parameters, a and b. The parameter a is flavor dependent, which isindicated with the index α corresponding to the ‘old’ flavor in the iteration process,and β to the ‘new’ flavor. Usually, the function is simplified by neglecting the flavor
dependence of a as,
f (z) ∝ z −1 (1 − z)a exp(−bm2t /z).

(2.15)• Peterson function
Charm and bottom data clearly indicate the need for a harder fragmentation function for heavy flavors. The best known of these is the Peterson/SLAC formula [50]f (z) ∝¡1²Q
z 1 − z1 − 1−z¢2 ,(2.16)

where ²Q is a free parameter, expected to scale between flavors like ²Q ∝ 1/m2Q .
The examples of the fragmentation functions are shown in Fig. 2.10.

Fragmentation Function f(z)2.4. NEUTRAL B MESON MIXING

17

10
9
8

Lund symm.

7(mt [GeV])6

ρ+ (0.76)

5

π+ (0.14)

4

K 0/+
1 (1.29)3

K 0/+ (0.49)

2Petersion

100c : ∈Q =0.031

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

b : ∈Q =0.0038

zFigure 2.10: Examples of fragmentation functions. The m2t in the Lund symmetric function is set to m2 .

In another aspect of the string fragmentation, the gluon is treated as a kink on thestring, carrying energy and momentum. For a qqg event, a string is stretched from the q
end via the g to the q.

2.4

Neutral B meson mixing

The neutral B meson mixing (B0 − B0 mixing) contributes to the AFB measurement
diluting the charge identification, since the B0 − B0 mixing inverts the relation between
the charge of the primary quark and the charges of the final state particles. The B0 − B0
mixing is predicted in the electroweak theory via box diagrams as shown in Fig. 2.11,
where the neutral B meson is represented as Bq0 (q = d, s).

18

CHAPTER 2. THEORETICAL FRAMEWORK       

             








      

Figure 2.11: Main diagrams causing B0 − B0 mixing.
The mass eigenstates are different from the flavor eigenstates, |Bq0 i and |Bq0 i. There
are a mass difference (∆mq ) and a total decay width difference (∆Γq ) between the twomass eigenstates. The time-dependent probability in which the |Bq0 i remains unchanged
is termed Pq+ (t) and the probability in which the |Bq0 i changes to |Bq0 i is termed Pq− (t).
If the CP violation is neglected, the Pq+ (t) and Pq− (t) can be written with the mean of
the total decay width, Γq , as
Pq± (t) =

·
µ
¶
¸
¡
¢
1
∆Γq
exp −Γq t cosht ± cos (∆mq t) .
22

(2.17)

Since ∆Γq is negligible compared to Γq , the above equation can be written as
Pq± (t) ≈¡¢1
exp −Γq t [1 ± cos (∆mq t)] ,2(2.18)

where the contributions of both the decay and the mixing are clearly seen. The timeintegrated mixing probability (mixing parameter), χq , is defined as,
R∞ −Pq (t) dt
R∞.
χq ≡ R ∞ + 0
Pq (t) dt + 0 Pq− (t) dt
0(2.19)At LEP, since both Bd0 and Bs0 can be generated, the average mixing parameter, χ, isused. It is defined as,χ ≡ fd0 χd + fs0 χs ,(2.20)

where fd0 and fs0 are the fractions of Bd0 and Bs0 respectively in a sample of semileptonic
b-hadron decays. The χ was measured to be 0.1257 ± 0.0042 [51]. The similar value
is used in the Monte Carlo samples and the above-mentioned value is used to evaluate
systematic uncertainty from the B0 − B0 mixing.

19

Chapter 3
Experimental Apparatus
Descriptions about LEP Collider and the OPAL detector are given in this chapter. The
events generated in the e+ e− collisions in the LEP Collider were recored with the OPAL
detector.

3.1

The LEP Collider

The Large Electron-Positron collider (LEP) [52, 53] at CERN (the European Laboratory
for Particle Physics) was an e+ e− storage ring with a circumference of approximately 27
km. The ring was built between 50 and 175 m below the surface. Electrons and positrons
collided at four places in the ring, where large detectors, ALEPH, DELPHI, L3 and OPAL
were located. The first beam at the LEP was injected in 1989 and the LEP was terminatedin 2000.
√
The LEP was operated at s ≈ mZ ≈ 91GeV until 1995 (LEP I phase). In the latterpart of 1995, the energy was raised to 130 GeV (LEP 1.5 phase). Then the center-of-massenergy was raised beyond 130 GeV and reached to a little under 209 GeV in the year 2000
(LEP II phase). The integrated luminosities versus the number of weeks between 1991
and 2000 are shown in Fig. 3.1.
Electrons and positrons were accelerated through the injector chains and LEP ring
step by step. The injector chains and the LEP accelerator are shown in Fig. 3.2.

20

CHAPTER 3. EXPERIMENTAL APPARATUS

Integrated Luminosity pb-1

OPAL Online Data-Taking Statistics
225

1991
1992
1993
1994
1995
1996
1997
1998
1999

200

175

150

125

2000
100

75

50

25

0

15

20

25

30

35

40

45

Week number

Figure 3.1: The LEP integrated luminosity

3.1. THE LEP COLLIDER

Figure 3.2: The LEP accelerator complex.

21

22

3.1.1

CHAPTER 3. EXPERIMENTAL APPARATUS

Injector chain

Electrons were produced by thermionic emission from a heated filament. Their energies
were raised to 200 MeV in passing through the LEP Injector LINAC (LIL). Some of theseelectrons were directed at a tungsten target and electron-positron pair production was
followed. These pairs were separated magnetically, and then accelerated further by the
LIL to 600 MeV. They were transferred to the Electron-Positron Accumulator (EPA),a small storage ring, 126 m in circumference. They were accumulated in the ring andwere transferred to the Proton Synchrotron (PS), 630 m in circumference. They wereaccelerated to 3.5 GeV there, then transferred to the Super Proton Synchrotron (SPS),
6.9 km in circumference and accelerated to about 22 GeV. At this stage, the beams wereready for injection into LEP.

3.1.2

LEP Accelerator

The LEP ring comprised eight curved sections (octants) equipped with bending magnets
and eight straight sections. The acceleration of the beams was performed in the straightsections and radio-frequency (RF) cavities were used to accelerate the beams. The beams
collide at interaction points situated at the center of four of these eight straight sections,where four large detector apparatus, ALEPH [52], DELPHI [54], L3 [55] and OPAL [56],were located. On either side of each experiment, strong superconducting quadrupole were
employed to reduce the transverse beam dimensions at the interaction points to about
5µm and 200µm in the vertical and horizontal planes respectively.
The bending of the beams in curved sections caused energy losses of the beams due
to synchrotron radiation. Additional accelerations with the RF cavities were performed
in order to compensate for the energy losses. As for the beam with the energy Eb , theenergy loss, δEb , due to synchrotron radiation par each revolution is given as,
4πe2 β 3 γ 4
,
δEb =
3R

(3.1)

where β is the speed of the beams divided by the speed of light, γ is Eb /me c2 (me is theelectron mass) and R is the radius of the curved path.
Four bunches of electrons were made to circulate anti-clockwise, and four bunches of
positrons clockwise, around the LEP ring. The collisions were occurred at each experimental site every 22 µs.

3.2. THE OPAL DETECTOR

3.2

23

The OPAL detector

The OPAL(Omni-Purpose Apparatus for LEP) detector [56] was located at the LEP at
CERN. It was a multipurpose apparatus designed to study all types of events occurring ine+ e− collisions, which enables accurate event reconstructions over an acceptance of nearlyfull solid angle. The OPAL detector is shown in Fig. 3.3 with its coordinate system1 .

Electromagnetic calorimeters
Muon detectors
Hadron calorimeters and return yoke

Jet chamber

Vertex detector

Microvertex detector

y
θ
z

ϕ

Z chambers
Solenoid and Pressure Vesselx
Presamplers
Original forward detector

Time of flight detector

Silicon-tungsten forward detector

Figure 3.3: The OPAL detector.

The OPAL detector consisted of a central tracking system, a time-of-flight detector,
electromagnetic calorimeters, hadron calorimeters, muon detectors and forward detectors, as shown in Fig. 3.3 and Fig. 3.4. The solid angles for these detector elements are shownin Table. 3.1. In the following sections these detector elements and their trigger system
are briefly described.
1

The OPAL employs a right-handed coordinate system in which the z axis is along the electron beam
direction and the x axis points toward the center of LEP. The polar angle, θ, is measured from the z axis,
and the azimuthal angle, φ, is measured from the x axis around the z axis. The origin of the coordinate
system is set at the nominal interaction point.

24

CHAPTER 3. EXPERIMENTAL APPARATUSy6m

BarrelMuon detector

Hadron calorimeter and return yoke

Electromagnetic calorimetera)Presampler
Time of flight detector
Solenoid and Pressure Vessel
Z chambers
Jet chamber
Vertex detector
Microvertex detector
1

2

3

4

5m

1

2

3

4

5m

x

Interaction region

Microvertex detector
Vertex detectorx

Jet chamber
Z chambers

Silicon-tungsten forward detector
Original forward detectorb)EndcapPresampler
Electromagnetic calorimeter
Pole tip hadron calorimeter
Hadron calorimeter
Muon detector

6mzFigure 3.4: The quarter of the OPAL detector in (a) the the front and (b) top views.

3.2. THE OPAL DETECTOR

25

Table 3.1: Angular coverage of the OPAL subdetectors.
Subdetector

Coverage
Polar angle
Solid angle
Central tracking system

Silicon microvertex detector
Vertex chamber
Jet chamberz-chambers| cos θ| < 0.89
| cos θ| < 0.95
| cos θ| < 0.98
| cos θ| < 0.72

86%
86%
98%
67%

Electromagnetic calorimeters
Barrel electromagnetic presampler| cos θ| < 0.81
Endcap electromagnetic presampler 0.83 < | cos θ| < 0.81
Barrel electromagnetic calorimeter| cos θ| < 0.82
Endcap electromagnetic calorimeter 0.81 < | cos θ| < 0.98

98%

Hadronic calorimeters
Barrel hadron calorimeter
Endcap hadronic calorimeter
Pole-tip hadronic calorimeter| cos θ| < 0.81
0.81 < | cos θ| < 0.91
0.91 < | cos θ| < 0.99

97%

Muon detectors
Barrel muon detector
Endcap muon detector| cos θ| < 0.72
0.67 < | cos θ| < 0.98

93%

26

CHAPTER 3. EXPERIMENTAL APPARATUS

3.2.1

Central Tracking System

Toward outward from the beam pipe, the central tracking system consisted of the silicon
microvertex detector and a series of drift chambers : the vertex chamber, the jet chamber
and the z-chambers. These drift chambers were placed inside a pressure vessel whose
inner pressure was kept at 4 bar by filling gas mixture of 88% argon, 9.4% methane and2.6% iso-butane. On the outside of the pressure vessel, a solenoid was mounted, providing
a uniform magnetic field of 0.435T along z axis throughout the central tracking system.
The main purpose of the tracking system was the reconstruction of the charged tracks.
Since the charged particle was bended in the r-φ plane in the magnetic field, B, the radiusof the curvature of the charged track, ρ, was connected with its momentum component
perpendicular to the z axis, p⊥ , asp⊥ [GeV/c] = 0.3 B[T] ρ[m].

(3.2)

Silicon Microvertex Detector(SI)The silicon microvertex detector (SI) [57] was positioned between the beam pipe, of outerradius 5.65 cm, and the pressure vessel, of inner radius 8.0 cm. It consisted of two layersof silicon wafer modules (‘ladders’), placed at radii of 6 and 7.5 cm, respectively. An
overall length of them were about 30 cm. The inner and outer layers consisted of 12 and15 ladders respectively and ladders were tilted not to make φ gaps. as shown in Fig. 3.5.

73.8mm 60.5mm
80mm

56.5mmBERYLLIUM
VACUUM TUBE
(BEAM PIPE)
ladderCARBON FIBRE
PRESSURE
TUBE

Figure 3.5: Silicon Microvertex Detector

3.2. THE OPAL DETECTOR

27

Each layer consisted of two types of single sided silicon wafers were used. One wastermed φ wafer with 25 µ pitch strips along the beam axis and the other was termed z
wafer with 25 µ pitch strips along φ direction. The readout strip pitch was 50 µm for theφ wafer and 100 µ for the z wafer. The two kinds of wafers were glued together.
The overall resolution of about 10µm in the r-φ plane and about 15µm in the z
direction. The SI played a significant role in reconstructing secondary vertexes, whichwere distinct for the particles with long lifetime which enable them to travel several
millimeters before decaying. The use of SI is essential in the b-tagging, since such longlived particles were typically B mesons.
Vertex Chamber (CV)The vertex detector (CV) was a high precision cylindrical jet drift chamber. It was 100
cm long with covered radii between 8.8 cm and 23.5 cm, outside the SI. The CV consistedof an inner and outer layers as shown in Fig. 3.6.

Figure 3.6: Vertex Chamber.

28

CHAPTER 3. EXPERIMENTAL APPARATUS
Each layer was divided into 36 azimuthal sectors. Each sector in the inner layer had 12‘axial’ sense wires running parallel to the beam direction. Each sector in the outer layer
had 6 ‘stereo’ sense wires inclined at an angle of 4◦ to the z axis. Precise measurementsof the drift times on the sense wires enabled the track positions to be determined in r-φ
to within 50 µm, and in z to within 700µm.
Tracks measured in the CV provided important links between those reconstructed in
SI closest to the interaction point, and those reconstructed in the jet chamber outside theCV.

Jet Chamber (CJ)
The jet chamber (CJ) was a cylindrical jet drift chamber of length 400 cm, with anouter radius of 185cm and inner radius of 25cm, placed outside the CV. It covered 98%of the solid angle and provided good spatial and double track resolution and particle
identification.
All wires used in the CJ were parallel to the beam direction. The CJ was subdivided
into 24 azimuthal sectors and each sector had 159 anode sense wires equally spaced by
10 mm, alternating with potential wires. The anode wires were slightly staggered withrespect to the potential plane to resolve left-right ambiguities. Cathode wire planes formed
the boundaries between adjacent sectors.
The coordinates of wire hits in the r-φ plane were determined from a measurementof drift time. The z coordinate was measured using a charge division technique at wire
ends. The resolution achieved was at the level of 135 µm in the r-φ plane and 6 cm
in the z direction. Its good space and double track resolution were essential for theefficient recording of jet-like events. The CJ provided the most detailed information on
the curvature of charged tracks, and hence their momentum and charge.
A further purpose of CJ was to provide a measurement of the ionization energy loss perunit length (referred to as “dE/dx”), of charged particles traversing the gaseous volumeof the chamber. The measurement of dE/dx was derived from the charge collected ateach hits along the track trajectory. For a given charged particle, its mass could be
calculated with the information of dE/dx and the track momentum, p. Hence, the particle
identification was enabled as shown in Fig. 3.7, where the hadronic tracks and muon pairs
with at least 130 hits used for the dE/dx calculation. A good dE/dx resolution of 2.8%
was obtained for the muon-pairs with 159 hits used for the dE/dx calculation.

dE/dx (keV/cm)

3.2. THE OPAL DETECTOR

29

18
p

16
K

dE/dx-resolution:(with 159 dE/dx hits)

14µ-pairs: 2.8 %12πµ-pairs

e

10
8

µ

6
10

-1

1

10

10

2

p (GeV/c)
Figure 3.7: Distribution of measured and predicted dE/dx for hadronic tracks and muonpairs as a function of the momentum (p). The dE/dx resolution for the muon-pairs with159 hits used for the dE/dx calculation was 2.8%.

Z chambers (CZ)
The z-chambers (CZ) were placed around the barrel region of CJ inside the pressure
vessel which also contains CV and CJ. The CZ provided a precise measurement of the z
coordinates of tracks emerging from the CJ. It was a planar drift chamber, 400 cm long,
50 cm wide and 5.9 cm thick. The CJ was surround by 24 z-chambers. Each of them wasdivided in z to 8 cells with 6 sense wires which ran perpendicular to the z axis. The time
during the drift in the z direction was used to measure the z coordinates of the hits.
Solenoid CoilThe solenoid coil was self-supporting, water-cooled aluminum coil which was mounted
around the pressure vessel. The solenoid coil generated the 0.435 T magnetic field, uniform
to within 0.5% throughout the volume of the central tracking system. It was designed
to be as thin as possible (actually about 96 mm of aluminum and 54 mm glass-epoxy),in order to minimize material in front of the calorimeters. The iron of the hadronic
calorimeter served as a return yoke for the magnetic field.

30

CHAPTER 3. EXPERIMENTAL APPARATUS

3.2.2

Time-of-Flight System (TOF)

The time-of-flight system (TOF) was a scintillator array just outside the solenoid coilfor the barrel region or outside the pressure vessel for the endcap region. It provided
charged particle identification in the energy range from 0.6 to 2.5 GeV and fast triggering
information. It also provided an effective rejection of both cosmic rays and the eventsfrom interactions between the beams and residual gas inside the beam pipe or the beam
pipe itself.
Barrel Time-of-Flight detector (TB)
The barrel time-of-flight detector (TB) consisted of a cylindrical layer of 160 scintillators,
45 mm thick, 80-91 mm wide and 6.84 m long, placed as its external radius was 2.4 m,
surrounding the solenoid coil. It achieved a timing resolution of about 460 ps.
Tile Endcap Detector (TE)The tile endcap detector (TE) complemented the TB by providing timing informationfrom the forward region. It consisted of 120 tiles containing a 10 mm thick scintillator
layer and was mounted between the endcap presamplers and the endcap electromagnetic
calorimeter. These were read out by wavelength-shifting fibers connected to photomultiplier tubes.

3.2.3

Electromagnetic Calorimeters

The OPAL electromagnetic calorimeter (ECAL) was designed to absorb and measure theenergy of electrons, positrons and photons with energies in excess of about 100 MeV. The
ECAL system consisted of main calorimeters and presampler chambers placed in frontof the main calorimeters. The main calorimeters were a barrel array (EB) and endcap
arrays (EE) of lead glass blocks, each with a cross-section of approximately 10 × 10cm2 .
The EB had a depth of 24.6 radiation length (X0 ) and each EE had a depth of 20 X0 .
Together with a forward lead scintillator calorimeter described in the subsection 3.2.6,
ECAL covered approximately 99% of the solid angle.
Since there was about 2 X0 of material ahead of the main calorimeters mainly due tothe pressure vessel and the solenoidal coil, the cascade shower could start before the lead
glass. Presampler chambers were therefore installed in front of the main calorimeter torecover the degradation of position and energy resolution caused by the shower initiationahead of the main calorimeters. They also gave additional γ/π 0 and electron/hadron
discrimination.

3.2. THE OPAL DETECTOR

31

Barrel and Endcap Electromagnetic Presamplers (PB & PE)
The barrel electromagnetic presampler (PB) consisted of 16 chambers forming a cylinderof radius 239 cm and length 662 cm, placed just out side the TB, covering | cos θ| <0.81. Each chamber consisted of two layers of limited streamer tubes with the anodewires running parallel to the beam direction. On either side of the tubes were cathodestrips oriented at 45◦ to the wire direction. Spatial positions were measured with thestrips and wires. The energy deposited in front of the presampler was evaluated with hit
multiplicities of the presamplers.
The endcap electromagnetic presampler (PE) was made up of 32 multiwire proportional chambers and was located between the pressure vessel and the endcap calorimeter,
covering the region 0.83 < | cos θ| < 0.95.
The PB and PE got spatial resolutions of about 5 mm. They assisted both in thespatial and energy resolution of the calorimetry system as a whole.
Barrel and Endcap Electromagnetic Calorimeters (EB & EE)
The barrel lead glass calorimeter consisted of a cylindrical array of 9440 lead glass blocks
at a radius of 246 cm covering the range | cos θ| < 0.82. The detection efficiency was
optimized by aligning the longitudinal axis of the lead glass blocks to point almost toward
the interaction region. The slight offset prevented the loss of the particles through the gaps
between the blocks. The Čerenkov lights produced by the relativistic charged particles
passing through the lead glasses were collected by photomultipliers placed at the base ofeach block. The intrinsic energy resolution for the EB was
δE
6.3%≈ 0.2% + p.
E
E[GeV](3.3)

The endcap electromagnetic calorimeter (EE) consisted of two umbrella shaped arrays,
each of which had 1132 lead glass blocks aligned coaxially around the beam pipe. The
Čerenkov lights were collected by vacuum photo triodes attached at the back of blocks.
The intrinsic energy resolution for the EE at low energies was
δE
5%
≈ p
.
E
E[GeV](3.4)

Gamma Catcher (FE)The gamma catcher was intended to complement the lead glass calorimeters by filling the
gap between the inner edge of EE and the outer edge of the forward detectors described in
the subsection 3.2.6. It was a ring of lead-scintillator sandwich modules with 7X0 thick.
The signals were transported to silicon photodiodes through wavelength shifters readouts.

32

CHAPTER 3. EXPERIMENTAL APPARATUS

Minimum ionizing particle (MIP) plug
The MIP plug was designed to provide good time resolution and detection efficiency forsingle minimum ionizing particles, such as muons, in the forward regions of OPAL. It
comprised four layers of 1 cm thick scintillator tiles, covering the | cos θ| between 43 and
220 mrad. The first two layers were positioned behind FE, the third, behind these and at
a shallower angle, and the fourth between the forward detector and the silicon-tungsten
luminometer described in the subsection 3.2.6.

2.36m
barrel scintillator
609m rad(cosθ=0.82)

CJ
cosθ=0.7(43 deg)

2mpressure bellpresampler

ECAL
lead-glass

1m

tracking
318m rad(cosθ=0.95)

TE
MIP Plug

MIP(noTE)
200mradcosθ=0.98
ForwardCalorimeter

Gamma catcher

MIP(with TE) 43 mrad
Beam pipe

1m

2m

SI-W luminometer

3m

3.3m

Figure 3.8: The forward region of OPAL detector..

3.2.4

Hadron Calorimeters

The hadron calorimeters (HCAL) were designed to measure the energies and positions ofhadrons, which could form hadronic showers due to the cascade process of the inelastic
nuclear collisions. The scale of the longitudinal development of the hadronic showers ischaracterized by the (λI ).
The hadron calorimeter was composed of three sections, the barrel (HB), the endcaps(HE) and the pole-tips (HP). They covered 97% of the solid angle. These detectors werepositioned between alternate layers of the iron return yoke of the magnet, to form sampling
calorimeters about 1 m thick. The total thickness of the iron yoke was about 4 ‘nuclear
interaction length’ (λI ). Essentially all hadrons are absorbed at this stage leaving only
muons to pass on into the surrounding muon chambers. Since there was a high probability

3.2. THE OPAL DETECTOR

33

of the hadronic showers being initiated in the 2.2 λI of material before the HCAL, theoverall hadronic energy had to be determined with the information from both the ECAL
and HCAL together.
Barrel and Endcap Hadron Calorimeters (HB & HE)
The barrel (HB) and endcap (HE) hadron calorimeters had the same structure, alternating
layers of iron slabs and limited streamer tubes. The barrel region (HB) contained 9 layersof 2.5 cm thick chambers alternating with 8 layers of 10 cm thick iron. The HB was placedat radii from 3.39 m to 4.39 m, covering the angular region | cos θ| < 0.81.
The HE consisted of 8 layers of chambers of 3.5 cm in thickness alternating with 7iron slabs of 10 cm in thickness. The HE extended the angular coverage to | cos θ| < 0.91.
The limited streamer tubes had anode wires with 1 cm spacing in a gas mixture ofisobutane(75%) and argon (25%). The signals were read with large pads and strips locatedon the surfaces of the tubes. The layers of pads were grouped together to form towersthat divide up the detector volume into 48 bins in φ and 21 bins in θ. They provided anenergy resolution as

δE
120%
.
≈ p
E
E[GeV](3.5)

The strips were 0.4cm wide providing the φ information of the hits, which were used inmuon tracking.
Pole-tip Hadron Calorimeters (HP)
The pole-tip (HP) hadron calorimeter extended the angular coverage down to | cos θ| <0.99. The HP consisted of 10 layers of thin multiwire chambers of 7 mm in thickness
alternating with 9 layers of 8 cm thick iron. The chambers were operated in high gainmode with a gas mixture of CO2 (55%) and n-pentane (45%). The spacing of the anodewires was 2 mm and the anode-to-cathode gap was 1.6 mm. The signals were read withpads on one side and strips on the other.

3.2.5

Muon Detectors (MU)

Muons were basically the only detectable particles capable of penetrating both the electromagnetic and hadron calorimeters. The muon detectors (MU) were therefore designed
to identify these particles unambiguously from other potential background. It was divided
into a barrel and two endcap regions, together covering 93% of the solid angle.
Barrel Muon Detector (MB)The barrel muon detector (MB) covered the angular range | cos θ| < 0.72. It was composedof 110 large-area drift chambers, 1.2 m in width and 90 mm in depth. Their lengths are

34

CHAPTER 3. EXPERIMENTAL APPARATUS

6.0, 8.4 or 10.4 m. They were mounted most outer layer of the OPAL detector surrounding
the HB. The MB achieved a spacial resolution of about 1.5 mm in the r-φ plane. Diamondshaped cathode pads determined the z coordinate within an accuracy of about 2 mm.
Endcap Muon Detector (ME)The endcap muon detector (ME) provided an angular coverage of 0.67 < | cos θ| < 0.98 bymeans of 2 layers of 4 quadrant chambers and 2 layers of 2 patch chambers. The quadrant
chambers were 6 m high and 6 m wide, whereas the patch chambers were 2.5 m high and
3 m wide. Each chamber contained 2 layers of limited streamer tubes. The tubes in one
layer were aligned horizontally and those in the other layer were aligned vertically in theplane perpendicular to the beam direction. Each chamber was placed at about ±5.5 mfrom the interaction point.

3.2.6

Luminometers

Since the cross-section for the Bhabha scattering process at small angles can be calculated
theoretically to high precision, it served as a reference reaction from which the luminosityof the colliding beams was obtained. The luminometers were installed in the region
between 25 and 150 mrad to measure the rate of such events.
Forward Detectors (FD)The two modules of the forward detector (FD) were located at ±2.6 m from the interaction point, covering angles 40 − 150 mrad from the beam pipe. Each comprised 35
sampling layers of lead-scintillator sandwich, corresponding to about 24 radiation lengthsof material. In addition to providing a measurement of the luminosity to a precision of
about 0.5%, this detectors also complemented the electromagnetic calorimeter, extending
the angular coverage to | cos θ| > 0.98, which gave an energy resolution of
δE
17%
= p
E
E[GeV](3.6)

Silicon Tungsten Detector (SW)The two modules of the silicon-tungsten detector (SW) [58,59] were placed at ±238.94 cm
from the interaction point, covering the angular range between 25 and 58 mrad from thebeam. Each had 19 layers of silicon detectors sandwiched between 18 layers of tungsten
absorbers. The SW achieved good energy resolution of approximately 4% and position
resolution on the radial coordinate of 130-170 µm for electrons and positrons scatteredinto the detector.

3.2. THE OPAL DETECTOR

3.2.7

35

Trigger

Bunch crossing occurred every 22 µs at LEP. The OPAL trigger system [60] was designed
to provide high efficiency for the various physics reactions, and good rejection of backgrounds arising from cosmic rays, from interactions of the beam particles with the gas
inside the beam pipe or the wall of the beam pipe, and from the detector noise. The trigger system used fast information from the subdetectors to select crossing with a possiblee+ e− interaction, which enabled the reset of the data acquisition system with no loss ofthe beam crossing in case of the negative trigger decision. The trigger system reduced the
45 kHz bunch crossing rate to an event rate of about 10Hz which can be handled by thedata acquisition system. The trigger system was briefly described as follows.
The trigger system used the information of subdetectors, CV, CJ, TOF, ECAL, HCAL,
MU, FD, and SW. The signals from the CV and CJ were combined to form the informationof “track trigger” . In addition, random trigger of the beam crossing was used. The trigger
system consisted of two complementary parts, “stand-alone” and “θ − φ” triggers. The
stand-alone trigger utilized the information from the single subdetector, such as track
multiplicity in the track trigger, total energy in the barrel ECAL. As for the θ − φ trigger,
24 elements were made from each subdetector subdivided into 6 in θ and 24 in φ (θ − φmatrix), whose information enabled to use the spacial correlations of hits within andbetween subdetectors. The final trigger decision was made from the combination of thestand-alone and θ − φ trigger signals, which made the trigger system highly redundant.
If an event was triggered, full data acquisition was followed which need about 15 ms.
The trigger efficiency for the multihadronic process (e+ e− → hadrons) used as a mainprocess in this thesis was 100% for the full acceptance. Some of the important trigger
conditions for the multihadronic process were described as follows.
• At least 3 tracks were obtained in the track trigger.
• At least 2 tracks were obtained in the track trigger in the barrel region.
• Collinear hits were obtained in the θ − φ bins in the track trigger.
• Coincident hits were obtained in the same θ − φ bin for the track and ECAL trigger.
• Coincident hits were obtained in the same θ − φ bin for the track and TOF trigger.
• At least 7 hits were obtained in the TOF θ − φ bins.
• Coincident hits were obtained in the same θ − φ bin for the TOF and ECAL trigger.
• Total energy in the barrel ECAL was at least 7 GeV.
• Total energy in one side of the endcap ECAL was at least 6 GeV.

36

CHAPTER 3. EXPERIMENTAL APPARATUS• Total energy in one side of the endcap ECAL was at least 6 GeV.
• At least 1 θ − φ bin with the energy of at least 2.6 GeV was obtained in the barrel
ECAL.
• At least 1 θ − φ bin with the energy of at least 3 GeV was obtained in one side ofthe endcap ECAL.

37

Chapter 4b,cMeasurements of the AFB
In this chapter, an overview of the physics processes at LEP2 is given at first. Some
additional conditions to define AFB is described, which is needed under such situation.
Then a brief description of the procedure to measure the Ab,c
FB is given. The descriptionsof the data and the Monte Carlo samples used in this thesis are followed. With thesematerials, the analyses are performed.

4.1

Overview of the Quark-Antiquark Pair Production at LEP2

The quark-antiquark pair production (e+ e− → qq) at LEP2 is the main process in thisthesis, which contains the signal processes, e+ e− → bb and e+ e− → cc. The quark andantiquark can be emitted in back-to-back direction, which form two back-to-back jets asshown in Fig. 4.2(b). (If hard gluon emissions are accompanied, it forms three or morejets.) At LEP2, large fraction of quark-antiquark pair production accompanies the initialstate radiation (ISR) (e+ e− → qq + nγ). The Feynman diagram for the case of single ISR
is shown in Fig. 4.1 and such a event with the ISR escaped in the beam pipe is shown in
Fig. 4.2(a).

$#"   "!

  ' (*) !

           "!

&&%Figure 4.1: A Feynman diagram of quark-antiquark production with single ISR.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

38
The

√

s of the quark-antiquark production is effectively decreased by the ISR to the√mass of the γ/Z propagator, as which the effective center-of-mass energy ( s0 ) is defined.
√√
The simulated distribution of s0 at s of 189 GeV is shown in Fig. 4.2(c). Approximately√
70% of events have the s0 around the Z mass. Such phenomena is called “radiative return√√to Z”. An event with lower s0 is termed “radiative event” and that with s0 close to√
the s is termed “non-radiative event”. Since the AFB is measured for the non-radiative
events in this analysis, the s0 evaluation is important.

(a)

(b)

Events

(c)

80000
70000
60000

radiative eventγ

non-radiative event

q

q

θ
-

θ
+

e

e

50000

-

+

e

e

q

q

40000
30000
20000

radiative return to Z

10000
0
0

20

40

60

80

100

120

140

160

180

200 220s’ [GeV]Figure 4.2: (a) A quark-antiquark pair production with ISR√escaped in the beam pipe.
0(b) A quark-antiquark pair production√ without ISR. (c) The s distribution of simulated
quark-antiquark pair events at the s of 189 GeV. A large peak of the radiative returnis seen around the Z mass.

4.1. OVERVIEW OF THE QUARK-ANTIQUARK PAIR PRODUCTION
AT LEP2
39
Some other physics processes than the quark-antiquark pair production exist at LEP2.
The cross-sections of representative physics processes at LEP are shown as a function of√
the center-of-mass energy ( s) in Fig. 4.3.
The cross-sections of the fermion-pair production (e+ e− → ff) in the figure are calculated with Zfitter except for the Bhabha process (e+ e− → e+ e− ), which contains at-channel diagram. As for the quark-antiquark pair production, contamination of largepfraction of radiative events can be seen by comparing the cross-sections of s0 /s > 0.85p
and s0 /s > 0.1. The s0 estimation is important to distinguish the non-radiative events
form the radiative events.
As for the lepton pair production, the cross-section of the muon pair production(e+ e− → µµ) is a few times smaller than that of the bb or cc production. The crosssection of the τ pair production is the same as that of the muon pair production. The
cross-sections of the Bhabha process is calculated with Bhwide [61]. Both the inclusive (| cos θ| of the electron smaller than 0.99961 ) cross-section and the large angle nonradiative (| cos θ| of the electron smaller than 0.7 with the collinearity angle2 (θacoll ) smaller
than 10 degree) cross-section are calculated. The large cross-section of the small angle
Bhabha process can be seen by comparing the two cross-sections. Such events are used tomeasure the luminosity. The cross-section of the large angle Bhabha process is similar tothat of bb or cc production. The lepton-pair events are distinct from the quark-antiquark
pair events in the number of tracks. The events with hadronic τ decay is thought to bea background.
The cross-section of the photon pair process (e+ e− → γγ) is calculated with QED
under the condition in which the | cos θ| of the emitted photon is smaller than 0.93. The
cross-section is a few times smaller than that of bb or cc and such events are thought tobe well separated from quark-antiquark pair events.
The cross-sections of W boson pair (e+ e− → W+ W− ) and Z boson pair production(e+ e− → ZZ), which are the main sources of the events with four fermion final state, arecalculated with YFSWW [62] and YFSZZ [63], respectively. The cross-section of the W√boson pair production is comparable to that of bb or cc production at s larger than 180
GeV, while that of the Z boson pair production is smaller by one order. They are thought
to be backgrounds due to their hadronic decay modes which form some jets.
The cross-section of the 2-photon process with hadronic final state (e+ e− → e+ e− γγ →e+ e− + hadrons) is calculated with Phojet [64] under the condition in which the minimum
center-of-mass energy of the 2-photon system (Wγγ ) is larger than 2 GeV. It may be abackground due to the large cross-section in spite of the large missing energy or theforward activity in the detector.
1
2

The maximum value of | cos θ| covered by SW.
The collinearity angle = 180[degree] − the opening angle among the emitted e+ e− .

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

Cross-section[pb]40e+e-→ q q

8

10

107

s’/s>0.1

e +e-→ q qs’/s>0.85

e +e-→ ccs’/s>0.1

e +e-→ bbs’/s>0.1

e+e-→ µ +µ -s’/s>0.1

e e→e e|cosθe|<0.9996

+ -

+ -

e +e-→ e+e-|cosθe|<0.7, θacoll<10[deg]e e → γ γ |cosθγ|<0.93
+ -

10

6

e+e-→ W+We+e-→ ZZ
e+e-→ e+e-γ γ → e+e-+hadrons,Wγγ>2GeV

105
104
103
102
10
1
80

100

120

140

160

180

200

220

s[GeV]
Figure 4.3: The cross-sections of main physics processes at LEP. The cross-sections of
the fermion-pair production except for the Bhabha process is calculated with Zfitter.
The cross-section of the Bhabha process is calculated with Bhwide. That of the photonpair process is analytically calculated with pure QED. Those of the W boson pair and Z
boson pair production are calculated with YFSWW and YFSZZ, respectively. That of
the 2-photon process with hadronic final state is calculated with Phojet.

4.2. DEFINITION OF THE AFB AT LEP2

4.2

41

Definition of the AFB at LEP2

The AFB is defined generally as Eq. 1.2. Since radiative events make up more than halfof the events at LEP2, some additional conditions to define the AFB are needed. The
AFB can be defined for the non-radiative events defined with the effective center-of-mass√energy ( s0 ). It means the ISR is excluded but the FSR is included in the calculation of√s0 , which is invalid for the case in which the ISR-FSR interference exists. Consideringsuch situation at LEP2, the AFB to be measured is defined as Eq. 1.2 under the conditions
as follows.
√• The effective center-of-mass energy ( s0 ) is defined as the mass of the γ/Z propagator as in the section 4.1.
• The effect of the ISR-FSR interference (IFI) is excluded, which assures the definition√of s0 mentioned above.
• Events withps0 /s larger than 0.85 (non-radiative events) are taken.

• The AFB is defined within the full angular acceptance.
• The direction of the final quark (after the hard gluon emission) is used to calculate
the AFB , where the QCD effect of the gluon radiation is included.
The theoretical values are calculated with Zfitter (v6.36). It can calculate the AFB
corresponding to the above definition with the configuration flags as follows.
• INTF=0
The initial-final interference term is ignored.
• FINR=0
The final state QED and QCD corrections are included.
• BOXD=2
The contributions of ZZ and WW box diagrams are added.
• CONV=2
Both the running α(s) and the electroweak radiative correction are calculated against
each s0 and then convoluted.
With the definition of the AFB and the configuration of the Zfitter, the experimental
result can be compared with the theoretical calculation consistently.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

42

4.3

Procedure of the AFB Measurement

The AFB for the fermion pair production in e+ e− annihilation is connected to the angular
distribution according to Eq.2.11. The equation is still valid for the AFB defined in the√√section 4.2 since the convolution of the AFB for the s0 smaller than s ensure the same
relation as Eq.2.11. With the relation, the procedure to measure the Ab,c
FB is essentially
to fit the angular distribution of b or c quark by a function of 1 + cos2 θ + 8/3 AFB cos θ.
Accordingly, there are 5 points to measure the AFB as follows.
1. To select quark-antiquark pair events (e+ e− → qq).
2. To determine the axis along which the quark-antiquark pair is emitted.
3. To determine to which direction along the axis the quark is emitted.
4. To select events with bb (bb events) or cc (cc events).
5. Fitting the angular direction to which the b-quark or c-quark is emitted with afunction of 1 + cos2 θ + 8/3 AFB cos θ.

Quark-Antiquark Pair

Thrust Axis

Jet = Track + Unassociated Cluster

q
-

-

e

-

e

e

+

+

e

+

e

e

Track
Unassociated Clusterq

Hemisphere
Thrust Axis

-

+

e

e

Which hemisphere q belongs?
(Determination of the"quark hemisphere")Hemisphere Charge I

Hemisphere-eThrust Axis+e

Lepton Charge
Lepton from Semileptonic Decay

Lepton Analysis(Lepton Charge Method)-

Inclusive Analysis(Hemisphere Charge Method)eEnrichment of the Events with b or c quark pair.
Fitting the angular direction of b or c quark is emitted.

Combination of the Lepton and Inclusive Analysis.

Figure 4.4: The procedure to measure the AFB .

Thrust Axis

Hemisphere Charge II+e4.3. PROCEDURE OF THE AFB MEASUREMENT

43

From here, the procedure to measure the Ab,c
FB practically with the information fromthe detector is briefly described. The procedure is schematically shown in Fig. 4.4.
The events with quark-antiquark pair can be seen as two jets or more jets in thecase with hard gluon emissions, where the jet is constructed from both the tracks formedin the central tracking system and ECAL clusters which have no association with the
tracks (“unassociated cluster”). The quark-antiquark-pair events can be selected withthat characteristics.
For such events with multi-jets, the axis along which the quark-antiquark pair isemitted can be determined with a “thrust axis” [65], which is defined in the relation witha quantity termed “thrust” as follows. The thrust (T ) is defined as,¶µP|~pi · ~n|iP
T = max
,
pi |
i |~

(4.1)

where the sum runs over all the particles (tracks and unassociated clusters) in a event,and where p~i is referred to as the momentum of the ith particle and ~n is referred to as
the unit vector to maximize the sum. The thrust axis is defined as the ~n, which is a good
approximation of the direction along which the primary quark-antiquark pair is emitted.
The angle of the thrust axis measured from the e− direction is termed θthrust . As regards
the thrust axis, each event is divided into two hemispheres defined by a plane that is
orthogonal to the thrust axis and is containing the origin of the coordinate system (thenominal e+ e− interaction point).
If the hemisphere to which the quark is emitted is termed a “quark hemisphere”, todetermine to which direction along the thrust axis the quark is emitted can be describedas the determination of the quark hemisphere. As regards the quark-hemisphere determination, two methods to utilize the charge information is taken. For each method, adifferent method to enrich bb or cc events is used.
One termed “lepton charge method” utilizes the charge of leptons from the semileptonic decays of b- or c-hadrons. For example, if a lepton with positive charge from thesemileptonic decay of a c-hadron is found, it is decided that the c-quark is emitted tothe hemisphere containing the lepton. For the case of the cc events, the probability tocorrectly determine the quark hemisphere with the lepton charge method is expected tobe approximately 100%, whereas a smaller probability is expected for the bb events dueto the B0 − B0 mixing. The analysis in which the lepton charge method is used is termed“lepton analysis”, where the information of the leptons from semileptonic decays is used
to enrich bb or cc events. This analysis can be applied to the events with leptons fromthe semileptonic decays.

CHAPTER 4. MEASUREMENTS OF THE AB,CFB

44

The other termed “hemisphere charge method” utilizes “hemisphere charge” [66],Qhemi , which is defined as

Phemi.

(pki )κ qi
Qhemi = Pi hemi.
,(pki )κ
i(4.2)

where the sum runs over all tracks in a hemisphere, and where pki is referred to as thelongitudinal momentum component with respect to the thrust axis and qi is referred toas the charge of track i. The value of the exponent, κ, is chosen in order to maximize
the power of the quark-hemisphere identification. As regards the identification, the hemisphere with larger hemisphere charge is determined as the quark hemisphere for up-type
quarks, whereas the hemisphere with smaller hemisphere charge is determined as thequark hemisphere for down-type quarks. The analysis in which the hemisphere charge
method is used is termed “inclusive analysis”, since this method can be applied to all
the events. In this analysis, a powerful b-tagging algorithm based on lifetime, lepton andevent-shape information is used to enhance events with bb or cc.
For both the analyses, the enrichment of the bb̄ or cc̄ event is performed so as to achieve
high purity, to keep high efficiency and to minimize the biases on the AFB . Accordingly,
several subsamples to hold different sensitivity to AbFB or AcFB are obtained and unbinned
likelihood fits are performed against the angular distributions for each subsample to get
Ab,cFB simultaneously.
Finally, some corrections are applied to the fitted values of Ab,c
FB . Biases on AFB due tothe event selections has to be corrected to recover the original AFB . Biases due to the useof the thrust axis to estimate the emitting direction of the primary quark-antiquark pairalso has to be corrected. The IFI effect is included in the fitted values, since the effectis naturally included in the real data. The effect is removed to get the AFB defined in
the previous section. The QCD effect (effect due to the final state gluon radiation) in thefitted value is already reduced since the thrust axis is insensitive to the gluon radiationand tended to indicate the direction of primary quark-antiquark pair. The QCD effecthas to be recovered to get the AFB defined in the previous section.
As regards the combination of the two methods, each analysis is performed avoiding
the event sharing between the two analyses. The two independent analyses are combinedwith a single likelihood function in the unbinned likelihood fit.

4.4

Data and Monte Carlo Samples

The data collected with the OPAL detector are divided into 12 coarse subsamples (18fine subsamples) according to the center-of-mass energy. The names of the division andthe Ecm ranges are shown in Table. 4.1. The analysis in this thesis is performed for eachcoarse subsample.

4.4. DATA AND MONTE CARLO SAMPLES

45

Table 4.1: The division of the data into 12 coarse subsamples (18 fine subsamples).
Name of the data subsamples
91 GeV
130 GeV133 GeV136 GeV
161 GeV
172 GeV183 GeV
189 GeV
192 GeV
196 GeV
200 GeV
202 GeV203 GeV
205 GeV
204 GeV
205 GeV206 GeV
207 GeV
207 GeV
208 GeV
209 GeV

4.4.1

Range of Ecm [GeV]
90
−
92
127
−
133
133
−
141
158
−
165
169
−
175
180
−
186
188
−
190
191
−
192
195
−
197
199
−
201
201
−
202.5
202.5
−
203.5
203.5
−
204.5
204.5
−
205.5
205.5
−
206.5
206.5
−
207.5
207.5
−
208.5
208.5
−
209.5

Luminosity Measurement

The luminosity is measured using small-angle Bhabha events, e+ e− → e+ e− , detectedin the SW. Since the small-angle Bhabha cross section is dominated by the t-channel
exchange of a photon, highly accurate calculation of the cross section is possible undera framework of nearly pure QED. The measurement of the luminosity is performed essentially by counting the number of the small-angle Bhabha events and comparing itwith the theoretical cross section. The statistical and systematic errors in the luminosity
measurements is less than 0.5%.

4.4.2

Data

The breakdown of the integrated luminosity is shown in Table. 4.2.The data collected at√s above 130 GeV are used for the measurement of the Ab,c
FB and the data collected at Z
resonance are used to check the methods used in the analysis.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

46

Table 4.2: A breakdown of the integrated luminosity
Ecm
[GeV]
1995∗
91
0.01
130
2.32
133
136
0.03
161
172
183
189
192
196
200
202
203
205 204
205
206
207
207
208
209
Sum
2.36

1996
1.14

Integrated Luminosity[pb−1 /year]
1997
1998
1999
2000
1.48
1.79
3.43
3.98
2.27
3.35

9.90
9.89
42.22
163.43
18.35
60.03
56.35
30.94

20.93

49.31

165.23169.10

0.82
0.66
1.47
7.97
67.54
64.09
61.96
7.74
0.08
216.32[pb−1 ]
Sum
11.83
4.59
7.97
3.38
9.90
9.89
42.22
163.43
18.35
60.03
57.17
31.60
1.47
7.97 76.99
67.54
64.09
61.96
133.877.74
0.08
623.24

mean Ecm
[GeV]
91.27
130.12133.29135.99161.36172.10182.68188.63
191.61
195.54199.53201.65202.67203.85 204.88
205.05206.31206.63206.56208.00208.64*) The integrated luminosity in the year 1995 is the integrated luminosity in the lastperiod of 1995 when the LEP energy increased beyond 91 GeV.

4.4.3

Monte Carlo Samples

The main physics process at LEP is described in section 4.1. The signal process is thequark-antiquark pair production. The 4-fermion process which has 4-fermion final states
mainly from W+ W− or ZZ decays, 2-photon and tau-pair process are the possible backgrounds. The contributions from other processes are checked and found to be negligible.
For the above-mentioned signal and background processes, the final-state particles were
simulated with a specialized event generator. Subsequently, the full simulation of the
OPAL detector was performed.
As for the multi-hadronic events at Z, JETSET 7.4 [67] was used for the quark-antiquark√
pair production and their fragmentation. On the other hand, for the events at s morethan 130 GeV, quark-antiquark pair was generated with KK2f v4 and fragmentation of
the quark and antiquark were treated with Pythia 6.125 or 6.158 [48]. The KK2f [45]includes first order O(α) QED and electroweak corrections and includes almost completeO(α2 ) QED corrections especially for the ISR with CEEX (coherent exclusive exponentiation) method. The accurate treatment of the ISR process is important since the ISR
is more significant at LEP2 than at LEP1. However the FSR was disabled and it was4.5. EVENT PRESELECTIONS

47

treated in the showering process in the Pythia instead. The ISR-FSR interference was√disabled in order to hold validity of the definition of s0 as the mass of the γ/Z propagator. The fragmentation was performed by the Pythia 6.150 (string model), and heavy
quark fragmentation was modeled according to the Peterson fragmentation scheme withparameters tuned according to [68].
The 4-fermion events were generated with grc4f v2.1 [69] or KORALW v1.42 [70].
Events with final states of eeee, eeµµ or eeqq were generated by grc4f and events withfinal states of qqqq, llqq or llll were generated by KORALW in which matrix elementsfrom grc4f were used. The fragmentation were treated by JETSET 7.4 for both cases.
The 2-photon events were generated by Pythia or Phojet and the τ + τ − pair eventswere generated by KK2f, interfaced to the tau decay package (Tauola [71]).

4.5

Event Preselections

From here, the analysis is performed with the data and the Monte Carlo (MC) describedabove. At first, the event preselection is described and the description of the inclusive
and lepton analysis are followed.
As regards the event preselection, the multi-hadronic events are selected at first witha selection termed “LEP2 multi-hadronic event selection”, which keeps 98% of the quarkantiquark pair events and events from almost all the other processes are rejected but for
4-fermion events. Then 4-fermion events rejection are performed and non-radiative event√√
selection, s0 > 0.85 s, and the acceptance cut, | cos θthrust | < 0.9 are applied.

4.5.1

LEP2 Multi-Hadronic Event Selection

The standard multi-hadronic events selection at LEP2 (L2MH) is constructed with theinformation of both tracks and electromagnetic clusters with good quality. The tracks
were constructed with the information of CV, CJ and CZ, and the electromagnetic clusters
were constructed with the information of EB and EE, where the detector statuses of thesesubdetectors were already required. The criteria to select good tracks are as follows.
• At least 20 hits measured in the tracking system of CV, CJ and CZ are required.
• The point of closest approach to the beam axis lies less than 2cm in the r − φ planefrom the nominal interaction point.
• The point of closest approach to the beam axis lies less than 40cm along the z axis
from the nominal interaction point.
• The distance between the primary vertex and the first hit associated with the track
must be less than 60cm.

48

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB
• The transverse momentum to the beam direction is more than 50MeV/c.
• The | cos θ| of the track is smaller than 0.995.

The criteria to select good electromagnetic clusters are as follows.
• For the clusters in EB, the energy deposit is larger than 0.1 GeV.
• For the clusters in EE, the energy deposit is larger than 0.2 GeV and are spreadover at least 2 adjacent lead glass blocks.
The hadronic events are characterized by both the large visible energy and the high multiplicities of the tracks and clusters in the final states. In order to select them efficiently,
the LEP2 multi-hadronic event selection is constructed as,
Ntrack ≥ 5,

(4.3)

Necal ≥ 7,
PE
√ ecal ≥ 0.14,
Rvis ≡sP| Eecal cos θecal |P
Rbal ≡≤ 0.75,
Eecal

(4.4)
(4.5)
(4.6)

where Ntrack is the number of charged tracks, Necal is the number of the electromagnetic
clusters, Eecal is the energy of the electromagnetic cluster, and θecal is the polar angle of√the cluster. The Rvis means the ratio of the visible energy to s and the Rbal means theenergy unbalance in z direction.
• The cuts on Ntrack and Necal are reflected the high-multiplicity nature of the hadronic
events which efficiently rejects the leptonic events.
• The cuts on Rvis is introduced in order to reject 2-photon background events whichtends to have large missing energy due to the escape of electron or positron in the
beam pipe. Some of the radiative hadronic events with large missing energy due tothe escape of the ISR photons in the beam pipe are also lost by this cut.
• The cuts on Rbal is introduced to further reject 2-photon background events whichtends to have large energy unbalance due to the escape of one of the electron or
positron into the beam pipe. Some of the radiative hadronic events with large energy
unbalance due to the escape of ISR photons are also lost.
√√
The selection efficiency of these cuts for the multihadronic events with s0 > 0.85 s isapproximately 98% form MC, while radiative events tends to be lost. The effect of the
L2MH selection on the hadronic events is shown in Fig. 4.5.

Events

4.5. EVENT PRESELECTIONS

49

103

102

no cut10

L2MH cut
1

0

20 40 60 80 100 120 140 160 180 200 220
s’[GeV]√√
Figure 4.5: The s0 distribution of multihadronic MC sample at s = 189 GeV before andafter LEP2 multihadronic√√ event selection (L2MH). The arrow indicates the non-radiative0
events with s > 0.85 s.

The distributions of Ntrack , Necal , Rvis and Rval are shown in Fig. 4.6, where a good
agreements between the data and MC are obtained. Almost all the 2-photon events andτ + τ − events are rejected. The main background to the multi-hadronic events is the 4fermion events. The fraction of the 4-fermion events is approximately 18% and that of
the 2-photon events is approximately 2% and that of the τ + τ − events is approximately
0.2%.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

50

(b)
EventsEvents

(a)
2200
2000
1800

1800
1600
1400

1600
1400

1200

1200

1000

1000

800

800

600

600
400

400

200

200
0
0

10

20

30

40

50

60

70

0
0

80

10

20

30

Ntrack

50

60

70

80
Necal

(d)
EventsEvents

(c)

40

1400

1000

1200
800

1000
800

Data

600

2 fermions b

600

400

2 fermions c

400

2 fermions uds

200

200
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

4 fermions0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

R vis

2 photons

1

2 tausR bal

Figure 4.6: The distributions of (a) Ntrack , (b)√Necal , (c) Rvis and (d) Rval for the remaining
events after the L2MH selection collected at s of 189 GeV.

4.5.2

s0 Determination and Non-Radiative Event Selection

The effective center-of-mass energy,√s0 , is determined as in [72]. The method is describedbriefly as follows. The isolated photon candidates are identified with the following criteria.
• Energy of the ECAL cluster is larger than 10 GeV.
• The selection of ECAL clusters is done by assigning the upper limits to both thenumber of lead glass blocks and the number of blocks which includes the 90% of thecluster energy.
• The sum of ECAL cluster energy in the cone with an angle of 0.2 radian is required
to be smaller than 1 GeV.
• Energy of associated HCAL cluster in the region | cos θ| < 0.95 is required to be lessthan 4 GeV.

4.5. EVENT PRESELECTIONS

51

The tracks and clusters except the isolated photon clusters are formed into jets using
Durham algorithm [73] with a jet resolution parameter ycut = 0.02. If more than four jets
were found the number was forced to be four by adjusting the jet resolution parameter.
The jets and observed photons are then subjected to a series of kinematic fits imposing theconstraints of energy and momentum conservation, in which zero, one, or two additional
photons emitted close to the beam direction are allowed. The fit with the lowest number of√extra photons which gives an acceptable χ2 is chosen. The value of s0 is then computedfrom the fitted four-momenta of the jets. If none of the kinematic fits gives an acceptable√χ2 , s0 is estimated directly from the angles of the jets.
A symbol of s0measured is used to represent the estimated s0 and a symbol of s0true is used

Eventsto represent the true s0 , if there is some ambiguity in using s0 . The distribution of the√measured s0 is shown in Fig. 4.7.

104

103
Data

102

2 fermions b
2 fermions c

10

2 fermions uds
4 fermions

1

2 photons

0

20 40 60 80 100 120 140 160 180 200 220

2 tauss’[GeV]√√
Figure 4.7: The measured s0 distribution of multihadronic MC sample at s = 189 GeV
after LEP2 multihadronic√√ event selection (L2MH). The arrow indicates the non-radiative0
events with s > 0.85 s.
√√
The non-radiative event selection is performed by applying a cut of s0 > 0.85 s.
This was set as a common cut among 4 experiments at LEP and this rather inclusive
selection reduces the effect of the ISR-FSR interference on the AFB .

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

52

p 0
smeasured for the 189 GeV multi-hadronic MC sample is shownpin Fig. 4.8, where the distribution of events with s0true /s > 0.85 is shown with a coloredp
positive-slope hatch and that of events with s0true /s < 0.85 is shown with a negativep
slope hatch. For the selection, the loss of the true non-radiative events (with s0true >√0.85 s) is approximately 5% and the contamination of the true radiative events (withp√s0true < 0.85 s) is approximately 5%.

EventsThe distribution of

105

104

103

2q MC All102

s’true/s >0.85

020 40 60 80 100 120 140 160 180 200 220

s’true/s <0.85

s’measured [GeV]p
Figure 4.8: The distribution of s0measured for the 189 GeV multi-hadronic MC sample,
p0
where the distribution of events withp strue /s > 0.85 is shown with a colored positiveslope hatch and that of events with s0true /s < 0.85 is shown with a negative-slope hatch.

4.5.3

4-fermion event rejection

The 4-fermion events are the main background after the multi-hadronic event selection.
In order to reduce them, the OPAL W+ W− event selection [74–76] is used. It is divided
into two types, W+ W− → qq`ν ` selection and W+ W− → qqqq selection. For both ofthem, likelihood based discriminants are constructed after cut based preselections.
W+ W− → qq`ν ` selection
Events with W+ W− → qqµν µ or W+ W− → qqeν e are characterized by two well-separated
hadronic jets, high momentum lepton, and missing momentum due to the undetected
neutrino, whereas events with W+ W− → qqτ ν τ are characterized by two well-separated

4.5. EVENT PRESELECTIONS

53

jets from hadronic W decay, one low multiplicity jet typically containing one or three
tracks, and low visible energy due to the undetected neutrinos from W and τ decays.
Such events are selected as follows.
• Preselection
Loose cuts on visible energy (Rvis ), number of charged tracks (Ntrack ) and numberof ECAL clusters (Necal ) are applied to reject obvious background events such as
low multiplicity events or 2-photon events. (Rvis > 0.2, Ntrack > 5 and Necal > 5)
• Identification of Lepton Candidates from W
For all tracks, a track most consistent with the lepton coming from W+ W− → qq`ν `decays is selected from information of the track isolation, the energy of the trackand e, µ or τ identification (hit multiplicity in the MU and HCAL, dE/dx, etc).
• Likelihood based selections
Likelihoods (Lqq̄`ν ` ) corresponding to each W+ W− → qq`ν ` decay to e, µ and τ areconstructed from the information of the kinematics and e, µ or τ identification.
The Lqq̄`ν ` distribution is shown in Fig. 4.9. Events with Lqq̄`ν ` larger than 0.5 are selected

Eventsas W+ W− events.

103

102

Data
2 fermions b

10

2 fermions c
2 fermions uds
4 fermions

1

2 photons0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2 tausLqqlν

Figure 4.9: The W+ W− → qq`ν ` likelihood discriminant (Lqq`ν ` ) distribution. The arrow
indicates the W+ W− -selection cut of Lqq`ν ` > 0.5.

W+ W− → qqqq selection
Any event selected in the W+ W− → qq`ν ` selection is rejected from a W+ W− → qqqq
candidate. Events with W+ W− → qqqq are characterized by 4 energetic, hadronic jetsand little missing energy. Such events are selected as follows.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

54
Preselection• The ratio of the visible energy to the√s, Rvis , is larger than 0.7.

• The single largest electromagnetic cluster, Rmax , is less than 30% of the√s to reject

events from the radiative Z/γ → qq process.
• The estimated invariant mass,p 0√
smeasured , is greater than 75% of the s.

• log10 WQCD [77] is less than 0, where the WQCD is the event weight calculated fromthe matrix elements for four jet production in Z/γ → qq process [78].
jet• The charged multiplicity of the lowest multiplicity jet, Nmin
, is greater than zero,

where the event is forced to four jets using the Durham algorithm [73].
• The charged multiplicity of the second lowest multiplicity jet, N2jet , is greater thanone, where the event is forced to four jets using the Durham algorithm.

Likelihood based selection The likelihood discriminant (Lqqqq ) for the W+ W− →qqqq selection is constructed from four input variables:• log10 WQCD .
• log10 WEXE , where WEXE is the event weight for the WW → qq̄qq̄ process calculated
from the EXCALIBUR [79] matrix element.
• Sphericity [65,80] of the event, S = 3/2 min(P2i pit /

P 2pi ), where the sum runs over

all tracks in the event, pi is the momentum of the ith track, and pit is the transverse
momentum of the ith track with respect to a vector to minimize the sum.
• log10 (y45 ), where the y45 is the value of jet resolution parameter at which the eventis reclassified from a four jet to a five jet event by Durham algorithm.
The distribution of log10 WQCD before the 4-fermion rejection is shown in Fig. 4.10(a).
The Lqqqq likelihood distribution is shown in Figure 4.10(b). The events with Lqqqq larger
than 0.24 are tagged as W+ W− events.

4.5. EVENT PRESELECTIONSEvents(b)
Events

(a)

55

350
300

120
100

250
80
200

Data

150

2 fermions b

2 fermions b

2 fermions c

40

2 fermions c

2 fermions uds

2 fermions uds

20
4 fermions

4 fermions

Data

60

100
50
0
-4

2 photons

-3

-2

-1

0

1

2

3

4

5

0

6

2 taus 0

2 photons

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2 taus

Lqqqq

log10WQCD

Figure 4.10: (a) the log10 WQCD distribution and (b)the W+ W− → √qqqq likelihood discriminant (Lqqqq ) distribution after non-radiative event selection at s of 189 GeV. The
arrows indicate the W+ W− selection.

4-fermion event rejection with W+ W− → qq`ν ` and W+ W− → qqqq selection
The events with Lqq`ν ` larger than 0.5 or Lqqqq larger than 0.24 are tagged as W+ W−
events and rejected for the Ab,c
FB measurements. The distribution of log10 WQCD after√√
the 4-fermion rejection at s = 189 GeV is shown in Fig. 4.11(a). The measured s0√
distribution at s = 189 GeV after 4-fermion rejection but non-radiative event selection
is shown in Fig. 4.11(b). Approximately 86% of the 4-fermion events are rejected and thecontamination of the 4-fermion events is approximately 6% after the 4-fermion rejection.

Events(b)
Events

(a)
300

104

250
103
200
Data

Data

2 fermions b

2 fermions b

2 fermions c

2 fermions c

2 fermions uds

2 fermions uds

4 fermions

4 fermions

102

150
100

10

50

1

0
-4

2 photons

-3

-2

-1

0

1

2

3

4

5

6

log10WQCD

2 taus 0

2 photons

20 40 60 80 100 120 140 160 180 200 220

2 tauss’[GeV]√
Figure
4.11:
(a)
The
log
W
distribution
and
(b)
The
measureds0 distribution at
QCD
10√s = 189 GeV after 4-fermion rejection but the non-radiative event selection. The arrow
indicates the non-radiative event selection.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

56

4.5.4

Acceptance Cut

The thrust axis is constructed from the tracks and unassociated ECAL clusters which
passed the quality criteria as follows. The quality criteria for the track are as follows.
• The number of CJ hits is equal or more than 20.
• The transverse momentum is larger than 0.15 GeV/c.
• The magnitude of the cosine of the polar angle is less than 0.99995.
• The magnitude of the z co-ordinate of the point of closest approach to the beamaxis in the r-φ plane (|z0 |) is less than 200 cm.
• The magnitude of the impact parameter (|d0 |) is less than 5 cm.
• The χ2 for the track fit is less than 100.
The quality criteria for the ECAL cluster are as follows.
• The raw energy of the cluster is larger than 50 MeV.
• The corrected energy of the cluster in EB is larger than 100 MeV.
• The corrected energy of the cluster in EE is larger than 200 MeV.
This axis is used to determine the axis along which qq are emitted. The polar angleof the axis is represented as θthrust . The range of the | cos θthrust | is limited within 0.9,

Eventswhich corresponded to the angle covered by SI. The distribution of | cos θthrust | is shownin Fig. 4.12.

250
200
Data

150

2 fermions b

100

2 fermions c
2 fermions uds

50
00

4 fermions
2 photons

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2 taus|cosθthrust |Figure 4.12: The | cos θthrust | distribution at
acceptance cut.

√s of 189 GeV. The arrow indicates the4.6. INCLUSIVE ANALYSIS

4.5.5

57

Effect of the Event Preselection

Fractions of the multi-hadronic, 4-fermion events after each event selection are shownp√in Table 4.3. The event-selection efficiencies of bb or cc event with s0true > 0.85 s
are also shown in Table 4.3. The L2MH selection keeps approximately 98% of multihadronic evetns. After all preselections are applied, about 78% of non-radiative bb eventsand 79% of non-radiative cc events are remained. The fraction of non-radiative multihadronic events is about 91% and the contamination of radiative multi-hadronic events
is about 4%, that of 4-fermion events is about 6% and contributions from other sources
are negligible. For example, the contribution from τ + τ − process is approximately 0.2%
afte L2MH selection and less than 0.1% after all the preselections. The contribution from
2-photon process is approximately 2% after L2MH selections and approximately 0.1%
after all the preselections.

Table 4.3: The event selection efficiencies and the fractions of selected events at
189 GeV, which are estimated with Monte Carlo samples.
cut

Number
Data

L2MH
√
s0
4f rejectioncos θthrust

4.6

17744
5230
3491
3153

fraction[%]qq
4-fermion
non-radiative radiative
19.4
61.0
17.6
64.9
3.1
31.6
89.8
4.1
5.7
90.6
3.6
5.7

√

s of

efficiency[%]bb
cc
non-radiative
98.1
98.2
90.9
93.2
84.9
87.1
77.6
79.4

Inclusive Analysis

The inclusive analysis is performed for the preselected events. The thrust axis is used
to determine the axis along which the qq are emitted. The quark hemisphere, which thequark is emitted to, is determined with the hemisphere charge. In this section, the description about the quark-hemisphere determination is given at first. Then the inclusive band c tagging is described. The description about the unbinned likelihood fit to obtain theAb,cFB with the quark-hemisphere determination, and b and c tagging is followed. Finally,some corrections applied to the fitted values of AFB are described.

4.6.1

Determination of the Quark-Hemisphere

The two hemispheres defined in section 4.3 are labeled as “hemisphere I” and “hemisphereII”, where one hemisphere arbitrarily selected is labeled as “hemisphere I” and the otherCHAPTER 4. MEASUREMENTS OF THE AB,C
FB

58

hemisphere is labeled as “hemisphere II”. The hemisphere charges for the “hemisphere I”
and the “hemisphere II” are termed QI and QII respectively as shown in Fig 4.13(a).
If a quark or antiquark with positive charge is emitted to the “hemisphere I”, the“hemisphere I” is determined as the quark hemisphere for up-type quarks, whereas the“hemisphere II” is determined as the quark hemisphere for down-type quarks. Thus the
charge identification of the hemisphere enables the quark-hemisphere determination.
The charge identification can be done with the hemisphere charge difference (QI −QII ).
If QI − QII > 0, the charge of a quark or antiquark emitted to the “hemisphere I” isidentified as positive. The distribution of the hemisphere charge difference (QI − QII ) isshown in Fig. 4.13(b), where the multi-hadronic events with an up-type quark emitting to“hemisphere I” or a down-type quark emitting to “hemisphere II” are labeled as “positive”.
Those with an up-type quark emitting to “hemisphere II” or a down-type quark emitting
to ‘hemisphere I” are labeled as “negative”.

(b)
Events(a)
Hemispher I (QI)
Thrust axis

250

200

150

Data
100

positive

Hemispher II (QII)50

0negative
-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

4-fermionsDifference of the Hemisphere Charges

Figure 4.13: (a) Labeling of the hemispheres and (b) the distribution of the hemispherecharge difference (QI −QII ). The multi-hadronic events with an up-type quark emitting to“hemisphere I” or a down-type quark emitting to “hemisphere II” are labeled as “positive”.
The multi-hadronic events with an up-type quark emitting to “hemisphere II” or a downtype quark emitting to ‘hemisphere I” are labeled as “negative”.

The performance of the charge identification is characterized by a probability thatthe quark charge in the “hemisphere I” is correctly determined (charge identification
probability). The charge identification probabilities for all quarks for various values ofκ are studied. The charge identification probabilities as a function of κ are shown in
Fig. 4.14 and the best performance is obtained at a κ of 0.4 for bottom quarks.

4.6. INCLUSIVE ANALYSIS

59

(b)
0.9

Charge.ID Prob.

Charge.ID Prob.

(a)

0.85
0.8
0.75

0.9
0.85
0.8
0.75

0.7

d

0.7

d

0.65

u

0.65

u

0.6

s

0.6

s

c

0.55

b

0.5
0

0.55
0.5
0

1

2

3

4

5

6

7

8

9

10

c
0.2

κ

0.4

0.6

0.8

b

1
κ

Figure 4.14: The charge identification probabilities as a function of κ.

If larger fraction of the momentum component of a quark along the quark direction
is transfered to the corresponding hadron (, which means harder fragmentation), better
performance of charge identification is expected since the charge of the hadron contributes
more to the hemisphere charge. The contributions of harder or softer momentum component can be tuned by changing the κ.
A higher charge identification probability is potentially expected for up-type quarks
due to their larger electric charge (2/3) compared to a charge of down-type quarks (1/3).
It is also because up-type quarks have more chance to form charged hadrons, for they cancouple anti-down or anti-strange quarks, while down-type quarks only have anti-up quarks
to form charged hadrons. It is reflected in the higher charge identification probabilities
for up-type quarks at a value of κ equal 0 in Fig. 4.14.
Charm quarks have lower charge identification probability in spite of their harder fragmentation function, since the decay products of the D+ or D0 mesons include significant
fraction of K − (27.5 ± 2.4% for D+ and 53 ± 4% for D0 ). Such opposite charge dilutes
the charge identification power.
Strange quarks have harder fragmentation function than that of down quarks andhave higher charge identification probability. Both down and strange quarks form stabler
hadrons and suffer smaller dilution due to their decays. Although bottom quarks have
hardest fragmentation function than others, the charge identification power is as high asthat of down quarks due to dilutions form both their decays and the B 0 − B¯0 mixing.
In the distribution of the hemisphere charge difference (Fig 4.13(b)), outer region
presents higher charge identification performance than inner region. Therefore events are
divided into 2 classes, “inner class” where |QI − QII | < 0.15 and “outer class” where|QI − QII | ≥ 0.15. The threshold of 0.15 is selected, which is determined to minimize theCHAPTER 4. MEASUREMENTS OF THE AB,C
FB

60

statistical error of the Ab,c
FB . The resulting charge identification probabilities are summarized in Table. 4.4.
Table
4.4: The charge identification probabilities for each quark and for each subclass at√
s of 189 GeV. The errors assigned are statistical error from the MC statistics.
allinnter classouter classdusc0.68 ±0.01 0.76 ±0.01 0.71 ±0.01 0.67 ±0.01
0.57 ±0.01 0.60 ±0.01 0.58 ±0.01 0.57 ±0.01
0.75 ±0.01 0.84 ±0.01 0.78 ±0.01 0.74 ±0.01b0.67 ±0.01
0.58 ±0.01
0.74 ±0.01

Practically, the quark hemisphere has to be determined with respect to the e− beam
direction. Accordingly, the hemisphere in the e− beam direction is labeled as “forward
hemisphere” and the other is labeled as “backward hemisphere”. The hemisphere charge
corresponding to the forward hemisphere is represented as QF and that corresponding to
the backward hemisphere is represented as QB as shown in Fig. 4.15.
Forward hemispher (QF)b

Thrust axis

θthrust-e beam directionb
Backward hemispher (QB)

Figure 4.15: The forward and backward hemisphere. The hemisphere charge corresponding to the forward hemisphere is represented as QF and that corresponding to the backward hemisphere is represented as QB .
The quark direction for down-type quarks can be evaluated with the polar angle ofthe thrust axis (θthrust ) as,−sign(QF − QB ) · | cos θthrust |,(4.7)

whereas that for up-type quarks can be evaluated as,sign(QF − QB ) · | cos θthrust |.

(4.8)

The angular distribution evaluated with −sign(QF −QB )·| cos θthrust | for the events after
the preselection is shown in Fig. 4.16(a), where the AcFB is observed to be negative sign,4.6. INCLUSIVE ANALYSIS

61

and where the dilution of the AbFB and the AcFB due to the charge identification probability
can be seen. Similar angular distributions corresponding to the inner charge identification
class (|QF − QB | < 0.15) and the outer charge identification class (|QF − QB | > 0.15)
are shown in Fig. 4.16(a) and (b), respectively, where the better charge identification
performance for the outer charge identification class can be seen.

Events

(a)
140
120
100
80

Data

60

2 fermions b

40

2 fermions c

20

2 fermions uds0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1

4 fermions-sign(QF-QB) |cosθthrust |Events(c)
Events

(b)

50

90
80

40

70

30

Data 50

60
Data

40

2 fermions b

20

2 fermions b

30
2 fermions c

2 fermions c

10 uds2 fermions

2 fermions uds

20

10
0
-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1-sign(QF-QB) |cosθthrust |0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8
4 fermions

1

4 fermions-sign(QF-QB) |cosθthrust |Figure 4.16: The angular distributions evaluated with −sign(QF −QB )·| cos θthrust | (a) after
the preselection, (b) with the inner class and (c) with the outer class of the hemispherecharge.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

62

4.6.2

Inclusive B and C Tagging

A high performance b-tagging algorithm termed “LB” [81] developed for the Higgs search
at LEP2 is used to purify the bb and cc events.
Overview
The b-tagging algorithm is based on three nearly independent properties of the b-hadron
and its decay products. The b-hadron has a long life and has a characteristic lepton inits semileptonic decay products. It has hard fragmentation and high multiplicity of its
decay products, which makes difference in the jet kinematics from that of uu, dd and ss
events. The algorithm is composed of three tags, lifetime tag, lepton tag and kinematics
tag, corresponding to the three properties. Finally, an unbinned likelihood is taken to
combine the three roughly independent tag. The scheme is shown in Fig. 4.17.
Hemisphere
Jet Clusteringin Hemisphere
Kinematic
Variables

ImpactParameter

Secondary
VertexKinematics ANN

Lifetime ANN

Lekelihood(kinematics)

Likelihood(lifetime)

LeptonTag

Likelihood(lepton pt)

Unbinned Likelihood
to Combine 3 Tags
in Each HemisphereB-tag Variable
Figure 4.17: The scheme of b-tagging algorithm, LB.

4.6. INCLUSIVE ANALYSIS

63

Jet Clustering
The cone algorithm is used for the jet clustering since it is better to include the tracks
from a b hadron forming a secondary vertex and exclude the tracks from the others,
which makes the jet sensitive to the nature of the b hadron. Such characteristics of thecone algorithm is described in the appendix C. The jet clustering is performed for eachhemisphere with the cone algorithm where the half angle of the cone is 0.5 radian and theminimum energy in forming the jet is 7 GeV.
Secondary Vertex Reconstruction
The secondary vertex reconstruction is one of the most important piece in the b-tagging.
One secondary vertex is formed from each jet constructed as described in the subsection 4.6.2. In order to increase the purity of the tracks from the secondary vertex, tracks
in the jet are ranked by an artificial neural net (ANN) termed “track-ANN” which tends
to give a lower value for the track coming from the secondary vertex using the impact
parameter of the tracks with respect to the primary vertex. The secondary vertex isreconstructed as follows.
• Tracks in a jet are sorted in a ascending order of the track-ANN discriminator.
• The first 6 tracks in the ordered tracks are selected as seed tracks which are used in
the vertex search for the first time. If the number of tracks in the given jet is lessthan 6, all tracks are used as the seed tracks.
• If there are other tracks with the track-ANN outputs larger than 0.4, these tracks
are additionally selected as the seed tracks.
• One vertex candidate is formed from the seed tracks as follows. A temporal vertex
is formed by a fit using all tracks at first and then the track with the highest
contribution to the χ2 of the vertex fit is removed. This procedure is repeateduntil no track contributes more than 5 to the χ2 and until at least two tracks areremained. Finally, one vertex is obtained, which is termed “seed vertex”.
• The remaining tracks not used for the seed vertex are sorted in the distance fromthe seed vertex.
• The tracks are added one by one from the nearest track into the seed vertex andvertex fit is performed again. If the contribution to the χ2 in the fit is smaller
than 5, the original seed vertex is replaced by the new vertex and the procedure isrepeated. Then one final vertex is formed.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

64

As regards each secondary vertex, a decay length significance is calculated with a distancefrom the primary vertex to the secondary vertex (l) and its error (σl ), as l/σl . This will
be used in the lifetime tag.
Lifetime Tag
The lifetime tag utilizes the information of the decay length significance, which is powerfulb-tagging variable reflecting the long lifetime of b hadrons. It utilizes also the informationof the impact parameter of the tracks which keeps the b-tagging efficiency in case no
secondary vertex is formed. Four such variables are constructed as follows.
At first, the secondary-vertex likelihood, LSV , is calculated for each hemisphere withall the decay length significances in the hemisphere as follows. The probability density
functions (PDF) of the decay length significance for bb events, cc events and uds events(uu, dd and ss events) are constructed with Monte Carlo samples. The combined estimator
Fq for each quark flavor q is computed by multiplying the PDF of the quark flavor for all
tracks. The LSV is obtained as Fb /(Fb + Fc + Fuds ).
Next, the impact-parameter likelihood, LIP , is calculated for each hemisphere with all
the tracks in the hemisphere as follows. The impact parameter significances in the r − φand r − z projections are formed by dividing the track impact parameters in the r − φplane3 and that in the r − z plane4 by their errors, respectively. The PDFs for the twovariables for each quark flavor are obtained from the Monte Carlo samples. The combined
estimator Fq for each quark flavor q is computed by multiplying the PDFs of two variables
for all tracks. The LIP is obtained as Fb /(Fb + Fc + Fuds ).
Then, the reduced secondary-vertex likelihood, RSV , is calculated as follows. To reduce
sensitivity to single mis-measured tracks, the track with the largest impact parametersignificance is removed from the secondary vertex candidate and the remaining tracks
are used to recompute the likelihood LSV , which is used as RSV . If the original vertex
has only two tracks, the impact parameter significance of the remaining track after theremoval of the track with the largest impact parameter significance is used as RSV .
Finally, the reduced vertex significance likelihood, RIP , is calculated as follows. The
track having the largest impact parameter significance is removed in the calculation of
LIP , which is used as RIP .
These highly correlating variables, LSV , RSV , LIP and RIP are combined with ANN
and a corresponding likelihood, lifetime likelihood (Llifetime ), is constructed. The distributions of the four inputs variables and Llifetime for b are shown in Fig. 4.18(a)-(e).
3The distance between the point of closest approach to the beam axis and the the nominal interaction
point in the r − φ plane.
4
The z coordinate of the point of closest approach to the beam axis.

4.6. INCLUSIVE ANALYSIS

65(b)
EventsEvents

(a)
103

102

3

10

102
Data

Data

2 fermions b

10

2 fermions b

10

2 fermions c

1

2 fermions uds0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2 fermions c

1

2 fermions uds

0

4 fermions

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

LSV

4 fermions

R SV

(d)
Events(c)
Events

1

103

102

103

102
Data

10

2 fermions b

Data

10

2 fermions b

2 fermions c

1

2 fermions uds0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

4 fermions

2 fermions c

1

2 fermions uds0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

LIP1

4 fermions

R IP

Events

(e)

103

102
Data
2 fermions b

10

2 fermions c

1

2 fermions uds0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

4 fermionsLblifetime

Figure 4.18: The input variables for the lifetime ANN and lifetime likelihood.
(a)LSV ,(b)RSV ,(c)LIP ,(d)RIP ,(e) Llifetime for b.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

66
Lepton Tag

The lepton identification is performed in the jets constructed in the subsection 4.6.2. The
transverse momentum (pt ) of the lepton is calculated with respect to the direction of thejet axis. The lepton likelihood (Llepton ) is constructed by combining all the leptons foundin the hemisphere. The distributions of pt and Llepton for b are shown in Fig. 4.19 (a) and(b), respectively.

(b)
EventsEvents

(a)
70
60

103

50
40

Data

30

2 fermions b

102

Data
2 fermions b

10
20

2 fermions c

1000

2 fermions uds

0.5

1

1.5

2

2.5

3

3.5

4

4 fermions

2 fermions c

1

2 fermions uds0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Lepton pT

1

4 fermions

Lblepton

Figure 4.19: (a)The lepton pt distribution. (b)The distribution of Llepton for b.
Kinematics Tag
Two kinds of alternative jets are constructed for both hemisphere using the cone jetclustering algorithm. One is constructed with relatively narrow cone size (0.4 radian) toenrich the tracks or clusters from b-hadrons, which is termed “narrow jet”. The otheris constructed with wider cone size of 1.5 radian to keep the jet shape stable, which istermed “broad jet”. If two or more jets are formed, the most energetic jet is used. Withthe two kinds of the jets, three kinematic variables are introduced as follows.
• Number of tracks or clusters in the narrow jet, where the double counting of theassociated clusters to tracks is corrected.
• Angle between the jet axis and sphericity axis for the narrow jet, which is calculated
at the rest frame of the jet (boosted sphericity axis).
• C-parameter [82] for the broad jet, which is calculated at the rest frame of the jet(boosted C-parameter). The C-parameter is one of the jet shape variables ranging
from 0 to 1. Events with C = 0 have a perfectly two-jet-like final state and eventswith C = 1 have an isotropic and coplanar distribution of final-state momenta.

4.6. INCLUSIVE ANALYSIS

67

These three variable are combined with ANN to from corresponding likelihood, kinematics
likelihood (Lkinematics ). Such three input variables for the ANN and Lkinematics for b are
shown in Fig. 4.20(a)-(d).

(b)
EventsEvents

(a)
600

600
500

500

400

400
Data

Data

300

300
2 fermions b

200

2 fermions b

200
2 fermions c

100

2 fermions c

100
2 fermions uds

0
0

5

10 15

20

25 30 35

40

45 50

2 fermions uds

00

4 fermions# of particles in narrow jet

1

4 fermions

Angle between jet and boosted sphericity axis(d)
Events(c)
Events

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

300

1000

250
800
200
Data

150

2 fermions b

100

600

Data
2 fermions b

400

2 fermions c

2 fermions c

200

50
2 fermions uds

00

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Boosted C parameter4 fermions

2 fermions uds

00

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

4 fermions

Lbkinematics

Figure 4.20: The input variables for the kinematics ANN and kinematics likelihood. (a)The number of tracks and clusters in the narrow jet. (b)The angle between the jet axis
and the boosted supercity axis. (c) The boosted C parameter. (d) Lkinematics for b.

Combined Likelihood Tag
Three kinds of likelihood variables corresponding to three kinds of tags of the lifetime,
lepton and kinematics tags for each hemisphere yield 6 likelihood variables. For eachlikelihood variable, probability densities for the bb, cc and the remaining uu, dd,ss eventsare calculated as fb , fc and fuds , respectively. Superscripts are used to indicated the kindof likelihood and hemisphere. For example, fblifetime,1 means probability density for bb
events in the hemisphere 1 corresponding to the lifetime tag. A combined probability is

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

68
defined as

fbcomb = fblifetime fblepton fbkinematics .

(4.9)

The 6 variables mentioned above are combined to form event likelihood (Levent ) withunbinned likelihood as,

Lbevent=

Lcevent =
Luds
event

=

Rb fbcomb,1 fbcomb,2
comb,1 comb,2
fuds
Rb fbcomb,1 fbcomb,2 + Rc fccomb,1 fccomb,2 + Ruds fudsRc fccomb,1 fccomb,2
comb,1 comb,2
fuds
Rb fbcomb,1 fbcomb,2 + Rc fccomb,1 fccomb,2 + Ruds fuds
comb,1 comb,2
Ruds fuds
fuds
comb,1 comb,2
fuds
Rb fbcomb,1 fbcomb,2 + Rc fccomb,1 fccomb,2 + Ruds fuds,

(4.10),(4.11)(4.12)The distributions of event likelihood for b (Lbevent ), for c (Lcevent ) and for uds (Luds
event )
are shown in Fig. 4.21(a), (b) and (c), respectively. The performance of each b-tagging
variable is shown in Fig. 4.22(a). The c-tagging and uds-tagging performance for c anduds event likelihood are shown in Fig. 4.22(b), (c), respectively. A significance of inclusive
analysis (inclusive significance) is defined as(inclusive)

Psig≡ (2Pb − 1)Lbevent ,(4.13)

where Pb is the charge identification probability defined in subsection 4.6.1 for b quarks.

4.6. INCLUSIVE ANALYSIS

69

Events

(a)

3

10

102
Data
2 fermions b

10

2 fermions c

1

2 fermions uds0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

4 fermions

Lbevent

Events(c)
Events

(b)

102

102

Data

10

Data

10

1

2 fermions b

2 fermions b

2 fermions c

2 fermions c

2 fermions uds0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Lcevent

4 fermions

1

2 fermions uds0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Luds
event

Figure 4.21: The event likelihood of (a) Lbevent , (b) Lcevent and (c) Luds
event . The lines indicate
cut position used in the event classification in the next section.

4 fermionsCHAPTER 4. MEASUREMENTS OF THE AB,C
FB

70

Purity

(a)

1
0.8

LSV
RSV
LIP
RIP
Lblifetime
Lblepton
Lbkinematics
Lbevent

0.6

0.4

0.2

0
0

0.2

0.4

0.6

0.8

1

b-tag Efficiency

Purity(c)
Purity

(b)
1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

0.2

0.4

0.6

0.8

1

Lcevent : c-tag Efficiency

0
0

0.2

0.4

0.6

0.8

1

Luds
event : uds-tag Efficiency

Figure 4.22: (a) The b-tag performance and (b) c-tag performance and (c) uds-tag performance of LB.

4.6. INCLUSIVE ANALYSIS

71

Event Classification with b-tagging variables
Event classification is performed with the Lbevent , Lcevent and Luds
event in order to obtain
event sub-samples with high sensitivities to the Ab,c
FB . The two thresholds of 0.8 and
0.7 are assigned to Lbevent and a threshold of 0.6 is assigned to Lcevent and a threshold of
0.9 is assigned to Luds
event , which are indicated in Fig. 4.21. With these thresholds, eventclassification is performed as,classI : 0.8 < Lbevent ,(4.14)

classII : 0.7 < Lbevent < 0.8,

(4.15)

classIII :Lbevent < 0.7 and 0.6 < Lcevent ,(4.16)

classIV :Lbevent < 0.7 and

Lcevent < 0.6 and 0.9 < Luds
event ,

(4.17)

classV :Lbevent < 0.7 and

Lcevent < 0.6 andLuds
event < 0.9. (4.18)

These threshold values are optimized to get accurate values of AFB . The angular distributions corresponding to each b, c-tagging subclass are shown in Fig. 4.23, where the quark
direction is evaluated with−sign(QF − QB ) · | cos θthrust |.

(4.19)The sample of class I is most sensitive to the AbFB with less correlations to other AFB .
The class II still has high sensitivity to the AbFB with small correlations to other AFB . The
class III is sensitive to the AcFB but included large correlations to other AFB . The class
IV is sensitive to the Auds
FB which is defined as
d
u
s
Auds
FB = −fd AFB + fu AFB − fs AFB .

(4.20)

where fu is the uu event fraction to the sum of uu, dd and ss events. The fd and fs aredefined similarly. The variable Auds
FB is meaningful only when the fu , fd and fs are stableagainst the b-tagging. The stabilities are natural because the b-tagging has little power
to separate uu, dd and ss events and it is checked with MC to find that the deviations arewithin the statistical fluctuations. The class V is less sensitive to AFB due to the large
contaminations of each flavor. It would also increase the correlations between AbFB , AcFB
and Auds
FB . Therefore AFB fits are performed to the class I,II,III and IV. The events in theclass V are rejected at the final event selection.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

72

Events

(a)
30
25
20
Data

15
2 fermions b

10
2 fermions c

5
2 fermions uds0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1

4 fermions-sign(QF-QB)| cosθthrust |(c)
EventsEvents

(b)
6
5

18
16
14
12

4

10
3

8
6

2

4
1

2

0
-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1-sign(QF-QB)| cosθthrust |(e)
22

EventsEvents(d)1-sign(QF-QB)| cosθthrust |

20

250

18
200

16
14

150

12
10

100

8
6

50

4
2
0
-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1-sign(QF-QB)| cosθthrust |0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1-sign(QF-QB)| cosθthrust |Figure 4.23: Angular distributions corresponding to each b-tag subclass, (a) class I, (b)class II, (c) class III, (d) class IV and (e) class V.

4.6. INCLUSIVE ANALYSIS

4.6.3

73

Unbinned Likelihood Fit

Finally, events in the b-tag event class V are rejected as described in the section 4.6.2.
Consequently, two charge identification class (inner and outer class) for each b-tag eventclass yield 8 event subclasses. Angular distributions corresponding to the 8 subclasses areshown in Fig. 4.24 and the numbers of events are shown in Table. 4.5. A probability density
function is constructed according as the AFB and unbinned likelihood fit is performed toeach subsample.

Table 4.5: The numbers of events in each subclass atcharge ID
b-tag
Data
MC
(b)
(c)
(s)
(u)
(d)
(4-fermion)√s of 189 GeV.

inner class
I II III IV
252
93 23 55 81

outer class
I II III IV
407
119 33 98 157

100 19

67

65

122

25

98

171

671

91
5
0
0
0
2

11
31
5
8
6
4

0
8
14
20
17
4

112 18
4 3
0 0
1 0
0 0
2 0

18
43
8
14
7
6

1
19
38
63
39
10

267
120
67
109
73
32

13
3
0
0
0
0

sum
659

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

74

(b)
EventsEvents

(a)
16
14

14
12

12

10

10

8

8
6
6
4

4

2

2
0
-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1-sign(QF-QB)| cosθthrust |(d)EventsEvents

(c)
5

5

4

4

3

3

2

2

1

1

0
-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1-sign(QF-QB)| cosθthrust |1-sign(QF-QB)| cosθthrust |(f)
EventsEvents

(e)
9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0
-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1-sign(QF-QB)| cosθthrust |1-sign(QF-QB)| cosθthrust |(h)Events(g)
Events1-sign(QF-QB)| cosθthrust |

9
8

18
16
14

7

12

6

10

5
4

8

3

6

2

4

1

2

Data
2 fermions b0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1-sign(QF-QB)| cosθthrust |2 fermions c
2 fermions uds0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1

4 fermions-sign(QF-QB)| cosθthrust |Figure 4.24: Angular distributions corresponding to each subclass constructed from eventclassifications concerning b-tag and charge identification. b-tag (a) b-tag class I, (c) b-tag
class II, (e) b-tag class III, (g) b-tag class IV in the inner charge identification class and(b) b-tag class I, (d) b-tag class II, (f) b-tag class III, (h) b-tag class IV in the outercharge identification class.

4.6. INCLUSIVE ANALYSIS

75

Probability Density Function
The observed angular distribution can be written with AFB s for bb, cc, ss, uu, dd and4-fermion events as,·¸Xdσ obs8b,c,d,u,s,4fq
0q2= C (AFB)
Rq ² (x) 1 + x + (2Pq − 1)sq AFB x ,dx
3
q=b,c,d,u,s,4f(4.21)

where• C 0 (Ab,c,d,u,s,4f) is the normalization factor possibly depend on the AFB ,FB• Rq is the initial fraction of sample q,• x is the estimator of the emitting angle of primary quark:x = −sign(QF − QB )| cos θthrust |,(4.22)• ²q (x) is the event selection efficiency depend on x,• Pq is the charge identification probability• sq is 1 for both down-type quark and 4-fermion samples, -1 for up-type quark samples.
The event selections are performed so as to have dependence on θthrust only in the formof | cos θthrust |. If not, it would induce obvious bias on the AFB . Hence ²q (x) is naturally
a symmetric function and it is checked with MC. Under such condition, the possible
dependence to Ab,c,d,u,s,4f
of normalization factor, C 0 is vanished and it can be rewritten
FBwith a constant normalization factor, C, as·¸
X8dσ obsqq2= C
Rq ² (x) 1 + x + (2Pq − 1)sq AFB xdx
3
q=b,c,d,u,s,4f(4.23)

If the function shapes of the ²q (x) are the same for all q, the common function shape,²(x)is extracted and the equation can be written as·¸
Xdσ obs8q2= C²(x)
1 + x + Fq (2Pq − 1)sq AFB x ,dx
3
q=b,c,d,u,s,4f≡ pdf(x, AqFB ),(4.24)(4.25)

where Fq is the event fraction of sample q after the final event selection. This function canbe used as the probability density function (pdf(x, AqFB )) to fit the AFB and the likelihood

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

76
L can written as,
X

log L =

log pdf(xi , AqFB ),(4.26)i∈all dataX=

log C²(xi )i∈all data

(

X

+

i∈all data

·
¸)
8
log
1 + x2i + Fq (2Pq − 1)sq AqFB xi(4.27)
3
q=b,c,d,u,s,4f
X

The first term is the constant term against the AFB and can be removed from the likelihood
function to fit the AFB by maximizing the log likelihood. Accordingly simpler likelihood
function is obtained aslog L =

X
i∈all data

(
log

X

·

b,c,d,u,s,4f

If Auds
FB is introduced as
Auds
FB ≡

8
1 + x2i + Fq (2Pq − 1)sq AqFB xi
3
X

¸)
.

Fq sq AqFB

(4.28)(4.29)

q=d,u,sand the equation can be written in terms of Auds
FB as,
(
¸)
X
X ·
8
q
,
log L =
log
1 + x2i + Fq (2Pq − 1)sq AFB xi
3b,c,uds,4f
i∈all data

(4.30)

where Pdus is defined to keep the identity between the two equation and sdus is set to 1.
Probability Density Function with Subsamples
The data have already divided into 8 subsamples with different sensitivities to AbFB , AcFB
and Auds
FB . In order to realize the condition that the function shapes of ²q (x) are the sameamong each qq and 4-fermion, the angular region are symmetrically divided into 3 regionsas,• Central

Region :| cos θthrust | < 0.65,• Transition Region : 0.65 < | cos θthrust | < 0.8,

• Endcap

Region : 0.8 < | cos θthrust |.

In such a region, the requirement of the same function shapes of ²q (x) is fulfilled wellchecked with MC and the division is altered to 2 or 4 to check the stability and to4.6. INCLUSIVE ANALYSIS

77

evaluate corresponding systematic error. Consequently, the AFB fitting is performed for
24 subsamples. The probability density function described in the previous section are
adjusted to the condition as,
X
log L =
log Ljsub(4.31)j∈all subsamples
where likelihood function in the subsample Ljsub is defined as,
(
¸)
X
X ·
8
j
q
log Lsub =log
1 + x2i + Fqj (2Pqj − 1)sq AFB xi.
(4.32)
3b,c,uds,4f
i∈subsample j
The definition of Auds is no longer valid since the event ratio among uū, dd¯ and ss̄ samples
FB

are differ between the two charge identification classes although it is stable against theb-tag classes. The Auds
FB is redefined separately in the two charge identification classes as,
X
Fqin sq AqFB
Adus,in≡(4.33)
FB
q=d,u,sAdus,out
FB

X≡

Fqout sq AqFB

(4.34)

q=d,u,s, where Fqin and Fqout are the event fraction in the inner or outer charge identification classrespectively.
Accordingly, the charge identification probabilities, flavor composition and 4-fermion
contamination are evaluated in each subsample and the fitting of the 4 parameters, AbFB ,AcFB , Auds,inand Auds,out
is performed simultaneously. As regards the AFB of the 4-fermion
FB
FB
obs,4fsample (A4f
) is defined as
FB ), an observed AFB (AFB
obs,4f
AFB≡ (2P4f − 1)s4f A4f
FB .

(4.35)is set to zero in the fitting. The possible contribution from the non-zero A4f
The Aobs,4f
FB
FB
is evaluated with Monte Carlo and is added to the systematic error.
Fitting
Fitting is performed to maximize the log L and the following values are obtained atof 189 GeV aswith a correlation matrix of

AbFB
AcFB
Auds,in
FB
Auds,outFB

1.00
 0.29 0.030.09+0.15= 0.56−0.16 ,
+0.47
= 0.59−0.48 ,
= −0.63 ±0.98,= −0.058 ±0.18,

0.29
1.00
0.16
0.51

0.03
0.16
1.00
0.08


0.09
0.51 
.
0.08 
1.00√

s(4.36)(4.37)

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

78

The contour plot corresponding to the 189 GeV result is shown in Fig. 4.25. The resultsfor each energy point are shown in Table. 4.6. If |AFB | exceeds 0.75, the AFB is unphysical.
However the unphysical fitted value of the AFB is possible, since the observed AFB in thephysical region as shown in Fig. 4.24 is converted according to the sample compositionc
AFB

and the charge identification probabilities.

2
1.5
1
0.5
1 σ contour0

2 σ contour-0.5

SM AbFB-1

SM AcFB-1.5(with QCD effect)

-2
-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(without IFI)

AbFB

Figure 4.25: A one σ and two σ contours of the Ab,c
FB result are shown.

Table 4.6: Results for the inclusive analysis with statistical errors.
Ecm
[GeV]
91
133
161
172
183
189
192
196
200
202
205
207

AbFB
(stat.)
0.10 ±0.03
0.61 +0.30
−0.32
+0.36
0.15 −0.37
1.53 +0.37−0.48
1.17 +0.23
−0.26
+0.15
0.56 −0.16
0.78 +0.37−0.42
0.46 +0.30−0.31
0.34 ±0.35
0.07 +0.46−0.47
0.48 +0.27−0.28
0.29 ±0.22

SM
Ab,
FB

0.10
0.48
0.55
0.56
0.58
0.58
0.58
0.58
0.59
0.59
0.59
0.59

AcFB
(stat.)
0.05 ±0.18
2.50 +0.93−1.21−0.65 +1.82−2.10−0.29 +1.33−1.23
2.30 +0.63−0.73
0.59 +0.47−0.48−0.21 +2.02−1.81−0.63 +0.92−0.89
0.87 +1.03−1.07−0.37 +1.30−1.27
0.60 +0.78−0.81
1.19 +0.64
−0.66

SM
Ac,
FB

Correlation

0.06
0.69
0.67
0.67
0.66
0.66
0.65
0.65
0.65
0.65
0.65
0.64

0.31
0.35
0.33
0.53
0.29
0.29
0.42
0.34
0.26
0.35
0.35
0.31

4.6. INCLUSIVE ANALYSIS

4.6.4

79

Corrections Applied to the Fitted Values of AFB

The fitted values of AFB correspond to the distribution of initial-quark hemisphere ×| cos θthrust |after the preselection. The shifts of the AFB in the preselection should be corrected. The
QCD effect should be recovered. The fitted values includes the effect of the ISR-FSR
interference, which should be removed finally.
Biases in the Event Preselection The thrust axis is the good estimator of the initialquark direction, which is shown in Table. 4.7. The shifts of the AFB at each stage of theevent preselection are also shown.

Table 4.7: The AFB shifts in the pre-selection for the initial-quark direction.
variablecos θinitialquark(initial-quark
hemisphere)
×| cos θthrust |cuttrue s’true s’measured s’+L2MH.
+4f rej.
+| cos θ| cut

AbFB
0.586 ± 0.005
0.583 ± 0.005
0.557 ± 0.005
0.558 ± 0.005
0.562 ± 0.005
0.569 ± 0.006

Sum

∆AbFB
-0.003-0.026
0.001
0.004
0.008
-0.016

AcFB∆AcFB
0.679 ± 0.003
0.674 ± 0.003 -0.005
0.657 ± 0.003 -0.017
0.657 ± 0.003 -0.000
0.660 ± 0.003 0.003
0.667 ± 0.004 0.007-0.012QCD Correction The QCD effect is evaluated with the Zfitter as shown in Table. 4.8.

Table 4.8: The QCD effect to the AFB evaluated with the Zfitter.
cutwith QCD, without IFI
without QCD, without IFI

AbFB
0.580
0.596

∆AbFB
0.016

AcFB
0.656
0.677

∆AcFB
0.021

IFI effect The IFI effect included in the fitted value is evaluated with the Zfitter asshown in Table. 4.9.

Table 4.9: The IFI effect evaluated with the Zfitter..
cut
AbFB ∆AbFB
AcFB ∆AcFB
without QCD, without IFI 0.596
0.677
without QCD, with IFI
0.601 0.005 0.668 -0.009

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

80

Total Correction The total corrections to be applied to the fitted AFB are shown in
Table. 4.10. These corrections are already included in the results on Table 4.6, since these
corrections have been applied in the fitting procedure.

Table 4.10: The total correction to be added to the fitted AFB . The error was due to the
MC statistics used for evaluating the shift in the event preselection.
item
IFI
QCD
pre-selection
sum

4.7

correction to the fitted AbFB
-0.005-0.0160.028 ± 0.007
0.007 ± 0.007

correction to the fittedAcFB
0.009-0.021
0.022 ± 0.004
0.010 ± 0.004

Lepton Analysis

The thrust axis is used to estimate the primary quark direction also in the lepton analysis. Leptons from semileptonic decay of bottom or charm hadrons (prompt leptons)
are characteristics in their high momentum and high transverse momentum, which areused to identify the bb or cc events. Their charges are used to distinguish whether the
leptons come from quarks or anti-quarks. In spite of the neutral B meson mixing, thequark-hemisphere identification with the lepton charge is still remarkable compared withthat in the inclusive analysis, which makes the lepton analysis significant compensating
the lower statistics due to the small branching ratio of the semileptonic decays.
In this section, the description about the prompt lepton is given at first. Then thedescription about the lepton analysis is followed. Lepton (electron or muon) tracks are
selected with the OPAL electron [83] and muon [42] identification from the preselected
events in the section 4.5. The conversion electrons are tagged with the OPAL conversionelectron finder [83] and are rejected. From the lepton tracks, the prompt-lepton tracks are
selected and are classified by their flavors with an ANN constructed to discriminate theprompt leptons. Then events with the prompt leptons are selected with the informationof the the prompt-lepton tracks and their angular distributions are evaluated with boththe lepton charge and the thrust axis, which are fitted with unbinned likelihood fitting toobtain the AbFB and AcFB simultaneously.

4.7.1

Prompt Leptons

Leptons (electrons and muons) produced in semileptonic decays of bottom and charm
hadrons usually are referred to as prompt leptons. Such leptons carry and hold the4.7. LEPTON ANALYSIS

81

information of the direction and the charge of the primary quark. In this analysis following
prompt leptons are treated as signals, where ` represents an electron or a muon.
• b→`
Leptons coming directly from the weak decays of b hadrons.
• b→τ →`
Electrons and muons from leptonic τ decays where the τ lepton comes from a directb decay.
• b→c→`
Leptons from semileptonic decays of c hadrons to which b hadrons decay.
• b→c→`
Leptons from semileptonic decays of anti-c hadrons to which b hadrons decay.
• c→`
Leptons coming directly from the weak decay of primary c hadrons.
Leptons from cc mesons, e.g. b → J/ψ → `+ `− decays, are included within the b → c → `or b → c → ` with the appropriate sign.
The relationships between the lepton charge and its parent primary quark or antiquark
are important issues. The b → `, b → τ → `, b → c → ` and c → ` have the same sign oflepton charge as the primary quark’s, while the b → c → ` have the opposite sign. Fromthe point of such view, b → `, b → τ → ` and b → c → ` are unified to bl− sample andb → c → ` and c → ` were newly termed bcl+ and cl+ , respectively.
The opposite sign of the lepton charge to that of the primary quark in the bcl+
sample dilutes the charge identification performance. The existence of B0 −B0 mixing also
introduces another dilution. Leptons from these mixed B mesons are classified according
to the decaying b quark. For example, b → b → c → ` is treated as b → c → `.
The B0 − B0 mixing at LEP is characterized by a mixing parameter (χ) which is the
expectation value of mixing rate including the contribution of both B0 and B0s . The χ
was measured to be 0.1257 ± 0.0042 [51]. The observed AFB is decreased by a factor of(1 − 2χ) effectively due to the mixing.

4.7.2

Lepton Identification

Electron Identification
The selection of electrons relies mainly on the dE/dx and the reconstructed energy in the
ECAL. After the preselection described in section 4.5, electron candidates are preselected
and then discriminated with an ANN.
Electron candidates are preselected with following criteria,

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

82

1. The momentum is larger than 2 GeV.
2. The number of hits used in the dE/dx measurements in the CJ is larger than orequal to 20.
3. The difference between the measured energy loss and that expected for an electron,
divided by the measurement error, is required to be between −2 and 4. Such quantity
is termed normalized energy loss ((dE/dx)norm ).
After the preselection, the electron identification ANN is performed separately in thebarrel region (| cos θ| < 0.8) and in the endcap region (| cos θ| > 0.8). The ANN input
variables in the barrel region are• the track momentum,• the absolute polar angle of the track,• the normalized energy loss ((dE/dx)norm ),• the dE/dx measurement error,• the energy deposited in the ECAL in a cone of half-angle 30 mrad around the track,
divided by the track momentum, (Such energy in the cone is termed Econe .)• the number of lead glass blocks used to calculate Econe .
In the endcap, two additional variables are used:• the difference in the polar angle between the track position extrapolated to the
ECAL and the center of the associated cluster,• the corresponding difference in φ.
The (dE/dx)norm and the output of the electron identification ANN (Nel ) are shown in
Fig. 4.26. In addition to the electron preselection, the Nel is required to be larger than0.9, with which the efficiency for selecting prompt electrons is approximately 25% at 189
GeV. The resulting sample is 75% pure in electrons.

4.7. LEPTON ANALYSIS

83(b)
TracksEvents

(a)
104

103
3

10

Data

102

102

Data

othersnon prompt µ

10othersnon prompt µ10

prompt µ
conversion e

1

prompt µ
conversion e

1

non prompt e

-15

-10

-5

0

5

10

15

prompt enon prompt e00.2

Normalized dE/dx

0.4

0.6

0.8

1

prompt e

Nel

Figure 4.26: (a) The normalized dE/dx before electron preselection and (b) Nel after thepreselection of the electron identification.

After this selection, electrons from photon conversions (conversion electrons) are adominant background to prompt electrons. A separate ANN is used to identify pairs ofconversion electrons (conversion ANN), where track pairs are made by associating a trackwith all the other tracks. The nine inputs of the conversion ANN, related to whether apair of tracks formed a vertex compatible with a photon conversion, are listed as follows.
• the distance in r-φ between the two tracks measured at points where the two tracks
where parallel in r-φ plane.
• the radius of the first measured hit of both tracks, as well as the radius of theircommon vertex;• the invariant mass of the pair assuming both tracks to be electrons and the impact
parameter of the parent photon with respect to the primary vertex of the event;• the two track momenta signed by their charge;• the Nel of the partner track.
The distribution of the ANN output (Ncv ) after the cut of Nel > 0.9 is shown in
Fig. 4.27. The contribution of the conversion electrons is reduced by requiring that the
Ncv is less than 0.4.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB
Tracks

84

102

Data

10

othersnon prompt µ
prompt µ
conversion e

1

non prompt e

0

0.2

0.4

0.6

0.8

1

prompt e

Ncv

Figure 4.27: The Ncv distribution after the cut of Nel > 0.9.

Muon Identification
The identification of muons is based mainly on the matching of track segments reconstructed in the muon chamber with extrapolated tracks from the central detector. In
addition, information both from the HCAL, where muons are expected to leave hits in alllayers, and from the dE/dx in the CJ is used.
The muon track preselection is performed as follows.
• The momentum of the track is larger than 2 GeV.
• For each track segment in the muon chambers, only one track with “best matching”
is selected, where the “best matching” means the track extrapolated to the muon
detector has the smallest angular separation to the muon track segment.
Then further identification with an ANN (muon ANN) is performed. The ANN inputs
are divided into 4 types as muon matching, HCAL, dE/dx and geometrical information.
They are listed below.
• Information from the matching:– The square root of the χ2 for the position match in θ and φ between the extrapolated track and the associated muon track segment in the muon chambers, as
described in [84];– the ratio of distances of the best and second best matching track to the muon
segment; this is a measure of how ambiguous the choice of the best matching
track is in the preselection;– the χ2 probability for the matching computed using both position and direction
information for the track in the central detector and the associated muon track
segment.

4.7. LEPTON ANALYSIS

85• Information from the hadron calorimeter:– The number of HCAL layers with hits in the cluster associated with the central
track;– the layer number of the outermost HCAL layer with hits associated with thecentral track;– the χ2 probability for the match in θ and φ between the track (extrapolated
to the HCAL) and the associated cluster.
• Specific energy loss:– The muon dE/dx weight for the track, which is a measure of the probability
that the track is compatible with a muon hypothesis;– σdE/dx , the error on the dE/dx measurement;– the momentum of the track.
• Geometrical information:– The position in | cos θ| and φ where the extrapolated track enters the muonchambers.
The distribution of the muon best matching flag and the neural network output (Nµ ) isshown in Fig. 4.28. The Nµ is required to be larger than 0.65, where the selection efficiency
for the prompt muons are evaluated to be 43% and the muon purity is evaluated to be√
73% with the MC at s of 189 GeV.

(b)
TracksTracks

(a)
105

103

104
102
3

10

Data

102

othersDataothers10

conversion e

10

conversion e

non prompt e
prompt e

1

non prompt e
prompt e

1

non prompt µ00.2 0.4 0.6 0.81

1.2 1.4 1.6 1.8

2µ best match flag

prompt µ

non prompt µ

0

0.2

0.4

0.6

0.8

1

prompt µ

Nµ

Figure 4.28: (a) The muon best match flag (1 means “best match”) for the tracks before
the muon preselection and (b) Nµ after the muon preselection.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

86

4.7.3

b and c Tagging and Flavor Separation

Three types of prompt leptons, bl− , bcl+ and cl+ , have different sensitivity to AbFB and
AcFB . Hence it is important both to enhance them and also separate them. For the purpose,three ANNs are constructed separately for electrons and muons, Nb to discriminate b → `tracks, Nbc to discriminate b → c → ` tracks and Nc to discriminate c → ` tracks.
Construction of Artificial Neural Networks
All three ANNs giving track-by-track discrimination use the same input variables related
to a track. As regards the track, three kinds of jets are used to construct some of the
input variables. One is the jet containing the track, which is labeled as “associated jet”.
The next is a jet with highest energy in the opposite hemisphere of the track, which islabeled as “other hemisphere jet”. The last is constructed with particles5 in the associatedjet which are nearer to the track than the jet axis, which is labeled as “subjet”. Eachjet is formed with the cone algorithm (the half-angle of the cone is 0.55 radian and theminimum energy of 5 GeV). The ANN inputs are listed as follows.
• p, the track momentum;• pt , the transverse momentum of the track with respect to the jet axis of the associated jet, where the track is excluded in the calculation of the jet axis;• Esubjet , the energy of the subjet;
vis• Ejet, the sum of the energies of the particles within the associated jet;•Ppt , scalar sum of the transverse momentum with respect to the ordinary jet axis

in the associated jet;
• d/σd , the impact parameter significance in the three-dimensional space of the trackwith respect to the primary vertex, where the impact parameter is defined as the
distance between the primary vertex and the track at the point of closest approachand its significance was defined as the impact parameter divided by its error;• Nfwd , the number of tracks with d/σd larger than 2 (forward multiplicity) in theassociated jet;
• L/σL , the decay length significances of the secondary vertexes (if existed) in theassociated jet, where secondary vertices’s are reconstructed in the three-dimensional
spaces;5

tracks and unassociated clusters.

4.7. LEPTON ANALYSIS

87• Lo /σLo , that corresponding to the other hemisphere jet;
• Qj × Ql , the jet charges of the associated jet multiplied by the track charge, where
the jet charge is defined as in Eq. 4.2 with κ = 0.4, but using only tracks associatedto the jet;
• Qoj × Ql , that corresponding to the other hemisphere jet;
• Nel and Ncv , the outputs of the electron identification ANN and the photon conversion ANN respectively, used only in the case of tracks identified as electrons;• Nµ , the output of the muon identification ANN, used only in the case of tracks
identified as muons;
The kinematic properties of the prompt leptons are best reflected to the two variables, p
and pt . In the direct decay (b → ` or c → `) the leptons tend to have large momentum dueto the hard fragmentation of the b or c hadrons. The large mass of the b quark yields highlepton energies in the rest frame of the decaying b-hadrons. The transverse component
is kept in the boost along the jet direction which resulted in the hard pt spectrum of
the b → `, b → c → ` and b → c → ` decays. The presence of a subjet is morecharacteristic of cascade decays (b → c → ` and b → c → `), since the fragmentation of
the secondary charm quarks is likely to form subjet near the lepton. The total jet energy,vis
Ejet, characterizes b-jets with semileptonic-decay leptons which are likely to have lowerPvisible energy due to the emission of an energetic neutrino. The variable,
pt , reflects
the hard fragmentation and high multiplicity of the particles in b-jets. The variables d/σd ,Nfwd , L/σL and Lo /σLo reflect the long life time, hard fragmentation and high multiplicity
in the decay of B mesons. The variables, Qj × Ql and Qoj × Ql can separate the bcl+
leptons from bl− and bcl+ leptons. The bcl+ leptons have the charge of opposite sign tothat of their primary quarks, while bl− and bcl+ leptons have the charge of the same sign.
Although the variables Nel , Ncv and Nµ have already been used to remove non-prompt
leptons, they are still used for the further separation of them.
With these input variables, 6 ANNs are constructed with JETNET [85]. These networks
have only 1 hidden layer and the number of nodes in the hidden layer is optimized. Moregeneral description of the ANN is given in appendix D, where the structure of the Nb isshown in Fig. D.4, for example. The training of the ANNs are done with events in the√multi-hadronic MC at s of 192 GeV after the lepton selection. For Nb , only b → `leptons are used as signal and other tracks are used as background. The input variables
for electrons are shown in Fig. 4.29 and those for muons are shown in Fig. 4.30. The
outputs of the ANNs are shown in Fig. 4.31.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB(b)
Tracks(c)
Tracks(a)
Tracks

88

102

10
10

1

Data 10

Data

Data

4 fermions

4 fermions

4 fermions

other 2 fermions

other 2 fermions

other 2 fermions

c→e+

c→e+

c→e+

1

b→c→e+

0.05

0.1

0.15

0.2 0.25(d)

0.3 0.35

0.4

(e)

1

2

3

4

5

6

(f)

Pt[GeV]
102

TracksTracksscaled momentum

b→c→e+

0b→e-Tracks

0

1

0

b→e-

b→c→e+

10

20 30

40

50 60 7080

90 100b→e-

Esubjet[GeV]

102

10
Data

1

10

Data

4 fermions

other 2 fermions

other 2 fermions

other 2 fermions

c→e+

c→e+

c→e+

1

40

60

80

100

(g)

120

140

0

b→e-

5

10

15

20

(h)

25

30

(i)

PtSum[GeV]102

TracksTracksEjet[GeV]

b→c→e+

Tracks20

4 fermions

1b→c→e+

0

Data

10

4 fermionsb→e-b→c→e+

-100

-50

0

50

100

150

b→e-

impact parmeter significance

102

10
Data

1

10

Data 10Data

4 fermions

4 fermions

4 fermions

other 2 fermions

other 2 fermions

other 2 fermions

c→e+

c→e+

c→e+

1

1

b→c→e+

(j)

2

4

6

8

10 12 14

16

18 20

-20

-10

0

10

(k)

20

30

40

50

60

(l)

decay length significanceTracksTracksforward multiplicity

b→e-

b→e-

1

10

20

30

40

50

60

b→e-

10
Data

Data

4 fermions

4 fermions

4 fermions

other 2 fermions

other 2 fermions

other 2 fermions

c→e+

c→e+

c→e+

b→e-

1

1

b→c→e+-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8(m)
TracksLepQ*JetQ

0

Datab→c→e+-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

b→c→e+-10

decay length significance2

10

10

1

-20

Tracks0b→c→e+

1

b→e-

0.9

b→c→e+

0.92

0.94

0.96

0.98

1

b→e-

eNN

LepQ*JetQ2

102

Data

10

4 fermions
other 2 fermionsc→e+

1
b→c→e+

0

0.05

0.1

0.15

0.2 0.25

0.3 0.35

0.4

b→e-

cvNN

Figure 4.29: (a)-(m) The distributions of input variables for electron Nb , Nbc and Nc
after electron selection. The name of the variable are shown in the x-axis in each figure.

4.7. LEPTON ANALYSIS

(c)

102

Tracks(b)
TracksTracks

(a)

89

102

10
10

Data 10

Data

Data

4 fermions

4 fermions

4 fermions

other 2 fermions

other 2 fermions

other 2 fermions

c→µ

c→µ

c→µ

+

1

+

1

b→c→µ

0.05

0.1

0.15

0.2 0.25(d)

0.3 0.35

0.4

-

1

2

3

4

5

(e)

6

Pt[GeV]

TracksTracksscaled momentum

0

b→µ

b→c→µ

+

102

0

b→µ

-

+

10

20 30

40

50 60 70(f)

80

90 100b→µ-Esubjet[GeV]

Tracks

0

+

1

b→c→µ

+

102

10
10

Data

Data

Data

10
4 fermions

4 fermions

other 2 fermions

other 2 fermions

other 2 fermions

c→µ

c→µ

c→µ

+

1

+

1

b→c→µ

40

60

80

100

(g)

120

140

b→c→µ

0

b→µ

-

5

10

15

20

(h)

25

30

PtSum[GeV]102

TracksTracksEjet[GeV]

10

b→µ

-

6

8

10 12 14

16

18 20

102

4 fermions

other 2 fermions

other 2 fermions

c→µ

c→µ

c→µ

+

1

b→µ

-

-20

b→c→µ

-10

(k)

0

10

20

30

40

50

60

decay length significance

b→µ

-

-20

b→c→µ

+

-10

(l)

0

10

20

30

40

50

60

b→µ

-

decay length significance2

10

Data

Data

Data

4 fermions

4 fermions

4 fermions

other 2 fermions

other 2 fermions

other 2 fermions

c→µ

c→µ

c→µ

+

1

b→c→µ
b→µ

-

+

1

b→c→µ

+

LepQ*JetQ

+

1
+

+

1

-

impact parmeter significance10-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

b→µother 2 fermions

10

1

150

Data

TracksTracksforward multiplicity

100

Tracks(j)

450

4 fermions

b→c→µ

2

0

Data 10+0-50

4 fermions

+

1

+

-100

(i)

10

Data

b→c→µ

+

Tracks

20

+

1

+

0

4 fermions-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1

LepQ*JetQ2

b→µ

-

+

b→c→µ

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

b→µ

+

muNN

Figure 4.30: (a)-(l) The distributions of input variables for muon Nb , Nbc and Nc after
electron selection. The name of the variable are shown in the x-axis in each figure.

-

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

90

(b)
TracksTracks

(a)

102

102

Data

Data

10

10
4 fermions

4 fermions

other 2 fermions

other 2 fermions

c→e+

c→µ

+

1

1
b→c→e+

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

b→c→µ

+

0

b→e-

NNb→e

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

NNb→µ

Tracks(d)

Tracks

(c)

102

102

Data

Data

10

10
4 fermions

4 fermions

other 2 fermions

other 2 fermions

c→e+

c→µ

+

1

1
b→c→e+

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

b→c→µ

+

0

b→e-

NNb→#c →e

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

b→µ

-

NNb→#c →µ(f)
Tracks(e)
Tracks

b→µ

-

102

Data

10

102

Data

10

4 fermions

4 fermions

other 2 fermions

other 2 fermions

c→e+

c→µ

+

1

1
b→c→e+

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

NNc→e

b→e-

b→c→µ

+

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

b→µ

-

NNc →µFigure 4.31: The distributions of ANN outputs,(a) electron Nb , (b) muon Nb , (c) electron
Nbc , (d) muon Nbc , (e) electron Nc (f) muon Nc .

4.7. LEPTON ANALYSIS

4.7.4

91

Event Selection

The output of the ANN is interpreted as the probability to discriminate the signal (as(`)

described in appendix D). Accordingly, significance of the prompt lepton, Psig , is definedwith the three ANN outputs, Nb , Nbc and Nc as,q(`)
Psig = (Nb )2 + (Nbc )2 + (Nc )2 .

(4.38)

This prompt lepton significance is used to select one lepton candidate par events, with(`)

which the event-base analysis is performed. The largest value of Psig in a event is selected

Eventsto construct the event-base distribution and it is shown in Fig. 4.32. Finally, the prompt(`)
lepton significance, Psig , is required to be larger than 0.1 to enhance the prompt lepton.

100
80
Data

60

4 fermions

40
other 2 fermions
c→l+

20

b→c→l+

0
0

0.2

0.4

0.6

0.8

1

b→l-

prompt lep event significanceFigure 4.32: The distributions of the lepton significance which is most significant in theevent.

4.7.5

Unbinned Likelihood Fit

Making Subsamples
The Ab,c
FB fit is performed in a similar way as in the inclusive analysis. The event subsamples are constructed in order to enhance the sensitivity to AbFB or AcFB . The variables ofNb , Nbc and Nc are used to divide the events into subsamples. The division is performed

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

92

with the values of each ANN output of 1/3 and 1/6 to make 33 = 27 subsamples. Further
divisions on | cos θthrust | at at 0.8 and 0.65 make 81 subsamples. The distributions of Nb ,Nbc and Nc for the most significant lepton are shown in Fig. 4.33(a), (b) and (c), respectively. Their angular distribution evaluated with the lepton charge, the hemisphere they
belong, and | cos θthrust | is shown in Fig. 4.33(d).

Events(b)

Events

(a)

102

102

Data

Data

10

10

4 fermions

4 fermions

other 2 fermions

other 2 fermions

c→l+

c→l+

1

1
b→c→l+

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1b→l

b→c→l+

0-

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

b→l-

(c)

(d)
EventsNc

Events

Nb

1

102

70
60
50
40

Data

10

4 fermions

30

other 2 fermions

20

c→l+

Data
4 fermions
other 2 fermions
c→l+

10

1
b→c→l+

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1b→c→l+

0-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

b→l-

1

b→l-

- Ql Hl |cosθthrust |Nbc

Figure 4.33: The distributions of ANN outputs,(a) Nb , (b) Nc , (c) Nbc , (e)The angular
distribution after the event selection with prompt lepton significance. The polar angle
estimator is constructed from the lepton charge, lepton hemisphere and | cos θthrust | .

Probability Density Function
In a similar manner as in the inclusive analysis, the observed angular distribution can be
represented with AFB as
8dσobs∼ 1 + y2 +
dy
3

Ã

X
i=0,··· ,3!
fi (1 − 2χi )AiFBy,(4.39)

4.7. LEPTON ANALYSIS

93

where fi is the fraction of each prompt lepton source. The subindex of 0, 1 and 2 meansthat the event has a track labeled as bl− , bcl+ and cl+ respectively. The subindex of 3means the event has no track with prompt-lepton label. The χi is the mixing parametercorresponding to the prompt lepton sources. The χ2 is set to 0 and χ3 is set to 0.5. The
y is the polar angle estimator defined as,
y = −ql · hl · cos θthrust .

(4.40)The ql is refereed to as the track charge. The hl is refereed as the hemisphere flag of the
lepton which is 1 if the lepton is in the forward and -1 otherwise. The AiFB (i=0,1,2,3)
can be written as
A0FB = AbFB , A1FB = −AbFB , A2FB = −AcFB , A3FB = 0.

(4.41)

The assumption of the 0 asymmetry on the non-prompt lepton source is almost trueand effect of possible small non-zero asymmetry is evaluated with MC and added into
the systematic errors. With the probability function, likelihood fit is performed and the√result at the s of 189 GeV is as follows.
AbFB = 0.54 +0.23−0.24c
AFB = 0.25 ±0.23,
with a correlation matrix ofµ1.00 0.26
0.26 1.00(4.42)¶,(4.43)

where the corrections discussed in the subsection 4.6.4 is already added in the result. The
1-σ and 2-σ contour plots corresponding to the 189 GeV analysis are shown in Fig. 4.34.

c
AFB

The results corresponding to the each energy point are listed in Table. 4.11.
2
1.5
1
0.5
1 σ contour0

2 σ contour-0.5

SM AbFB-1

SM AcFB-1.5(with QCD effect)

-2
-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(without IFI)

AbFB

Figure 4.34: A one σ and two σ contours of the Ab,c
FB result of 189 GeV lepton analysis.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

94

Table 4.11: Results for the lepton analysis with statistical errors.
Ecm
[GeV]
91
133
161
172
183
189
192
196
200
202
205
207

AbFB
(stat.)
0.05 ±0.05-0.24 +0.35
−0.30
+0.38
0.29 −0.42
0.45 +0.36−0.44
0.67 +0.35
−0.38
+0.23
0.54 −0.24
1.44 +0.35−0.53
1.07 +0.27
−0.32
+0.40
0.44 −0.42-0.16 +0.63−0.62
0.26 +0.32−0.36
0.35 ±0.27

SM
Ab,
FB

0.10
0.48
0.55
0.56
0.58
0.58
0.58
0.59
0.59
0.59
0.59
0.59

AcFB
(stat.)−0.04 ±0.07
0.57 +0.36−0.39
0.79 +0.56−0.64
0.80 +0.52−0.62
0.63 +0.36−0.38
0.25 ±0.23
0.65 +0.72−0.75
0.78 +0.35−0.38
0.36 +0.41−0.42
0.87 +0.46−0.53
0.55 +0.36−0.38
0.84 +0.26
−0.27

SM
Ac,
FB

Correlation

0.06
0.69
0.67
0.67
0.66
0.66
0.65
0.65
0.65
0.65
0.65
0.64

0.33
0.13
0.13
0.10
0.23
0.26
0.14
0.29
0.23
0.14
0.23
0.25

4.8. COMBINED ANALYSIS

4.8

95

Combined Analysis

All events selected for the AFB fits in the inclusive and lepton analyses are used in the
combined analysis. These events are divided into two samples, where one is called as
an inclusive sample and the other is called as a lepton sample. Basically, the eventstreated in the inclusive analysis is assigned to the inclusive sample and the events treatedin the lepton analysis is assigned to the lepton sample. For the events shared between(inclusive)the inclusive and lepton analysis, events with Psig(`)

> Psig (Eq. 4.13, Eq. 4.38) are(inclusive)

assigned to the inclusive sample, whereas events with Psig(`)

< Psig are assigned to

the lepton sample.
For the inclusive sample, the same likelihood function as that used in the inclusive
analysis is used and the corresponding likelihood is constructed as Linclusive , whereas forthe lepton sample, the same likelihood function to that used in the lepton analysis is used
to construct the corresponding likelihood (Llepton ). The combined likelihood, Lcombined , isdefined as
Lcombined = Linclusive × Llepton .

(4.44)

With the combined likelihood, an unbinned maximum likelihood is performed to get the

c
AFB

Ab,c
FB simultaneously. The 1-σ and 2-σ contours are shown in Fig. 4.35.
2
1.5
1
0.5
1 σ contour0

2 σ contour-0.5

SM AbFB-1

SM AcFB-1.5(with QCD effect)

-2
-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(without IFI)

AbFB

Figure 4.35: The results of the Ab,c
FB measurement with 1-σ and 2-σ contours.
.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

96

The values obtained for the 189 GeV data are as follows.
+0.15= 0.53−0.16= 0.32 ±0.22,= −0.63 ±1.1,= −0.08 ±0.16,

AbFB
AcFB
Auds,in
FB
Auds,out
FB
with a correlation matrix of

1.00
 0.22 −0.01−0.001(4.45)


0.22 −0.01 −0.001
1.00 0.04
0.18 
.
0.04 1.00
0.08 
0.18 0.01
1.00(4.46)The result corresponding to each energy point is shown with the statistical error in Table. 4.12, where the values predicted by the standard model are also shown. The evaluationof the systematic errors is given in the next section.

Table 4.12: Results for the combined analysis with statistical errors.
Ecm
[GeV]
91
133
161
172
183
189
192
196
200
202
205
207

AbFB
(stat.)
0.08 ±0.03
0.21 +0.28−0.29
0.06 ±0.33
0.70 +0.39
−0.46
+0.25
0.90 −0.26
0.53 +0.15−0.16
1.14 +0.31
−0.37
+0.26
0.83 −0.28
0.31 +0.30−0.31
0.11 ±0.41
0.26 +0.24−0.25
0.26 ±0.20

SM
Ab,
FB

0.10
0.48
0.55
0.56
0.58
0.58
0.58
0.59
0.59
0.59
0.59
0.59

AcFB
(stat.)−0.01 ±0.07
0.63 +0.35−0.38
1.07 +0.71−0.74
1.08 +0.43−0.54
0.87 +0.31−0.34
0.32 ±0.22
0.65 +0.71−0.74
0.61 +0.35−0.37
0.30 +0.40−0.41
0.81 +0.48−0.55
0.49 +0.35−0.36
0.74 +0.26
−0.27

SM
Ac,
FB

Correlation

0.06
0.69
0.67
0.67
0.66
0.66
0.65
0.65
0.65
0.65
0.65
0.64

0.27
0.15
0.15
0.19
0.22
0.22
0.18
0.26
0.20
0.12
0.23
0.24

4.9. SYSTEMATIC UNCERTAINTIES

4.9

97

Systematic Uncertainties

This analysis relies heavily on the correct modeling by the Monte Carlo simulation of
the b-tagging and charge identification. Uncertainties which affect the charge identification probability and b-tagging variable are considered to be the important source of thesystematic errors.
The hemisphere charge can be affected by the the fragmentation, semileptonic modeland decay multiplicity of b or c hadrons. The B0 − B0 mixing can affect the hemispherecharge for b hadrons and it also affects the charge dilution factor in the lepton analysis.
As for the b-tagging variable, some change in the b-tagging variable affect the fractionsof b, c and light-quark (u, d and s) events in the selected samples and change the values ofthe fitted Ab,c
FB . The b-tagging variables for the inclusive and lepton analysis are sensitive
to the nature of b and c hadrons, such as the lifetime, fragmentation, decay multiplicity,and the momentum spectrum of the leptons from semileptonic decay. The b-tagging
depends strongly on the secondary vertex reconstruction and the impact parameter where
the resolution of the reconstructed tracks in the detector is essential. Accordingly, theuncertainties in the resolution for the track reconstruction should be considered as thesystematic error. In this analysis, b-tagging variables are constructed with ANN, which
is based on the validity of modeling of the Monte Carlo sample used in the ANN learning.
The uncertainties in the modeling of the input variables for the ANN should be considered.
Furthermore, the rate of the gluon splitting to bb or cc, the rate of long-lived light
hadrons (K0 and Λ), and the four-fermion rate can also affect the fractions. As regards
the fitting procedure, the possible bias in the AFB fitting should be taken into account.
Uncertainties in some corrections to be added to the fitted values of AFB should beconsidered too.
Most of the uncertainties mentioned above is evaluated by reweighting the concerning
quantities in the Monte Carlo sample within the range of their uncertainties. Then the AFB
fit is repeated and deviation of the resulting value from the original one is treated as thesystematic error. Since most of the systematic errors are common to all the energy points,
such common systematic errors are evaluated using combined data and corresponding√
Monte Carlo samples with s larger than 180 GeV. The evaluated systematic errors forthe combined data are shown in Table. 4.13, where a systematic error due to the Monte
Carlo statistics is that for the 189 GeV sample, which is not a common systematic error.
The others are common to all the energy points. More detailed description is given as
follows.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

98

Uncertainty sourceb fragmentationb decay multiplicity
b hadron compositionb lifetime
B0 − B0 mixing
c fragmentationc decay multiplicity
c hadron compositionc lifetime
semileptonic branching ratio
semileptonic decay model
Total b,c physics modeling
ISR-FSR interference
Final state QCD effect
gbb rate
gcc rate
K0 , Λ rate
Four-fermion rate
Other physics modeling
Track reconstruction
Event pre-selection
Lepton ID
Modeling of ANN Inputs
Total detector modelingMonte Carlo statistics
Background AFB
Fitting Procedure
Total AFB fit
Total systematic uncertainty

∆AbFB
0.010
0.003
0.012
0.001
0.018
0.002
0.010
0.013
0.001
0.007
0.006
0.031
0.005
0.016
0.004
0.008
0.008
0.001
0.021
0.076
0.001
0.005
0.011
0.077
0.004
0.002
0.032
0.032
0.09

∆AcFB
0.012
0.021
0.005
0.001
0.002
0.012
0.015
0.017
0.001
0.029
0.003
0.046
0.009
0.021
0.001
0.017
0.006
0.001
0.029
0.029
0.001
0.021
0.017
0.040
0.006
0.047
0.037
0.060
0.09

√
Table 4.13: Systematic uncertainty breakdown for the combined data with s larger than
180 GeV. A systematic error due to the Monte Carlo statistics shown here is that for the
189 GeV sample, which is not a common systematic error. The others are common to all
the energy points.

4.9.1

Bottom and Charm Physics Modeling

Uncertainties in bottom and charm fragmentation and decay properties are evaluated byreweighting the Monte Carlo samples in the ranges of their uncertainties as follows.
• b fragmentation
The Peterson fragmentation parameter (²b ) which models the b-quark fragmentation√
process described in subsection 2.3.2 is considered not to vary with the s, since the4.9. SYSTEMATIC UNCERTAINTIES

99

energy scale of the hadronization phase where the fragmentation function is used istypically below 1 GeV. Therefore, the uncertainties from the study of the b-quark
fragmentation at LEP1 [68] can be used in this thesis and the simulated bb events
are reweighted within the range of 0.0030 < ²b < 0.0048 [68].
In addition, other fragmentation functions are applied to the MC to evaluate theuncertainties from the fragmentation model. The b-quark fragmentation function
is changed to that suggested by Collins and Spiller [86] and to that of Kartvelishvili [87]. The best-fit values of the fragmentation parameters of the two models
in the study of the b-quark fragmentation at LEP1 [88] are used. The largest difference in the AFB due to the above tests is taken as the systematic uncertainty.
• b decay multiplicity
The charged decay multiplicity of b hadrons is varied in the MC according to the
combined results of the measurements at LEP [68] as 4.955 ± 0.062. The largest
difference in the AFB is taken as the systematic uncertainty.
• b hadron composition
The tagging efficiency differs for the various b hadron species. The fractions ofb hadrons and their errors have been calculated from the best values of mean lives,
mixing parameters, and branching fractions measured at LEP and CDF [51]. The
fractions of b baryon and B0s are varied independently within their errors, as 9.9 ±
1.7% and 10.7 ± 1.1%, respectively. Their variation is compensated by the fractionsof B0 and B+ , where the fraction of B0 and B+ is the same.
• b hadron lifetimes
The lifetimes of the different b hadrons are varied in the Monte Carlo by their errors
according to the results mainly from the measurements at LEP [68], as 1.576 ±0.016 ps.
• B0 − B0 mixing
The mixing parameter, χ, is varied in the Monte Carlo as 0.1257 ± 0.0042 from themeasurements mainly at LEP as shown in [51].
• c fragmentation
The Peterson fragmentation parameter ²cP is varied in the Monte Carlo within therange of 0.022 < ²cP < 0.039, which is taken from the study of the c-quark fragmentation at LEP1 [68].
• c decay multiplicity
The average charged track multiplicities of D+ , D0 and D+
s decays are varied in theCHAPTER 4. MEASUREMENTS OF THE AB,C
FB

100

Monte Carlo within the ranges of the experimental measurements [89], which are
2.38 ± 0.06 for D+ , 2.56 ± 0.05 for D0 , and 2.69 ± 0.33 for D+s .
• c hadron composition
The D0 fraction is written as f (D0 ) = 1 − f (D+ ) − f (D+s ) − f (cbaryon ). The last
three parameters are varied independently by their uncertainties according to [68],which are obtained through fitting the electroweak observables as, 23.3 ± 2.7 % for
D+ , 10.3 ± 2.9 % for D+s , and 6.3 ± 2.8 % for c baryons.
• c hadron lifetimes
Charmed hadron lifetimes are varied within their experimental uncertainties according to the several measurements [68], where the lifetime of D+ is 1.057 ± 0.016 ps,that of D0 is 0.415 ± 0.004 ps and that of D+
s is 0.467 ± 0.017 ps.
• semileptonic decay modeling and branching ratios
The semileptonic branching ratios of bottom- and charm-hadrons are varied within
their experimental uncertainties shown in [68,90], where the uncertainties of branching ratio for b → ` is 0.1056 ± 0.0021, that for b → τ → ` is 0.00419 ± 0.00055,
that for b → c → ` is 0.0801 ± 0.0026, that for b → c → ` is 0.0162+0.0044−0.0036 , and thatfor c → ` is 0.098 ± 0.005. The alternatives of semileptonic decay models shownin [68] are used to evaluate the systematic uncertainty, which makes the momentum
spectrum of the lepton softer or harder.

4.9.2

Effect of the ISR-FSR Interference

The effect of the ISR-FSR interference to the AFB is evaluated with Zfitter, which isremoved from the measured AFB which is already described in Section 4.6.4. The fulldeviations of AFB evaluated with or without the ISR-FSR interference with Zfitter are
added to the systematic errors.

4.9.3

Final State QCD Corrections

Since the thrust axis is the good estimator of the initial quark direction checked withMC as seen in Table 4.7, the AFB for the initial quark direction is set to the value to be
fitted. The charge identification with either the hemisphere or lepton charge technique isperformed with respect to the initial quark direction. Hence the flip of the quark direction
due to the hard gluon emission is treated as the dilution of the charge identification. The
QCD effects are evaluated with Zfitter, which are finally added to the fitted values ad
described in Section 4.6.4. The full corrections are assigned as systematic errors.

4.9. SYSTEMATIC UNCERTAINTIES

4.9.4

101

The Rates of Gluon Splitting to bb or cc

The rates of gluon splitting to bb or cc (gbb , gcc ) were measured at LEP1 [68, 90] as
0.00254 ± 0.00051 and 0.0296 ± 0.0038 respectively. The rates of gbb and gcc are increased√
as s increase according to the theoretical calculation in the next-to-leading-logarithmic
approximation shown in [91]. The measured values of gbb and gcc are extrapolated to thehigher energies using the analytical formula in [91], where the uncertainties of αs , mb andmc are added with in their errors in [51]. The rates of gbb and gcc in the Monte Carlo are√varied with in the errors. At s of 200 GeV, the rate of gbb is evaluated as 0.0059±0.0016
and that of gcc as 0.051 ± 0.014.

4.9.5

Total K0 and Λ Rate

The rate of long lived light hadrons affects the background in a b-tagged samples. The
total production rates of K0 and Λ in the Monte Carlo are varied by 10% according
to [68, 90].

4.9.6

Four-fermion Background

Above the WW production threshold, the four-fermion background is largely dominatedby W pairs. The uncertainty in the W pair production cross-section is taken into accountby changing the rate in the MC according to the results from the measurements of WW
cross-section [35].

4.9.7

Track Reconstruction

The effect of the detector resolution on the track parameters is estimated by degrading
or improving the resolution of all tracks in the Monte Carlo simulation. This is done by
applying a single multiplicative scale factor to the difference between the reconstructed
and true track parameters. A ±5% variation is applied independently to the rφ and rz
track parameters, which fits the data distributions of the d0 significance (d0 divided by
its error) and z0 significance (z0 divided by its error), respectively.
The number of SI hits along a track is compared between data and Monte Carlosamples and the difference is found to be small. To fit the small difference, the matchingefficiency for assigning measurement points in the silicon microvertex detector to thetracks is varied by 1% in rφ and 3% in rz. The systematic errors resulting from theindividual variations are summed in quadrature.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

102

4.9.8

Event Pre-Selection

As for the uncertainty in the non-radiative event selection, the analysis is repeated using
a different method for estimating s0 , described in [72], which is a less sophisticated alternative of that used in the section 4.5.2. In the method, only single photon radiated either
to the beam pipe direction or in the ECAL is considered. The difference of AFB between
the two methods is taken as the systematic error. The biases from other items of event
selections are consistent to be zero within the statistics of the MC, where no systematic
errors are assigned.

4.9.9

Lepton Identification

The fractions of misidentified electrons and muons are taken from Monte Carlo simulation.
The systematic errors for the electron efficiency (15%) and fake rate (30%) described in [42]are used to reweight the Monte Carlo. The deviations of the AFB due to the reweighting
are treated as the systematic errors. The systematic errors in the muon identification isevaluated as in [42], where the scaling factors to reweight the Monte Carlo for the muontracks is 0.953 ± 0.047 and that for the other background tracks is 0.971 ± 0.041.

4.9.10Modeling of Artificial Neural Network Inputs

The ANNs are used in the inclusive (the LB b-tagging method) and lepton analysis (Nb ,Nbc and Nc ). Each of the input distributions used in those ANNs are compared between
data and Monte Carlo. The simulated distributions are reweighted for each input variablein turn to agree with the corresponding data distributions, and the analysis is repeatedwith the weighted events. The observed differences from the original fit result are added inquadrature to yield the systematic uncertainty due to the modeling of the input variables.

4.9.11Monte Carlo Statistics

The fractions of b, c-tagging events and charge identification probabilities are varied by
the statistical error arising from the finite number of Monte Carlo simulated events.

4.9.12

Background Asymmetries

The AFB of the backgrounds introduces the biases on the measured AFB . In the inclusive
analysis, the observed asymmetry of the 4-fermion events is assumed to be zero. In the
lepton analysis, the observed asymmetry of the background events is assumed to be zero.
The effects of the possible non-zero asymmetry evaluated with the Monte Carlo are addedin the fitting and the deviations of the results are added as the systematic errors.

4.9. SYSTEMATIC UNCERTAINTIES

4.9.13103

Fitting Procedure

The large Monte Carlo sample is used as the input data and the sample is fitted with
the Monte Carlo to evaluate the fitting bias. The obtained values from the fitting is
consistent with the input value of AFB for the Monte Carlo within the statistical errors.
The variation of the fitted values from the input values are treated as the systematic
errors, where the contributions from the background asymmetries are removed which are
already evaluated in the section 4.9.12. No local minimum is exist in the fitting, which
is checked by scanning the value of the likelihood in the AbFB -AcFB plane as shown in-Log(L)

Fig. 4.36.

c B

AF

-678
-680
-682
-684
-686
-688
-690
-692
-694
0.3
0.4
0.5
0.6
0.7
0.8

0.8
AbFB

0.7

0.6

0.5

0.4

0.3

Figure 4.36: The distribution of − log L in AbFB -AcFB plane for the combined data.
.

4.9.14

Cross-Checks

The data corrected at√s of Z mass in the LEP2 is used to check the fitting method. The

same procedure is applied to the data and the consistent results to the standard model
predictions are obtained. Since the measurements at LEP1 shows good agreements withthe standard model predictions, it is a good cross-check of the methods of the measurement
used in this thesis.

104

4.10

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

Result and Discussion

The result of the combined analysis is summarized here. The same analysis for combined√
data with s larger than 180 GeV is performed. The both results are compared to the
LEP results including previous OPAL result. They are also compared to the results for√s below LEP2 energy. The obtained results can be used to constrain some new physics.
In a general framework, the existence of the contact interaction between electron and b
or c quarks are investigated. It can be interpreted as the effective theory for the exchangeof the heavy particle between electron and b or c quarks. In some specific cases, the limiton the mass of some leptoquarks or R-parity violating squarks are placed. Finally, newphysics due to the effect of graviton exchange in large extra dimensions is also studied.

The result of the combined analysis is shown with the statistical and systematic errors
in Table. 4.14 and Fig. 4.37, where the prediction of the standard model is also shown. For
all measured points, good agreements with the standard model prediction are observed.
The values of χ2 for b and c with respect to the standard model prediction for the energy
points from 130 to 207 GeV are calculated to be 14.6 and 4.68, respectively, where thenumber of degree of freedom is 11.
√
The data with s larger than 180 GeV (the data samples of 183, 189, 192, 196, 200,
202, 205, and 207 GeV) are combined, where the expected variation of AFB as a function√of s is less than 2% in the snatdard model prediction. With the combined data, the AbFB
and AcFB are also measured with the same method, where the luminosity-weighted mean√of s is 197 GeV. The result is shown in Table. 4.15 and Fig. 4.37 with open markers.
The result is consistent with the standard model prediction.

4.10. RESULT AND DISCUSSION

105

Table 4.14: Result for the combined analysis with statistical ans systematic errors.
Ecm
[GeV]
91
133
161
172
183
189
192
196
200
202
205
207

AbFB
(stat.)
0.08 ±0.03
0.21 +0.28−0.29
0.06 ±0.33
0.70 +0.39−0.46
0.90 +0.25−0.26
0.53 +0.15−0.16
1.14 +0.31−0.37
0.83 +0.26−0.28
0.31 +0.30−0.31
0.11 ±0.41
0.26 +0.24−0.25
0.26 ±0.20

SM
Ab,
FB

(syst.)
±0.09
±0.09
±0.09
±0.10
±0.09
±0.09
±0.09
±0.09
±0.09
±0.09
±0.09
±0.09

0.10
0.48
0.55
0.56
0.58
0.58
0.58
0.59
0.59
0.59
0.59
0.59

−0.01
0.63
1.07
1.08
0.87
0.32
0.65
0.61
0.30
0.81
0.49
0.74

AcFB
(stat.) (syst.)±0.07 ±0.09
+0.35
±0.09
−0.38
+0.71
±0.10
−0.74
+0.43
±0.10
−0.54
+0.31
±0.09
−0.34
±0.22 ±0.09
+0.71
±0.09
−0.74
+0.35
±0.09
−0.37
+0.40
±0.09
−0.41
+0.48
±0.09
−0.55
+0.35
±0.09
−0.36
+0.26
±0.09
−0.27

Table 4.15: Result for the combined data with
Ecm
[GeV]
197

AbFB
(stat.) (syst.)
0.52 ±0.09 ±0.09

SM
Ab,
FB

0.59

0.59

√

SM
Ac,
FB

Correlation

0.06
0.69
0.67
0.67
0.66
0.66
0.65
0.65
0.65
0.65
0.65
0.64

(stat.)
0.27
0.15
0.15
0.19
0.22
0.22
0.18
0.26
0.20
0.12
0.23
0.24

s larger than 180 GeV.

AcFB
(stat.) (syst.)±0.12 ±0.09

SM
Ac,
FB

0.65

Correlation(stat.)
0.23

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

b
AFB

106

2
1.5
1
0.5
0
-0.5
-1
-1.5
-2

80

100

120

140

160

180

200

220

c
AFB

s[GeV]
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2

80

100

120

140

160

180

200

220

s[GeV]
Figure 4.37: The measured AbFB and AcFB for each energy point where the values of thestandard model expectation
are indicated with lines. The measured AbFB and AcFB for the√
combined data with s larger than 180 GeV are also plotted with open markers.

4.10. RESULT AND DISCUSSION

107

In addition to the results of the Ab,c
FB measurements performed by OPAL (the previousb,cresults [42] and this measurement), results of AFB
measurements at LEP2 performed byother experiments (ALEPH [92], DELPHI [93], L3 [94]) are also shown in Fig. 4.38. Inp
the ALEPH analysis, multi-hadronic events with s0 /s larger than 0.9 were used andthey were clustered into two jets. The AbFB was measured for b-enriched sample basedon the mean difference between the forward jet charge and backward jet charge, where
the standard-model values of AFB for u , d, s, and c were assumed. Delphi utilized
the similar method to this inclusive analysis to measure the Ab,c
FB by fitting the angular
distributions of b, c, and light-quark (u, d, and s) enriched samples. The obtained valuesof this measurement are more accurate than all results reported by other experiments.
The measurement of the AcFB is mainly performed by OPAL and is most precise. The√
result of this measurement at s from 130 GeV to 189 GeV is consistent with and is√comparable to the previous OPAL result. The result at s larger than 189 GeV is uniqueto this measurement.
b,cThe results of the AFB
performed at PEP, PETRA, TRISTAN, and LEP are summa√
rized in Fig. 4.39, where the results at the s of 133, 161 and 172 GeV, and the combined

result for the energy points from 183 GeV to 207 GeV are shown for this measurement.
The combined result for the energy points of 183 GeV and 189 GeV, and that for theenergy points between 192 GeV to 209 GeV are also shown in the figure, where the latteris the combined result for the energy points newly measured in this thesis. The result of
this measurement is important, since the measurement reported in this thesis is performedat the highest energy in e+ e− collisions and gains the best accuracy at LEP2 with efficient
analysis methods using all the data collected by OPAL during LEP2.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

b
AFB

108

2
1.5
1
0.5
SM prediction

0

ALEPH

-0.5

DELPHI

-1
L3

-1.5
OPAL

-2

130 140 150 160 170 180 190 200 210

this measurementc
AFBs[GeV]

2
1.5
1
0.5
0

SM prediction

-0.5
OPAL

-1
DELPHI-1.5-2

130 140 150 160 170 180 190 200 210

this measurements[GeV]

Figure 4.38: The measured AbFB and AcFB by ALEPH, DELPHI, L3, and OPAL at LEP2
where the values of the√standard model expectation √are indicated with lines. DELPHI
obtained the result at s of 167 GeV where data at s of 161 GeV and 172 GeV werecombined.

b
AFB

4.10. RESULT AND DISCUSSION

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

109

New Point
192-209 GeV

combined

SM prediction

183-189 GeV

PEP combined

PETRA combined

TRISTAN combined

LEP I+SLD combined

-2

20

40

-2
60 20
80 10040 12060 14080 100160 120180 140200 160
220 180 200 220

this measurementc
AFBs[GeV]

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

New Point
192-209 GeV

combined

SM prediction

183-189 GeV

PEP combined

PETRA combined

TRISTAN combined

LEP I+SLD combined

-2

20

40

-2
60 20
80 10040 12060 14080 100160 120180 140200 160
220 180 200 220

this measurements[GeV]
Figure 4.39: The results of the AbFB and AcFB measurements in e+ e− collisions at the
PEP, PETRA,√ TRISTAN, SLC and LEP are shown, where the combined result for thedata with s larger than 180 GeV is used for this measurement. The standard model
predictions are given by lines. The combined reslut only for the energy points newly
measured in this thesis is also indicated.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

110

The result can be used to constrain some new physics beyond the standard model.
For example, it can place a limit on contact interaction between electron and b or cquarks. It is a general framework of a new physics, since the contact interaction can beinterpreted as the effective interaction of the exchange of a very heavy particle concerning
to a new physics at a higher energy scale. In the context of the composite models of
leptons and quarks, the contact interaction can also be interpreted as a remnant of thebinding force between the substructures of fermions. The study of the contact interaction
between electron and b or c quarks at e+ e− colliders is particularly important since it isinaccessible to pp̄ or ep colliders.
Following [95], the Lagrangian6 for the four-fermion contact interaction in the processe+ e− → ff (f 6= e) is defined by
L=

2
X
£
¤
gCI
µ
η
[e
γ
e
]
f
γ
f
,
ij
i
i
j
µ
j
Λ2 i,j=L,R(4.47)

which is added to the standard model Lagrangian. Here eL and eR (fL and fR ) are chirality
projections of electron (fermion) spinors. The coefficients ηij determine the type of chiralcouplings of the four fermions, and the Λ is the energy scale of the contact interaction.
2
By convention, the coupling constant, gCI , is set to gCI/4π = 1 and the ηij is set to ±1

or 0. The models of the contact interaction can be defined with sets of values of ηij andsome models used here are shown in Table 4.16 with the corresponding sets of valuesof ηij . For example, LL± means a model with left-handed couplings for both electrons
and fermions. The + (−) sign means constructive (destructive) interference between thestandard model and the contact-interaction model.
Table 4.16: The models of the contact interaction and their corresponding values of ηij
Model
LL±
RR±
VV±
AA±
LR±
RL±

ηLL
±1
0
±1
±1
0
0

ηRR
0
±1
±1
±1
0
0

ηLR
0
0
±1
∓1
±1
0

ηRL
0
0
±1
∓1
0
±1

According to [97], the helicity amplitude, Mij (i, j = L, R), for e+ e− → ff including
the contact interaction can be written as
2
gie gjf
4παgCI
4παQe Qf+·
+ ηij 2 . (4.48)
Mij (s) =ΓZsΛsin2 θW cos2 θW s − MZ2 + is M
Z

6

Here only vector type contact interaction is considred, since other scalar and tensor type interactions
are strongly limitted [96].

4.10. RESULT AND DISCUSSION111

The first and sectond terms in Eq. 4.48 correspond to γ and Z exchanges respectively, andthe thired one corresponds to the contact interaction, where Qf is the electric charge of thefermion in units of the elementary charge. The subscripts i and j label the chiralities of
the initial electron and final fermion, respectively (i, j = L, R), and gif is the electroweak
couplings of the fermion as follows.
gLf = 2T3f − Qf sin2 θW ,gRf =(4.49)−Qf sin2 θW .

The differential cross-section in the presence of the contact interaction can be written asdσ11= SM0 (s, t) + C20 (s, t) 2 + C40 (s, t) 4 .
d cos θΛ
Λ(4.50)

where SM0 (s, t) is the standard model cross-section and t = − 21 s(1−cos θ). The measured
AFB can be fitted with the AFB calculated from Eq. 4.50, where the fitting parameter isthe energy scale, Λ. In the calculation, the improved Born calculation of the contactinteraction part is corrected to take into account QED radiation as in [97]. The standardmodel part is calculated with Zfitter. Hereafter only the contact interaction between
electron and b quark or that between electron and c quark is considered. For the modelpof RL+ with Λ of 1 and 5 TeV, the σ 0 and A0FB for b quarks with a cut of s0 /s > 0.85√
as a function of s are shown in Fig. 4.40, where the standard-model prediction is also
shown. Deviations from the standard-model prediction due to the contact interaction canbe seen in the figure.

(b)
10

4

b
AFB

-1

σb [pb ]

(a)

1
0.8
0.6

103

0.4
0.2

102

SM

-0

SM

-0.2
-0.4

10

Λ=1 TeV -0.6

1

Λ=5 TeV -1

Λ=1 TeV-0.8
0 20 40 60 80 100120140160180200+

Λ=1,5TeV RL

s[GeV]

0 20 40 60 80 100120140160180200+

Λ=1,5TeV RL

Λ=5 TeVs[GeV]

Figure 4.40: The calculation
p of the (a) cross-section and (b)√ forward-backward asymmetryof b quarks with a cut of s0 /s > 0.85 as a function of s, where a contact interactionof RL+ model with Λ = 1 or 5 TeV is added.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

112

As regards the fitting, the parameter, ² ≡ 1/Λ2 , is used practically instead of Λ. The ²
is allowed to take both positive and negative values, which can treat the constructive anddestructive models simultaneously since the ηij always appears with 1/Λ2 according to
Eq. 4.47. As regards the result of the combined data whose luminosity-weighted mean of√s is 197 GeV, the AbFB is shown as a function of ² for RL+ model in Fig. 4.41(a) and theχ2 measured from its minimum (∆χ2 ) is shown in Fig. 4.41(b). The ∆χ2 corresponding√
to the measured values of AbFB at 11 different s points is alos calculated as shwon in
Fig. 4.41(b), where the systematic correlations are taken into account. The latter non√combined case is more sensitive to ² due to the s dependence of AFB , which is used to
set the limits on the contact interactions.

(b)
0.8

∆χ2

b
AFB

(a)
0.6

6
5

0.4
4

0.2
-0

3

-0.2

2

A bFB combined

-0.4
1

-0.6
-0.8
-2 -1.5 -1 -0.5 0

0.5

1

1.5

-2

2

RL ∈[TeV ]0-0.3

A bFB 11 points

-0.2

-0.1

0

0.1

0.2

0.3

-2

RL ∈[TeV ]√Figure 4.41: (a) The AbFB at s of 197 GeV as a function of ² for RL model. The
measrued values of Ab,c
FB for the combined data are indicated with horizontal lines. (b)2The χ measured from its minimum (∆χ2 ) for the AbFB corresponding to the combined
data and 11 measured points.

The cross-sections of e+ e− → bb (σb ) are calculated with the results of the Rb measurements [98] and total hadronic cross-section (σhad ) measurements at LEP2 [72, 99] at√s beyond 180 GeV, where correlations between Rb and σhad measurements are neglected.
The result of σb are also used to set the limit on the contact interaction. The ∆χ2 corresponding to the AbFB and σb results as a function of ² are shown in Fig. 4.42. The
χ2 corresponding to the combination of AbFB and Rb results is calculated and is shownin Fig. 4.42, where Rb dependence of AbFB is treated but the systematic correlations are
neglected.

∆χ2

4.10. RESULT AND DISCUSSION

113

10
8
6

AbFB

4

σb

2
0
-0.3

-0.2

-0.1

0

0.1

0.2

0.3

AbFB+σb-2

RL ∈[TeV ]Figure 4.42: (a) The ∆χ2 for AbFB , σb , and their combined results as a function of ² for
RL model.
With the results of AbFB , σb , and their combination, χ2 fits for ² are performed andthe results are shown in Fig. 4.43 (a) with one standard deviation errors. Similar fits for
AcFB are performed and the results are shown in Fig. 4.43(b). They shows that the ² isconsistent to be zero, which means no contact interaction.

(a)

(b)
LL

LL

RR

RR

VV

VV

AA

AbFB

AA

LR

σb

LR

RL
-0.2

0

0.2

-2

∈ [TeV ]AbFB+σb

AcFB

RL
-1

-0.5

0

0.5

-2

1

∈ [TeV ]Figure 4.43: (a) The results of the χ2 fits for AbFB , σb , and their combinaion. (b) The
result of the χ2 fits for AcFB .
For both positive and negative ², the minimum of the χ2 is calculated and 95% confidence level (95%CL) limits are set at ² where ∆χ2 is 3.84. The Λ corresponding to thelimit for positive (negative) ² is denoted as Λ+ (Λ− ), which means the 95% CL limit for
the energy scale of the constructive (destructive) model of the contact interaction. The
evaluated limits are shown in Fig. 4.44 and Table. 4.17. In Fig. 4.44, the current limits

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

114

are also shown, which are evaluated by ALEPH [92] for b quarks, by DELPHI [93] andPETRA [20] for c quarks.

(a)

LL
RR
VV

AbFB
AA

σb

LR

AbFB+σbRL
105Λ [TeV]
-

0

5

10

Λ [TeV]+

ALEPH b

(b)

LL
RR
VV
AcFB

AA
LR

DELPHI c

RL
105

Λ- [TeV]0

5

10

PETRA cΛ+ [TeV]Figure 4.44: The excluded region of energy scale of the contact interactions in 95% conficence level (a) for b quark and (b) for c quarks.

4.10. RESULT AND DISCUSSION115

For the limits with only AbFB result, the limit on LR+ is better than the ALEPH’s
result. All limits with σb measurements and the combination of AbFB and σb measurements
exceed the ALEPH’s results and they are the highest limits for the contact interaction
between electron and b quark. As regards the limits from AcFB , the limits on LL and RR
models are not accessible. The limits on LR± and RL+ model exceed that for DELPHI
or PETRA and they are the highest limits for the contact interaction between electron
and c quark.
Table 4.17: The 95% conficnece level limits on the energy scale of contact interaction.
model
LL
RR
VV
AA
LR
RL

+
−
+
−
+
−
+
−
+
−
+
−

95% CL limits on Λ [TeV]b quarkc quarkwith AbFB with σb with AbFB +σb with AcFB
1.3
7.4
7.4
∗
2.4
8.5
8.4
1.1
5.1
5.1
∗
∗
0.98
4.8
4.7
1.6
8.5
8.6
2.9
4.2
9.2
8.8
1.8
1.7
9.3
9.3
4.0
3.6
11
11
1.3
2.8
4.0
3.9
1.8
2.2
2.7
2.7
2.6
2.0
3.8∗
5.1∗
2.1
3.1
4.7
4.6
2.3

∗) There are some regions not excluded at Λ smaller than the value listed in the table asshown in Fig. 4.44.
Next, leptoquarks are considered for examples of heavy particles exchanged between
electron and b or c quark, which can be regarded as contact interaction effectively. The
leptoquarks are motivated by the symmetry between leptons and quarks in the standard model and they appearer in theories of grand unification [100] and in compositenessmodels [101]. The leptoquarks mediate quark-lepton transition carrying both color andfractional electric charge and a lepton number, which can be classified with BuchmüllerRückl-Wyler (BRW) model [38] as shown in Table. 4.18. An S(V) denotes a scalar(vector)
leptoquark and the subscript denotes the weak isospin. When the leptoquark can couple to both right- and left-handed leptons, an additional superscript indicates the lepton
chirality. A tilde is introduced to differentiate between leptoquarks with different hypercharge. Some squarks in supersymmetric theories with broken R-parity couple to quarklepton pair in the same way as leptoquarks. In the minimal supersymmetric extensionof the standard model, the renormalizable gauge invariant operator that couples squarks

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

116

to quarks and leptons is given by λ0ijk Li Qj Dk , where i, j, or k denotes the generation,
L is the left-handed lepton, Q is the left-handed quark, D is the right-handed singlet ofdown-type quark, and one of the three is the superparticle. The down-type squark (deR )mediates left-handed electron (e−L ) and left-handed up-type quark (uL ) transition in thesame way as S0L leptoquarks. The up-type squark (euL ) mediates left-handed electron (e−L)
and right-handed down-type quark (dR ) transition in the same way as Se1/2 leptoquarks.
Such assignments of the squark to the leptoquarks are also shown in Table. 4.18. The
leptoquark model used here is the same as the original BRW model but the leptoquark
couplings between generations are allowed. For example, couplings between electron (1stgeneration) and b quark (3rd generation) is possible. Such leptquarks can be exchanged
in e+ e− → qq as shown in Fig. 4.45(b,c). The Feynman diagrams corresponding to the
R-parity violating squarks are also shown in Fig. 4.45(d,e), where the cases with bb andcc final states are separately shown.
Model

Fermion Charge
number F
Q

S◦L
S◦R
Se◦
L
S1/2

2
2
2
0

R
S1/2

0

Se1/2

0

S1

2

V◦L
V◦R
Ve◦
L
V1/2

0
0
0
2

R
V1/2

2

Ve1/2

2

V1

0

Coupling

−1/3
−1/3
−4/3
−5/3

eL u
eR u
eR d
eL ū

−5/3
−2/3
−2/3
−4/3
−1/3

eR ū
eR d¯
eL d¯
eL d
eL u νdeL d¯ ν ū
eR d¯

−2/3
−2/3
−5/3
−4/3
−4/3
−1/3
−1/3
−5/3
−2/3

νd

Squark
type
df
R

u
eL

eR ū
eL d
eR d
eR u
eL u
eL ū
eL d¯ ν ūTable 4.18: A general classification of leptoquark states in the Buchmüller-Rückl-Wyler
model. Possible squark assignments to the leptoquark states in the minimal supersymmetric theories with broken R-parity are also also shown.

4.10. RESULT AND DISCUSSION

117

(a)






(b)





 

(c)





 

   



 





(d)

(e)

 


 




 




            





  




 

Figure 4.45: A Feynman diagram of e+ e− → qq. (a) standard model. (b), (c) leptoquarks.
(d), (e) R-parity violating squarks.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

118

According to [96], the helicity amplitude of the e+ e− → qq with leptoquarks, Mij (i, j =L, R) can be written as
gie gjf
4παQe Qf
4πα
Mij (s) =+·+ ηij (t, u) , (4.51)
ΓZssin2 θW cos2 θW s − MZ2 + is M
Z

where t = − 12 s(1 − cos θ), u = − 12 s(1 + cos θ) and ηij (t, u) is defined as follows according
to the fermion number, F , of the final state quark.

aij · λ2LQfor F = 0 ;

2

 MLQ
−t
ηijeq (t, u) =
,
2


a
·
λ
ij

LQ

for |F | = 2 .

2
MLQ
−u
where the mass and coupling of the leptoquark are MLQ and λLQ respectively. The type2of the leptoquark is defined with aij as shown in Table. 4.19. In the limit of MLQ
À t, u,the interaction is the same as the contact interaction, which can be seen by comparing
Eq. 4.48 and Eq. 4.51. At such limit, corresponding energy scale of the contact interactionp
MLQ. The corresponding contact-interaction model is seen incan be written as 4π|aik | λLQ
R
Table. 4.19. For example, V1/2
corresponds to LR+ model.

Table 4.19: The leptoquark types corresponding to the coefficients aik .
aik
S0L
S0R
S̃0
S1
L
V1/2

up-type final state
LL RR LR
RL
1/2
1/2

1/2
1/2

1
1

R
V1/2

Ṽ1/2

1

L
S1/2
R
S1/2

−1/2

S̃1/2
V0L
V0R
Ṽ0
V1

down-type finale state
LL RR LR
RL

1

1

−1/2

−1/2
−1/2
−1
−1

−1
−2

−1

4.10. RESULT AND DISCUSSION

119

With the helicity amplitude, the theoretical cross-section and AFB for e+ e− → bb or
cc is calculated in a similar way to the case of the contact interaction. The λLQ is used asa fitting parameter and the fitting is performed at various MLQ points for each leptoquark
types. The 95% CL exclusion plots of λLQ as a function of MLQ are shown in Fig. 4.46
(Fig. 4.46) for leptquarks which couple electron and b (c) quark. In Fig. 4.46, exclusions
R, the information ofwith AbFB only, σb only and their combination are shown. For V1/2bb
AFB is efficiently used in the exclusion with combination of AFB and σb , which can be

expected from the result of the contact interaction. In Fig. 4.47, figuress corresponding
to S0L , S0R and S1 are not shown, for no space is excluded. The 95% CL limits on MLQ at√
λ = 4πα are calculated and the results from e+ e− → bb and cc are shown in Table. 4.20
and Table. 4.21, respectively.

(b)
0.9

1
0.9

0.8

0.8

0.7

0.7

0.7

0.6

0.6
b0.5
AFB+σb
0.4

0.6
b0.5
AFB+σb
0.4
0.3
σb0.2
0.1
AbFB0

0.4

σb0.2
0.3

0.2
0.1
0

500

(d)

1000

1500

0.1
AbFB0

2000

~
(e)
S
0 MLQ[GeV]

1000

1500

2000

1
0.9

0.8

0.7

0.7

0.6

0.6
b0.5
AFB+σb
0.4

0.6
b0.5
AFB+σb
0.4
0.3
σb0.2
0.1
AbFB0

σb0.2
0.3

0.2
0.1
0

500

1000

1500

0.1
AbFB0

2000

500

VR1/2 MLQ[GeV] (h)
λLQ

1
0.9

1500

2000

1
0.9
0.8

0.8

0.7

0.7

0.6

0.6
b0.5
AFB+σb
0.4

0.6
b0.5
AFB+σb
0.4
0.3
σb0.2
0.1
AbFB0

σb0.2

0.3

0.3

0.2
0.1
0

0.1

500

1000

1500

2000

VL0 MLQ[GeV]

AbFB0

500

1000

1500

2000

VR0 MLQ[GeV]

AbFB

σb
500

1000

1500

2000

AbFB

~
S
1/2 MLQ[GeV]

1

0.7

0.4

2000

0.9

0.8

0.5

1500

AbFB+σbS1/2 MLQ[GeV] (i)
λLQ

(g)

1000

1000

1

0.8

0.3

500

0.9

0.7

0.4

σb

VL1/2 MLQ[GeV]

0.8

0.5

AbFB+σb

S1 MLQ[GeV] (f)

λLQ

1
0.9

500

λLQ

0.3

λLQ

1
0.9

0.8

0.5

λLQ

(c)

λLQ

1

λLQ

λLQ

(a)

AbFB+σb
σb
500

1000

1500

2000

AbFB

V1 MLQ[GeV]

Figure 4.46: The 95% CL exclusion plots of λLQ as a function of MLQ for leptoquarks
which couple electron and b quark.

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

120

(b)

(c)

λLQ

1
0.9

1
0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1
500

1000

1500

AcFB V1/2 MLQ[GeV]

(d)

0

2000

1
0.9

0.1
500

1000

1500

~
AcFB V
1/2 MLQ[GeV]

(e)

0

2000

1
0.9
0.8

0.8

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1
1000

1500

2000

AcFB SR1/2 MLQ[GeV]

0

1500

2000

1

0.7

500

1000

0.9

0.8

0

500

AcFB SL1/2 MLQ[GeV]

(f)

λLQ

0

λLQ

λLQ

1
0.9

λLQ

λLQ

(a)

0.1
500

1000

1500

2000

~
AcFB V
0 MLQ[GeV]

0

500

1000

1500

2000

AcFB V1 MLQ[GeV]

Figure 4.47: The 95% CL exclusion plots of λLQ as a function of MLQ for leptoquarks
which couple electron and c quark.

4.10. RESULT AND DISCUSSION121

The results of leptoquarks which couple electron and b quark are compared to previR
ously obtained ALEPH [92], DELPHI [93] and OPAL [99] results. This result for V1/2
isbbbetter than them with only AFB information. All results with σb and AFB are better thanthe previous results. As regards the results of leptoquarks which couple electron and cquark are new for OPAL. The result of the Se1/2 is equivalent to that of R-parity violatingsquarks (euL ).
√4πα from e+ e− → bb√
95%CL Limit on MLQ [GeV] at λLQ = 4παfrom e+ e− → bb
LQ type with AbF B with σb with AbF B + σb
Se0
—
253
256
S1
—
603
612
L
V1/2
233
333
330
R
V1/2
171
308
422
—
230
227
S1/2
e
S1/2
—
—
—
L
V0
201
720
707
V0R
131
401
393
V1
201
720
707

Table 4.20: 95%CL Limit on MLQ [GeV] at λLQ =√4πα from e+ e− → cc√
95%CL Limits on MLQ [GeV] at λLQ = 4παfrom e+ e− → ccwith AcF B
LQ type
S0L
—
R
S0
—
S1
—
174
V1/2
Ve1/2
146
L
S1/2
—
R
S1/2
—
Ve0
—
V1
109

Table 4.21: 95%CL Limits on MLQ [GeV] at λLQ =122

CHAPTER 4. MEASUREMENTS OF THE AB,C
FB

Finally, new physics due to the effect of graviton exchange in large extra dimensions
is studied. There are many orders of difference between the scale of the gravitational andelectroweak interaction, which are the Planck scale (Mpl ∼ 1019 GeV) and the electroweak
scale (Mew ∼ 102 GeV), respectively. The unexplained large difference is known as “hierarchy problem”. As a solution, it has been pointed out that the quantum-gravity scale
could be as low as the electroweak scale with graviton propagating in the compactified
higher dimensional space [102]. According to this theory, the Planck mass in D = n + 4
dimensions (MD ) is chosen to be the electroweak scale, where MD is connected to thecompactification radius (R) of the n extra dimension as Mpl2 = Rn MDn+2 . The gravitons
can contribute to e+ e− → ff process via the process of virtual exchange of the the possible
exitation mode of the graviton (G∗ ) as e+ e− → G∗ → ff. Although the contribution fromsingle graviton mode is small, the large number of possible excitation modes can make theeffect measurable [103]. According to [103], the differential cross-section with the virtual
graviton exchange in the s-channel can be writen asdσλλ2= A(s, t) + 4 · B(s, t) + 8 · C(s, t),d cos θMs
Ms

(4.52)

where t = − 21 s(1 − cos θ), Ms is the mass scale of the order of MD , and λ is of O(1)which dependes on the details of the theory but weakly depends on the number of extra
dimensions. With an assumption of λ = ±1, the Ab,c
FB and σb are calculated accordingto [103], 95% CL limits on Ms are obtained as shown in Table. 4.22 in a similar way tothe case of the contact interaction. Although this limits are not superior to limits fromother measurements [104–109], this resutls can be used with them to improve the limits.
Table 4.22: The 95% CL limits on Ms .
95% CL limits on Ms [GeV]
λ = −1 λ = 1bwith AFB
300
420
with σb
490
490
500
490
with AbFB + σbcwith AFB
430
250
b,c
with AFB + σb
510
490

123

Chapter 5Conclusion
In this thesis, the forward-backward asymmetries for b and c quarks have been measured
in e+ e− collisions at center-of-mass energies between 130 GeV and 209 GeV, which is themeasurement at the highest energy in e+ e− collisions.
Two methods are used to evaluate the production angle of the primary quark andto identify the bb or cc events. One utilizes the hemisphere charge with a powerful btagging algorithm based on lifetime, lepton and event-shape information. The other uses
the information of the leptons from semileptonic decays of heavy hadrons both to evaluate
the primary-quark direction and to enhance events with primary b or c quarks. Finally,
the two independent methods have been combined with a single likelihood function withwhich the AbFB and AcFB have been measured simultaneously by fitting the distribution of
the evaluated primary-quark direction.
The forward-backward asymmetries for b and c quarks have been obtained at 11
different center-of-mass energies between 130 and 209 GeV, which are in good agreementswith the standard model predictions. The corresponding χ2 values are 14.6 and 4.68 for
the AbFB and AcFB , respectively.

√
The same analysis has been performed for the combined data with s larger than√
180 GeV, where the expected variation of AFB as a function of s is less than 2% in the√snatdard model prediction, and where the luminosity-weighted mean of s is 197 GeV.
The following results have been obtained.
AbFB = 0.52 ± 0.09(stat.) ± 0.09(syst.),AcFB = 0.59 ± 0.12(stat.) ± 0.09(syst.).
This analysis gains the best accuracy at LEP2 due to the development of the efficientmethods and use of the full data collected by the OPAL detector during LEP2. This
result is one of the verifications of the correctness of the standard model in the highest
energy region currently available in e+ e− collisions.
The result can be used to constrain some new physics beyond the standard model.
Here, new physics due to contact interactions, leptoquarks, R-parity violating squarks,

124

CHAPTER 5. CONCLUSION

and graviton in extra dimensions have been studied. New limit is obtained on the contact
interaction between right-handed electron and left-handed b quark with constructive interference to the standard model (RL+ model). New limit on the RL+ contactinteraction
for c quark is also obtained. New limits on that between left-handed electron and righthanded c-quark with both constructive and destructive interference to the standard model(LR± ) are also obtained. As regards the leptoquarks, new limit on the vector leptoquarkwith weak isospin of 1/2 which couples right-handed electron and left-handed b quarkR) is also obtained.
(V1/2

125

Appendix A
The Standard Model
The standard model is a theory to describe the constituents of matter and their interactions. There are two groups of particles, ’fermions’ and ’bosons’. The fermions have
odd-half-integer (1/2, 3/2, ...) intrinsic angular momentum (spin)1 . The bosons haveinteger spin. The matter consists of two kinds of fermions, quarks and leptons. The interactions between fermions are described as exchanging of a boson between them. There are
four kinds of such bosons, photon, W and Z bosons, and gluons. The electromagnetic andweak interactions are unified into the electroweak theory [4–6]. The strong interaction is
described by quantum chromodynamics (QCD). The standard model is the union of theelectroweak theory and QCD.
There are one more interaction known as gravity which is not included in the standard
model. The gravitational interaction is too weak at the scale of the elementary particles
and at the energy scale of this thesis, so it is negligible and ignored in subsequent consideration.

A.1

Elementary Particles in the Standard Model

A.1.1

Fermions

There are two groups of elementary fermions with spin 1/2, quarks and leptons. Bothare further divided into two classes with their electric charge. The quarks are divided in
up-type quarks (electric charge2 is 2/3) and down-type quarks (electric charge is −1/3).
The leptons are divided in charged leptons with -1 electric charge and neutrinos withneutral charge. Each of them is composed of three kinds of particles, manifesting three
generations as shown in Table A.1. There are also anti-particles for each of them. The
masses and the electric charges of the particles [51] are listed together.
1

The spin is always measured in the unit of reduced Plank constant (~ = 1.054571596(82) × 10−34 Js).
The electric charge is always measured in the unit of the electron charge magnitude(e = 1.602176462(63) × 10−19 C).
2

126

APPENDIX A. THE STANDARD MODEL

Table A.1: Three generations of fermions in the standard model. The mass (in brackets)and the electric charge of the particles are listed together.
1st(mass)

2νee(0.5110 MeV/c )(< 0.003 MeV/c2 )

u
d

2

A.1.2

(1.5 − 4.5 MeV/c )(5 − 8.5 MeV/c2 )Fermion Generation
2nd(mass)

3rd(mass)

Cha
rge

νµ
µ

Leptons(< 0.19 MeV/c2 )(105.7 MeV/c2 )νττ

(< 18.2 MeV/c2 )(1777 MeV/c2 )

0
-1

c
s

Quarks(1.0 − 1.4 GeV/c2 )(80 − 155 MeV/c2 )tb(174 ± 5 GeV/c2 )(4 − 4.5 GeV/c2 )

+ 23
− 13

Bosons

The interaction between fermions is described as exchanging of a spin-1 boson (vectorboson) between them. Such bosons are photon (γ), W± and Z bosons and eight kinds ofgluons (g). The exchanging of photon corresponds to the electromagnetic interaction, thatof W± or Z bosons corresponds to the weak interactions and that of gluons corresponds
to the strong interaction, as shown in Table A.2. Their masses and charges [51] are also
listed together.
Table A.2: The vector bosons and their corresponding interactions are shown. Theirmasses and charges are also listed together.
Vector Boson(mass)

chargeinteraction

γ
W±
Z
g

< 2 × 10−16 eV
80.42 ± 0.04 GeV
91.188 ± 0.002 GeV
0a

0
±1
0
0

electromagnetic
weak
weak
stronga

A.1.3

Theoretical value.

Chirality and Quark Mixing

One of the characteristics of the weak interaction is left-right asymmetric nature, which
violate parity conservation. The term ,’left’ or ’right’ means the chirality, which correspond to the eigen value for the chirality operator, γ 5 . Left-handed state is the eigen statewith eigen value of −1. The relations of a fermion’s field Ψ and its left(ΨL ) or right(ΨR )A.1. ELEMENTARY PARTICLES IN THE STANDARD MODEL

127

fields are as follows.
µ

ΨL
ΨR

¶
1 + γ5=Ψ.
2µ¶1 − γ5
=
Ψ.
2

(A.1)
(A.2)

The chirality is invariant under the proper Lorentz transformation and it is equivalent tohelicity for a massless particle.
For weak interaction, Left-handed and right-handed fermions have different quantum
number and behave differently. For example, W± boson couples only left-handed doublets,
µ

νe
e−

¶

µ
,
L

u
d0

¶
,

(A.3)

L

where the doublet of (u, d0 ) is the eigen state of weak interaction. This is differ from theeigen state of mass. Such weak eigen state like d0 is related to mass eigen state with 3 × 3
unitary matrix Vas,




 
d0
d
 s0  = V  s  .
b0
b

(A.4)

This matrix, V, is called Cabbibo-Kobayashi-Masakawa matrix [110, 111] and is represented as,
Vud Vus Vub
V ≡  Vcd Vcs Vcb  ,Vtd Vts Vtb(A.5)

and the 90% confidence limits on the magnuitude of the elements are,


0.9741 to 0.9756, 0.219 to 0.226, 0.0025 to 0.0048
 0.219 to 0.226, 0.9732 to 0.9748, 0.038 to 0.044  ,0.004 to 0.014, 0.037 to 0.044, 0.9990 to 0.9993(A.6)

Since V is almost diagonal, weak interaction occurs within the single generation dominantly. The non-zero off-diagonal elements generate the couplings across quark generation.
V has 3 parameters corresponding to the rotation angles and one complex phase, whichbrakes the CP invariance3 for the weak interaction.
For lepton doublet such as (νe , e− )L , there are no couplings across the generation in
the framework of the standard model. The mass of the neutrinos is assumed to be zero4
and right-handed neutrino is not exist in the framework.
3

the invariance under the series operation of the parity (P ) transformation and the charge conjugation(C).
4
The neutrino oscillation was observed at the Super-Kamiokande [112], therefore neutrinos have mass.
Since its mass is very small, the standard model is still valid for almost all the subjects.

128

APPENDIX A. THE STANDARD MODEL

A.2

Interactions (Gauge Theory)

All interactions in the standard model can be understood in one theoretical framework,
a gauge theory.

A.2.1

Gauge Theory

The gauge theory is a theory which require the invariance under certain symmetry transformations, called ’local gauge transformations’. (The term ’local’ means the transformation is space-time dependent.)
The local gauge transformation transforms a fermion’s field ψ as ψ → U ψ, where U
is parameterized in the formÃU = exp iX!
αB (x)T B

.

(A.7)

B

This is a presentation of a Lie group where T B are the generators. Its lie algebra is thespace yielded with the generators. The sum for the T B is performed through the numberof the group’s dimension, which are decided from the kind of the group. For example, thenatures of the U (n), SU (n) groups are shown in the following table.
Table A.3: Examples of lie groups.
lie group

representationn × n complex
Unitary matrix

dimensionU (1)unit complex number

1

purely imaginary
complex number

SU (n)n × n complex
Unitary matrix
with determinant 1n2 − 1n × n complex
traceless anti-Hermite
matrixSU (2)

3

Pauli spin matrices

SU (3)

8

U (n)

n2

generatorn × n complex
anti-Hermite matrix

In the framework of the gauge theory, interactions are naturally introduced to freefermions through following manner.
1. Select one symmetry group to reflect the symmetry related to the interaction.

A.2. INTERACTIONS (GAUGE THEORY)

129

2. Add an interaction term to the free Lagrangian in order that it is invariant under
the local gauge transformation corresponding to the symmetry group.
This introduce massless vector bosons, gauge bosons, which mediate interactions
among fermions. The invariance under the symmetry transformation is related to conserved quantity according to the Noether’s theorem. Such conserved quantity corresponds
to the ’charge’, which is the measure of the strength of the coupling between a fermion
and a gauge boson.

A.2.2

Quantum Electrodynamics(QED)

quantum electrodynamics (QED) is the gauge theory for the electromagnetic interaction.
It is based on the U (1) symmetry group, which introduce invariance under complex phasetransformation of the fermion’s field. A free Lagrangian for a Dirac fermion field, Ψ, canbe written as
L0 = iΨ(x)γ µ ∂µ Ψ(x) − mΨ(x)Ψ(x).

(A.8)

The U (1) local gauge transformation of the fermion’s field, Ψ(x), can be written as
Ψ(x) → Ψ0 (x) ==U Ψ(x)(A.9)

exp (iθ(x)Q) Ψ(x).

(A.10)The number of the generator is one because the dimension of the U (1) gauge group isone. Therefore one new spin-1 gauge field, Aµ (x), is required from the gauge principle
and it is transformed as
1
Aµ (x) → A0µ (x) = Aµ (x) + ∂µ θ(x).
e(A.11)

Accordingly, the new Lagrangian is written as
L ==iΨ(x)γ µ Dµ Ψ(x) − mΨ(x)Ψ(x)(A.12)

L0 + eQAµ (x)Ψ(x)γ µ Ψ(x),(A.13)

where Dµ is covariant derivative defined asDµ ≡ [∂µ − ieQAµ (x)].

(A.14)

This new Lagrangian is invariant under the local U (1) transformation and the interaction
term between the fermion and the gauge field is naturally introduced. The gauge boson
correspond to the photon and Q correspond to the electric charge. Thus this U (1) gauge
symmetry related to the electromagnetic interaction is written as U (1)EM . The electriccharge is conserved quantity under the electromagnetic interaction, which is the ’Noether
conserved quantity’ for the U (1)EM gauge symmetry.

130

APPENDIX A. THE STANDARD MODEL

In addition, a gauge invariant kinetic term,

1
Lkin ≡ − Fµν F µν
4(A.15)

is required for the gauge boson field to propagate, where Fµν ≡ ∂µ Aν − ∂ν Aµ . The
complete QED Lagrangian can be obtained adding the kinetic term. The gauge boson ismassless because a possible mass term for the gauge field, 12 m2 Aµ Aµ , would violate thegauge invariance.

A.2.3

Charge

The conserved quantity related to the gauge symmetry is the ’charge’ for the interaction.
This charge acts as the source of the force, deciding the coupling to the gauge boson
corresponding to the interaction. There are three kinds of charge corresponding to thethree kinds of interactions, electric charge, weak isospin and color charge.
Each quark has three kinds of color charge such as ’R’,’G’ and ’B’ which is the sourceof the strong interaction. This makes the quark a color triplet.
The left-handed fermions compose weak isospin doublets as shown in the section A.1.3.
On the other hand, the right-handed fermions are singlets without weak isospin. Since
W boson carrying the weak isospin has electric charge, electric charge contains weakisospin. In the electroweak theory, another charge, ’weak hypercharge’ is used as more
fundamental charge instead of the electric charge. The weak hypercharge (Y ) is related
to the third component of weak isospin (T3 ) and electric charge (Q) as,

Y /2 ≡ Q − T3 .

Then the fermions can be arranged with their charge as shown in Table A.4.

(A.16)

A.2. INTERACTIONS (GAUGE THEORY)

131

Table A.4: Fermion’s charge, where Q is the electric charge, and T and T3 denote theweak isospin and its third component Y is the hypercharge, C is the color charge. Y isrelated to Q with T3 asQ = T3 + Y /2. d0 , s0 , b0 are the weak eigen states.
Fermion Generation
1
2
3

Q

Quantum Numbers
T
T3
Y
C

Leptons
µ

νe
e−

¶

µ
L

eR

¶

νµ
µ−

µ

L

µR

¶

ντ
τ−

L

τR

0
−1

1
2

+ 12
− 12

−1

0

−1

0

0

-2

0

+ 23
− 13

1
2

+ 12
− 12

+ 13

(R, G, B)

+ 23
− 31

0
0

0
0

+ 43
− 23

(R,G,B)(R,G,B)

Quarks
µ

u
d0

¶

µ
L

uR
dR

c
s0
cR
sR

¶

µ
L

t
b0

¶

tR
bR

L

The quarks compose color triplet and the strong interaction is invariant under exchanging among the color triplet. Based on the symmetry, The strong interaction is introducedwith the SU (3)C 5 gauge transformation. It requires 8 gluons, where the number of gluons
is equal to the dimension of the SU (3) group. The QCD Lagrangian LQCD can be writtenwith a single parameter, a strong coupling constant (αs ) at certain energy scale. The αsdepends on the energy scale and tends to decrease at small distance or high energy scale.
This nature is called “asymptotic freedom”. Therefore, the quarks and gluons have thenature of “color confinement”, which means quarks and gluons can not exist in statesother than color singlet.
The left-handed fermions compose weak isospin doublet and related to SU (2)L 6 gauge
symmetry. All of the fermions have weak hypercharge and related to U (1)Y 7 gauge
symmetry. The electroweak interaction is introduced applying the invariance under theSU (2)L ⊗ U (1)Y gauge transformation.
5

The subscript ’C’ shows this SU (3) group is related to the color charge.
The subscript ’L’ shows this SU (2) group is related to the weak isospin doublet.
7
The subscript ’Y’ shows this U (1) group is related to the weak hypercharge.
6

132

APPENDIX A. THE STANDARD MODEL

A.3

Electroweak Theory

A.3.1

Massless Electroweak Theory

The electroweak theory is based on the SU (2)L ⊗ U (1)Y gauge theory. The corresponding
symmetry group is
G ≡ SU (2)L ⊗ U (1)Y .

(A.17)A single family of quarks in the weak eigen state is considered for simplicity without lossof generality. The fermion field is represented asµ¶uψ1 (x) =, ψ2 (x) = uR , ψ3 (x) = d0R .
d0 L(A.18)

The free Lagrangian of the fermion field is written as
L0 =

3
X

iψj (x)γ µ ∂µ ψj (x),(A.19)

j=1

where no mass term exist. For fermion field Ψ, a possible mass term can be written asmΨΨ = m(ΨL + ΨR )(ΨL + ΨR ) = m(ΨL ΨR + ΨR ΨL ),(A.20)

which mixes the left-handed and right-handed component of the fermion field. In theelectroweak theory, left-handed and right-handed field are transformed differently (thesymmetry is chiral). Hence no mass term is allowed in the electroweak theory because
the mass term break the chiral symmetry.
According to the gauge principle, the Lagrangian is required to be invariant under theSU (2)L ⊗ U (1)Y local gauge transformation,
n
o
G
0
~
ψj (x) −→ ψj (x) ≡ exp i~τ α(x)/2 exp {iyj β(x)} ψj (x),(A.21)
where~α(x) is a three vector of real functions, ~τ is the vector of the Pauli spin matrices,
β(x) is the real function and yj is the weak hypercharge for ψj . The Pauli spin matrices
are written as,
µ
¶
µ
¶
µ
¶
0 1
0 −i
1 0
τ1 =
, τ2 =, τ3 =,(A.22)
1 0
i 0
0 −1
and they only act on the doublet field ψ1 as a SU (2) generator.
It requires covariant derivatives with four new gauge bosons as,
·
¸
~τ →
g0
Dµ ψj (x) ≡ ∂µ − ig · W µ −i yj Bµ ψj (x),2
2

(A.23)

where g and g 0 are the coupling constants between fermions and gauge bosons for SU (2)L→and U (1)Y respectively. The W µ = (Wµ1 , Wµ2 , Wµ3 ) are new vector bosons correspondingA.3. ELECTROWEAK THEORY

133

to SU (2)L and Bµ is that for U (1)Y , where the number of the new gauge bosons is assame as the dimension of the symmetry groups. To keep the gauge invariance, theirtransformation properties are fixed as,
GBµ (x) −→ Bµ0 (x) ≡ Bµ (x) +
→

→

G

2
∂µ β(x),g0→~τ · W µ −→ ~τ · W 0 µ ≡ U (x) ~τ · W µ U † (x) +(A.24)
2i
U (x) ∂µ U † (x),g(A.25)

where U (x) ≡ exp {i~τ α~ (x)/2}.
Then new Lagrangian
L

=

3
X

i ψj (x)γ µ Dµ ψj (x)

j=1

=

3
X

·
i ψj (x)γ

µ

j=1

=
=

¸
~τ →
g0
∂µ − ig · W µ −i yj Bµ ψj (x)
2
2

3
h
→ i
g
g0 X
µ
L0 + ψ1 (x)γ ~τ · W µ ψ1 (x) +ψj (x)γ µ [yj Bµ ] ψj (x)
2
2 j=1
µ
¶
g
Wµ3
Wµ1 − iWµ2µ
L0 + ψ1 (x)γψ1 (x)
Wµ1 + iWµ2
−Wµ3
2
3
X
g0
yj ψj (x)γ µ ψj (x)
+ Bµ
2
j=1

≡ L0 + Lfermion(A.26)(A.27)(A.28)(A.29)(A.30)

is invariant under the SU (2)L ⊗ U (1)Y local gauge transformation G. The newly introduced term, Lfermion represents interactions between the gauge bosons and fermions.
→

The gauge boson fields, W µ and Bµ should be converted to the fields for existingbosons, Wµ+ , Wµ− , Zµ , Aµ photon). The relations between them are defined as,
Wµ+
µ

Wµ−
¶
3

Wµ
Bµ

√
≡ (Wµ1 − iWµ2 )/ 2,√≡ (Wµ1 + iWµ2 )/ 2,
¶
¶µ
µ
Zµ
cos θW sin θW,≡Aµ− sin θW cos θW(A.31)(A.32)(A.33)

where θW is called weak mixing angle (’Weinberg angle’) to decide the mixing between
Wµ3 and Bµ . The θW will be fixed to realize the QED Lagrangian from Aµ piece. With
these relations, the fermion part of the Lagrangian, Lfermion can be composed of threeparts. The first One is LCC from Wµ± piece representing the charged current. The second
one is LZNC from Zµ piece representing the neutral current of Z. The third one is LEM

134

APPENDIX A. THE STANDARD MODEL

from Aµ piece representing the electromagnetic current. LEM can be written as¡¢= ψ1 γ µ g sin θW τ 3 /2 + g 0 cos θW y1 /2 ψ1

LEM+ ψ2 γ µ (g 0 cos θW y2 /2) ψ2+ ψ3 γ µ (g 0 cos θW y3 /2) ψ3(A.34)In order to match LEM to the QED Lagrangian , the θW is defined asg sin θW = g 0 cos θW = e.

(A.35)The electric charge operator Qj (j = 1, 2, 3) is defined as follows.
µ 1

+ y21
0
2
0
− 12 + y21

Q1 =

¶
,

Q2 =

y2
,
2

Q3 =

y3
,
2

(A.36)

which corresponding to Eq. A.16. With these relations, the fermion part of the Lagrangian, Lfermion can be written as,
Lfermion = LCC + LZNC + LEM ,(A.37)

where,e(uL (x)γ µ d0L (x) Wµ+ + h.c.),2 sin θW3Xe
τ3=
ψj (x)γ µ ( − Qj sin2 θW )ψj (x) Zµ ,sin θW cos θW j=1
2

LCC = √

(A.38)LZNC(A.39)

LEM = e3Xψj (x)γ µ Qj ψj (x) Aµ .

(A.40)

j=1

The Lagrangian LCC is further converted with the fermion fields u(x), d0 (x) as,e(u(x)γ µ (1 − γ 5 )d0 (x) Wµ+ + h.c.).
LCC = √
2 2 sin θW(A.41)

It can also be written with all quark fields in mass eigen stats ((d, s, b) and (u, c, t)) andthe CKM matrix (V) as,






d
e
(u, c, t)γ µ (1 − γ 5 )V  s  Wµ+ + h.c. .
LCC = √
2 2 sin θWb(A.42)The Lagrangian LEM is further converted with the electric charge of u (Qu ) and that ofd (Qd ) as,¡¢
LEM = e u(x)γ µ Qu u(x) + d0 (x)γ µ Qd d0 (x) Aµ .
(A.43)

A.3. ELECTROWEAK THEORY135

It also can be written with all the three generation and the CKM matrix V,

 
 
u
d
¡
¢
¡
¢
LEM = e  u, c, t γ µ Qu  c  + d, s, b V† γ µ Qd V  s  Aµ
t
b
X
£ µ
¤
= e
f γ Qf f A µ .

(A.44)(A.45)

f =d,u,s,c,b,tThis reproduce the QED interaction. For the case of the leptons (νe , νµ , ντ ) and (e, µ, τ ),their Lagrangian can be written in similar way.
For the case of the Lagrangian LZNC , the weak eigen states for the quarks d, s and b
can be converted with the CKM matrix in the same way and it can be converted as
LZNC =

Xe

2 sin θW cos θW f =d,u,s,c,b,t

£

©
ª
¤
f γ µ vf − af γ 5 f Zµ ,(A.46)

where vf and af are the vector and axial vector couplings for the fermions f and can be
written with its electric charge (Qf ) and the third component of the weak isospin for theirleft-handed part (T3f ) asvf

= T3f (1 − 4 |Qf | sin2 θW ),(A.47)af= T3f ,

(A.48)and their values are shown in Table A.5.
Table A.5: Fermion’s vector and axial vector couplings (v and a respectively),where Q
is the electric charge, and T3 denote the third component of the weak isospin for theleft-handed part.
Fermions

Q

T3

v

a

neutrinos
leptons(νe , νµ , ντ )(e, µ, τ )

0
−1

+ 21
− 12

+ 12
− 12

+2 sin2 θW

+ 12
− 12

up-type quarks
down-type quarks(u, c, t)(d, s, b)

+ 23
− 13

+ 21
− 12

+ 12
− 12

− 43 sin2 θW+ 32 sin2 θW

+ 12
− 12

In addition, gauge invariant kinetic terms Lkin for the gauge bosons should be added
to the Lagrangian. It is given by1
1 → →Lkin = − Bµν B µν − W µν W µν ,44(A.49)

136

APPENDIX A. THE STANDARD MODEL→

where Bµν and W µν are the field strengths:
→

→

→

→

→

W µν ≡ ∂µ W ν −∂ν W µ + g W µ × W ν .

Bµν ≡ ∂µ Bν − ∂ν Bµ ,(A.50)

With the existing bosons,Wµ+ , Wµ− , Zµ , Lkin represents the kinetic terms for the gaugebosons and the triple and quartic couplings between the gauge bosons.

A.3.2

Gauge Boson Masses and Higgs Mechanism

The gauge symmetry forbids the mass term of the gauge bosons. The fermion’s massterm is also forbidden because of the chiral symmetry of the electroweak theory. They are
introduced in gauge symmetric way through spontaneous electroweak symmetry breaking
with a new SU(2)L doublet of complex scalar (spin-0) fields (Higgs fields),
µ + ¶
φ
Φ ≡
.
φ0

(A.51)

The weak hypercharge of the doublet is 1. With the covariant derivative,~τ ~0Y
Dµ ≡ ∂µ + ig · W
Bµµ + ig2
2(A.52)

the Lagrangian
Lscalar = Dµ Φ† Dµ Φ − µ2 (Φ† Φ) − λ(Φ† Φ)2(λ > 0, µ2 < 0) ,(A.53)

is invariant under local SU (2)L ⊗ U (1)Y transformations. There is an infinite set ofdegenerate vacuum states with the vacuum expectation value ofµ¶< φ+ >0< Φ >0 =,< φ0 >0(A.54)

where

−µ2
.
(A.55)
2λIf a vacuum expectation value is selected among them, the SU (2)L ⊗ U (1)Y symmetry of(< φ0 >0 )2 + (< φ+ >0 )2 =the vacuum is broken. The U (1) symmetry for the electromagnetic force (U (1)EM ) should
be kept in the vacuum after the symmetry breaking. In order to keep this U (1)EM symmetry, the invariance of the vacuum expectation value against the U (1)EM transformationis required as,< eiα(x) Q Φ >0 =< Φ >0 ,(A.56)

where α(x) is real function and Q is the electric charge operator. This means the operator
Q annihilates the vacuum asµ
Q < Φ >0 =

1 0
0 0

¶µ

< φ+ >0< φ0 >0¶=< φ+ >0 = 0.

(A.57)

A.3. ELECTROWEAK THEORY

137

Accordingly, in order to break the symmetry of the vacuum as SU (2)L ⊗U (1)Y → U (1)EM ,the vacuum expectation value is selected asµ< Φ >0 =
where

0
√v
2

¶
,

(A.58)r−µ2.
λThe Higgs doublet can be represented with v as¶
µ i ¶µτ χi0 √.
Φ = exp i(v + H)/ 2
2 v
v =(A.59)(A.60)The three fields, χi (i = 1, 2, 3), are related to massless Goldstone bosons. Due to theSU (2)L invariance of the Lagrangian, these Goldstone bosons disappear through fixing
the gauge as,µ¶µ¶τ i χi0 √Ψ = exp −iΨ =.
(A.61)(v + H)/ 2
2 v
The fixed gauge is called as “unitary gauge” and the Lagrangian written with the new0

Higgs fields lose the freedom against SU (2)L transformation. The Lagrangian Lscalar canbe written in terms of the v and H as,
¯µ
¶
µ ¶¯2
0
i
¯
g
v
+
H
τ
0 ¯¯
i
√
Lscalar = ¯¯ ∂µ + ig Wµ + i Y Bµ
1 ¯
2
2
2
2
(v + H)42 (v + H)
−λ
−µ
2
4
±
In terms of the physical fields W and Z, the Lscalar can be written as,
¯µ
µ
¶
¶¯2
+
¯
¯
g
W
0
µ
¯
√
√
Lscalar = ¯¯+ i (v + H)∂µ H/ 2(−1/ 2 cos θW )Zµ ¯
2(v + H)4(v + H)2
−λ
−µ2
2
µ4
¶
2
1
g
1
−µ
+
µ
2
µ
∂µ H∂ H + (v + H) Wµ W +
Zµ Z
=
2
4
2 cos2 θW(v + H)4(v + H)2
−µ2
−λ
2
4
1
µ
=
∂µ H∂ H2
1
g 2 v 2 + −µg2v2
− (−2µ2 )H 2 +Wµ W+
Zµ Z µ2
4
8 cos2 θW
g2v
g2v
H Wµ+ W −µ +HZµ Z µ+2
4 cos2 θW
g2
g2
H 2 Zµ Z µ+ H 2 Wµ+ W −µ +48 cos2 θW
µ2
µ2
µ2 2
+ H3 + 2 H4 −
v
v
4v
4

(A.62)(A.63)(A.64)(A.65)

138

APPENDIX A. THE STANDARD MODEL

The first term is the kinetic term of the Higgs. The terms from fifth to tenth represent
the triple and quartic couplings between the gauge bosons and the Higgs and the triple
and quartic self coupling of the Higgs. The second to fourth terms can be interpreted as
mass terms of the Higgs(MH ), W ± (MW ) and Z(MZ ) bosons respectively as,
1
g 2 v 2 + −µg2v2
− (−2µ2 )H 2 +Wµ W+
Zµ Z µ2
4
8 cos2 θW
1
1
2= − MH2 H 2 + MW
Wµ+ W −µ + MZ2 Zµ Z µ .
22
Accordingly the Higgs, W and Z masses can be written as,
p
−2µ2
MH =
gv
ev
MW =
=
2
2 sin θW
gv
ev
MZ =
=
2 cos θW
2 sin θW cos θW(A.66)(A.67)(A.68)(A.69)

From the low-energy phenomenology,
g
√ =
2 2

µ

2
MW
G
√ F
2

¶1/2
.

(A.70)

From the measured value of GF , v can be calculated as,
³√
´1/2
v=
2GF
∼ 246GeV.

A.3.3(A.71)

Fermion Masses

The masses of the fermions can be generated in a gauge symmetric way through Yukawa
couplings with the Higgs boson. With the Higgs doublet, the following gauge-invariant
Lagrangian, LY , with fermion-scalar Yukawa couplings can be written as,µ + ¶µ 0† ¶
¡
¢
¡
¢
φ
φ
¯
¯
LY = c1 ū, d L
dR + c2 ū, d L
uR
0
φ
−φ+†
µ + ¶
φ
eR + h.c.
+ c3 (ν̄e , ē)L
φ0

(A.72)In the unitary gauge after spontaneous symmetry breaking, the Yukawa-type Lagrangiantakes the simple form as,ª©
1¯ + c2 ūu + c3 ēe .
LY = √ (v + H) c1 dd
2(A.73)

Therefore, the mechanism of the spontaneous symmetry breaking also generates fermion
masses as,√
md = −c1 v/ 2 ;√mu = −c2 v/ 2 ;√me = −c3 v/ 2 .

(A.74)

Since we do not know the parameters ci , the values of the fermion masses are arbitrary.

A.4. THE STANDARD MODEL PARAMETERS

A.4

139

The Standard Model Parameters

The standard model Lagrangian LSM can be written as
LSM = LQCD + LEW(A.75)

LEW = L0 + LCC + LZNC + LEM + Lkin + Lscalar + LY .

(A.76)The parameters of the standard model (SM) are the couplings (αs , g and g 0 ), the vacuum
expectation value (v), the Higgs mass (mH ), all fermion’s masses (mf ) and the 4 parameters (3 rotation angles and 1 phase) of the CKM matrix (V). The set of variables, g and
g 0 can be replaced with θW and αEM . The variable v can be replaced with the GF . Oncethese parameters are set, the couplings between particles and their propagation in thespace-time are known with the standard model Lagrangian and physics quantities related
to the standard model process can be calculated. Particularly, a kind of diagram called“Feynman diagram” is used, which represents the couplings and the propagation of theparticles in the SM Lagrangian.

141

Appendix B
Massless Tree-Level AFB Calculation
The tree level Feynman diagram for the process e+ e− → f f¯ is written in Fig. B.1.
           

Figure B.1: The tree level diagram for e+ e− → f f¯.

This is a superposition of the two diagrams below, where propagators and vertex
coefficients are shown.
      $  %  $ &'() +*-, .

         / () *-01  !"# !

1  !)3 #%$& /  !)3 #+*( !-, #.$)0-/  !-, #"45$ 6 7 8?9@ : A ; B< C  D < 8 = >            E      

  !" #%$'&( !)" #"$ 6 7 8 ?+F @ : A ; B GC  D G 8 = >/1  !)2#%$ 0- !)2 #Figure B.2: The tree level diagrams for e+ e− → f f¯.

The invariant amplitude for the diagram with γ propagator is defined as Mγ and thatfor the diagram with Z propagator is defined as MZ . The invariant amplitude for thereal process as shown in Fig. B.1 is defend as M and is related with MZ and Mγ as,
M = MZ + Mγ .

(B.1)

142

APPENDIX B. MASSLESS TREE-LEVEL AFB CALCULATION

The interaction rate Γ is given as,ZZ
d3 p~cd3 p~d4Γ = (2π) Nc · · ·h|M|2 i δ 4 (pa + pb + pc + pd ),33(2π) 2Ec (2π) 2Ed(B.2)

where Nc is the degree of freedom due to the color charge of the final state fermion(1 for
leptons, 3 for quarks), pi ≡ (Ei , p~i ) is 4-momentum of particle i, mi is mass of particle
i and h|M|2 i means that |M|2 is averaged over the initial number of the spin statesand is summed over the final number of the spin states, which means that electron andpositron are unpolarized and the polarization of the final fermion and anti-fermion are
not measured. The h|M|2 i can be written as
h|M|2 i = h|MZ |2 i + h|Mγ |2 i + 2 Re{hMZ M∗γ i}.

(B.3)

The third part means the contribution of the Z − γ interference.
√
If the 4-momentums of initial electrons are given with the center-of-mass energy, s,
as
pa

=

pb

=

¡√
¡√

s/2, 0, 0, |~pe |¢¢s/2, 0, 0, −|~pe | ,

(B.4)
(B.5)

he Γ can be written as

Z Z
Nc |~pf |√Γ=h|M|2 i dΩ2
16π swith constraints on the 4-momentums of the outgoing fermions as(B.6)

p~c = p~f ,(B.7)

p~d = −~pf ,
q
q
√
2
2
|~pf | + mc + |~pf |2 + m2d = s.

(B.8)

With the incident fluxµF = 2 Ea 2 Eb|~pa | |~pb |
+
Ea
Eb

¶

√
= 4|~pe | s,

(B.9)

(B.10)The differential cross-section of the e+ e− → f f¯ can be written asdσ
1 dΓ
Nc |~pf |==h|M|2 idΩ
F dΩ
64π 2 s |~pe |The invariant amplitudes can be written with the diagrams shown in Fig. B.2,µ¶−igµ ν
µ−iMγ = v̄(pb ) (ieγ ) u(pa )ū(pc ) (−ieQf γ ν ) v(pd ),
2
q
µ
¶
µ
¶
µ
−igγ (ve − ae γ 5 )−i(gµ ν − qµ qν /m2Z )−iMZ = v̄(pb )u(pa )
2 cos θW
q 2 − m2Z − imZ ΓZµ¶−igγ ν (vf − af γ 5 )ū(pc )v(pd ).
2 cos θW

(B.11)(B.12)(B.13)

143
In the massless limit for the fermions and with θ which is defined as the polar angle ofthe fermion f , h|M2γ |i,h|M2Z |i and hMZ M∗γ i can be written as
h|M2γ |i=h|M2Z |i=hMZ M∗γ i=

Q2f e4 (1 + cos2 θ)(B.14)£¤|χ(s)|2 e4 (ve2 + a2e ) (vf2 + a2f ) (1 + cos2 θ) + 8 ve ae vf af cos θ (B.15)£¤−Qf e4 χ(s) ve vf (1 + cos2 θ) + 2 ae af cos θ ,(B.16)

where χ(s) is defined as
χ(s) ≡

³s´
e2

1
g2
2
4 cos2 θW s − mZ + imZ ΓZ(B.17)The differential cross-section also in the massless limit for the fermions can be written as¤dσ
Nc α 2 £=G(s)(1 + cos2 θ) + H(s) cos θ ,(B.18)dΩ4s2e, and where G(s) and H(s) are defined aswhere α is defined as 4π

G(s)

≡ Q2f

+ |χ(s)|2 (ve2 + a2e )(vf2 + a2f ) − 2 Qf ve vf Re{χ(s)}(B.19)H(s)≡ |{z}+8|χ(s)|2 ve ae vf af{z|
Z exchange(B.20)

γ,−4Qf ae af Re{χ(s)} .
} |
{z
}
,
γ − Z interference ,The total cross-section σ can be written as
4Nc πα2
σ =
G(s),
3s
and the forward-backward asymmetry (AFB ) can be written as,
AFB ≡(B.21)

3 H(s)
σF − σB=,
σF + σB
8 G(s)(B.22)

where σF and σB means the cross-section with cos θ > 0 and cos θ < 0, respectively.

0.8

c
AFB

b
AFB

In Eq. B.20, B.21, contributions of γ exchange, Z exchange and γ − Z interference are√
separately written. The AFB for b and c quarks are shown as a function of s in Fig. B.3.

0.6

0.8
0.6

0.4

0.4

0.2

0.2

-0

-0

-0.2

AFB(Z)-0.2

AFB(Z)-0.4

AFB(Z+γ )-0.4

AFB(Z+γ )-0.6

AFB(total)-0.6

AFB(total)-0.8
0 20 40 60 80 100 120 140 160 180 200 220-0.8
0 20 40 60 80 100 120 140 160 180 200 220

s[GeV]

s[GeV]

Figure B.3: The√s dependence of Ab,c
FB .

145

Appendix C
Jet Clustering
The emission of a parton (quarks or gluons) can form a jet at LEP. The jet can be used
for many purposes, such as to determine the direction of a parton which initiates the jet,to evaluate the charge of such parton with the jet charge, and to construct secondary
vertex from the tracks in the jet. The multiplicity of the jet is used to study the event
topology. Some variables to characterize the shape of the jet are introduced, which areused to study the nature of the QCD. For those purposes, some jet-clustering algorithms
are developed. In this thesis, the evaluations of the primary-quark direction and theidentification of the primary-quark charge, and the secondary-vertex reconstruction withthe jet are important. As regards the subjects, the performance is compared amongsome jet-clustering algorithms. The performance in the evaluation of the primary-quark
direction with the jet is compared with the performance with the thrust axis. That in
the identification of the primary-quark charge with the jet charge is compared with thatwith the hemisphere charge.

C.1

Jet-Clustering Algorithm

There are mainly two types of jet-clustering algorithms. One is based on successive
recombinations of pairs of particles, such as the JADE algorithm [25, 113], the Durham
algorithm [73]. The other is based on geometrical assembly, such as the cone algorithm,
which is developed in the hadron-collider experiments [114–118] and imported to the
OPAL experiment [119]. These jet-clustering algorithms are frequently used in LEP
experiments and they are briefly described here.

C.1.1

JADE algorithm with the E0 recombination scheme

The jet clustering is started with a list of jets that are just the observed particles (tracks
and unassociated clusters). Then an iterative process of successive recombinations of pairsof particles is followed as described below. At each stage of iteration, two jets, i and j,

146

APPENDIX C. JET CLUSTERING

are considered as candidates for combination into a single jet according to the value of“jettiness” variable, xij or yij (xij scaled by the visible energy, Evis ), defined as,
xij = 2Ei Ej (1 − cos θij ),

(C.1)

2
yij = xij /Evis
,

(C.2)

where Ei and Ej are the energy of the jets i, j, and cos θij is the opening angle of thetwo jets. Among all the possible pairs, the pair with the smallest value of xij (or yij ) is
combined first, where the four-momentum of the combined jets is formed as the vector
sum of the four-momentum of the original two jets. The jet list is updated by subtracting
the original pair and adding the new jet. This process is iteratively continued until every
remaining xij (or yij ) is larger than a preset cut-off parameter xcut (or yrmcut ), which iscalled as “jet resolution parameter”. This has been a very useful tool enabling the testingof many perturbative QCD calculations.

C.1.2

Durham algorithm

This algorithm is similar to the JADE algorithm, but the xij is defined differently as,
xij = 2 min(Ei2 , Ej2 )(1 − cos θij ),(C.3)

which is interpreted as the relative transverse momentum of the pair. As regards the
QCD study, this benefits from smaller hadronization corrections than the JADE schemes,
and is calculable in resummed perturbation theory [73].

C.1.3

Cone algorithm

In the cone algorithm, a jet is formed with the particles (tracks and unassociated clusters)
included in a cone with the interaction point as its apex, where the cone size is defined byits half-angle R. In addition to the R, the minimum jet energy to form the jet (ε) is usedas the parameters to characterize the cone algorithm. The cone (jet) axis (pJ ) is definedas the sum of the momentum of the particles (pi ) inside the cone. The condition to froma jet can be written as,pJ =Xi ∈ jetpi ;pi · pJ≤ cos(R) ∀ i ∈ jet|pi ||pJ |(C.4)

The detail of the algorithm is described as follows. At first, for the given R, thealgorithm seeks as many sets of particles to fulfill the condition as possible, where suchset of particles is called as “proto-jet”. Then proto-jets with the energy less than εis eliminated. If the two (or more) proto-jets share the same particles, the sharing iseliminated as follows. A proto-jet with its all particles included in another proto-jet isC.2. EVALUATION OF PRIMARY-QUARK DIRECTION

147

eliminated. If the proto-jets share finite number of particles, If the sum of the energy
for the shared particles is larger than 75% of the energy of the proto-jet which is theleast energetic one among the proto-jet concerning the sharing, the shared particles are
assigned to the proto-jet with the highest energy and the other proto-jets are eliminated.
If the sum of the energy for the shared particles is less than 75% of the energy of the
proto-jet with the smallest energy, the shared particles are assigned to the nearest protojet measured with the jet axis. Finally, proto-jets with the energy less than ε is again
eliminated. The resulting proto-jets are the final jets.
The cone algorithm cluster the particle with definite opening angle, whereas the jadelike algorithm can cluster less energetic particle with large opening angle. It enables tostudy the kinematic information in a given jet with finer cone. For example, the tracks
coming from the secondary vertex can be enriched with finer cone, the isolation for agiven track can be studied with comparing the jets constructed with finer and broader
cones.

C.2

Evaluation of Primary-Quark Direction

The performances in the evaluation of the primary-quark direction with the thrust axis
and with the jet axises are compared. The jets are constructed with the above-mentioned
three kinds of jet-clustering algorithms and they are labeled as follows.
Cone0.25/5 Cone algorithm with R = 0.25, ε = 5[GeV].
Cone0.3/5 Cone algorithm with R = 0.3, ε = 5[GeV].
Cone0.4/5 Cone algorithm with R = 0.4, ε = 5[GeV].
Cone0.55/5 Cone algorithm with R = 0.55, ε = 5[GeV].
Cone0.55/7 Cone algorithm with R = 0.55, ε = 7[GeV].
Cone0.7/5 Cone algorithm with R = 0.7, ε = 5[GeV].
Cone0.77/7 Cone algorithm with R = 0.7, ε = 7[GeV].
Durham49 Durham algorithm with xcut = 49[GeV2 ].
Durham0.001 Durham algorithm with ycut = 0.001.
Durham0.01 Durham algorithm with ycut = 0.01.
Durham0.02 Durham algorithm with ycut = 0.02.
Durham0.03 Durham algorithm with ycut = 0.03.

148

APPENDIX C. JET CLUSTERING

JadeE0/25 Jade algorithm(E0) with xcut = 25[GeV2 ].
JadeE0/49 Jade algorithm(E0) with xcut = 49[GeV2 ].
JadeE0/0.03 Jade algorithm(E0) with ycut = 0.03.
By means of∆ ≡ | cos θthrust/jet | − | cos θprimary quark |,(C.5)

where cos θthrust/jet is the polar angle of the thrust or jet axis and cos θprimary quarkis the polar angle of the primary quark direction, the resolution of the primary-quark
direction is evaluated with the ratio of the events with |∆| < 0.1, which means the ratioof the entries in the colored histogram shown in Fig. C.1(a). The resolutions obtained withthe thrust axis or the jet axises for each jet-clustering algorithm are shown in Fig. C.1(b).
The use of the thrust axis marks the best performance, since it is not sensitive to thehard gluon emission.

(b)

103

102

10

1
-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8

1|cosθthrust |-|cosθprimary quark |

0.92
0.9
0.88
0.86
0.84
0.82
0.8

ThrustAxis
Cone0.25/5
Cone0.3/5
Cone0.4/5
Cone0.55/5
Cone0.55/7
Cone0.7/5
Cone0.77/7
Durham49
Durham0.001
Durham0.01
Durham0.02
Durham0.03
JadeE0/25
JadeE0/49
JadeE0/0.03

104

Resolution of the primary-quark direction

Events(a)

Figure C.1: (a)The distributions of | cos θthrust/jet | − | cos θprimary quark | and the resolutions of the primary-quark direction obtained with the thrust axis or the jet axises foreach jet-clustering algorithm.

C.3

Evaluation of Primary-Quark Charge

The performances in the identification of the primary-quark charge with the hemispherecharge and the jet charges constructed with the above-mentioned three kinds of jetclustering algorithms are compared as follows, where the labeling of the jet-clustering
algorithms in the section C.2 are used. The charge identification probabilities with thehemisphere charge and with the jet charges are shown in Fig. C.2, where the use of theC.3. EVALUATION OF PRIMARY-QUARK CHARGE

149

cone algorithm with broader cone marks best performance among the jet charges. The
performance for the hemisphere charge is comparable to that for the jet charge with thecone algorithm. Furthermore, the performance with the hemisphere charge is close to
optimal when only the track momentum and charge information is used [120].

150

APPENDIX C. JET CLUSTERING

Charge ID Prob.[%]

(a)

0.68

HemisphereCharge

0.66

Cone0.25/5

0.64

Cone0.3/5

0.62

Cone0.4/5

0.6

Cone0.55/5

0.58

Cone0.55/7

0.56

Cone0.7/5

0.54

Cone0.7/7

0

0.2

0.4

0.6

0.8

1
κ

(c)

0.68
HemisphereCharge

0.66
Durham49

0.64
0.62

Durham0.001

0.6Durham0.01

Charge ID Prob.[%]Charge ID Prob.[%]

(b)

0.68
HemisphereCharge

0.66
0.64

JadeE0/0.03

0.62
0.6
JadeE0/25

0.58

0.58
Durham0.02

0.56

0.56

JadeE0/49

Durham0.03

0.54

0.54
0

0.2

0.4

0.6

0.8

1

0
κ

0.2

0.4

0.6

0.8

1
κ

Figure C.2: The charge identification probabilities with jet charges constructed with (a)the cone algorithms, (b) the Durham algorithms, and (c) Jade (E0) algorithms, wherethat for the hemisphere charge is shown in each figure.

C.4. B-TAGGING

C.4

151

b-Tagging

Here the performances of jet-based b-tagging from the jets constructed with the abovementioned three kinds of jet-clustering algorithms are compared. The jet-based b-tagging
is done with two kinds of b-tagging methods as follows.
1. cut-based tag on the decay length significance.
The decay length significances (l/σl ) are constructed from each secondary vertex
which is reconstructed from each jet. The most significant l/σl is selected in a eventand the b-tagging is done by applying a cut on the l/σl .
2. BT NN based tag. The ANN-based discriminant termed “BT NN” [40] is constructed from each jet with |l/σl | larger than 4. In addition to the l/σl , the ANN
have 4 more inputs derived from vertex multiplicity and invariant mass information.
Then most significant value of the ANN outputs are selected and the b-tagging is
performed by applying a cut on the ANN output.
The distributions of the decay length significance and the BT NN are shown in Fig. C.3(a)and (b), respectively.

(b)

10
10

3

2

b
c
uds
4f
2p

Number of Events

Number of Events

(a)

10
1
10

-1

-20

-10

0

10

20

30

40

50

60

Decay Length Significance

b
c
uds
4f
2p

10

1

10

-1

-8

-6

-4

-2

0

2

4

6

8

BT NN

Figure C.3: The distributions of the decay length significance and the BT NN for theof 189 GeV.

√s

With these discriminants, the b-tagging are preformed for each jet-clustering algorithm. The performances for each jet-clustering are shown in Fig. C.4(a) for the cut-based
tag on the decay length significance. Those for the b-tagging with BT-NN are shown in
Fig. C.4(b). For the both case, the cone algorithm marks the best performance. Amongthe cone algorithms, that with relatively finer cone marks better performance, which is

152

APPENDIX C. JET CLUSTERING

due to the higher efficiency and better purity of the tracks and clusters coming from theb hadrons in selecting them in the jet with the cone algorithms. It is shown in Fig. C.5,
where a b-particle efficiency is defined as the ratio of the number of tracks or clusters
originating from the b hadrons (b-particles) included in the jets to the total number of bparticles, and where a b-particle purity is defined as the ratio of the number of b-particles
included in the jets to the total number of tracks or clusters included in the jets. A
superior performance of the cone algorithms is observed.

Purity(a)Cone0.25/5

0.9

Cone0.3/5
Cone0.4/5
Cone0.55/5

0.8

Cone0.55/7
Cone0.7/5
Cone0.7/7

0.7Durham0.001
Durham0.01

0.6

Durham0.02
Durham0.03
Durham49

0.5

JadeE0/25
JadeE0/49
JadeE0/0.03

0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Efficiency

Purity(b)
Cone0.25/5

0.9

Cone0.3/5
Cone0.4/5
Cone0.55/5

0.8

Cone0.55/7
Cone0.7/5
Cone0.7/7

0.7Durham0.001
Durham0.01

0.6

Durham0.02
Durham0.03
Durham49

0.5

JadeE0/25
JadeE0/49
JadeE0/0.03

0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Efficiency

Figure C.4: (a) The purity vs. efficiency for the b-tagging with the cut-based b-tagging
on the decay length significance for each jet-clustering algorithm. (b) That with the
ANN-based b-tagging (BT NN)(b).

C.4. B-TAGGING

153

b-particle Purity(a)Cone0.25/5

0.8

Cone0.3/5

0.75

Cone0.4/5

0.7

Cone0.55/7

0.65

Cone0.7/7

Cone0.55/5
Cone0.7/5
Durham0.001
Durham0.01

0.6

Durham0.02

0.55

Durham0.03
Durham49

0.5

Jade25
Jade490.45

Jade0.03

0.8

0.85

0.9

0.95

1

b-particle Efficiency

Figure C.5: The b-particle purity vs. efficiency for each jet-clustering algorithm.

155

Appendix D
Artificial Neural Network (ANN)A measurement of a physical quantity of a particular class of events (signal) is performed
generally under the existence of the background, where the discrimination of the signal
from the background is important. For example, the b-tagging is important for themeasurement of the AbFB .
The properties of the signal and background can be described by a set of variables(a input vector, x), such as the momentum, the transverse momentum, number of hits,
etc. For the case of n such variables, the discrimination can be done by constructing aintersecting hypersurface, which divide the n dimensional space of the variables into thesignal and the background region. If the probability density function of the signal andbackground are known as f s (x) and f b (x), respectively, such intersecting hypersurface
can be obtained as

f s (x)= r,
f b (x)(D.1)

where the r determines the signal-selection efficiency and the hypersurface gives the highest signal purity for the given signal-selection efficiency according to the Neyman-Pearsonlemma [121]. In other words, the r is the optimal scalar test statistics on which a cut isapplied giving maximum purity for the given efficiency.
Such scalar test statistics can be obtained with an Artificial Neural Networks (ANNs)[122], which is described as follows. The Artificial Neural Network is a term referring
to numbers of mathematical models which have some distributed nodes connected witheach other. Among them, one of the most important type of ANNs is a network termed“feedforward multilayer perceptron” which processes numbers of data inputs into some
outputs through the layered network structure with nodes connected with each other asshown in Fig. D.1. Here the networks with only one output are described. Furthermore,
the networks with only three layers (one input, one hidden, and one output layer) aredescribed here, since it is proved by [123], [124] and [125] that one hidden layer is inprinciple sufficient for arbitrary recognition task which means, for example, any continuousfunction can be represented by the ANN with one hidden layer basically.

156

APPENDIX D. ARTIFICIAL NEURAL NETWORK (ANN)

input
layer
(node:xi)

hiddenlayer(node:hj)

outputlayer(node:o)threshold:φjweight:v
ijweight:w

jthreshold:θ

Figure D.1: The structure of the feedforward multilayer parceptron with one hidden layer.

The value of the output node of o is connected with the values of nodes in the hidden
layer (hj ) and in the input layer (xi ) through the weights (wij , v j ), using a function (g(z))
and the thresholds (θj , φ), as,

o = g

Ã
X
j

= g

Ã
X

!
v j hj + φ

(D.2)

!
X
wij xi + θj ) + φ ,v j g(

(D.3)

j

i

where the g(z) is termed an activation function and can be written asg(z) =

1,
1 + exp(−2z)(D.4)

which is called as a sigmoid function giving a value from 0 to 1 as shown in Fig. D.2.

sigmoid function

157

1
0.8
0.6
0.4
0.2
0
-10

-8

-6

-4

-2

0

2

4

6

8

10
z

Figure D.2: The sigmoid function.

The feedforward multilayer perceptron can be used in the discrimination of the signal
from the background by tuning the weights and thresholds to be fitted to the discrimination problem. Such tuning can be done through an iterative fitting with many sets of an
input vector of signal or background and a desired feature value which is 1 for the signal
or 0 for the background. The fitting is performed by minimizing an error measure of thefit, e.g. a mean square errorNp

1 X (p)(o − t(p) )2E=
Np p=1(D.5)

between o(p) (o for the pth input vector) and the desired feature value t(p) (1 for the signal
or 0 for the background for the pth input vector), where Np is the number of samples(sets of an input vector and an output) used for the fitting. This process is called as a“supervised learning”. Among numbers of the supervised learning methods, a learning
process termed “back propagation” learning [126, 127] is frequently used, which minimize
the E updating the weight vector, ω (vector of all weights and thresholds), with gradientdescent as,ω t+1 = ω t + ∆ω t ,(D.6)

where t represents the iteration number and ∆ω t is represented as,∆ω t = −η

∂Et
.
∂ω

(D.7)

158

APPENDIX D. ARTIFICIAL NEURAL NETWORK (ANN)Practically, the back-propagation learning is performed with JETNET [85] using the MC
sample containing the signal and background. The sample is divided into two independent
samples termed “learning sample” and “validation sample”. The learning sample is used
for the learning and the tuned networks is applied to the the validation sample to checkthe “overlearning” which means the ANN learns the characteristics specific to the learning
sample which is not the global nature of the signal. The learning process has to be stopped
before the overlearning starts. For example, the error (E) as a function of the numberof the ANN updates (learning curve) for the prompt b → µ discriminating ANN (Nb formuon) described in subsection 4.7.3 is shown in Fig. D.3. After 114 updates, the error forthe validation data starts to increase although the error for the learning data decrease,which shows the start of the overlearning and the learning is stopped there. The structureof the the Nb for muon is shown in Fig. D.4, where the magnitudes of the weights areindicated with the line width.

Error

graError_Trai
0.03
0.028

learning sample

0.026

validation sample

0.024
0.022
0.02
0.018
0.016
0.014
0.012
0

200

400

600

800

1000

Epoch:
min:1140.018081)
numberof(error=
updatesFigure D.3: The learning curve.

159

p
pt

Esub-jetEvis
Sum ptI.P.Sig
Nfwd
jet

Nb

DLS
DLSo
QjQl
QjoQl
Nµ
Figure D.4: An example of the constructed ANN (Nb for muon). The width of the line
indicated the strength of the coupling for the two nodes.

The output (o) of the feedforward multilayer perceptron after the learning can beused to discriminate the signal from the background for other samples. For example, thedistribution of the output of the Nb for muon is shown in Fig. D.5(a).

S/(S+B)(b)
Events

(a)

3

10

1.2
1
0.8

102

0.6
0.4

10

0.2
1

0

0.2

0.4

0.6

0.8

1
NNout

0
0

0.2

0.4

0.6

0.8

1

1.2
NNout

Figure D.5: The distribution of the output of the Nb for muon and that of the signal ratio
as a function of the ANN output.

160

APPENDIX D. ARTIFICIAL NEURAL NETWORK (ANN)The output of the such feedforward multilayer perceptron after the learning can beinterpreted as the Bayesian a posteriori probability [128], which can be written with theprobability density function of the signal and background used in Eq. D.1 as,o=

Ns f s (x),Ns f s (x) + Nb f b (x)(D.8)

where Ns and Nb are the numbers of signal and background, respectively. The relation
can be seen in Fig. D.5(b), where the signal ratio is drawn at each bin for the histograms
shown in Fig D.5(a). The Eq. D.8 can be written with the r (the ratio between theprobability density functions of signal and background) defined in Eq. D.1 as,
o=

1
1 ,
1 + R·r(D.9)

where R is the signal to background ratio (Ns /Nb ). Since the Eq. D.9 is the monotonic
function of r, the output of the ANN can also be used as the optimal scalar test statistics.
More proof about the fact can be found in [129] and [130]. Since the r can be obtained
from the output of the ANN with the relation in Eq. D.9, it can be transformed to the
output (o0 ) corresponding to other sample with different signal to background ratio (R0 )
as,
R0 o
.
(D.10)o0 = 0R o + R(1 − o)For example, the output corresponding to the sample including the same numbers of

Eventssignal and background is shown in Fig. D.6.

103

102

10

1

0

0.2

0.4

0.6

0.8

1
NNout

Figure D.6: The distribution of the output corresponding to the sample including thesame numbers of signal and background, which is made from Fig. D.5(a).

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169

Acknowledgment
At first, I would like to thank Prof. Tomio Kobayashi, for his warm guidance and encouragement during over six years of my research. He guided me to the OPAL experiments forthe theme of my doctor thesis. He allowed me to stay at CERN for the study, gave manysuggestions about the analysis and read and corrected this draft many times. I learned
much from his words, “Do exhaustively!” and “Be experimental physicist”.
I would like to thank Prof. Tatsuo Kawamoto, for his great help on this analysis.
He gave many realistic and suggestive advices when I brooded over problems. He also
corrected this draft many times patiently and gave me valuable suggestions in searchingfor new physics with the measurements. I learned much from him about the collider
physics, analysis methods, the way of thinking in general, through many discussions. I
really enjoyed discussions about many subjects (not only high-energy physics) with a cupof coffee or a glass of beer or wine.
I also appreciate Dr. Ikuo Ueda and Dr. Masaya Ishino for their daily maintenanceand improvements of the computing environment at CERN. They accepted my annoying
requests. I also thank Prof. Shoji Asai. They also gave me various suggestions about
my analysis which made me think deeper on this analysis. I also thank them for their
help on living at CERN for example, housing, French conversation and eating. I enjoyedconversation with them and learned there are many ways of thinking. I also thank Mrs.
Terue Ishino for her help in obtaining the French visa and her inviting me to the dinnermany times.
I would like to thank Dr. Daisuke Toya for his kind help about the OPAL analysis
and about the life at CERN. I would like to thank Dr. Koichi Nagai, Dr. Keiichi Nagai,
Dr. Kunihiro Nagano, Dr. Tadashi Maeno, and Dr. Naoko Kanaya. I enjoyed my life at
CERN with them.
I am indebted to all the members of the OPAL Collaboration, for their construction of
the detector and successful operation. They gave me the chance for the analysis and manysuggestions at OPAL plenaries. I express my gratitude especially to Dr. David Plane, Dr.
Pippa Wells, Dr. Patricia Ward, Dr. Richard Hawkings and Dr. Frank Fiedler. I also
thank Ms Dawn Hudson who is the OPAL secretary. I couldn’t get French visa without
her help. I also thank Ms Mette Stuwe who is the previous OPAL secretary. She invitedus to the dinner many times.
I would like to express my gratitude to all members in ICEPP, especially to Prof. Sachio Komamiya, Prof. Toshinori Mori, Prof. Satoru Yamashita, Prof. Hiroshi Sakamoto,Prof. Tetsuro Mashimo, Dr. Junichi Tanaka, for useful comments at daily discussions,
meetings and at the defense examination. I would like to thank secretaries in ICEPP: Ms.
Emi Bandai, Ms. Chiyo Ito, Ms. Ryoko Ono, Ms. Ikue Tsukamoto, and Ms. Eiko Yuno,
for supporting our researches. I would like to thank Dr. Kenji Ozone and Ms. Yasuko
Hisamatsu for various discussions at night. I also enjoyed discussions with Dr. Toshio
Namba, Dr. Toshiyuki Iwamoto, Dr. Shuei Yamada, Mr. Hajime Nishiguchi, Mr. Ryu
Sawada, Mr. Yousuke Kataoka, Mr. Hiroshi Nomoto, Mr. Yusuke Fujii, Mr. Hiroaki
Natori, Mr. Yusuke Uchiyama, Mr. Takayuki Sasaki and Mr. Yoshiki Yamaguchi.
Finally I thank my parents Osamu and Yoshi Nanjo for their continuous help, encouragement and understandings.
Hajime Nanjo