COLLIDER PHENOMENOLOGY OF HEAVY CHARGED GAUGE BOSONS
By
Jiang-Hao Yu

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics and Astronomy
2012

ABSTRACT
COLLIDER PHENOMENOLOGY OF HEAVY CHARGED GAUGE BOSONS
By
Jiang-Hao Yu

New vector resonances, such as heavy W and Z gauge bosons, may be one of the cleanest
and most important signals of new physics that can be discovered in the early data from the
CERN Large Hadron Collider (LHC). Such new heavy gauge bosons arise in numerous gauge
extensions of the Standard Model (SM) of particle interactions. Their discovery would help
to elucidate the nature of the spontaneous symmetry breaking at the weak scale. In this
thesis, we focus on extended gauge models with these novel electroweak gauge bosons in the
framework of an effective theory. Based on breaking patterns and fermion assignments, we
classify a number of models with SU(2)1 × SU(2)2 × U(1)X gauge symmetry (the so-called
G(221) models) and write down the general Lagrangian and Feynman rules in a linearized
effective field theory.
The current constraints on G(221) models are studied in detail. This is done by combining the indirect constraints from the precision electroweak tests (EWPT) at the Large
Electron Positron (LEP) collider with other low energy observables. Moreover, direct search
constraints from the Fermilab TeVatron and the LHC are also included. Our results indicate
that, although the light W and Z bosons are allowed by EWPTs, direct constraints require
the masses of the W and Z bosons in G(221) models to be at the TeV scales.
In this thesis, we also consider the prospect for discovering these bosons in the coming
data from the LHC. We focus on searches for heavy resonances in final states with a lepton
plus large amounts of missing energy. Having a lepton and large missing energy in the final

state ensures the experimental collaborations can readily identify these potential signatures
of new physics. We explore the potential for discovering the heavy charged gauge bosons in
the G(221) models in the coming data, as well as the ability to place limits on their masses
and couplings. It is shown that in some models it is more efficient to use the W leptonic
decay channel for discovery or exclusion than the Z leptonic decay channel. Also, we note,
for various G(221) phobic models, observing a Z alone cannot rule out the possibility of a
non-abelian gauge extension of new physics.
To help unravel the nature of the new gauge bosons, it is necessary to study the properties
of the charge gauge bosons in the top quark channels. We show that it is possible to probe the
chiral structure of the charged gauge bosons using top quark polarization with the upcoming
data from the LHC. Furthermore, a flavor-violating W model might explain the anomalously
large forward-backward asymmetry of the top pairs. An anomalously large forward-backward
asymmetry was observed at both the CDF and DØexperiments at the TeVatron. Finally, we
explore the searches for a top-philic W in the top quark pair plus a hard b-jet final states
at the LHC. This provides a sensitive test for exotically charged gauged bosons.

Copyright by
JIANG-HAO YU
2012

Dedicate to my parents and Nan.

v

ACKNOWLEDGMENTS

First, I would like to thank my advisors, Prof. C.–P. Yuan and Prof. Carl Schmidt for
their inspiration and support during my years at Michigan State University. I am deeply
indebted to Prof. Yuan, who opened the door for me to enjoy the beauty and simplicity of
particle physics. I have benefited continuously from sharing many of his ideas and intuitions
with me, which helped me grow as a physicist. I owe my deep appreciation to Prof. Schmidt,
who taught me how to do research as a physicist, and how to see the simplicity behind the
complex. Both of them gave me free times, supported my interests, and allowed me to
collaborate with others.
I am also grateful to Prof. R. Sekhar Chivukula and Prof. Elizabeth H. Simmons for
their invaluable advise and suggestions. I am indebted to Prof.Edmond Berger at Argonne
National Lab, who offered me the chance to work at Argonne, and he has also provided
invaluable guidance on my research. I thank Prof. Wayne Repko for his advise and suggestions, and Prof. Reinhard Schwienhorst for his endless patience to answer my questions on
experimental techniques. I have further benefited from the following professors, Raymond
Brock, Wad Fisher, Joey Huston and James Linnemann, who have given me important advice and suggestions. I also want to thank my committee members Prof. Bhanu Mahanti
and Prof. Scott Pratt for their guidance.
My appreciation also goes to current and former post-docs: Anupama Atre, Neil Christensen, Roshan Foadi, Ken Hsieh, Zhao Li, for many illuminating discussions and collaborations. I especially thank to Prof. Yuan’s former students, Qing-Hong Cao and Chuan-Ren
Chen, who gave me lots of help when I started to learn the techniques of calculation and
numerical codes. I also thank Devin Walker for his patience to polish the English in my
vi

thesis. Thanks to Baradhwaj Coleppa, Stefano DiChiara, Arsham Farzinnia, Mohammad
Hussein, Pawin Ittisamai, Jing Ren, and Kai Schmitz for the discussions in our office. Thank
my friends, He Huang, Liyuan Jia, Xu Lu, Bai Nie, Chen Sun, Zhensheng Tao, Ding Wang,
Hui Wang, Kaijie Xu and He Zhang for the great times during my years at Michigan State
University.
I am sincerely grateful to Debbie Simmons and Brenda Wenzlick and all of my friends
who gave me lots of help during my year at Michigan State University. Finally, my sincere
gratitude goes to my parents and my wife, who bring joy to each day of my life.

vii

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . .
List of Figures

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xi

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Chapter 1 INTRODUCTION . . . . . . . . . . . . . . .
1.1 Charged Gauge Bosons in the Standard Model . . . . .
1.1.1 Fermi Theory of Weak Interaction and Unitarity
1.1.2 Intermediate Charged Weak Boson Model . . .
1.1.3 The Standard Model and Moose Diagram . . .
1.2 New Physics and Heavy Charged Gauge Bosons . . . .
1.2.1 Motivation for New Physics . . . . . . . . . . .
1.2.2 Phenomenology of New Charged Gauge Bosons

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Chapter 2 MODELS OF HEAVY CHARGED GUAGE BOSONS .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Model Classification by Moose Diagrams . . . . . . . . . . . . . . . .
2.3 G(221) Lagrangian in the Gauge Basis and Physical Basis . . . . . .
2.3.1 Lagrangian in the Gauge Basis . . . . . . . . . . . . . . . . .
2.3.1.1 The Scalar Sector . . . . . . . . . . . . . . . . . . . .
2.3.1.1.1 Scalar Multiplets in the Breaking Pattern I
2.3.1.1.2 Scalar Multiplets in the Breaking Pattern II
2.3.1.2 The Gauge Boson Sector . . . . . . . . . . . . . . . .
2.3.1.3 The Fermion Sector . . . . . . . . . . . . . . . . . .
2.3.1.4 The Yukawa Sector . . . . . . . . . . . . . . . . . . .
2.3.2 Symmetry Breaking and Mass Eigenstates . . . . . . . . . . .
2.3.2.1 Higgs Potential and VEV . . . . . . . . . . . . . . .
2.3.2.2 Gauge Boson Mass Mixing in Breaking Pattern I . .
2.3.2.3 Gauge Boson Mass Mixing in Breaking Pattern II . .
2.3.3 Lagrangian in the Physical Basis . . . . . . . . . . . . . . . .
2.3.3.1 Gauge couplings of fermions . . . . . . . . . . . . . .
2.3.3.2 Couplings of Gauge Bosons and Higgs Boson . . . .
2.4 Model Parameters and Mass Spectrum . . . . . . . . . . . . . . . . .
2.4.1 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . .

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viii

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Violation
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Chapter 3 DISCOVERY OF HEAVY CHARGED GAUGE BOSONS
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Drell-Yan Production and Decay at Next-to-Leading Order . . . . . . . .
3.2.1 Drell-Yan Production at the NLO . . . . . . . . . . . . . . . . . .
3.2.2 V Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Event Simulation and Search Limits . . . . . . . . . . . . . . . . . . . . .
3.3.1 Signal and Backgrounds . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Object Reconstructions and Event Selections at the LHC . . . . .
3.3.3 Upper Limits on W masses at the LHC . . . . . . . . . . . . . .
3.4 Discovery Potential and Significance . . . . . . . . . . . . . . . . . . . . .
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4 PROPERTIES OF HEAVY CHARGED GAUGE BOSONS
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 s-channel Production of W : Resonances in Single Top Channel . . . . . .
4.2.1 New Physics Models . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Cross Section and Event Simulation . . . . . . . . . . . . . . . . . .
4.2.3 Lepton Angular Distribution and Chiral Structure . . . . . . . . . .
4.3 t-channel Production of W : Top Quark Forward-backward Asymmetry . .
4.3.1 Forward-backward Asymmetry of Top Quark . . . . . . . . . . . . .
4.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 At B and AF B . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.5

2.6

4.4

2.4.1.1 Electric Charge . . . . . . . . . . . . . . . .
2.4.1.2 The Fermi Constant . . . . . . . . . . . . .
2.4.1.3 Z-Mass . . . . . . . . . . . . . . . . . . . .
2.4.2 Mass Spectra . . . . . . . . . . . . . . . . . . . . . .
2.4.2.1 Breaking Pattern I with Doublet Higgs . . .
2.4.2.2 Breaking Pattern I with Triplet Higgs . . .
2.4.2.3 Breaking Pattern II . . . . . . . . . . . . .
Indirect and Direct Constraints on Parameter Spaces . . . .
2.5.1 Indirect Constraints from Electroweak Precision Tests
2.5.1.1 Precision Measurements at the Z-Pole . . .
2.5.1.2 The Tau Lifetime . . . . . . . . . . . . . . .
2.5.1.3 νN Deep Inelastic Scattering . . . . . . . .
2.5.1.4
νe Scattering . . . . . . . . . . . . . . . . .
2.5.1.5 Parity Violation Experiments . . . . . . . .
2.5.2 Direct Searches at the Tevatron and LHC . . . . . .
2.5.3 Parameter constraints . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3.4 Flavor-changing W Model . . . . . . . . . . . . . . . . . . . . . . . . 184
4.3.5 Correlations on AF B and At B . . . . . . . . . . . . . . . . . . . . . 187
F
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
ix

Chapter 5 EXOTIC HEAVY CHARGED GAUGE BOSONS .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Collider phenomenology . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Selection cuts . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 χ2 -template and extra-jet tagging . . . . . . . . . . . . .
5.3.3 Mass window ∆M cut . . . . . . . . . . . . . . . . . . .
5.3.4 Discovery potential . . . . . . . . . . . . . . . . . . . . .
5.4 W -t-b coupling and t-polarization . . . . . . . . . . . . . . . . .
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 6

CONCLUSION

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191
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Appendix A Statistics for Upper limit and Significance
A.1 Upper Limit . . . . . . . . . . . . . . . . . . . . . . .
A.1.1 Bayesian Approach . . . . . . . . . . . . . . .
A.1.2 Frequentist Approach . . . . . . . . . . . . . .
A.2 Discovery Significance . . . . . . . . . . . . . . . . .
A.2.1 Bayesian Approach . . . . . . . . . . . . . . .
A.2.2 Frequentist Approach . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . .

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LIST OF TABLES

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Table 2.6

These tables display the model-specific Higgs representations and
VEVs that achieve the symmetry breaking of G(221) models. . . .

27

Assignment of SM fermions under the G(221) symmetry: (TL , TR )X
in breaking pattern I while (Tl , Th )Y in breaking pattern II. Unless otherwise specified, the fermion doublet represents three generations of SM fermions. LRD (LRT) denotes the left-right doublet
(triplet) model, where the G(221) model is broken by a scalar doublet (triplet). Similarly, LPD (LPT) denotes the lepto-phobic doublet (triplet) model, HPD (HPT) the hadro-phobic doublet (triplet)
model, FPD (FPT) the fermio-phobic doublet (triplet) model, SQD
the sequential doublet model, TFD the top-flavor doublet model, and
UUD the un-unified doublet model. . . . . . . . . . . . . . . . . . .

33

The left-handed fermion couplings of the heavy gauge boson in Breaking Pattern I and II. For the fermion couplings, the quantum numbers
(TL , TR ) in BP-I and (Tl , Th ) in BP-II are implied in Tab. 2.2. In
BP-II, the fermion notation f means the fermions listed in the column
SU(2)1 , while F means the fermions listed in the column SU(2)2 in
Tab. 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

The right-handed fermion couplings of the heavy gauge boson in
Breaking Pattern I and II. For the fermion couplings, the quantum numbers (TL , TR ) in BP-I and (Tl , Th ) in BP-II are implied
in Tab. 2.2. In BP-II, the fermion notation f means the fermions
listed in the column SU(2)1 , while F means the fermions listed in
the column SU(2)2 in Tab. 2.2. . . . . . . . . . . . . . . . . . . . .

49

The triple boson couplings of the heavy gauge boson in Breaking
Pattern I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

The triple boson couplings of the heavy gauge boson in Breaking
Pattern II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

xi

Table 3.1

Expected and observed numbers of events in 1.04 fb−1 in ref. [54]
from the various background sources in each decay channel for MT >
891GeV, the region used to search for a W with a mass of 1500 GeV.
In the table, the signal selection efficiency, sig , and the prediction for
the number of signal events, Nsig , obtained with this efficiency. The
last two columns are the expected number of background events, Nbg
, and the number of events observed in data, Nobs . The uncertainties
are statistical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Table 3.2

The current lowest limits and discovery reaches at the LHC14 with
100 fb−1 luminosity on W masses. . . . . . . . . . . . . . . . . . . 153

Table 3.3

The current lowest limits and discovery reaches at the LHC14 with
100 fb−1 luminosity on W and Z masses. . . . . . . . . . . . . . 153

Table 4.1

Cross sections for the signal and various background processes at the
7 TeV LHC in different cut levels are listed. . . . . . . . . . . . . . . 169

Table 5.1

The numbers of signal and background events at 14 TeV with an
integrated luminosity of 100 fb−1 before and after cuts, with fL = 1,
for seven values of m
(GeV). The top quark decay branching ratio
W
2/27 is included in the “no cut” column, and the b tagging efficiency
is included in the fifth column. The cut acceptances cut are also
listed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Table 5.2

The efficiency for extra jet reconstruction with the χ2 -template method.
The net b-tagging efficiencies ( extra−j−tag ), calculated with Eqs. (5.17)
and (5.19), are shown in the last column. . . . . . . . . . . . . . . . 208

xii

LIST OF FIGURES

Figure 1.1

The Moose diagram for the standard model. For interpretation of the
references to color in this and all other figures, the reader is referred
to the electronic version of this thesis. . . . . . . . . . . . . . . . . .

11

√
+
−
+
−
Scattering cross sections for WL WL → WL WL versus sW W . Various
values of δ are shown, where δ denotes the size of the Higgs amplitude
relative to the SM one. An angular cut of | cos θW W | < 0.8 is applied
and the light Higgs boson mass mh = 200 GeV is assumed. Figure taken
from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

The scale Λ at which the two-loop RGEs drive the quartic SM Higgs
coupling non-perturbative (upper curves), and the scale Λ at which
the RGEs create an instability in the electroweak vacuum (lower
curves). Figure taken from [9]. . . . . . . . . . . . . . . . . . . . .

17

Figure 2.1

The Moose diagram for the breaking pattern I (Left-Right Model).

25

Figure 2.2

The Moose diagram for the breaking pattern II (Sequential Model).

26

Figure 2.3

The Moose diagram for the breaking pattern I (Lepton-phobic Model
with vector-like fermions). . . . . . . . . . . . . . . . . . . . . . . .

36

The Moose diagram for the breaking pattern II (Top-Flavor model
with vector-like fermions). . . . . . . . . . . . . . . . . . . . . . . .

37

Allowed parameter space of the LRD and LRT models at 95% CL in
the M
−M plane after including indirect and direct constraints:
W
Z
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

77

Allowed parameter space of the LPD and LPT models at 95% CL in
the M
−M plane after including indirect and direct constraints:
W
Z
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

78

Figure 1.2

Figure 1.3

Figure 2.4

Figure 2.5

Figure 2.6

xiii

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

Allowed parameter space of the HPD and HPT models at 95% CL in
the MW −MZ plane after including indirect and direct constraints:
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

79

Allowed parameter space of the FPD and FPT models at 95% CL in
the M
−M plane after including indirect and direct constraints:
W
Z
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

80

Allowed parameter space of the SQD and MSQ models at 95% CL in
the M
−M plane after including indirect and direct constraints:
W
Z
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

81

Allowed parameter space of the TFD and UUQ models at 95% CL in
the M
−M plane after including indirect and direct constraints:
W
Z
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

82

Allowed parameter space of the LRD and LRT models at 95% CL in
the M
− cφ plane after including indirect and direct constraints:
W
EWPTs (green), Tevatron (red) and LHC7 (blue). The dashed black
lines in LRD represent MLR models. . . . . . . . . . . . . . . . . . .

83

Allowed parameter space of the LPD and LPT models at 95% CL in
the MW − cφ plane after including indirect and direct constraints:
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

84

Allowed parameter space of the HPD and HPT models at 95% CL in
the M
− cφ plane after including indirect and direct constraints:
W
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

85

Allowed parameter space of the FPD and FPT models at 95% CL in
the M
− cφ plane after including indirect and direct constraints:
W
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

86

Allowed parameter space of the SQD models at 95% CL in the M −
W
cφ plane after including indirect and direct constraints: EWPTs
(green), Tevatron (red) and LHC7 (blue). The dashed black lines
represent MSQ models. . . . . . . . . . . . . . . . . . . . . . . . . .

87

Allowed parameter space of the TFD and UUQ models at 95% CL in
the M
− cφ plane after including indirect and direct constraints:
W
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

88

xiv

Figure 2.17

Figure 2.18

Figure 2.19

Figure 2.20

Figure 2.21

Figure 2.22

Allowed parameter space of the LRD and LRT models at 95% CL in
the MZ − cφ plane after including indirect and direct constraints:
EWPTs (green), Tevatron (red) and LHC7 (blue). The dashed black
lines in LRD represent MLR models. . . . . . . . . . . . . . . . . . .

89

Allowed parameter space of the LPD and LPT models at 95% CL in
the MZ − cφ plane after including indirect and direct constraints:
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

90

Allowed parameter space of the HPD and HPT models at 95% CL in
the M − cφ plane after including indirect and direct constraints:
Z
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

91

Allowed parameter space of the FPD and FPT models at 95% CL in
the M − cφ plane after including indirect and direct constraints:
Z
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

92

Allowed parameter space of the SQD models at 95% CL in the M −
Z
cφ plane after including indirect and direct constraints: EWPTs
(green), Tevatron (red) and LHC7 (blue). The dashed black lines
represent MSQ models. . . . . . . . . . . . . . . . . . . . . . . . . .

93

Allowed parameter space of the TFD and UUQ models at 95% CL
in the MZ − cφ plane after including indirect and direct constraints:
EWPTs (green), Tevatron (red) and LHC7 (blue). . . . . . . . . . .

94

Figure 3.1

Upper panel: the LO and NLO cross sections of pp → W process
with a SM like coupling as a function of new heavy gauge boson mass
(mW ) in hadron collisions. Lower panel: the K-factor as a function
of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
W

Figure 3.2

Upper panel: the LO and NLO cross sections of pp → Z process
with a SM like coupling as a function of new heavy gauge boson mass
(m ) in hadron collisions. Lower panel: the K-factor as a function
Z
of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Z

Figure 3.3

Upper panel: the LO and NLO cross sections of pp → Z process
with a SM-like couplings as a function of m
in hadron collision:
Z
induced by up-type quark initial state Lower panel: the K-factor as
a function of mZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
xv

Figure 3.4

Upper panel: the LO and NLO cross sections of pp → Z process
with a SM-like couplings as a function of mZ in hadron collision:
induced by down-type quark initial state. Lower panel: the K-factor
as a function of m . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Z

Figure 3.5

The pT distribution for 1500 GeV W in the charged lepton and
missing transverse energy final states. . . . . . . . . . . . . . . . . . 112

Figure 3.6

miss distribution for 1500 GeV W in the charged lepton and
The ET
missing transverse energy final states. . . . . . . . . . . . . . . . . . 113

Figure 3.7

The MT distribution for 1500 GeV W in the charged lepton and
missing transverse energy final states is shown. . . . . . . . . . . . 114

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

5σ Discovery potential (fb−1 ) in the LRD and LRT models for different luminosity at LHC7 via W leptonic decay channel. The current
combined constraints are within solid black contour. The dashed
black lines in LRD represent MLR models. . . . . . . . . . . . . . . 125
5σ Discovery potential (fb−1 ) in the LPD and LPT models for different luminosity at LHC7 via W leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 126
5σ Discovery potential (fb−1 ) in the HPD and HPT models for different luminosity at LHC7 via W leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 127
5σ Discovery potential (fb−1 ) in the FPD and FPT models for different luminosity at LHC7 via W leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 128
5σ Discovery potential (fb−1 ) in the SQD models for different luminosity at LHC7 via W leptonic decay channel. The current combined
constraints are within solid black contour. The dashed black lines in
SQD represent MSQ models. . . . . . . . . . . . . . . . . . . . . . . 129
5σ Discovery potential (fb−1 ) in the TFD and UUD models for different luminosity at LHC7 via W leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 130
xvi

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

Figure 3.18

Figure 3.19

Figure 3.20

Figure 3.21

Figure 3.22

Figure 3.23

5σ Discovery potential (fb−1 ) in the LRD and LRT models for different luminosity at LHC7 via Z leptonic decay channel. The current
combined constraints are within solid black contour. The dashed
black lines in LRD represent MLR models. . . . . . . . . . . . . . . 132
5σ Discovery potential (fb−1 ) in the LPD and LPT models for different luminosity at LHC7 via Z leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 133
5σ Discovery potential (fb−1 ) in the HPD and HPT models for different luminosity at LHC7 via Z leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 134
5σ Discovery potential (fb−1 ) in the FPD and FPT models for different luminosity at LHC7 via Z leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 135
5σ Discovery potential (fb−1 ) in the SQD models for different luminosity at LHC7 via Z leptonic decay channel. The current combined
constraints are within solid black contour. The dashed black lines in
SQD represent MSQ models. . . . . . . . . . . . . . . . . . . . . . . 136
5σ Discovery potential (fb−1 ) in the TFD and UUD models for different luminosity at LHC7 via Z leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 137
5σ Discovery potential (fb−1 ) in the LRD and LRT models for different luminosity at LHC14 via W leptonic decay channel. The current
combined constraints are within solid black contour. The dashed
black lines in LRD represent MLR models. . . . . . . . . . . . . . . 138
5σ Discovery potential (fb−1 ) in the LPD and LPT models for different luminosity at LHC14 via W leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 139
5σ Discovery potential (fb−1 ) in the HPD and HPT models for different luminosity at LHC14 via W leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 140
5σ Discovery potential (fb−1 ) in the FPD and FPT models for different luminosity at LHC14 via W leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 141
xvii

Figure 3.24

Figure 3.25

Figure 3.26

Figure 3.27

Figure 3.28

Figure 3.29

Figure 3.30

Figure 3.31

5σ Discovery potential (fb−1 ) in the SQD models for different luminosity at LHC14 via W leptonic decay channel. The current combined constraints are within solid black contour. The dashed black
lines in SQD represent MSQ models. . . . . . . . . . . . . . . . . . . 142
5σ Discovery potential (fb−1 ) in the TFD and UUD models for different luminosity at LHC14 via W leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 143
5σ Discovery potential (fb−1 ) in the LRD and LRT models for different luminosity at LHC14 via Z leptonic decay channel. The current
combined constraints are within solid black contour. The dashed
black lines in LRD represent MLR models. . . . . . . . . . . . . . . 145
5σ Discovery potential (fb−1 ) in the LPD and LPT models for different luminosity at LHC14 via Z leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 146
5σ Discovery potential (fb−1 ) in the HPD and HPT models for different luminosity at LHC14 via Z leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 147
5σ Discovery potential (fb−1 ) in the FPD and FPT models for different luminosity at LHC14 via Z leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 148
5σ Discovery potential (fb−1 ) in the SQD models for different luminosity at LHC14 via Z leptonic decay channel. The current combined
constraints are within solid black contour. The dashed black lines in
SQD represent MSQ models. . . . . . . . . . . . . . . . . . . . . . . 149
5σ Discovery potential (fb−1 ) in the TFD and UUD models for different luminosity at LHC14 via Z leptonic decay channel. The current
combined constraints are within solid black contour. . . . . . . . . 150

Figure 4.1

Feynman diagrams of the s-channel and t-channel single top production with exchanging a W . . . . . . . . . . . . . . . . . . . . . . . . 159

Figure 4.2

Total cross section of W for fL = 1, fR = 0 at the LHC. . . . . . . 160

Figure 4.3

The normalized PT distributions of the leading jet for signal and
backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
xviii

Figure 4.4

The normalized PT distributions of the subleading jet for signal and
backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure 4.5

The normalized HT distributions of the system for signal and backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Figure 4.6

After all the cuts, the reconstructed invariant mass distributions of
the system for W signal and backgrounds. . . . . . . . . . . . . . . 167

Figure 4.7

Before and after basic cuts, the cos θ angular distributions of the final
state leptons for left-handed W and right-handed W . . . . . . . . 168

Figure 4.8

(a) cos θhel distribution in the top quark rest frame for both tL and
tR . (b) cos θt distribution in the boosted frame for a top quark with
Et = 200 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Figure 4.9

The ratio RF as a function of yt for a top quark with fixed energy:
(a) Et = 200 GeV and (b) Et = 600 GeV. . . . . . . . . . . . . . . 176

Figure 4.10

The ratio of the charged lepton in the forward and backward region
as a function of the top quark rapidity for top quarks with fixed
transverse momentum pT = 10, 50 GeV. . . . . . . . . . . . . . . . 178

Figure 4.11

The ratio of the charged lepton in the forward and backward region
as a function of the top quark rapidity for top quarks with fixed
transverse momentum pT = 100, 300 GeV. . . . . . . . . . . . . . . 179

Figure 4.12

Correlation between AF B and At B in the W models. The point
F
corresponding to the D0 data is also shown. The numbers within the
parentheses label the lower and upper limits of the mass of the W .

185

Figure 4.13

The statistics for the ratio of predicted AF B to At B in the W
F
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Figure 5.1

Feynman diagrams for the associated production of a W and a top
¯
quark: (a, b) W − t, and (c, d) W + t. . . . . . . . . . . . . . . . . . 194

Figure 5.2

¯
The inclusive cross section of the tW − and tW + associated production at a 14 TeV LHC with fL = 1 and fR = 0. . . . . . . . . . 195
xix

Figure 5.3

Normalized pT distribution of (a) the leading jet, (b) the second
leading jet distributions. The black-solid curves represent the signal
¯
(m
= 1 TeV), the red-dashed curves the ttb background, while the
W
¯
blue-dotted curves the ttj background. . . . . . . . . . . . . . . . . 200

Figure 5.4

Normalized pT distribution of (a) the third leading jet, as well as (b)
the normalized HT distribution. The black-solid curves represent the
¯
signal (m
= 1 TeV), the red-dashed curves the ttb background,
W
¯
while the blue-dotted curves the ttj background. . . . . . . . . . . 201

Figure 5.5

(a) Reconstructed invariant mass distribution of the leptonic decaying
top and extra jet; (b) reconstructed hadronic decaying top and extra
jet invariant mass distribution. . . . . . . . . . . . . . . . . . . . . 209

Figure 5.6

The discovery potential for the top-philic W 14 TeV with an integrated luminosity of 100 fb−1 . . . . . . . . . . . . . . . . . . . . . 212

Figure 5.7

The angular distributions of the final lepton cos θl for the left-handed
and the right-handed W . . . . . . . . . . . . . . . . . . . . . . . . . 214

Figure 5.8

The angular distributions of the final lepton cos θl for the SM background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

xx

Chapter 1

INTRODUCTION

1.1

Charged Gauge Bosons in the Standard Model

The standard model (SM) provides a very successful description of the properties and interactions of the existing particles without serious discrepancy with almost all existing data.
The discovery of the charged gauge bosons W ± and the neutral massive gauge boson Z
with expected masses is a great triumph of the model. During the development of the SM,
the charged gauge bosons W ± played an important role in the discovery of the electroweak
gauge symmetry SU(2)L × U(1)Y , in which the electromagnetic and weak interaction are
unified into a renormalizable field theory. Here we briefly review the history of charged gauge
bosons in the standard model, and refer to refs. [1, 2, 3] for details.
1

1.1.1

Fermi Theory of Weak Interaction and Unitarity Violation

The weak interaction among elementary particles was first discovered in the β-decay of
neutron
n → p + e− + ν .
¯

(1.1)

The theory of weak interactions was originally formulated by Fermi through the charged
current-current interactions in quantum field theory

¯
¯
Hweak = GF [ψp γµ ψn ] [ψe γ µ ψν ],

(1.2)

where GF is the weak coupling constant which has a much smaller value than the electromagnetic coupling constant e. After discovery of parity violation in the weak interaction,
the weak interaction Hamiltonian was reformulated to a more general form

¯
¯
GF i [ψp Oi ψn ] [ψe Oi (1 + ci γ5 )ψν ],

Hweak =

(1.3)

i
where ci is taken to be −1 since the parity violation is maximum and the neutrino is lefthanded, and the index i represents the type of the Lorentz structure O with

OS = 1

(scalar),

OV

= γµ

(vector),

OP

= γ5

(axial-scalar),

OA = γµ γ5
OT

(axial-vector),

= σµν

(tensor).
2

Later experimental evidence on the V − A Lorentz structure and universality of the
fermi constant GF in various weak processes, suggested that the Hamiltonian of the weak
interaction was described by the V − A current-current interactions
G
¯
¯
Hweak = √F [ψp γµ (1 − gA γ5 )ψn ] [ψe γ µ (1 − γ5 )ψν ],
2
where the Fermi constant is GF = 1.166364 × 10−5 GeV−2 , and gA

(1.4)

1.26.

Although the Fermi theory described the low energy weak interaction processes successfully, it encountered difficulty in explaining theory-motivated problems, i.e., unitarity
violation in the weak scattering process, and non-renormalizability beyond the lowest-order
approximation. Consider the neutrino-electron scattering process

νe + e− → νe + e− ,

(1.5)

G
¯
¯
Hweak = √F [ψνe γµ (1 − γ5 )ψe ] [ψe γ µ (1 − γ5 )ψνe ].
2

(1.6)

where the Hamiltonian is given by

The differential cross section for this process is
G2
dσ
= F s,
dΩ
4π 2

(1.7)

where s is the center of mass energy squared of the system. On the other hand, the differential
3

cross section can be decomposed into partial wave amplitudes

1
dσ
=| √
dΩ
2 s

∞

(2 + 1)a P (cos θ)|2 ,

(1.8)

=0

where a is the partial wave amplitude for angular momentum . In the case of Fermi
interaction, only a0 contributes to the differential cross section
1
1
dσ
= |a0 |2 ≤ ,
dΩ
s
s

(1.9)

where |a0 | < 1 from the unitarity property in quantum mechanics. From the upper bound
of the cross section, one find that the Fermi theory violates unitarity at an energy above
√
E=

1.1.2

s
≥
2

π
2GF

370 GeV.

(1.10)

Intermediate Charged Weak Boson Model

The difficulty of unitarity violation can be avoided if the analogy with the electromagnetic
interaction is extended to include a virtual intermediate charged weak boson as the mediator
of the weak processes. This is a successful phenomenological model to describe the charged
weak processes even at the high energy scale. Consider again the process νe + e− → νe + e−
as an example. Assuming the charged gauge boson exchange between two charged currents,
the matrix element describing this process is

M=

gW
1 − γ5
√ ue γµ
uνe
¯
2
2

qµqν
2
MW
2
q 2 − MW

−g µν +

4

gW
1 − γ5
√ uνe γν
ue ,
¯
2
2

(1.11)

where MW is the mass of the charged gauge bosons W , q is the four-momentum of the
W -boson, and gW is the dimensionless weak coupling constant. At low energy where q 2 is
far smaller than the gauge boson mass MW , the propagator term reduces to
qµqν
µν
M2
W → g ,
q2 − M 2
M2
W
W

−g µν +

(1.12)

and the above matrix element reduces to the Fermi Hamiltonian of Eq. 1.6, with
GF
g2
√ =
.
2
8M 2
W

(1.13)

Because the range of the weak interaction is very small, this indicates the mass of the
intermediate gauge bosons is large, and can be estimated to be of the order of O(100 GeV).
In this model, the unitarity problem in the scattering process νe + e− → νe + e− is solved,
because of the existence of the weak boson mediator.

Although this immediate charged weak boson model cures the unitarity violation problem
in this particular channel, it still has problems of non-renormalizablity. Loop calculation
involving in the charge gauge boson can not be renormalized to get the finite results in this
model. Furthermore, the unitarity violation still occurs in other scattering processes like
ν + ν → W + + W − . Writing down the matrix element in this process, one can calculate
¯
the differential cross section
G2 sin2 θ
dσ
= F
s,
dΩ
3π 2

(1.14)

where s and θ are the center of mass energy sqaured of the system and the polar angle in
the center of mass frame. Notice that the cross section increases with s and hence violates
5

the unitarity bound at some high energy.

1.1.3

The Standard Model and Moose Diagram

Although the intermediate charged boson model describes the phenomenology relevant to
the charged-current processes successfully, it is still an effective theory, due to the nonrenormalizability and unitarity violation. A beautiful renormalizable theory was finally
formulated based on the unification of the electromagnetic and weak interactions, which
is the so-called electroweak standard model. Let us review the essential elements of the
electroweak theory following the ref. [4].
The essential ingredient of the SM is the non-abelian gauge group with SU(2)L × U(1)Y
symmetry, where the subscript L represents the weak isospin for the left-handed fermion
fields, and Y denotes the weak hyper-charge. The electroweak gauge group, SU(2)L ×U(1)Y ,
implies two sets of gauge fields: a weak isovector Wµ , including two charged components
±
3
Wµ and one neutral component Wµ , with coupling constant g, and a weak isoscalar Bµ ,
with coupling constant g . Corresponding to these gauge fields are the field-strength tensors

a
a
a
b c
Fµν = ∂ν Wµ − ∂µ Wν + gεabc Wµ Wν ,

(1.15)

for the weak-isospin symmetry, and

Bµν = ∂ν Bµ − ∂µ Bν ,

(1.16)

for the weak-hypercharge symmetry. From the non-abelian gauge theory, the kinematic term
6

in the gauge boson sector can be written as

1 a
1
Lgauge = − Fµν F aµν − Bµν B µν .
4
4

(1.17)

For the fermion sector, the electroweak theory takes the crucial clues from experiment:
the existence of left-handed weak-isospin doublets and massless neutrinos. To conserve the
electromagnetic interaction in the gauge group, the quantum numbers satisfy the relation
1
between the weak hypercharge Y and weak isospin, Q = I3 + Y . The fermions are specified
2
based on the quantum numbers: a left-handed weak isospin doublet

I3




 ν 
Ll =  
 
e
L

Q

Y = 2(Q − I3 )

1
+0

2
−

(1.18)
−1 ,

−1

1
2

and right-handed weak isospin singlets

I3

Q

Y = 2(Q − I3 )

Rν = νR

0

+0

+0

Re = eR

0

(1.19)
,

−1

−2

where we have induced the possibility of a right-handed neutrino. To extend our theory to
include the electroweak interactions of quarks, we observe that each generation consists of a
7

left-handed doublet

I3




 u 
Lq =  
 
d
L

Q

1

2
+

2
−

Y = 2(Q − I3 )

1

−

2

(1.20)

1

3

3

1

,

3

and two right-handed singlets,

I3

Y = 2(Q − I3 )

Q

4

2

Ru = u R

0

Rd = d R

0

+
−

+

3
1

−

3

(1.21)

3 ,
2
3

The fermion Lagrangian can be written as

Lfermions = R iγ µ ∂µ + i
+ L iγ µ ∂µ + i

g
Aµ Y
2

R

(1.22)

g
g
Aµ Y + i τ · bµ L.
2
2

The SU(2)L × U(1)Y gauge symmetry forbids a mass term for the fermions. Moreover,
the theory we have described contains four massless electroweak gauge bosons, namely Bµ ,
1
2
3
Wµ , Wµ , and Wµ , whereas nature has but one: the photon. To give masses to the gauge
bosons and constituent fermions, we must hide the electroweak symmetry. The Higgs mechanism can give masses to the gauge bosons and fermions through the spontaneous breakdown
of the gauge symmetry. Since nature has one massless photon, the following breaking pattern
8

is taken
SU(2)L × U(1)Y → U(1)em .

(1.23)

To fulfill this symmetry breaking, we introduce a complex doublet of scalar fields


φ≡

φ+
φ0





(1.24)

with weak hyper-charge Yφ = +1. The gauge-invariant Lagrangian for the interaction and
propagation of the scalars is given by,

Lscalar = (D µ φ)† (Dµ φ) − V (φ† φ),

(1.25)

where the gauge-covariant derivative is

g
g
Dµ = ∂µ Y + i τ · Aµ + i Bµ ,
2
2

(1.26)

and the potential interaction has the form

V (φ† φ) = µ2 (φ† φ) + |λ| (φ† φ)2 ,

(1.27)

where µ and λ are real constant parameters. To give masses to the fermions, it is necessary
to add a Yukawa interaction between the scalar fields and the fermions,

LYukawa = − R(Y φ† L) + (LY φ)R ,

9

(1.28)

where Y are called Yukawa coupling matrices, and cannot be predicted within the model.

To summarize the above, the full Lagrangian in the electroweak standard model can be
given by

1
/
L = − Fa Faµν + iψD ψ + h.c.
4 µν
¯
+ ψi yij ψj φ + h.c. + |Dµ φ|2 − V (φ) ,

where we have summed over the different fermions and gauge bosons. However, the above
simple Lagrangian does not contain the information on the quantum numbers of the gauge
bosons, fermions and scalars. The Moose notation [5] is introduced to summarize the SM
Lagrangian with full information diagrammatically. In Fig. 1.1, we plot the Moose diagram
for the standard model. Each circle represents a gauge group, and the links between circles
are the Higgs doublet which is charged under two symmetry groups. The dashed lines
emerging downward and upward from the circles represent the left-handed and right-handed
leptons, respectively, and the solid lines emerging downward and the upward represent the
left-handed and right-handed quarks, respectively. The diagonal dashed green lines represent
the Yukawa couplings through the Higgs doublet.
The electroweak symmetry is spontaneously broken if the parameter µ2 < 0. The minimum energy, or vacuum state, may then be chosen to correspond to the vacuum expectation
value


φ =

where v =





0 
,
√ 
v/ 2

(1.29)

−µ2 / |λ|. In the unitary gauge, the Lagrangian for the scalars and gauge boson
10

φ ∼v
SU(2)

U(1)

Figure 1.1: The Moose diagram for the standard model. For interpretation of the references
to color in this and all other figures, the reader is referred to the electronic version of this
thesis.

masses becomes,

1 2
1
λ
∂µ H∂ µ H − MH H 2 − λvH 3 − H 4
2
2
4
2
1
g
+
(H 2 + 2Hv)
Zµ Z µ + 2Wµ W −µ
+
2θ
8
cos W

Lscalar =

1 2
+
2
+ MW Wµ W −µ + MZ Zµ Z µ ,
2

(1.30)

where the new gauge bosons Z and A are admixtures of the gauge bosons W 3 and B as
follows

Z = cos θW W 3 − sin θW B ,

(1.31)

A = cos θW B + sin θW W 3 ,

(1.32)

11

and the masses of the W -boson and Z-boson are

2
MW

=

2
MZ =

v 2 e2
,
4 sin2 θW
e2
v2
4 sin2 θW cos2 θW

(1.33)
.

(1.34)

With the wave-function mixing, the charged current and neutral current are

g
−
+
LCC = √ (Jµ W −µ + Jµ W +µ ),
2
Y
3 W 3µ + g Jµ B µ )
LNC = gJµ
2
Y
3 + g cos θ Jµ )Aµ
= (g sin θW Jµ
W 2
Y
Jµ
3
+ (g cos θW Jµ − g sin θW
)Z µ ,
2

(1.35)

(1.36)

where the currents are
τa
L,
2

(1.37)

= Lγµ Y L + Rγµ Y R.

(1.38)

a
Jµ = Lγµ
Y
Jµ

In the standard model, the unitarity violation is recovered even at high energy scales.
+ −
+ −
Consider the charged gauge boson scattering process WL WL → WL WL [6], in which the
Feynman diagrams are the t- and s-channels of γ and Z exchanges, the 4-point vertex, and
the s- and t-channels of Higgs exchanges. At high energies, the longitudinal polarization of
the W boson can be expressed as

µ
(p) = pµ /mW + v µ (p), which is used in the equivalent
L

theorem [7]. Because the longitudinal polarization of the charged gauge boson dominates in
12

the scattering process, the high energy behavior of individual Feynman diagram grows with
the center of mass energy squared s like
√
M(s, t) = A

s

√

4
+B

MW

s

2
+ C,

MW

(1.39)

where the coefficients A, B and C are functions of the polar angle θ in the CMS frame and the
coupling constant g. On summing over diagrams except the Higgs exchange, the coefficient
A cancels, but B has contributions

B=−

g2
(1 + cos θ) .
8M 2
W

(1.40)

Including in the Higgs contribution, there is full cancellation of the B term. Thus, the
bad energy-growing term is delicately canceled between the gauge diagrams and the Higgs
diagrams. On summing over all diagrams, the final result is

M(s, t) = −

2
g 2 MH
2
4MW

s
2
s − MH

+

t
2
t − MH

.

(1.41)

Performing the partial wave analysis, we obtain that the dominate S-wave amplitude is
1
1
d cos θ M W + W + → W + W +
|a0 |2 =
L L
L L
64π −1
2
2
2
MH
MH
GF MH
s
ln(1 +
−
)
2+
= − √
s
8 2π
s − M2
M2
H
H
√
4 2π
If the Higgs mass is smaller than
GF

.

(1.42)

1 TeV, the unitarity violation is avoided. However,

if the Higgs couplings to the charged gauge boson deviate from the standard model couplings

13

to a fraction

√

δ of the SM value, the unitarity cancellation is only partial. In the high energy

limit, the total amplitude becomes

iMgauge + iMHiggs

i

g2
(u + t) (1 − δ) .
4m2
W

(1.43)

One can then obtain [10] the cross section of W W scattering as a function of δ, as shown in
Fig.1.2.

Cross Section (pb)

4
10
3
10

δ =0
0.25
0.5

2
10

0.75
1
10

0
10

0.9

SM
200

400

600

1000

2000 3000

5000

√ s WW (GeV)
√
sW W . Various values of δ
are shown, where δ denotes the size of the Higgs amplitude relative to the SM one. An angular cut
of | cos θW W | < 0.8 is applied and the light Higgs boson mass mh = 200 GeV is assumed. Figure
taken from [10].
+
−
+
−
Figure 1.2: Scattering cross sections for WL WL → WL WL versus

14

1.2
1.2.1

New Physics and Heavy Charged Gauge Bosons
Motivation for New Physics

With the start of the CERN Large Hadron Collider (LHC), a new era of discovery has just
begun. It will revolutionize particle physics by opening the TeV energy region to direct
experimental exploration. One of the primary motivations for LHC experiments is searching
for the Higgs boson, and understanding its role in the origin of electroweak symmetry breaking. Recently, the ATLAS and CMS Collaborations have reported their updated searches
for the SM Higgs boson: they find a local 3σ level excess in γγ final states at 125 GeV, and
a small excess in ZZ final states.
Although the 3σ evidence of a 125 GeV Higgs boson indicates another success of the
SM, it also guides us towards the physics beyond the standard model. If it is a non-SM
Higgs boson, which means the Higgs couplings to gauge bosons deviate from the SM value,
we showed that unitarity violation will happen in the W W scattering process at some high
energy scale. New particles are needed to remedy this problem. On the other hand, if it
is the SM Higgs, there are still some theoretic issues in the SM: the hierarchy problem and
vacuum stability.
Let us consider the hierarchy problem first, following ref. [4]. Beyond the classical approximation, gauge boson, fermion and scalar mass parameters receive quantum corrections
1
from loops that contain particles of spins J = 0, , and 1. Because of gauge symmetry and
2
chiral symmetry, the loop corrections of the gauge boson and fermion mass parameters are
proportional to ln Λ, where Λ defines a reference scale where the SM is still valid. However, there is no symmetry to protect the Higgs mass parameters in the loop calculation.
15

Symbolically we write down the loop corrections to the Higgs mass parameters

2
2
MH (p2 ) = MH (Λ2 ) +

,

where the dashed lines represent the Higgs boson, solid lines represent fermions and antifermions, and wavy lines represent gauge bosons. The quantum corrections can be written,
with the divergence Λ included, as follows,
Λ2
2 = M 2 (Λ2 ) + Cg 2
MH
dk 2 + · · ·
H
0
G Λ2
2
2
2
2
= MH (Λ2 ) + F √ (6MW + 3MZ + MH − 12m2 ),
t
2 2
4π

(1.44)

where C is a finite coefficient. The loop integrals are quadratically divergent, ∝ Λ2 . In the
absence of new physics, a natural reference scale Λ is the Planck mass, Λ ∼ MPlanck =
c/GNewton 1/2 ≈ 1.2 × 1019 GeV. The challenge of preserving a modest value of the
2
Higgs mass MH at about the electroweak scale through the fine-tuning cancellation between
2
the huge loop corrections and MH (Λ2 ) is known as the hierarchy problem. Some new
physics, a new symmetry or new dynamics, should reduce the loop correction from the
quadratic divergence to a reasonable value.
Another inconsistency comes from the vacuum stability [8]. Shown in Fig. 1.3, for a light
Higgs with about 125 GeV mass, there is a band below which the present electroweak vacuum
becomes unstable against decay into a vacuum with |φ| ∼ 1010 GeV. This instability is
due to the negative renormalization of λ by the top quark, which overcomes the positive
renormalization due to λ itself, and drives λ < 0. One could avoid this vacuum instability
by introducing some new physics at an energy scale < Λ to overcome the negative effects of
16

renormalization of λ.
Although there are several existing theoretical problems with the SM, the experimental
measurements at the LEP, Tevatron and LHC are very consistent with the SM. Since there
are no obvious hints of new physics beyond SM, many possible theories of new physics have
been proposed to solve the hierarchy problem and to explain the electroweak symmetry

M [GeV]
H

breaking mechanism, such as supersymmetry, extra dimensions, strong dynamics, and so on.

350

300

Perturbativity bound
Stability bound
Finite-T metastability
Zero-T metastability

λ = 2π
λ=π

Shown are 1 σ error bands

250

200

Tevatron exclusion at >95% CL
150

100

LEP exclusion
at >95% CL
4

6

8

10

12

14

16

18

lo g10 ( Λ / GeV)
Figure 1.3: The scale Λ at which the two-loop RGEs drive the quartic SM Higgs coupling
non-perturbative (upper curves), and the scale Λ at which the RGEs create an instability in
the electroweak vacuum (lower curves). Figure taken from [9].

17

1.2.2

Phenomenology of New Charged Gauge Bosons

There are two approaches to the theoretical exploration of new physics: top-down model
building and bottom-up effective theory. The effective theory approach offers us a way
to explore the unknown physics before it is finally formulated. Like the intermediate gauge
boson model of the weak interactions, we would like to study phenomenology of new particles,
such as new gauge bosons, new fermions, new scalars, and the graviton, in the framework of
an effective field theory.
As remnants of electroweak symmetry breaking, extra gauge bosons exist in many new
physics (NP) models beyond the standard model of particle physics. Importantly, their
discovery would help to elucidate the nature of the spontaneous symmetry breaking at the
weak scale. According to their electromagnetic charges, extra gauge bosons are usually
separated into two categories: charged gauge bosons W and neutral gauge bosons Z . While
a Z boson could originate from an additional abelian U(1) group, the W bosons are often
associated with an extra non-abelian group.
Such new heavy gauge bosons arise in numerous gauge extensions of the standard model
of particle interactions. The simplest gauge extension involves an additional U(1)X gauge
symmetry (and thus an extra gauge boson Z ). One of the next-simplest extensions is the
model with a SU(2)1 × SU(2)2 × U(1)X gauge group, obtained by adding an additional
SU(2) group to the SM. The left-right model [11, 12, 13] is perhaps the most widely-studied
example of such a model. Motivated from the unification of gauge couplings, grand unified
models are proposed by extending the gauge symmetry to the SU(5) or SO(10) group [14].
However, the above extensions of the gauge symmetry can not solve the hierarchy problem.
To resolve the hierarchy problem, new symmetries or new dynamics are needed in addition
18

to the extended gauge groups. An example of a class of extended gauge models with new
global symmetries is the little Higgs scenario[15][16][17], in which the Higgs is a psuedoGoldstone boson. Another scenario is the extra-dimensional models with Kaluza-Klein W
bosons [18], such as Higgsless models [19][20], and Gauge-Higgs models [21][22]. However,
we will not address the hierarchy problem in the extended gauge models, but only focus on
the phenomenological consequences of the heavy charged gauge bosons.
In this thesis, we focus on the collider phenomenology of heavy charged gauge bosons.
The minimal extension of the standard model, which consists of heavy charge gauge bosons,
exhibits a gauge structure of SU(2) × SU(2) × U(1), named the G(221) in ref. [24]. Searches
for the heavy charged gauge bosons in the lepton final states, may yield one of the cleanest
and most important signals of new physics that can be discovered in the early data from
the LHC. On the other hand, investigating these new gauge bosons in the top quark final
states, and determining their quantum numbers would shed light on the gauge structure of
new physics.
This thesis is organized as follows. Chapter 2 introduces the G(221) models within the
framework of an effective field theory with the gauge symmetries SU(2)1 × SU(2)2 × U(1)X
models. We discuss the Lagrangians, the particle spectrum, and the experimental observables
in these G(221) models. The combined constraints from the electroweak precision data, and
direct searches at the Tevatron and LHC are discussed.
Chapter 3 presents the Drell-Yan production of the heavy charged gauge bosons in G(221)
models and its decay channels. We perform the next-to-leading order calculation of the DrellYan production of the heavy gauge bosons. Then we analyze the signal and backgrounds in
the lepton plus missing transverse energy final states at the LHC. We show how to calculate
19

the upper limits using Bayesian statistics. Finally, we discuss the discovery reach of the
heavy gauge bosons at the LHC.
Chapter 4 describes the study of the heavy charged gauge bosons in the top quark final
states. A Monte-Carlo simulation is performed on the signal and backgrounds. We show that
it is possible to probe the chiral structure of the charged gauge bosons using the top quark
polarization. We also investigate the possible origin of the forward-backward asymmetry in
the top pair final states in the flavor-violating W model.
Chapter 5 discusses the discovery and identification of a possible exotic W , in which
W only couples to the third generation quarks, in the top pair plus bottom jet final states.
We also study the chiral structure of the W -bosons using the final state lepton angular
distribution of the top quark.
In appendix A, we list the complete sets of Feynman rules of the G(221) models. In
appendix B, we discuss the statistics for the upper limit and significance in the Bayesian
and frequentist approaches.

20

Chapter 2
MODELS OF HEAVY CHARGED
GUAGE BOSONS

2.1

Introduction

Although the Standard Model has tremendous success in describing a wide range of physical
phenomena, there are still open questions that are unanswered and motivate further modelbuilding. Since the electroweak gauge group SU(2)L × U(1)Y is the cornerstone of this
model, one of the most common model-building tools is to extend the gauge structure of
the Standard Model. The simplest extension involves an additional U(1)X gauge symmetry
(and thus an extra gauge boson Z ). One of the next-simplest extensions is the model with
SU(2)1 ×SU(2)2 ×U(1)X gauge group by adding an additional SU(2) group. The left-right
model [11, 12, 13] is perhaps the most widely-studied case of such models. On the other hand,
given the extended gauge group SU(2)1 × SU(2)2 × U(1)X in the electroweak sector, there
are many other kinds of models besides the left-right model which can be constructed, and
21

which may have very different collider phenomenology. Motivated by the unification of gauge
couplings, grand unified models are proposed by extending the gauge symmetry to SU(5) or
SO(10) group. To incorporate a global symmetry in which the Higgs is a psuedo-Goldstone
boson, there are various Little Higgs models with extended gauge groups. In the case that
SU(2)1 × SU(2)2 × U(1)X symmetry is the sub-group of the new physics models, one can
establish an effective theory with SU(2)1 × SU(2)2 × U(1)X symmetry by integrating out
the heavy particles in the higher scale.

Based on the effective theory framework, we present a unified, systematic study of many
such models, which are commonly called G(221) models in the literature. In this section
we briefly review the G(221) model and the masses and couplings of W and Z bosons.
In particular we consider various G(221) models categorized as follows: left-right (LR) [11,
12, 13], lepto-phobic (LP), hadro-phobic (HP), fermio-phobic (FP) [27, 25, 26], un-unified
(UU) [28, 29], and Top-flavor (TF) [31, 30, 32]. We also considered a widely-used reference
model in the experimental searches: the sequential W model (SQ). In the LR model and SQ
models, if the gauge couplings are assigned to be the same for the two SU(2) gauge groups,
the models are considered as the manifest left-right model (MLR), and manifest sequential
model (MSQ). In the MSQ, the W couplings to fermions are the same as the standard
model W couplings to fermions, which served as the reference model in the experimental
searches. We focus our attention on the full Lagrangian of the extra gauge bosons in the
G(221) model.
22

2.2

Model Classification by Moose Diagrams

We focus on the so-called G(221) models having a SU(2)1 ×SU(2)2 ×U(1)X gauge structure
that ultimately breaks to U(1) em . Relative to the Standard Model, these models have three
additional massive gauge bosons, and their phenomenology depends on the specific patterns
of symmetry breaking as well as the charge assignments of the SM fermions.
The SU(2)1 × SU(2)2 × U(1)X gauge symmetry can be broken through mass mixing
or/and kinetic mixing of the gauge bosons. In this note, we would like to discuss the models
with mass mixing only, which can be categorized by three patterns of symmetry breaking:
• Breaking pattern I (BP-I):
We identify SU(2)1 as SU(2)L of the SM. The first stage of symmetry breaking then
is SU(2)2 × U(1)X → U(1)Y , giving rise to three heavy gauge bosons W ± and Z at
the TeV-scale. The second stage is SU(2)L × U(1)Y → U(1) em at the electroweak
scale.
• Breaking pattern II (BP-II):
We identify U(1)X as U(1)Y of the SM. The first stage of symmetry breaking is
SU(2)1 × SU(2)2 → SU(2)L . The second stage is SU(2)L × U(1)Y → U(1) em at
the electroweak scale.
• Breaking pattern III (BP-III):
We assume there is a hidden SU(2) gauge symmetry, which comes from dynamical
symmetry breaking sector. There is only one stage of symmetry breaking SU(2)L ×
SU(2) hidden × U(1)Y → U(1) em at the electroweak scale.
The symmetry breaking is assumed to be induced by fundamental scalar fields, except in the
23

breaking pattern III, in which the model has no Higgs, called the Higgs-less model [27]. The
quantum number of the scalar fields under the G(221) gauge group depends on the breaking
pattern. It is possible to choose several kinds of Higgs multiplets in first or second breaking
stages, as shown in Table 2.1. In the BPI, the symmetry breaking of SU(2)2 ⊗ U(1)X →
U(1)Y at the TeV scale could be induced by a scalar doublet field Φ ∼ (1, 2)1/2, or a
triplet scalar field (1, 3)1 with a vacuum expectation value (VEV) u, and the subsequent
symmetry breaking of SU(2)1 ⊗ U(1)Y → U(1)Q at the electroweak scale could be via
another scalar field H ∼ (2, ¯ 0 with two VEVs v1 and v2 , which can be redefined as a VEV
2)
v=

2
2
v1 + v2 and a mixing angle β = arctan(v1 /v2 ). In the BPII, the symmetry breaking

of SU(2)1 ⊗ SU(2)2 → SU(2)L at the TeV scale is owing to a Higgs bi-doublet Φ ∼ (2, ¯ 0
2)
with only one VEV u, and the subsequent breaking of SU(2)L ⊗ U(1)Y → U(1)Q at the
electroweak scale is generated by a Higgs doublet H ∼ (1, 2)1/2 with the VEV v. Since the
precision data constraints (including those from CERN LEP and SLAC SLC experimental
data) pushed the TeV symmetry breaking higher than 1 TeV, we shall approximate the
predictions of physical observables by taking a Taylor expansion in 1/x with x = u2 /v 2 . We
will only focus on models with physical Higgs multiplet. Thus, the Higgsless model is not
discussed.

The G(221) models can be further classified based on the charge assignments of the SM
fermions. In the breaking patterns I and II, the SM fermions can be assigned to be the
electroweak doublets under SU(2)1 and/or SU(2)2 symmetry, such as the left-right model
(LR) in BP-I and the sequential model (SQ) in BP-II. If the SM fermions with different
flavors or generations belong to doublets in different SU(2) gauge symmetries, it is called a
flavor or generation non-universal model, such as the lepto-phobic (LP), hadro-phobic (HP),
24

and fermio-phobic (FP) model in BP-I, and the top-flavor (TF), and un-unified (UU) model.
We display the moose diagrams for BP-I models and BP-II models, in Figs. 2.1 and 2.2. In
Figs. 2.1 and 2.2 Each circle represents a gauge group, and the links between circles are the
Higgs multiplets which is charged under two symmetry groups. The dashed lines emerging
downward and upward from the circles represent the left-handed and right-handed leptons,
respectively, and the solid lines emerging downward and the upward represent the left-handed
and right-handed quarks, respectively. The diagonal dashed green lines represent the Yukawa
couplings through the Higgs doublet. For LP, remove the dashed line in the middle SU(2),
and attach to U(1). For HP, remove the black line in the middle SU(2) , and attach to
U(1). For FP, remove both the black line and dashed line in the middle SU(2), and attach
to U(1). In Table 2.2, we list the possible assignments for ΨL and ΨR .

Φ ∼v
SU(2)

Σ ∼f
SU(2)

U(1)

Figure 2.1: The Moose diagram for the breaking pattern I (Left-Right Model).

25

Σ ∼f
SU(2)

Φ ∼v
SU(2)

U(1)

Figure 2.2: The Moose diagram for the breaking pattern II (Sequential Model).

26

Rep.
LR-D, LP-D
HP-D, FP-D
LR-T, LP-T
HP-T, FP-T

SQ, TF, UU

First stage breaking
Multiplet and VEV

1
Σ ∼ (1, 2, )
2

Σ ∼ (1, 3, 1)

φ+
φ0

Σ=

δ+
√ 0
2δ

1
Σ= √
2

Σ ∼ (2, 2, 0)

1
, Σ =√
2

√ ++
2δ
−δ +

0
uD

1
, Σ =√
2

0

0
uT 0

√ +
1 u 0
σ0 + π0
2π
˜
√ −
0 − π 0 , Σ = √2 0 u
˜
2π
σ

Σ=

Second stage breaking
Rep.
Multiplet and VEV
LR-D, LP-D
HP-D, FP-D

Φ ∼ (2, 2, 0)

LR-T, LP-T
HP-T, FP-T

Φ ∼ (2, 2, 0)

SQ, TF, UU

Φ ∼ (1, 2, 1 )
2

Φ=

φ0
1
φ−
2

φ+
v
˜
1 , Φ =√
0
φ2
2

φ0
1
φ−
2

φ+
1
φ0
2

Φ=

Φ=

v
˜
φ+
0 , Φ = √2
φ

v
˜
, Φ =√
2

cβ 0
0 sβ
cβ 0
0 sβ
0
1

Table 2.1: These tables display the model-specific Higgs representations and VEVs that
achieve the symmetry breaking of G(221) models.

27

2.3

G(221) Lagrangian in the Gauge Basis and Physical
Basis

2.3.1

Lagrangian in the Gauge Basis

The classical gauge invariant G(221) Lagrangian can be decomposed as

L = L Higgs + L Gauge + L Fermion + L Yukawa ,

(2.1)

where the terms on the right hand side are identified as the Higgs scalar part, the gauge
boson part, the fermion part and the Yukawa part respectively.

2.3.1.1

The Scalar Sector

The Lagrangian for the scalar fields are written as

L Higgs = Tr

Dµ Σ † D µ Σ

+ Tr

Dµ Φ † D µ Φ

− V (Σ, Φ) ,

(2.2)

where Σ is the scalar multiplet in the TeV symmetry breaking stage, and Φ is the Higgs
multiplet in the electroweak symmetry breaking stage. In the Moose diagram notation, two
link fields connect the three gauge groups. One link field with double lines represents Σ in
the first stage, while another link field with single line represents Φ in the second stage. The
Higgs potential V (Σ, Φ) is introduced to break the TeV and electroweak symmetries.

2.3.1.1.1

Scalar Multiplets in the Breaking Pattern I At the TeV breaking stage,

there are two kinds of typical scalar multiplet Σ, doublet and triplet, to generate masses for
28

heavy gauge bosons W and Z . In the doublet case, we assign the Higgs doublet Σ as


Σ = 

φ+





√



  (φ3 + iφ4 ) / 2 
.
≡
√ 
0
φ
(φ1 + iφ2 ) / 2

(2.3)

The covariant derivative in the Eq. 2.2 is defined as follows

aµ a
Dµ Σ = ∂ µ Σ + igR W TR Σ + igX B µ XΣ.
R

(2.4)

To generate the neutrino mass, it is popular to assign a scalar triplet,


++
+ / √2
δR

δ
,
Σ= R
√ 
+/ 2
0
δR
−δ
R

where δ =

(2.5)

δr + iδi
√
. The covariant derivative in the Eq. 2.2 is defined as follows
2

Dµ Σ = ∂µ Σ + igR

τ
W , Σ + igX XBµ Σ,
2 Rµ

(2.6)

where the quantum number X is 1.

In order to generate the masses for the W and Z bosons and SM fermion at the second
breaking stage, one introduces the bi-doublet

0 φ+
 φ1
1
Φ=

− φ0
φ2
2


29

: (2, 2, 0) ,

(2.7)

with




0∗ −φ+
 φ
2 
˜
Φ = τ2 Φ∗ τ2 =  2

−φ− φ0∗
1
1

(2.8)

1
where φi = √ φia + iφib . The covariant derivative in Eq. 2.2 can be written as
2
bµ b
bµ b
D µ Φ = ∂ µ Φ + igL W TL Φ − igR W ΦTR .
L
R

2.3.1.1.2

(2.9)

Scalar Multiplets in the Breaking Pattern II To generate masses for the

heavy gauge bosons W and Z at the TeV symmetry breaking stage, one introduces the
Higgs bi-doublet with the quantum number (2, 2)0

1 
τ

Σ=σ+i ·π = √ 
2
2

σ + iπ 3
√
2
π 1 − iπ 2
− √
2

π 1 + iπ 2
√
2
σ − iπ 3
√
2




,


(2.10)

1
where the coefficient √ comes from the normalization of the kinetic term in Eq. 2.2. The
2
covariant derivative in Eq. 2.2 is defined as follows

µa
µa
D µ Σ = ∂ µ Σ + igL WL T a Σ − igR WR ΣT a .

(2.11)

At the electroweak breaking stage, we use the Higgs doublet


Φ = 

30

φ+
φ0



.

(2.12)

The covariant derivative is written as

aµ
Dµ Φ = ∂ µ Φ + igL W T a Φ + igX B µ XΦ.
L

2.3.1.2

(2.13)

The Gauge Boson Sector

µ
µ
We define W1 , W2 , and B µ as the SU(2)1 , SU(2)2 , and U(1)X gauge fields, respectively.
The Lagrangian of the gauge bosons with SU(2)1 × SU(2)2 × U(1)X gauge invariance is
given as

1 aµν a
1
1 aµν a
L = − W1 W1µν − W2 W2µν − B µν Bµν ,
4
4
4

(2.14)

aµν
µ
bµ cν
cµ bν
ν
W1
= ∂ µ W1 − ∂ ν W1 + f abc g1 (W1 W1 − W1 W1 ),

(2.15)

aµν
µ
bµ cν
cµ bν
ν
W2
= ∂ µ W2 − ∂ ν W2 + f abc g2 (W2 W2 − W2 W2 ),

(2.16)

where

B µν = ∂ µ B ν − ∂ ν B µ .

(2.17)

Notice that the gauge bosons are still massless at this stage to ensure a gauge invariant
Lagrangian.

2.3.1.3

The Fermion Sector

The generic fermion terms can be written as

¯
ψR iγ µ Dµ ψR .

¯
ΨL iγ µ Dµ ΨL +

L =
ΨL

ψR
31

(2.18)

In Table 2.2, we list the possible assignments for ΨL and ΨR .
In the breaking pattern I, the covariant derivatives are

Dµ Ψ L =

a a
∂µ + ig1 TL W1µ + igX XBµ ΨL ,

(2.19)

Dµ ψR =

a a
∂µ + ig2 TR W1µ + igX XBµ ψR .

(2.20)

where ΨL can be






 u 
QL =   ∼ (2, 1) 1 ,
d
6
L



 ν 
LL =   ∼ (2, 1) 1 ;
−
e
2
L

(2.21)

and ψR are different in different models in breaking pattern I. If the ψR is the right-handed
doublet QR , or LR , one will obtain






 u 
QR =   ∼ (1, 2) 1 ,
d
6
R



 ν 
LR =   ∼ (1, 2) 1 .
−
e
2
R

(2.22)

One the other hand, if ψR is the right-handed singlet, the possible ψR would be

uR ∼ (1, 1) 2 ,
3

dR ∼ (1, 1) 1 ,
−
3

eR ∼ (1, 1) 1 ,
−
2

νR ∼ (1, 1)0 .

(2.23)

In the breaking pattern II, the covariant derivatives are

Dµ ΨL =

a
a a
∂µ + ig1 Tla W1µ + ig2 Th W2µ + igX XBµ ΨL ,

(2.24)

Dµ ψR =

∂µ + igX XBµ ψR .

(2.25)

32

Models

SU(2)1 (TL , Tl )

SU(2)2 (TR , Th )

Xq = 1/6
Xl = −1/2

LRD
LRT

uL
dL

,

νL
eL

LPD
LPT

uL
dL

,

νL
eL

uR
dR

Xq = 1/6
Xl = YSM

HPD
HPT

uL
dL

,

νL
eL

νR
eR

Xq = YSM
Xl = −1/2

FPD
FPT

uL
dL

,

νL
eL

Xf = YSM

SQD

uL
dL

,

νL
eL

Xf = YSM

,

νL
eL

TFD

UUD

uL
dL

1st,2nd

uR
dR

U(1)X (X, Y )

uL
dL

1st,2nd

uL
dL

,

,
3rd
νL
eL

νR
eR

νL
eL

3rd

Xf = YSM

Xf = YSM

Table 2.2: Assignment of SM fermions under the G(221) symmetry: (TL , TR )X in breaking
pattern I while (Tl , Th )Y in breaking pattern II. Unless otherwise specified, the fermion
doublet represents three generations of SM fermions. LRD (LRT) denotes the left-right
doublet (triplet) model, where the G(221) model is broken by a scalar doublet (triplet).
Similarly, LPD (LPT) denotes the lepto-phobic doublet (triplet) model, HPD (HPT) the
hadro-phobic doublet (triplet) model, FPD (FPT) the fermio-phobic doublet (triplet) model,
SQD the sequential doublet model, TFD the top-flavor doublet model, and UUD the ununified doublet model.

33

where ΨL can be






 u 
QL =   ∼ (2, 1) 1 ,
d
6
L



 ν 
LL =   ∼ (2, 1) 1 ;
−
e
2
L

(2.26)

and ψR are the right-handed singlets, given by

dR ∼ (1, 1) 1 ,
−
3

uR ∼ (1, 1) 2 ,
3
2.3.1.4

eR ∼ (1, 1) 1 ,
−
2

νR ∼ (1, 1)0 .

(2.27)

The Yukawa Sector

To generate the SM fermion masses, we introduce the Yukawa Lagrangian

q
q˜
¯
QiL Yij Φ + Yij Φ QjR ,

l
¯
˜
LiL Yij Φ + Yijl Φ LjR −

L = −

(2.28)

ij

ij

˜
˜
where Φ = σ2 Φ σ2 in the breaking pattern I, and Φ = σ2 Φ in the breaking pattern II.
In the flavor universal models, the dashed lines in the Moose diagrams correspond to the
Yukawa terms in above.
q
l
Notice that in the breaking pattern I, Yij and Yij are defined as the matrix form

l
l 
Yij = Yij 




1 0 
,
0 1



q 1 0 
q
Yij = Yij 
,
0 1

(2.29)

q
l
where the Yukawa couplings Yij and Yij are the same for the up- and down-type fermions.
In this case, after symmetry breaking the mass splitting of the up- and down-type fermions
comes from the different VEVs of the Φ for the up- and down-type fermions. while in the
34

q
l
breaking pattern II, Yij and Yij are defined as the matrix form



ν 0
 Y

l
Yij =  ij
,
e
0 Yij




u 0
 Y

q
Yij =  ij
,
d
0 Yij

(2.30)

where the Yukawa couplings are different for the up- and down-type fermions.

In the flavor non-universal models, such as the LP and HP models in breaking pattern I,
and the TF and UU models in breaking pattern II, one Φ field is not enough to give masses
to all the SM fermions. There are two simple ways to solve this problem by adding new
particle content to the Lagrangian.

Usually, we can add heavy vector-like fermions to generate the SM fermion masses
through the see-saw mechanism, as displayed in Figs. 2.3, and 2.4. For example, in the
LP model, the Yukawa terms can be re-written as

q˜
q
¯
QiL Yij Φ + Yij Φ QjR −

L = −

l
¯
˜
LiL Yij Σ + Yijl Σ LjR ,

¯
LiL Mij LjR −
ij

(2.31)

ij

ij
−

l
¯
˜
LiL Yij Φ + Yijl Φ LjR

(2.32)

ij

where L is a vector-like doublet




 N 
LL,R = 
,

E
L,R
35

(2.33)

with mass matrix M and the Yukawa couplings are

ν 0

 Y
l
Yij =  ij
,
e
0 Yij







q
q 1 0 
Yij = Yij 
.
0 1

(2.34)

Similarly, one can write the Yukawa Lagrangian for UU and TF models in the second breaking
pattern. If the vector-like fermions are heavy enough, they will decouple at the low energy
scale.

Alternatively, one can add another Higgs multiplet φ to generate the SM fermion masses.
In this case, the Higgs scalar Lagrangian will be modified, and the mass spectrum of gauge
bosons will be affected. In our current study, we don’t consider this case in detail.

Φ ∼v
SU(2)

Σ ∼f
SU(2)

U(1)

Figure 2.3: The Moose diagram for the breaking pattern I (Lepton-phobic Model with vectorlike fermions).

36

Σ ∼f

Φ ∼v

SU(2)

SU(2)

U(1)

Figure 2.4: The Moose diagram for the breaking pattern II (Top-Flavor model with vectorlike fermions).

2.3.2

Symmetry Breaking and Mass Eigenstates

2.3.2.1

Higgs Potential and VEV

In the first breaking pattern, at the scale u, there are two kinds of Higgs multiplets. The
Higgs doublet gets the VEV





 0 

Σ =
 u .
√
2

(2.35)

Another possibility is the Higgs triplet which gets the VEV




 0 0
.
Σ =

 u
√ 0
2

(2.36)

At the scale v, the Higgs bi-doublet gets the VEV

Φ

√



0
 κ1 / 2

= 
.
√ 
0
κ2 / 2
37

(2.37)

To relate these VEV parameters κ1 , κ2 to the conventional parameter v, we define

v 2 = κ2 + κ2 ,
1
2
κ
tan β = 2 .
κ1

(2.38)
(2.39)

In the breaking pattern II, at the scale u, the Higgs bi-doublet gets the VEV

<Σ> =



1 u 0 

.
2
0 u

(2.40)

At the scale v, the Higgs doublet gets the VEV




 0 

Φ =
 v .
√
2

(2.41)

The Higgs potential can be decomposed into three parts, VΦ , VΣ and V(Φ,Σ) . If no CP
violation originates from the Higgs potential, all the coefficients in the Higgs potential will
be real. In this study, we only consider Φ and Σ as doublet, bi-doublet, or triplet scalars.
The general Higgs potential VΦ and VΣ are given by

VΦ = V(Φ),

VΣ = V(Σ),
38

(2.42)

where V has the form as follows
˜
˜
V(φ) = −µ2 Tr(φ† φ) − µ2 Tr(φφ† ) + Tr(φ† φ)
1
2
2
2
2
˜
˜
+ λ1 Tr(φ† φ) + λ2 Tr(φφ† ) + Tr(φ† φ)
˜
˜
˜
˜
+ λ3 Tr(φφ† )Tr(φ† φ) + λ4 Tr(φ† φ) Tr(φφ† ) + Tr(φ† φ) .

(2.43)

If φ is a scalar doublet, one can reduce V by
˜
˜
Tr(φ† φ) = Tr(φφ† ) = Tr(φ† φ)

(2.44)

to otain
V(φ) = −µ2 Tr(φ† φ) + λ Tr(φ† φ)

2

(2.45)

with µ2 = µ2 + µ2 and λ = λ1 + 2λ2 + λ3 + 2λ4 .
2
1

The Higgs potential V(Φ,Σ) has the following typical form
˜
˜
V(φ, ∆) = α1 Tr(φ† φ)Tr(∆∆† ) + α2 Tr(φ† φ)Tr(∆∆† ) + Tr(φφ† )Tr(∆∆† )
+ α3 Tr(φ† φ∆∆† ).

(2.46)

In the breaking pattern I, the Higgs potential V(Φ,Σ) is defined as

V(Φ,Σ) = V(Φ, Σ).
39

(2.47)

While in the breaking pattern II, the Higgs potential V(Φ,Σ) is defined as

V(Φ,Σ) = V(Σ, Φ).

(2.48)

After spontaneous symmetry breaking, a full analysis of the Higgs mass spectrum can be
performed in detail. However, we would like to do a simple analysis focusing on the Higgs
potential in which only the Higgs scalar Φ is included, through replacing the scalar Σ with
its VEV u in the Higgs potential. The reason is in the following. We know the scalar Σ
is mainly giving masses to the W and Z bosons, but not giving masses to the SM gauge
bosons and fermions. So the scalar Σ mainly couples to W and Z , and the couplings to the
SM gauge bosons are suppressed by the order of O(v 2 /u2 ). When the VEV u is much larger
than the VEV v, the scalar Σ becomes heavy and almost decouple from the Higgs scalar Φ.

In the breaking pattern I, the Higgs potential, only involving the bi-doublet scalar Φ and
VEV u, is

˜
˜
VΦ = −µ2 Tr(Φ† Φ) − µ2 Tr(φφ† ) + Tr(φ† φ)
1
2
2
2
2
˜
˜
+ λ1 Tr(φ† φ) + λ2 Tr(φφ† ) + Tr(φ† φ)
u
˜
˜
˜
˜
+ λ3 Tr(φφ† )Tr(φ† φ) + λ4 Tr(φ† φ) Tr(φφ† ) + Tr(φ† φ) + V(Φ, √ ). (2.49)
2

After electroweak symmetry breaking, the charged fields φ± form the Goldstone bosons
1,2
of W ± and two charged Higgses. The Goldstone bosons are given by

G ± = cβ φ ± − sβ φ ± .
1
2
40

(2.50)

The physical charged Higgs states are the linear combinations

H ± = sβ φ ± + cβ φ ± .
1
2

(2.51)

The mass of the lightest charged Higgs is given to a very good approximation by
α3 u 2
.
MH+ =
c2 − s2
β
β

(2.52)

The neutral Goldstone boson is

G 0 = cβ φ 0 − sβ φ 0 .
1
2

(2.53)

In addition there are three neutral Higgs fields in the spectrum, given by

h0 = cβ φ0 + sβ (φ0 )
1
2

(2.54)

0
H1 = −sβ φ0 + cβ (φ0 )
1
2

(2.55)

0
H2 = sβ φ0 + cβ (φ0 ),
1
2

(2.56)

0
0
where h0 can be identified as the light Higgs with a mass of O(v), while H1 and H2 are
two new neutral Higgses with masses O(u). To leading order their masses are equal to each
other
α3 u 2
2
.
MH ≡ M 2 0 = M 2 0 =
H1
H2
c2 − s2
β
β

(2.57)

In the breaking pattern II, there is only one Higgs doublet. So after electroweak symmetry
breaking, only one light Higgs is left with three Goldstone bosons eaten by the W and Z
41

gauge bosons.

2.3.2.2

Gauge Boson Mass Mixing in Breaking Pattern I

In this breaking pattern, the gauge couplings are related to the U(1)em gauge coupling by
the relation

1
1
1
1
=
+
+
,
e2
g2
g2
g2
L
R
X

(2.58)

with

gL =

e
,
sin θ

gR =

e
,
cos θ sin φ

gX =

e
,
cos θ cos φ

(2.59)

3
where θ is the usual weak mixing angle, and φ is the new mixing angle between WR and B
in this model.
To get the gauge boson mass eigenstates, there are two steps. For the charged gauge
bosons, there is no rotation. The mass matrix is

2
2
MW = M0 


1
tan θ
sin 2β
−
sin φ


tan θ
sin 2β 
−
sin φ
,

tan2 θ
(1 + x)
sin2 φ

(2.60)

where
1 2 2 v 2 e2
2
,
g v =
M0 =
4 L
4 sin2 θ

and x =

u2
2u2
for Higgs doublet, and x =
for Higgs triplet.
v2
v2
42

(2.61)

3
3
For the neutral gauge bosons WL , WR , B, one rotates the gauge fields by the angle φ
and re-parametrize them by θ:

3
3
WH = cos φWR − sin φB ,

(2.62)

3
3
3
WZ = cos θWL − sin θ(sin φWR + cos φB)

(2.63)

3
3
A = sin θWL + cos θ(sin φWR + cos φB)

(2.64)

3
3
The mixing mass matrix in the (A, WZ , WH ) basis reduces to



 0
2 
M0 
2
 0
MZ =
cos2 θ 


0

where x =

0

0
−

1
−

sin θ
tan φ

sin θ
tan φ

sin2 θ
(x + cos4 φ)
2 φ cos2 φ
sin




,




(2.65)

4u2
u2
for Higgs doublet, and x =
for Higgs triplet.
v2
v2

2
2
To get the mass eigenstates, one can further diagonalize the mass matrices MW and MZ .
Here we focus on the region x

1. (one can impose further constraints on gR and gX , of

course.) Up to the leading order of x, the wave-function mixing is given by the ratio of the
non-diagonal term and the diagonal term. To order 1/x the eigenstates of the charged gauge
bosons are

sin φ sin 2β ±
±
±
WR ,
Wµ = WL +
µ
µ
x tan θ
sin φ sin 2β ±
±
+ W± ,
W
W µ = −
Rµ
Lµ
x tan θ
43

(2.66)
(2.67)

while for the neutral gauge bosons
sin φ cos3 φ 3
3
WH µ ,
Zµ = WZ µ +
x sin θ
sin φ cos3 φ 3
3
WZ µ + WH µ .
Zµ = −
x sin θ

(2.68)
(2.69)

To the same order, the gauge boson masses are
sin2 2β
2
M 2 ± = M0 (1 −
),
W
x
2
2
2 x tan θ + tan θ
M 2 ± = M0
W
sin2 φ
sin2 φ

(2.70)
,

(2.71)

and
2
4
2 = M0 (1 − cos φ ) ,
MZ
x
cos2 θ
2
M0
x tan2 θ tan2 θ cos4 φ
+
M2 =
Z
cos2 φ
sin2 φ
sin2 φ

2.3.2.3

(2.72)
.

(2.73)

Gauge Boson Mass Mixing in Breaking Pattern II

In this breaking pattern, the gauge coupling relation is

gl =

e
,
sin θ cos φ

gh =

e
,
sin θ sin φ

gX =

e
,
cos θ

(2.74)

with the relation

1
1
1
1
=
+
+
,
2
e2
g 2 g 2 gY
l
h
44

(2.75)

µ
and
l

where θ is the usual weak mixing angle, and φ is the new mixing angle between W

µ
W in this model. If φ is small, the couplings of the third generation will become large.
h

There are two steps to get the gauge boson mass eigenstates: For the charged gauge
bosons, one can rotate the gauge fields by φ:

W ± = − sin φW ± + cos φW ± ,
H
l
h

(2.76)

W ± = cos φW ± + sin φW ± .
L
l
h

(2.77)

The mass matrix reduces to

2
2
MW = M0 



1

− tan φ

− tan φ

x
+ tan2 φ
sin2 φ cos2 φ


,


(2.78)

where
2 2
1 2
2 = v e
= gL v 2 .
M0
4 sin2 θ
4

(2.79)

3
3
For the neutral gauge bosons WL , WR , B, one rotates the gauge fields by the angle φ and
re-parametrize them by θ:

3
3
WH = − sin φWl3 + cos φWh ,

(2.80)

3
3
WL = cos θ(cos φWl3 + sin φWh ) − sin θB

(2.81)

3
A = cos θB + sin θ(cos φWl3 + sin φWh )

(2.82)

45

3
3
The mixing mass matrix in the (A, WZ , WH ) basis reduces to



 0
2 
M0 
2
 0
MZ =
cos2 θ 


0

0

0

1

− cos θ tan φ

− cos θ tan φ

cos2 θ(

x
+ tan2 φ)
2 φ cos2 φ
sin




.




(2.83)

2
To get the mass eigenstates, one can further diagonalize the mass matrices MW and
2
MZ . Here we focus on the region x
1. Up to the leading order of x, the wave-function
mixing is given by the ratio of the non-diagonal term and the diagonal term. To order 1/x
the eigenstates of the charged gauge bosons are
cos φ sin3 φ ±
±
W
,
Wµ = W ± +
Hµ
Lµ
x
cos φ sin3 φ ±
±
W µ = −
W
+ W± ,
Lµ
Hµ
x

(2.84)
(2.85)

while for the neutral gauge bosons
cos φ sin3 φ 3
3
WH µ ,
Zµ = WL µ +
x cos θ
cos φ sin3 φ 3
3
WL µ + WH µ .
Zµ = −
x cos θ

(2.86)
(2.87)

To the same order, the gauge boson masses are
4
2
2 (1 − sin φ ) ,
M ± = M0
W
x
2
M0
x
+ sin2 φ
M2 ± =
2 φ sin2 φ
W
cos
46

(2.88)
,

(2.89)

and
2
4
2 = M0 (1 − sin φ ) ,
MZ
x
cos2 θ
2
M0
x
M2 =
+ sin2 φ
Z
cos2 φ sin2 φ

2.3.3

(2.90)
.

(2.91)

Lagrangian in the Physical Basis

After the symmetry breaking and mass mixing, one can obtain the Lagrangian involving the
physical fields. Here we listed the Feynman rules of the Lagrangian in the physical basis in
all the models. In the Feynman rules, all particles are assumed to be outgoing, and we adopt
the conventional Feynman rule, iL.

2.3.3.1

Gauge couplings of fermions

The most general interaction of the Z and W to SM fermions is

¯
¯
Lf = g2 Zµ f γ µ (gL PL + gR PR )f + g2 Wµ f γ µ (gL PL + gR PR )f + h.c. ,

(2.92)

where g2 = e/ sin θ is the weak coupling strength and PL,R = (1 ∓ γ5 )/2 are the usual
chirality projectors. On the other hand, the most general interaction of the Z and W to
SM fermions is

¯
¯
Lf = g2 Zµ f γ µ (gL PL + gR PR )f + g2 Wµ f γ µ (gL PL + gR PR )f + h.c. ,

(2.93)

For simplicity, we use gL and gR for both Z and W bosons from now on. Detailed
expressions of gL and gR for each individual NP model are listed in Tables 2.3.3.1 and 2.3.3.1.
47

Couplings

¯
W +µ f f (BPI)

¯
Z f f (BPI)

¯
W ±µ f f (BPII)

gL
c2 s s2
em
+ W 2β φ
γρ TL
−√
2cW sφ
xs2
W


c4 s2 T3L − Qs2
em
W 
φ φ
γρ  T3L − Q s2 −
φ
2
cW cφ sφ
xs
W


s2 c2
em sφ
φ φ
γ µ T ± 1 +
−√
l
x
2sW cφ

em cφ
√
γ µ T ± 1 −
h
2sW sφ

¯
W ±µ F F (BPII)



¯
Z F F (BPII)

x





2 2
sφ sφ cφ
em µ 
−
T3l 1 +
s2 Q
−
γ
2
2 W
sW
cφ
cφ xc
xcW
W




4
s4
cφ sφ
cφ
em µ 
φ 
T3h 1 −
+
s2 Q
γ
2
sW
sφ
sφ xc2 W
xc
W
W


¯
Z f f (BPII)

s4
φ

sφ

s2 c2
φ φ



Table 2.3: The left-handed fermion couplings of the heavy gauge boson in Breaking Pattern
I and II. For the fermion couplings, the quantum numbers (TL , TR ) in BP-I and (Tl , Th ) in
BP-II are implied in Tab. 2.2. In BP-II, the fermion notation f means the fermions listed in
the column SU(2)1 , while F means the fermions listed in the column SU(2)2 in Tab. 2.2.

48

Couplings

gR
e
√ m γρ T +
R
2cW sφ

¯
W +µ f f (BPI)


¯
Z f f (BPI)

c4 s2
φ φ



em

γρ  T3R − Qs2 + Q
φ
cW cφ sφ
x

¯
W ±µ f f (BPII)

0

¯
W ±µ F F (BPII)

0


¯
Z f f (BPII)

¯
Z F F (BPII)

2 2
sφ sφ cφ



em µ 
γ
s2 Q
sW
cφ xc2 W
W


4
cφ sφ
em µ 
s2 Q
γ
sW
sφ xc2 W
W

Table 2.4: The right-handed fermion couplings of the heavy gauge boson in Breaking Pattern
I and II. For the fermion couplings, the quantum numbers (TL , TR ) in BP-I and (Tl , Th ) in
BP-II are implied in Tab. 2.2. In BP-II, the fermion notation f means the fermions listed in
the column SU(2)1 , while F means the fermions listed in the column SU(2)2 in Tab. 2.2.

49

Couplings
+ −
H Wν Wρ

H Wν Wρ

H Zν Zρ

H Zν Zρ

BPI


2s2
2
i em
2β 
vgνρ 1 −
2 s2
x
W
2
c2 s2 − s2
i em s2β
W
W φ
vgνρ 1 +
−
2
2 cW sW sφ
xs
W


2c4
2
i em
φ
vgνρ 1 −
2 s2 c2
x
W W
2
c2 c2 s2 − s2
i em cφ
φ
φ φ W
vgνρ 1 −
−
2
2 c2 s s
xs
W W φ
W

+ −
Wµ Wν Zρ

cW em
sW

+
Wµ Wν− Zρ

i

em s2β sφ
xs2
W

em cW sφ c3
φ
i
2
xs
W

+ −
Wµ Wν Zρ

Table 2.5: The triple boson couplings of the heavy gauge boson in Breaking Pattern I.

50

Couplings
+ −
H Wν Wρ

H Wν Wρ

H Zν Zρ

H Zν Zρ

BPII


2s4
2
i em
φ
vgνρ 1 −
2 s2
x
W
2
s2 c2 − s2
i em sφ
φ φ
φ
vgνρ 1 +
−
2 c
2s
x
W φ


2s4
2
i em
φ
vgνρ 1 −
2 s2 c2
x
W W
2
s2 s2 c2 − c2
i em sφ
φ
φ φ W
vgνρ 1 −
−
2
2 c s2 c
xc
W W φ
W

+ −
Wµ Wν Zρ

cW em
sW

+
Wµ Wν− Zρ

em cφ s3
φ

i

+ −
Wµ Wν Zρ

i

xsW cW
em cφ s3
φ
xsW

Table 2.6: The triple boson couplings of the heavy gauge boson in Breaking Pattern II.

51

2.3.3.2

Couplings of Gauge Bosons and Higgs Boson

For the bosonic sector, we are only interested in the triple gauge boson couplings and gauge
boson - gauge boson -Higgs couplings. So we write down the most general Lagrangian
involving in triple couplings of gauge bosons and Higgs, as follows,

a
L = −gV V V f abc (∂µ Vν )V bµ V cν + gV V H g µν HVaµ Vaν ,

(2.94)

where the coefficients gV V V and gV V H are listed in the Table. For the triple gauge boson
couplings, the Lorentz index in the Feynman rules [g µν (k1 − k2 )ρ + g νρ (k2 − k3 )µ + g ρµ (k3 − k1 )ν ]
is implied. For the gauge boson - gauge boson -Higgs couplings, the Lorentz index in the
Feynman rules g µν is implied. Feynman rules of the triple gauge boson couplings and gauge
boson - gauge boson -Higgs couplings are shown in Tabs. 2.3.3.1 and 2.3.3.1.

2.4

Model Parameters and Mass Spectrum

From Tables 2.1, 2.3.3.1 and 2.3.3.1, we see that the G(221) models contain six (five) parameters for the first (second) breaking pattern: three (two) VEV’s {uD,T , v sin β, v cos β}
in Table 2.1 and three gauge couplings {g1 , g2 , gX } in Table IV. (For breaking pattern II,
there are only two VEV’s {u, v}.) Compared to the gauge sector of the SM, which contains
only three parameters (two gauge couplings and one VEV; gL , gY and v), there are three
(two) additional parameters. We would like to find a useful parameterization of these three
additional parameters, so as to parameterize the effects of new physics. We discuss these in
detail in turn.
52

2.4.1

Input Parameters

As stated above, the G(221) models contain six (five) parameters in the gauge sector:

{g1 , g2 , gX , uD (uT , or u), v 2 , sβ },

(2.95)

where the parameter β only exists in models with breaking pattern I. An equivalent set of
parameters is

{αe , sin θ, cos φ, x, v 2 , s2β },

(2.96)

where θ is the weak mixing angle, φ is the new mixing angle, and x is defined as


 2 2

u /v
(for LRD, LPD, HPD, FPD)
 D




x ≡ u2 /v 2 (for LRT, LPT, HPT, FPT)
 T





 2 2
u /v

(for SQ, TF, UU).

As we expect x to be large (x

(2.97)

100), we work to leading order in x−1 .

In addition to these parameters, the loop-level predictions will require the values of the
masses of the top quark (mt ) and the Higgs boson (MH ). The bar (¯) over mt indicates
that we will use the top quark mass as defined in the MS-scheme.
We take as reference observables the experimental measurements of
−1 SM
• the fine structure constant (αe (MZ ) = 127.918 at the scale MZ ).
• the Fermi constant (GSM = 1.16637 × 10−5 GeV−2 ), determined from the lifetime of
F
53

the muon,

SM
• the mass of the Z boson (MZ = 91.1876 GeV), determined from the Z-line shape at
LEP-I.

Our task then is to express the model parameters, cf Eq. 2.96

{αe , sin θ, v 2 , x, cos φ, s2β , mt , MH },

in terms of the reference and fit parameters

SM
SM
{αe , MZ , GSM , x, cos φ, s2β , mt , MH }.
F

(2.98)

That is, we want the relationships
reference parameters

model parameters

SM
SM
{αe , sin θ, v 2 , x, cos φ, s2β , mt , MH } ⇔ { αe , MZ , GSM
F
fit parameters
x, cos φ, s2β , mt , MH }

(2.99)

Since {x, cos φ, s2β , mt , MH } appear in both the model and fit parameters (by construction), we only have to solve for {αe , sin θ, v 2 , } in terms of the reference and fit
parameters. This can be done by analyzing how the reference parameters are related to the
model parameters.
54

2.4.1.1

Electric Charge

The electric charge in the G(221) models is the gauge coupling of the unbroken U(1)em
group, which we have parameterized as e defined as

1
1
1
1
+
+
=
,
2 g2 g2
2
e
g1
2
X

(2.100)

and αe ≡ e2 /4π. There are no tree-level modifications to the wavefunction renormalization
of the photon, so we then simply have the relationship

SM
αe = α e .

2.4.1.2

(2.101)

The Fermi Constant

The Fermi constant, GF , is experimentally determined from the muon lifetime as [39]
2 5
−1 = GF mµ 1 + O
τµ
192π 3

m2
e
m2
µ

1+O

m2
µ
2
M
W

1+O

1
16π 2

,

(2.102)

where the precise forms of the higher-order corrections are given in Ref. [39]. Neglecting
these higher-order corrections, the SM contribution to the muon lifetime is

−1
τµ =

4
gL
192 · 32π 3 M 4
W

m5 ,
µ

(2.103)

2
2
and, using the SM relation 4MW = gL v 2 , we obtain
GSM = √
F

1
2
2v SM

55

.

(2.104)

In the standard model, the effective four fermion Lagrangian gives us

LSM =

e SM
√
2 sin(θ SM )

2

√
2
J +J − =
J + J − = 2 2GF J + J − . (2.105)
L L
L L
L L
M2
v2
W SM
SM
1

In the G(221) models, we have extra contributions to the four-fermion charged-current effective theory below the electroweak scale. The effective four fermion Lagrangian is

Leff = cLL J + J − + cLR J + J − + cRL J + J − + cRR J + J −
L L
L R
R L
R R

(2.106)

where the charged currents are

J

−µ
= νL,R γ µ lL,R ,
¯
L,R

J + = (J − )†

(2.107)

and the coefficients are
2
2
gW L gW P L
,
+
cLL =
2
MW
M2
W
gW L gW R gW P L gW P R
cLR = cRL =
+
,
2
MW
M2
W
2
2
g
g
cRR = W R + W P R .
M2
M2
W
W

(2.108)
(2.109)

(2.110)

These contributions will modify the SM relation in Eq. 2.104. In principle, the fermionic
contributions can have both left- and right-handed components and differ among the different
generations. However, for the G(221) models we consider here, the charged currents couples
universally to the first two generations. Furthermore, the charged current is either purely
right-handed (the LR, HP, LP, FP models) or purely left-handed (the UU and TF models).
56

We therefore focus on these special cases instead of performing the general analysis.

In the BP-I, the contributions to the amplitude do not interfere with one another in the
limit of neglecting the masses of electrons and neutrinos. From the lifetime to the

1
order
x2

we can obtain

8G2
F

= c2 + 2c2 + c2
LL
LR
RR
2
4
sin2 (2β)
4
+2
1+
=
4
x
v
v4

sin 2β 2
4
+
x
v4

1 2
1
+ O( ),
x
x3

(2.111)

in BP-I with doublet Higgs, and

GF = √

1

1+

2v 2

sin2 2β
2x

,

(2.112)

in BP-I with triplet Higgs. Though the left-right and right-right current operators do not
contribute to the total muon decay rate at the order O(˜−1 ) , they do contribute at leading
x
order to the Michel parameters (for a detailed discussion of the Michel parameters, see the
Muon Decay Parameters article in the Particle Data Group (PDG) [39]). The expression of
GF , which depends on the details of the Higgs representation, is written in terms of model
parameters as

GF =


 1

√

 2v 2
 1

√


2v 2

1+
1+

sin2 2β
x
sin2 2β
2x

,

(for LRD, LPD, HPD, FPD)
(2.113)

,

(for LRT, LPT, HPT, FPT)
57

In the BP-II, the expression of the charged-current operators is
2
2
gW L gW P L
+
,
cLL =
M2
M2
W
W
cLR = cRL = 0 ,
cRR = 0 .

From the lifetime, we can obtain, to the

8G2
F

(2.114)
(2.115)
(2.116)

1
order
x2

= c2 + 2c2 + c2
LL
LR
RR
1
4
+ O( ) .
=
4
v
x3

(2.117)

So we are simply left with

GF = √

1
2˜2
v

(for SQ, TF, UU).

(2.118)

We can rewrite our results in a more suggestive manner by defining the SM VEV through
the Fermi constant

v2 ≡ √

1
.
2GF

58

(2.119)

We then have


sin2 2β
 SM
v

1+
, (for LRD, LPD, HPD, FPD)


2x




2
v = v SM 1 + sin 2β , (for LRT, LPT, HPT, FPT)


4x





 SM
v

. (for SQ, TF, UU)

2.4.1.3

(2.120)

Z-Mass

The expression of the Z mass can be written as



ev



 2 sin θ cos θ





ev
2 =
MZ
 2 sin θ cos θ






ev



 2 sin θ cos θ

cos4 φ
1−
,
2x
cos4 φ
1−
,
8x
sin4 φ
1−
,
2x

(for LRD, LPD, HPD, FPD)
(for LRT, LPT, HPT, FPT)

.

(2.121)

(for UU and TF)

Thus we have

 e SM v SM




 2 sin θ cos θ




 SM SM
e
v

2
MZ =
 2 sin θ cos θ



 SM SM
e

v



 2 sin θ cos θ

sin2 2β cos4 φ
−
, (for LRD, LPD, HPD, FPD)
2x
2x
sin2 2β cos4 φ
.
−
1+
, (for LRT, LPT, HPT, FPT) (2.122)
4x
8x
sin4 φ
1−
, (for UU and TF)
2x
1+

While in the SM Z mass can be written as
e SM v SM
SM =
MZ
.
2 sin θ SM cos θ SM
59

(2.123)

Here sin θ SM is the SM weak coupling angle, and its value can be obtained by
πα(MZ ) SM
,
sin2 θ SM cos2 θ SM = √
SM
SM
2GF (MZ )2

(2.124)

1

= 128.91 ± 0.02, yields sin2 θ SM = 0.23108 ± 0.00005. Since MZ is
SM
α(MZ )
SM
defined in both the SM and the proposed model through the Z peak at LEP MZ = MZ ,

Using

we can solve for θ in terms of the reference and fit parameters

 2 SM
cos4 φ sin2 2β

sin θ
cos2 θ SM 1 −
+



x
x






(for LRD, LPD, HPD, FPD)







 2 SM
cos4 φ sin2 2β
sin θ

+
cos2 θ SM 1 −

4x
2x
2 θ cos2 θ =
sin



(for LRT, LPT, HPT, FPT)






4

 2 SM
sin θ
2 θ SM 1 − sin φ ,

cos


x







(for SQ, TF, UU).

,

,
(2.125)

Hereafter, solving for sin2 θ = 1 − cos2 θ in terms of sin2 θ SM (the solutions of x(1 − x) = a
√
1
are x = (1 ± 1 − 4a)), one finds that
2
sin2 θ = sin2 θ SM 1 −

cos2 θ SM
∆θ ,
cos2 θ SM − sin2 θ SM

cos2 θ = cos2 θ SM 1 +

sin2 θ SM
∆θ ,
cos2 θ SM − sin2 θ SM

60

(2.126)

where the correction ∆θ is

 cos4 φ sin2 2β



−
, (for LRD, LPD, HPD, FPD)


x
x





cos4 φ sin2 2β
∆θ =
, (for LRT, LPT, HPT, FPT)
−

4x
2x




 sin4 φ



, (for SQ, TF, UU).


x

2.4.2

(2.127)

Mass Spectra

With the input parameters defined as above, we can calculate the corrections to the experimental observables. We take the mass spectrum as an example and collect the formula of
the masses of the W -boson, W -boson and Z -boson.

2.4.2.1

Breaking Pattern I with Doublet Higgs

The mass spectrum in the Breaking Pattern I has the form of


s2
2 v2
e
2β 
2
,
MW = m 1 −
2
x
4sW


c4
2 v2
em
2
1 − φ  ,
MZ =
2 c2
x
4sW W

(2.128)

(2.129)

and

M2
W
M2
Z

=

e2 v 2 x
m
4c2 s2
W φ

1+

1
x


2 v2x
em
1 +
=
4c2 s2 c2
W φ φ
61

,
c4
φ
x

(2.130)

.

(2.131)

More specifically, in terms of the input parameters, we can convert all the model parameters
to reference and fit parameters



sm2
s2
c4
sm2 v sm2
cW
e
2
1 +
 φ − 2β  ,
MW = m
sm2
sm2 − s sm2
x
x
4sW
cW
W
sm2
em v sm2
2
2
= MZ SM ,
MZ =
sm2 c sm2
4s
W
W

(2.132)

(2.133)

and

M2
W
M2
Z

2.4.2.2


sm2 v sm2 x
e
1 +
= m
4c sm2 s2
W
φ

sm2 v sm2 x
e
1 +
= m
4c sm2 s2 c2
W
φ φ


sm2
sm2
cW s2 − sW c4
1
2β
φ
+
,
sm2 − s sm2 )x
x
(c
W
W

4
sm2 s2 − s sm2 c4
cφ cW
W
2β
φ
.
+
sm2 − s sm2 )x
x
(c
W
W

(2.134)

(2.135)

Breaking Pattern I with Triplet Higgs

The mass spectrum has the form of


s2
2 v2
e
2β 
2
MW = m 1 −
,
2
2x
4sW


c4
2 v2
em
2
1 − φ  ,
MZ =
2 c2
4x
4sW W

(2.136)

(2.137)

and

M2
W
M2
Z

1
e2 v 2 2x
m
1+
,
2x
4c2 s2
W φ


c4
2 v 2 4x
em
1 + φ  .
=
2 s2 c2
4x
4c
W φ φ

=

62

(2.138)

(2.139)

More specifically, in terms of the input parameters, we can convert all the model parameters
to reference and fit parameters



sm2
s2
c4
sm2 v sm2
cW
e
2
1 +
 φ − 2β  ,
MW = m
sm2
sm2 − s sm2 4x
2x
4sW
cW
W
sm2
em v sm2
2
2
= MZ SM ,
MZ =
sm2 c sm2
4s
W
W

(2.140)

(2.141)

and

M2
W
M2
Z

2.4.2.3


sm2 v sm2 x
2em
1 +
=
4c sm2 s2
W
φ

sm2 v sm2 x
4em
1 +
=
4c sm2 s2 c2
W
φ φ


sm2
sm2
2cW s2 − sW c4
1
2β
φ
+
,
sm2 − s sm2 )x
2x
4(c
W
W

4
sm2 s2 − s sm2 c4
cφ 2cW
W
2β
φ
.
+
sm2 − s sm2 )x
4x
4(c
W
W

(2.142)

(2.143)

Breaking Pattern II

The mass spectrum in the Breaking Pattern II have


s4
2 v2
e
φ
2
MW = m 1 −
,
2
x
4sW


s4
2 v2
em
2
1 − φ  ,
MZ =
2 c2
x
4sW W

(2.144)

(2.145)

and

M2
W
M2
Z


2 v2x
em
1 +
=
4s2 s2 c2
W φ φ

2 v2x
em
1 +
=
4s2 s2 c2
W φ φ
63

s4
φ
x
s4
φ
x


,

(2.146)


.

(2.147)

More specifically, in terms of the input parameters, we can convert all the model parameters
to reference and fit parameters


sm2
s4
sm2 v sm2
sW
e
φ
2
1 +
,
MW = m
sm2
sm2 − s sm2 x
4sW
cW
W
sm2 v sm2
em
2
2
= MZ SM ,
MZ =
sm2 c sm2
4s
W
W

(2.148)

(2.149)

and

M2
W
M2
Z

2.5


sm2 v sm2 x
e
1 +
= m
4s sm2 s2 c2
W
φ φ

sm2 v sm2 x
e
1 +
= m
4s sm2 s2 c2
W
φ φ

sm2
sm2
(2 cW − sW )s4
φ



,
(c sm2 − s sm2 )x
W
W

sm2 − s sm2 )s4
(2 cW
W
φ
.
sm2 − s sm2 )x
(c
W
W

(2.150)

(2.151)

Indirect and Direct Constraints on Parameter Spaces

Even though the W and Z bosons are not observed yet, they could contribute to a few
observables, which can be measured precisely at low energy, via quantum effects. In this
section we perform a global-fit analysis of 37 electroweak precision test observables (EWPTs)
to derive the allowed model parameter space of those NP models of our interests. In addition,
we also include direct search limits from the Tevatron and LHC.
Note that m
and m are not independent in the G(221) model. In this study we
W
Z
choose M
as an input parameter. In addition, other independent parameters are the
W
gauge mixing angle φ, and the mixing angle β in the EWSB scale between two Higgs VEVs
s2β = sin(2β) that only exists in BP-I. Our parameter scan is not sensitive to the parameter
β as it contributes to physical observables only at the order of 1/x = v/u. We then present
64

our scan results in the plane of (M , cφ ) or (M , M ).
W
W
Z

2.5.1

Indirect Constraints from Electroweak Precision Tests

For a measured observable O exp , the SM prediction can be broken down into the tree- and
loop-level components

th,tree
th,loop
th
O SM = O
+O
(mt , MH ),
SM
SM

(2.152)

where O th is expressed in terms of the reference parameters. Since the top quark mass (mt )
and the mass of the Higgs boson (MH ) enter into the loop-calculations in the SM, a global
analysis of precision data and direct detection data can be used to constrain MH . In the
G(221) models, we can express the theoretical prediction as

th,tree
th,tree
th,loop
(x, φ, β),
O th = O
+O
(mt , MH ) + O
NP
SM
SM

where O

(2.153)

th,tree
is of the order O(1/x), and we assume that
NP
x−1 ∼

1
16π 2

∼O

th,loop
.
SM

(2.154)

That is, the Born-level new physics contributions from the G(221) models are numerically of
one-loop order, and loop corrections involving new physics are of two-loop order O

1
,
16π 2 x

which we discard in our analysis.
To compare with precision data (from LEP-1 and SLD) and low-energy observables,
th,tree
we calculate the shifts in observables O NP
(x, φ, β), as in the previous examples of the
65

partial decay widths of the Z-boson and the mass of the W -boson, and we adapt these
corrections into a numerical package GAPP [37]. GAPP then computes O

th,tree
and
SM

th,loop
th,tree
O SM
(mt , MH )1 , together with the O NP (x, φ, β) to find the best-fit values of the
¯
fit parameters and the confidence level contours using the CERN library MINUIT [38].
We perform a global fit over the following classes of observables
• LEP-I Z-pole observables: the total Z-width (ΓZ ), left-right asymmetries (ALR ), and
related observables,
• the mass (MW ) and decay width (ΓW ) of the W -boson,
• the tau lifetime ττ ,
• the ratios of neutral-to-charged current cross sections measured from neutrino-hadron
NC CC
deep-inelastic scattering (DIS) experiments (Rν ≡ σνN /σνN and similarly defined
for ν),
νe
νe
• effective vector and axial-vector neutrino-electron couplings (gV and gA ),
• weak charges (QW ) of atoms and the electron measured from atomic parity experiments.
Detailed information on these observables can be found in PDG [39], and here we only briefly
summarize the observables. The set of the observables included in our analysis is the same
as that used in the PDG analysis [39], with two exceptions.
• First, we do not include the anomalous magnetic moment of the muon and the decay
branching ratio b → sγ. At leading order, these observables are of one-loop order, and
1 The higher order SM corrections included in our analysis are the same as those in the
default GAPP code used for the PDG analysis.
66

they depend on the details of the extended flavor structure of the G(221) models. In
this work, we assume W bosons only couple to fermions in the same generation.

• Second, we include the measurements of the decay width of the W -boson, which are
not included in the PDG analysis. However, because of the comparatively low precision
of these measurements, this observable turns out to be insensitive to the new physics
contributions from the G(221) models.

In total, we include a set of 37 experimental observables in our global-fit analysis.
Before we give a brief discussion on each of these classes of observables, we note that
for some low-energy observables, such as the measurements from the atomic parity violation
and neutrino-neucleus DIS experiments, we implement the shifts in the coefficients of the
relevant four-fermion interactions, and rely on GAPP to compute the theoretical predictions
based on these modified coefficients.
For the ease of typesetting in the following subsections, we introduce the abbreviation
for the various forms of the fermionic currents

µ
¯
¯
f1 f2 L ≡ f1 γ µ (1 − γ5 ) f2 ,
µ
¯
¯
f1 f2
≡ f1 γ µ (1 + γ5 ) f2 ,
R

(2.155)

and

µ
¯
f1 f2
V

¯
≡ f1 γ µ f2 ,

µ
¯
¯
≡ f1 γ µ γ5 f2 .
f1 f2
A
67

(2.156)

2.5.1.1

Precision Measurements at the Z-Pole

The precision measurements at the Z-pole (including LEP-1 and SLD experiments) fall into
two broad classes: observables that can be constructed from the partial widths and the
asymmetry. We discuss these two classes in turn.

In the first class, we consider the Z → f f partial width, which at tree-level has the
expression in the Standard Model

Γ(Z → f f ) =

nc
2
2
MZ gV + gA ,
12π

(2.157)

where nc = 3 if f is s quark, and nc = 1 for leptons, and gV (f ), and gA (f ) depend on
the details of the model. In addition to the total width ΓZ , there are also the following
measurements:

12π
· ΓZ e− e+ ΓZ (had.) ,
2 Γ2
MZ Z
Γ (had.)
, for = e, µ, τ,
R( )
= Z
ΓZ ¯
Γ (q q)
¯
R(q) = Z
, for q = u, d, c, s, b,
ΓZ (had.)
R(s)
,
R(s)
=
R(u) + R(d) + R(s)

σ had =

(2.158)
(2.159)
(2.160)
(2.161)

where ΓZ (f f ) is the partial decay width Γ(Z → f f ), and

ΓZ (qq).

ΓZ (had.) =
q=u,d,c,s,b
68

(2.162)

The left-right asymmetry ALR (f ) is defined as
2
Z
− gR (f )
,
ALR (f ) ≡
Z (f ) 2 + g Z (f ) 2
g
R
L
Z
gL (f )

2

(2.163)

Z
Z
where gL (f ) and gR (f ) are the couplings of the fermion f to the Z-boson:
L ⊃ Zµ (g Z (f )f L γ µ f L + g Z (f )f R γ µ f R ).
L
R

(2.164)

From the quark branching ratios R(q) defined above, the hadronic left-right asymmetry QLR
can be defined as [37] [40]

QLR ≡

R(q)ALR (q) −
q=d,s,b

q=u,c

R(q)ALR (q).

(2.165)

A second class of asymmetries, the forward-backward asymmetries AF B (f ), emerges from
the convolution of the ALR (f ) asymmetries with the polarization asymmetry ALR (e) of the
electron. The hadronic charge asymmetry QF B is defined accordingly [37] [40]
3
AF B (f ) ≡ ALR (e)ALR (f ),
4
3
QF B
≡ ALR (e)QLR .
4

2.5.1.2

(2.166)
(2.167)

The Tau Lifetime

In terms of model parameters, the expression of the tau (τ ) lifetime is similar to the muon
(µ) lifetime in the G(221) models, with the obvious replacement of mµ in the µ lifetime by
mτ in the τ lifetime. This is true even in the top-flavor model, in which third generation
69

fermions transform under a different gauge group compared to the first two generations. In
the four-fermion effective theory of the TF model, only interactions involving two pairs of
third-generation fermions receive new physics contributions, and the interactions involving
one pair of third-generation fermions with one pair of light-flavor fermions (those responsible
for the decay of the τ ) are the same as those between two pairs of first two generations of
fermions (those responsible for the decay of µ). This is similar to the case of the un-unified
model, where only interactions involving two pairs of quarks (qq)(qq) receive new physics
contributions, while the (qq)( ) interactions are the same as the ( )( ). The lifetime ττ
can be calculated at tree level as

−1
ττ

G2 m5
F τ
192π 3

1+3

m2
τ
2
MW

,

(2.168)

in the SM. The dominant new physics contribution from G(221) models can be captured in
the shift of MW .

2.5.1.3

νN Deep Inelastic Scattering

The νN deep inelastic scattering experiments probe the coefficients εL (q) and εR (q) (for q
being u or d) that parameterize the neutral current ννqq interactions below the electroweak
scale

G
¯
L ⊃ − √F ν ν L,µ
2

µ
µ
εL (q) q q L + εR (q) q q R .
¯
¯
q=u,d
70

(2.169)

The DIS experiments measure the ratios of neutral-to-charged current cross sections

NC CC
Rν ≡ σνN /σνN ,

NC CC
Rν ≡ σνN /σνN ,

(2.170)

which can be written in terms of εL (q) and εR (q) as

Rν = (1 − δ) aL (u)ε2 (u) + aL (d)ε2 (d) + aR (u)ε2 (u) + aR (u)ε2 (d) , (2.171)
L
R
R
R
¯
Rν = 1 − δ
¯

aL (u)ε2 (u) + aL (d)ε2 (d) + aR (u)ε2 (u) + aR (u)ε2 (d) . (2.172)
¯
¯
¯
¯
R
L
R
R

The coefficients δ and aL,R are related to the nuclei form factors that are experiment specific.
These coefficients are included in GAPP, and we implement only the corrections to εL (q)
and εR (q) .

2.5.1.4

νe Scattering

The most precise data on neutrino-electron scattering comes from the CHARM II [41] experiment at CERN that utilized νµ and ν µ . The relevant parameters εL (e) and εR (e) are
defined similarly as in the νN scattering

G
µ
µ
¯
¯
¯
L ⊃ − √F ν ν L,µ εL (e) ee + εR (e) ee
.
L
R
2

(2.173)

We can further define

νe
gV ≡ εR (e) + εL (e) ,

(2.174)

νe
gA ≡ εR (e) − εL (e) ,

(2.175)

71

NC
NC
which are related to the measured total cross sections σνe and σνNC or their ratio σνe /σνNC .
¯e
¯e
In the limit of large incident neutrino energies, Eν

NC
σνe

=

G2 me Eν
F

σνNC
¯e

=

G2 me Eν
F

2π
2π

me , the cross sections are given as

νe
νe 2 + 1 g νe − g νe 2 ,
gV + gA
A
3 V

(2.176)

νe
νe 2 + 1 g νe + g νe 2 .
gV − gA
A
3 V

(2.177)

We implement corrections to the couplings due to new physics in GAPP and compute the
cross sections that are used in the global-fit analysis.

2.5.1.5

Parity Violation Experiments

We consider observables from three different measurements: atomic parity violation (APV),
Møller scattering (e− e− → e− e− ) [42], and eN DIS. These experiments measure the weak
charge (QW ) of the electron [42], caesium-133 [43][44] and thallium-205 nuclei [45][46]. Before defining the weak charge, it is useful to parameterize the coefficients of the (ee)(qq) and
(ee)(ee) interactions in terms of C1q , C2q , and C1e as
G
L ⊃ − √F
2 q

G
µ
µ
µ
¯
¯
¯
¯
¯
¯
− √F C1e ee A,µ ee (2.178)
C1q ee A,µ qq + C2q ee A,µ q q
A
A
A
2

The weak charges of the quark and electron are defined as

QW (q) = 2C1q ,

QW (e) = 2C1e.
72

(2.179)

Z
We can express the SM tree-level couplings of quarks to the Z-boson as L ⊃ Z µ Jµ , where

Z
Z
Jµ = gA (q) · QW (q) q q V ,µ ± q q A,µ ,
¯
¯

(2.180)

3q
and the ± on the axial-vector term is the opposite sign of the T . Hence QW (q) can be
L
interpreted as the ratio of the vector current to axial-vector current coupling of quark q to
the Z-boson:

QW, SM (q) =

Z
gV (q)
g Z (q)
A

.

(2.181)

The weak charges of the nucleons and nuclei can be built up from those of the quarks

QW (p) = 2QW (u) + QW (d),

(2.182)

QW (n) = QW (u) + 2QW (d),

(2.183)

and for nucleus A Z (with atomic number Z and mass number A), which contains Z protons
and N(= A − Z) neutrons,

QW A Z

= Z · QW (p) + N · QW (n)

(2.184)

= 2 (Z + A) · C1u + (2A − Z) · C1d .

(2.185)

There are also measurements of certain linear combinations of the coupling coefficients
C1u and C1d from polarized electron-hadron scattering data [47]. The particular linear
73

combinations, determined by the experimental data,

C1 = 9C1u + 4C1d ,
C2 = −4C1u + 9C1d ,

(2.186)

are included in our global analysis.

2.5.2

Direct Searches at the Tevatron and LHC

Another important bound on the G(221) model originates from direct searches at the Tevatron and LHC. Searches for the W and Z bosons as a s-channel resonance have been carried
out at the Tevatron and LHC in leptonic decay modes, quark decay channels and diboson
decays. For the constraints from Tevatron, we use the latest Tevatron data:
• DØ: p¯ → Z → e+ e− ( Ldt=5.4 fb−1 ) [59];
p
• CDF: p¯ → W ± → eν ( Ldt=5.3 fb−1 ) [50];
p
• CDF: p¯ → W ± → t¯ ( Ldt=1.9 fb−1 ) [51];
p
b
¯
• CDF: p¯ → Z → tt ( Ldt=955 pb−1 ) [53].
p
and LHC7 data:
• ATLAS: pp → W ± → ν ( Ldt=1.04 fb−1 ) [54];
• ATLAS: pp → Z → l+ l− ( Ldt=1.1 fb−1 ) [55];
¯
• CMS: pp → Z ± → tt in the electron + jets channel ( Ldt=4.33 fb−1 ) [56].
We will discuss the direct searches in the next chapter in more details.
74

2.5.3

Parameter constraints

Using all the indirect and direct searches mentioned above, we scan over the parameter
space of a few G(221) models to find allowed parameter contours at the 95% confidence level
(CL). The NLO QCD corrections to new heavy gauge boson productions are included in the
approach described in Sec. III. For each individual NP model the total width is calculated
with all the possible decay channels included; see the discussion in Sec. III.
The parameter scan results are plotted in Figs. 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, and 2.11, 2.12,
2.13, 2.14, 2.15, 2.16, and 2.17, 2.18, 2.19, 2.20, 2.21, 2.22. In order to better understand the
impact of various experiment data on the parameter space of the G(221) model, we separate
the indirect and direct search constraints into three categories: the electroweak indirect
constraints (green region) and the direct search constraints from the Tevatron (red region)
and the LHC7 (blue region). In Figs. 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, we note the following
points:
• For the LRD (LRT) model, the LHC7 data gives stronger constraints on W and Z
masses than both EWPT and Tevatron data, and excludes the region where the W
mass is smaller than 1.7 TeV (1.8 TeV) and Z mass is smaller than 2.3 TeV (3.3 TeV);
• For the SQD model, although the W and Z with degenerate masses 500 GeV can be
allowed by the EWPTs at large cφ , the limits from the Tevatron and the LHC exclude
the region where the W and Z masses are smaller than 1.5 TeV.
• For all the models except the flavor universal models, such as LRD(T) and SQD, the
EWPT data still gives the strongest constraints on the W and Z masses, because of
the non-universal flavor structure in these models.
75

• In BP-I, with combined constraints, all the phobic models, in which the couplings of
W to either quarks or leptons are suppressed, can still have relatively light W around
500 GeV, but heavier Z (about 1.5 TeV);
• For the non-universal models, such as TFD and UUD, the electroweak indirect constraints are tighter than Tevatron and LHC7 direct search constraints, and push the
new gauge boson mass up to more than 2 TeV (TFD) and 3 TeV (UUD), respectively.
In Figs. 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, and 2.17, 2.18, 2.19, 2.20, 2.21, 2.22, we also want
to point out:
• In BP-I, the M −cφ plane shows that small cφ is favored by direct search constraints
W
because the W coupling is proportional to 1/sφ , which leads to a small W production
rate. However, in the M −cφ plane, small cφ is disfavored by direct search constraints
Z
because the mass relation M
Z

M /cφ , push the exclusion region of small cφ to
W

larger M .
Z
• In BP-II, the shape of the excluded bounds in the small cφ region are very similar for
all models because the production cross section of W and Z are proportional to tan φ
in all models such as SQD, TFD, and UUD. Because quarks and leptons are un-unified
in UUD, the gauge couplings to leptons are proportional to cot φ, which implies the
large cφ region is also disfavored.
• Within the direct searches, for LRD(T) the most sensitive constraint comes from the
W leptonic decay channel, while for phobic models, the tightest constraints comes
from the Z leptonic decay channel. This explains why the contours in the phobic
models have similar shapes, but different from those in the LRD(T) models.
76

MZ’ [TeV]
MZ’ [TeV]

5
4.5
4
3.5
3
2.5
2
1.5
LRD
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
5
4.5
4
3.5
3
2.5
2
1.5
LRT
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.5: Allowed parameter space of the LRD and LRT models at 95% CL in the M
−
W
M plane after including indirect and direct constraints: EWPTs (green), Tevatron (red)
Z
and LHC7 (blue).

77

MZ’ [TeV]
MZ’ [TeV]

5
4.5
4
3.5
3
2.5
2
1.5
LPD
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
5
4.5
4
3.5
3
2.5
2
1.5
LPT
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.6: Allowed parameter space of the LPD and LPT models at 95% CL in the M
−
W
M plane after including indirect and direct constraints: EWPTs (green), Tevatron (red)
Z
and LHC7 (blue).

78

MZ’ [TeV]
MZ’ [TeV]

5
4.5
4
3.5
3
2.5
2
1.5
HPD
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
5
4.5
4
3.5
3
2.5
2
1.5
HPT
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.7: Allowed parameter space of the HPD and HPT models at 95% CL in the M
−
W
M plane after including indirect and direct constraints: EWPTs (green), Tevatron (red)
Z
and LHC7 (blue).

79

MZ’ [TeV]
MZ’ [TeV]

5
4.5
4
3.5
3
2.5
2
1.5
FPD
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
5
4.5
4
3.5
3
2.5
2
1.5
FPT
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.8: Allowed parameter space of the FPD and FPT models at 95% CL in the M
−
W
M plane after including indirect and direct constraints: EWPTs (green), Tevatron (red)
Z
and LHC7 (blue).

80

MZ’ [TeV]
MZ’ [TeV]

5
4.5
4
3.5
3
2.5
2
1.5
SQD
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
5
4.5
4
3.5
3
2.5
2
1.5
MSQ
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.9: Allowed parameter space of the SQD and MSQ models at 95% CL in the M
−
W
M plane after including indirect and direct constraints: EWPTs (green), Tevatron (red)
Z
and LHC7 (blue).

81

MZ’ [TeV]
MZ’ [TeV]

5
4.5
4
3.5
3
2.5
2
1.5
TFD
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
5
4.5
4
3.5
3
2.5
2
1.5
UUD
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.10: Allowed parameter space of the TFD and UUQ models at 95% CL in the
M
−M plane after including indirect and direct constraints: EWPTs (green), Tevatron
W
Z
(red) and LHC7 (blue).

82

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
LRD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
LRT
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.11: Allowed parameter space of the LRD and LRT models at 95% CL in the
MW − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue). The dashed black lines in LRD represent MLR models.

83

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
LPD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
LPT
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.12: Allowed parameter space of the LPD and LPT models at 95% CL in the
MW − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue).

84

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
HPD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
HPT
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.13: Allowed parameter space of the HPD and HPT models at 95% CL in the
MW − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue).

85

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
FPD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
FPT
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.14: Allowed parameter space of the FPD and FPT models at 95% CL in the
MW − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue).

86

cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
SQD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.15: Allowed parameter space of the SQD models at 95% CL in the MW − cφ plane
after including indirect and direct constraints: EWPTs (green), Tevatron (red) and LHC7
(blue). The dashed black lines represent MSQ models.

87

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
TFD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
UUD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MW’ [TeV]

Figure 2.16: Allowed parameter space of the TFD and UUQ models at 95% CL in the
MW − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue).

88

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
LRD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
LRT
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]

Figure 2.17: Allowed parameter space of the LRD and LRT models at 95% CL in the
MZ − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue). The dashed black lines in LRD represent MLR models.

89

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
LPD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
LPT
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]

Figure 2.18: Allowed parameter space of the LPD and LPT models at 95% CL in the
MZ − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue).

90

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
HPD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
HPT
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]

Figure 2.19: Allowed parameter space of the HPD and HPT models at 95% CL in the
MZ − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue).

91

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
FPD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
FPT
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]

Figure 2.20: Allowed parameter space of the FPD and FPT models at 95% CL in the
MZ − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue).

92

cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
SQD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]

Figure 2.21: Allowed parameter space of the SQD models at 95% CL in the MZ − cφ plane
after including indirect and direct constraints: EWPTs (green), Tevatron (red) and LHC7
(blue). The dashed black lines represent MSQ models.

93

cosφ
cosφ

1
0.9
0.8
0.7
0.6
0.5
0.4
TFD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]
1
0.9
0.8
0.7
0.6
0.5
0.4
UUD
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
MZ’ [TeV]

Figure 2.22: Allowed parameter space of the TFD and UUQ models at 95% CL in the
MZ − cφ plane after including indirect and direct constraints: EWPTs (green), Tevatron
(red) and LHC7 (blue).

94

2.6

Summary

We classified the G(221) models (including the left-right (LR), leptophobic (LP), hadrophobic (HP) and fermiophobic (FP) as well as the ununified (UU) and top-flavor(TF)) models
in a unified view in terms of the patterns of symmetry breaking and the gauge couplings
of fermions. We analyzed the constraints on the masses of the heavy gauge bosons of the
G(221) models. We performed a global-fit analysis about a set of 37 electroweak observables,
including Z pole data, the mass and the width of the W ± boson, and various low-energy
observables. Moreover, the direct search for the W and Z bosons at the Fermilab Tevatron
and CERN LHC could further constrain the G(221) model parameter space. The experimental precision with which these observables have been measured allows us to put strong
bounds on the parameter space of the G(221) models and to constrain the masses of the Z
and W ± bosons. We presented our key results in terms of 95% C.L. contours of the allowed
regions both on the MW -cos φ plane, as well as on the MZ -MW plane, from which we
can readily give the lower bounds on the masses of the W and Z , which can be used as a
guide for future collider searches.

95

Chapter 3
DISCOVERY OF HEAVY
CHARGED GAUGE BOSONS

3.1

Introduction

At the Large Hadron Collider (LHC), it is very promising to search for the heavy Z and
W bosons through their single production channel as an s-channel resonance with their
subsequent leptonic decays [65]. It yields the simplest event topology to discover the Z
and/or W with a large production rate and clean experimental signature. These channels
may be one of the most promising early discoveries at the LHC [48, 49, 54, 55]. There
are many theoretical studies of searching for the Z boson at the LHC [33, 62, 64, 63, 67].
However, the study the discovery potential of the W boson at the LHC has fared somewhat
less well [66]. In many NP models with extended gauge groups, the W boson emerges
together with the Z boson after symmetry breaking, and usually, the W boson is lighter
than, or equal to, the Z boson. It is therefore possible to discover the W prior to the Z .
96

More often, the masses of the W and Z bosons are not generally independent, and so as
their couplings to the SM fermions. Hence, the discovery potential of the W and Z at the
LHC could be highly correlated. In this paper we present a comprehensive study of discovery
potentials of both the W and Z bosons in the G(221) model at the LHC.

The G(221) model is the minimal extension of the SM gauge group to include both the
W and Z bosons. Its gauge structure is SU(2) ⊗ SU(2) ⊗ U(1) [11, 12, 13, 27, 25, 26,
28, 29, 31, 30, 32, 24]. The model can be viewed as the low energy effective theory of
many NP models with extended gauge structure when all the heavy particles other than the
W and Z decouple. In this chapter, based on a linearized effective theory including the
SU(2) × SU(2) × U(1) gauge group, we present the collider phenomenology related to the
simplest event topology in the resonance Z and W processes aimed at the early discovery.

In the TeV scale, different symmetry breaking patterns will induce different Z and
W mass relations. In breaking pattern I, which has the SU(2) ⊗ U(1) breaking down to
U(1)Y , the W mass is always smaller than the Z mass; while in breaking pattern II, the
SU(2) ⊗ SU(2) breaking down to SU(2)L requires the W and Z have the same mass at
tree level. This feature could assist us to distinguish these two breaking patterns after the
W and Z are discovered.
97

3.2

Drell-Yan Production and Decay at Next-to-Leading
Order

At the hadron collider, the total cross section can be written as the convolution of the parton
cross section and parton distribution functions, as follows,

σ=

fi/A (x1 )fj/B (x2 ) · σˆ (x1 x2 s),
ij

dx1 dx2

(3.1)

i,j
with s is the Lab energy of (A, B), and the c.m. energy of (a, b) is determined by s = x1 x2 s.
ˆ
τ
. With this variable change, the
One can also keep the x2 unintegrated, with x1 =
x2
Jacobian is
∂(τ, x)
=
∂(x1 , x2 )

x2 x1
0

= x2 .

(3.2)

1

So the integration will be
1
dτ
τ0
Taking into account the

1 dx
2.
τ x2

(3.3)

1
behavior of the hard-scattering processes as sσ (ˆ), the parton
ˆˆ s
s
ˆ

luminosity L is
σ=

1 dτ
· L(τ ) · [ˆσ (ˆ)]
sˆ s
τ0 τ

(3.4)

with
L(τ ) =

1 dx
1
τ
s 1 + δij τ x
ˆ

(fi (τ /x)fj (x) + fi (x)fj (τ /x)).

(3.5)

(a)
where i and j denote the initial state partons and fi (x) is the parton distribution of the
parton i inside the hadron a with a momentum fraction of x = pi /pa . The parton luminosity
98

has dimensions of a cross section, is a convenient measure of the reach of a collider of given
energy and hadron-hadron luminosity.
¯
At the LHC, the cross section of pp → V → f f (V = W /Z ) is

σ
¯ =
pp→V →f f

1 dτ
· Lij (τ ) · [ˆ σ
sˆ
s
¯ (ˆ)] ,
ij→V →f f
τ0 τ
{ij}

(3.6)

where τ ≡ s/s. The lower limit of the τ variable is determined by the kinematic threshold
ˆ
of the V production, i.e. τ0 = M 2 /s. Using the narrow width approximation (NWA) one
V
¯
can factorize the pp → V → f f process into the V production and the V decay,




1 d τ 1 d Lij


¯
σ
=
·
· [ˆ σ
sˆ
(ˆ)] × Br(V → f f ),
s
¯
pp→V →f f
ij→V
τ s dτ
τ
{ij} 0

(3.7)

¯
¯
where the branching ratio (Br) is defined as Br(V → f f ) = Γ(V → f f )/Γtot . As to be
shown later, the decay widths of Z and W bosons in most of the allowed parameter space
are much smaller than their masses, which validates the NWA adapted in this work.

3.2.1

Drell-Yan Production at the NLO

At the next-to-leading-order (NLO) the partonic cross section of the V production is



M2
π 2 2
2
σ
ˆ
(ˆ) = g2 (gL + gR ) Hij  V  ,
s
ij→V
6ˆ
s
s
ˆ
99

(3.8)

where the functions Hij (z) for different parton flavors ij = (¯q , qg, qg) are
q
¯

H

qq
¯

(z) = δ(1 − z)

αs
C
2π F

+

2(1 + z 2 )
2π 2
− 8 δ(1 − z) −
log(z)
3
1−z

+4(1 + z 2 )

log(1 − z)
,
1−z
+

(3.9)

and

Hqg (z) = Hq g (z) =
¯

αs
T
2π F

z 2 + (1 − z)2 log

(1 + z)2 1
7
+ + 3z − z 2 .
z
2
2

(3.10)

Here, CF and TF are the color factor defined as CF = 4/3 and TF = 1/2. The plus function
has the properties such as
1
dzf (z)
z0

log(1 − z)
=
1−z
+

1
dzf (z)
z0
−

1
z0

log(1 − z)
1−z

f (z)δ(1 − z)

1
dz
z0

log(1 − z )
,
1−z

(3.11)

where f (z) is a function including parton luminosity. In the soft limit z → 1 , the plus
function is still finite.

It is convenient to parametrize the V production cross section into one model-dependent
√
V
V
piece Cq and another model-independent piece Fq (M , s). The first piece consists of
V
model couplings, while the second piece, which includes all the hadronic contributions [62],
depends on m and
V

√

s only. We separate the up-quark and down-quark contributions in

the Z production because Z couples quite differently to up- and down-quarks in a few NP
100

models. The NLO cross sections of Z and W production then can be expressed as
√
√
π Z Z
Z Z
[Cu Fu (M , s) + Cd Fd (M , s)],
σ
=
V
V
pp→Z →f f
6
√
π W W
[Cq Fq (M , s)] ,
σ
=
V
pp→W →f f
6

(3.12)

where

V
Cq

2 2
2
= g2 fL + fR × Br(V → f f ),
(3.13)
1 dτ
√
V
q
· Lq q (τ ) · Hqq (z) + Lqg (τ ) · Hq g (z) + (¯ → q) , (3.14)
Fq (MV , s) =
¯
¯
¯
τ0 τ
1 dz
M2
M2
= M2
· Lq q (
) · H (z) + Lqg (
) · Hq g (z)
¯ zs
¯
qq
¯
z
zs
s
+(¯ → q)]
q
(3.15)

V
Note that the decay branching ratio is allocated to the model-dependent piece Cq . After
V
convoluting with PDFs, the model-independent piece Fq is merely a function of mV and
√
the collider energy s.

As the model dependent couplings can be factorized out, the total cross section in the
sequential W and Z models can be used as the reference cross section. The upper panels
of Fig. 3.1 show the LO and NLO production cross sections of the sequential W (left) and
Z boson (right) as a function of the extra gauge boson mass at the Tevatron, the 7 TeV
and 14 TeV LHC. The lower panels display the K-factor, defined as the ratio of the NLO
and LO cross sections. In the upper panels of Fig. 3.3 we plot the cross section of Z
¯
production induced by u¯ (left) and dd (right) initial state, respectively. Again, the lower
u
panels show the corresponding K-factors. Note that the K-factors are model independent
101

once one separates the up-quark and down-quark contributions in the Z production. The
K-factor is defined as

√
V
Fq (M , s)NLO
σNLO
V
.
Kq =
=
σLO
Vseq
√
(MV , s)LO
Fq

(3.16)

Here we adopt the CTEQ6.6M parton distribution package [68] for both the LO and NLO
calculations. Both the factorization and renormalization scales are set to be M .
V
The NLO cross section of other NP models can be obtained easily from the sequential
W and Z cross sections plotted in Figs. 3.1, 3.2 and 3.3, 3.4 by:
Zseq Z
Zseq W
Wseq
Z
• scaling the model-dependent C V -coefficients (Cu /Cu
, Cd /C
, Cq /Cq
),
d
• including the NLO QCD correction with the inclusive K-factors (Ku , Kd and Kq ).
To be more specific, the NLO cross sections of new gauge boson productions in the G(221)
model are

σ
W

σ
Z

3.2.2

=

W
Cq
Wseq
Cq

Z
Cu
=
Zseq
Cu

W
Fq

Z
Fu

LO

LO

× Kq ,

× Ku +

Z
Cd
Zseq
C
d

Z
Fd

LO

× Kd .

(3.17)

V Decay

In the G(221) model the W and Z bosons can decay into SM fermions, gauge bosons, or
a pair of SM gauge boson and Higgs boson. In this subsection we give detailed formula of
partial decay widths of the extra gauge bosons.
102

total rate σW’ [pb]

105

LHC 14 TeV
LHC 14 TeV NLO
LHC 7 TeV
LHC 7 TeV NLO
Tevatron
Tevatron NLO

10

K factor

10-3

1.6
1.4
1.2
1

0

1

2
3
MW’ [TeV]

4

5

Figure 3.1: Upper panel: the LO and NLO cross sections of pp → W process with a SM like
coupling as a function of new heavy gauge boson mass (mW ) in hadron collisions. Lower
panel: the K-factor as a function of m .
W

103

total rate σZ’ [pb]

105

LHC 14 TeV
LHC 14 TeV NLO
LHC 7 TeV
LHC 7 TeV NLO
Tevatron
Tevatron NLO

10

K factor

10-3

1.6
1.4
1.2
1

0

1

2
3
MZ’ [TeV]

4

5

Figure 3.2: Upper panel: the LO and NLO cross sections of pp → Z process with a SM like
coupling as a function of new heavy gauge boson mass (mZ ) in hadron collisions. Lower
panel: the K-factor as a function of m .
Z

104

total rate σZ’(u) [pb]

105
LHC 14 TeV
LHC 14 TeV NLO
LHC 7 TeV
LHC 7 TeV NLO
Tevatron
Tevatron NLO

10

K factor

10-3

1.6
1.4
1.2
1

0

1

2
3
MZ’ [TeV]

4

5

Figure 3.3: Upper panel: the LO and NLO cross sections of pp → Z process with a SM-like
couplings as a function of mZ in hadron collision: induced by up-type quark initial state
Lower panel: the K-factor as a function of m .
Z

105

total rate σZ’(d) [pb]

105
LHC 14 TeV
LHC 14 TeV NLO
LHC 7 TeV
LHC 7 TeV NLO
Tevatron
Tevatron NLO

10

K factor

10-3

1.6
1.4
1.2
1

0

1

2
3
MZ’ [TeV]

4

5

Figure 3.4: Upper panel: the LO and NLO cross sections of pp → Z process with a SM-like
couplings as a function of mZ in hadron collision: induced by down-type quark initial state.
Lower panel: the K-factor as a function of m .
Z

106

¯
First, consider the fermionic mode. The decay width of V → f1 f2 is

Γ

¯
V →f1 f2

=

mf mf
MV
2
2
β0 (gL + gR )β1 + 6gL gR 1 2 Θ(M − mf − mf ) , (3.18)
V
1
2
24π
MV

where
2 2
2
m2 + m2
f2 (mf1 − mf2 )
f1
+
,
1−2
β0 =
M2
M4
V
V
2 + m2
2 − m2 )2
mf
(mf
f2
f2
1
1
β1 = 1 −
−
.
2
4
2M
2M
V
V

(3.19)

Note that the color factor is not included in Eq. 3.18 and the third generation quark decay
channel opens only for a heavy Z and W .

Second, consider the bosonic decay mode, e.g. W and Z decay to gauge bosons and
Higgs bosons. Such decay modes are induced by gauge interactions between the extra
gauge boson and the SM gauge boson after symmetry breaking. Even though the couplings
g

and g
are suppressed by the gauge boson mixing term 1/x, the bosonic
V V1 V2
V V1 H
decay channel could be the major decay channel in certain models, e.g. fermo-phobic model
in which the extra gauge boson does not couple to fermions at all.

The decay width of V → V1 V2 is
M5
V
Γ
=
g2
β 3 β Θ(M − MV − MV ) ,
V →V1 V2 192πM 2 M 2 V V1 V2 0 1
V
1
2
V1 V2
107

(3.20)

where
2
2
2
2
(MV − MV )2
MV + MV
2 ,
2 +
1
1
β0 =
1−2
2
4
M
M
V
V
4 + 10M 2 M 2 + M 4
2
2
MV 1 + MV 2 MV1
V 2 V2
V1
+
.
β1 = 1 + 10
2M 2
M4
V
V

(3.21)

The width of V → V1 H (where V1 = W or Z boson and H is the lightest Higgs boson) is
2
MV gV V1 H
Γ
=
β0 β1 Θ(M − MV − MV ) ,
V →V1 H
V
1
2
192π M 2
V1

(3.22)

where
2
2
(MV − m2 )2
MV + m2
H
H
1
1
β0 =
+
,
1−2
M2
M4
V
V
2 − m2 )2
2 − 2m2
(MV
10MV
H
H
1
1
+
.
β1 = 1 +
2
4
2M
M
V
V

The couplings g

V V1 V2

and g

V V1 H

(3.23)

for various models are listed in Table II for reference.

In this study only left-handed neutrinos are considered while the possible right-handed neutrinos are assumed to be very heavy. In addition we also assume all the heavy Higgs bosons,
except the SM-like Higgs boson, decouple from the TeV scale. As a result, the total decay
width of the W boson is

Γ

W ,tot

= 3Γ

W →¯ν
e

+2NC Γ
+NC Γ
¯ +ΓW →W Z ++ΓW →W H , (3.24)
W →¯d
u
W →tb
108

while the width of the Z boson is

Γ

Z ,tot

= 3Γ

Z →¯e
e

+ 3Γ

Z →¯ν
ν

+ 2NC Γ

Z →¯u
u

+ 3NC Γ

¯
Z →dd

+ NC Γ
¯ + ΓZ →W W + ΓZ →ZH ,
Z →tt

(3.25)

where NC = 3 originates from summation of all possible color quantum number.

3.3
3.3.1

Event Simulation and Search Limits
Signal and Backgrounds

At hadron colliders, searches for a heavy resonance decaying to a charged lepton and missing
transverse energy are always a primary task because of their simple final states. In the final
states, the charged lepton can be electron or muon. The tau lepton is not considered here,
because of the complicated hadronic decays of the tau lepton. The neutrino in the final
states results in missing transverse energy in the detector.
One can not reconstruct the invariant mass of the final states, due to the missing transverse energy in the final states. The kinematic variable reconstructed instead is the transverse
mass

MT =

(E + E miss )2 − (p + pmis )2 =
T
T
T
T

2pT E miss (1 − cos ϕlν ),
T

(3.26)

miss is the magnitude of the missing transwhere pT is the lepton transverse momentum, ET
miss vectors. The
miss
verse momentum (ET ), and ϕlν is the angle between the pT and ET
transverse mass distribution displays a Jacobian peak that falls sharply above the resonance
109

mass, as can be seen by the differential cross section

dσ d cos θ
dσ
dσ
=
=
dMT
d cos θ dMT
d cos θ 2M
W

MT
,
M2 − M2
T
W

(3.27)

where θ is the polar decay angle in the center of mass (COM) frame of the system. The
Jacobian peak will become broad when the width of the W boson and the effect of detector
smearing are taken into account. Since there is a relation between pT and the polar angle θ
in the COM frame with COM energy s,
ˆ
√
pT =

s
ˆ
sin θ
2

(3.28)

a similar Jacobian peak will show up in the transverse momentum distribution of the final
state lepton, as can be seen by the differential cross section

dσ
dp

T

=

dσ d cos θ
dσ
=
d cos θ dp
d cos θ
T

where the peak position is at pT =

2pT
M
W

M2
W −p 2
T
4

,

(3.29)

M
W . Measuring the Jacobian peak is complicated
2

when the width of the W boson and the effect of detector smearing are taken into account.
From the above kinematic consideration, one would like to consider three variables, pT , MT ,
miss
and ET , to describe the characteristic feature of this process.
e
We choose the benchmark point as 1500 GeV mass, and coupling fL = √
and
2 sin θW
fR = 0. The total decay width is Γ
= 51 GeV. Other couplings can be scaled using the
W
scaling formula discussed in above. The CTEQ6L1 parton distribution functions are used in
110

our calculation with the renormalization and factorization scales chosen as m . The final
W
state of this channel is
pp → W → ν.

(3.30)

For the signal, we choose the renormalization and factorization scales as the W mass M .
W
The following SM backgrounds are considered:

p p → W → ν,

main background

(3.31)

p p → Z/γ ∗ → ¯,

(3.32)

p p → V V → X,

(3.33)

¯
p p → tt → X,

(3.34)

and, of course, the QCD background estimated from experimental data. The main background comes from the high energy tail of the SM W decays to the same lepton and missing
energy. After basic event selection, other backgrounds than W decays are only estimated
miss distributions for the signal are shown in Figs. 3.5
to be less than 10%. The pT , and ET
and 3.6. The MT distribution for the signal is shown in the left panel of the Fig. 3.7.

For the purpose of our study, we will only simulate the signal process using the benchmark
point as a template to justify the simulation procedure. Then we will adopt the event
simulations at the Tevatron and LHC for both signal and backgrounds to reproduce the
search limits on the W resonance. To illustrate the procedure, we summarize the results at
the LHC with the energy 7 TeV and the luminosity 1 fb−1 using the ATLAS detector. We
refer to the ATLAS paper in ref. [54] for all the details.
111

Event Weight

0.015
Wprime (1.5 TeV)

0.01

0.005

0
0

200

400

600

800
1000
p (lepton)
T

Figure 3.5: The pT distribution for 1500 GeV W in the charged lepton and missing transverse energy final states.

112

Event Weight

0.015
Wprime (1.5 TeV)

0.01

0.005

0
0

200

400

600

800
1000
ET (missing)

miss distribution for 1500 GeV W in the charged lepton and missing
Figure 3.6: The ET
transverse energy final states.

113

Event Weight

0.015
Wprime (1.5 TeV)

0.01

0.005

0
0

500

1000

1500
2000
MT (W-prime)

Figure 3.7: The MT distribution for 1500 GeV W in the charged lepton and missing transverse energy final states is shown.

114

3.3.2

Object Reconstructions and Event Selections at the LHC

The ATLAS detector has three major components: the inner tracking detector, the calorimeter and the muon spectrometer. Events are required to have their primary vertex reconstructed from at least three tracks with pT > 0.4 GeV and longitudinal distance less than
200 mm from the center of the collision region.
Events are required to have exactly one candidate electron or one candidate muon satisfying the requirements described below:
• Electron candidate:
– tracks with pT > 20 GeV
– electromagnetic compartment of calorimeter with ET > 25 GeV and |η| < 1.37
or 1.52 < |η| < 2.47 if matches with an inner detector track;
– The resolution of the energy measurement is 2% for ET ≈ 50 GeV and approaches
1% in the high-ET range relevant to this analysis.
– the isolation energy is measured with the calorimeter in a cone ∆R < 0.4 (∆R ≡
(∆η)2 + (∆ϕ)2 ) around the electron track, and the requirement is

ET < 9 GeV,

where the sum includes all calorimeter energy clusters in the cone excluding the
core energy deposited by the electron.
• Muon candidates:
– matching tracks in the muon spectrometer and inner detector with combined
pT > 22 GeV. Muon tracks can be reconstructed independently in both the
inner detector and muon spectrometer, and the muons used in this study are
required to have matching tracks in both systems. The muons are required to
115

have pT > 25 GeV, where the momentum of the muon is obtained by combining
the inner detector and muon spectrometer measurements.
– approximately |η| < 1.0 and 1.3 < |η| < 2.0.
– The average momentum resolution is currently about 15% at pT = 1 TeV.
– the isolation energy is measured using inner detector tracks with ptrack > 1 GeV
T
in a cone ∆R < 0.3 around the muon track.

The isolation requirement is

ptrack < 0.05 pT , where the muon track is excluded from the sum.
T
miss in the electron channel is obtained from a vector sum over calorimeter cells
The ET
miss = E miss (calo).
ET
T

(3.35)

Muons only deposit a small fraction of their energy in the calorimeter, and so, in the muon
miss is obtained from
channel, the ET
miss = E miss (calo) − p (µ).
ET
T
T

(3.36)

The second term in this vector sum subtracts the muon transverse momentum. The missing
miss
miss > 25GeV in both channels. In the electron
ET requirements are imposed to be ET
channel, to avoid the misidentification of hadronic jets as a lepton, a tighter cut is imposed
miss > 0.6 p .
with ET
T
After the lepton reconstruction and the basic event selections, one will notice that the
MT distributions, shown in the right panel of the Fig. 3.7, accumulate more data for the
signal than the backgrounds in the high-MT tails. To search for a W with a mass of 1500
116

eν

µν

92.248

92.248

0.541 ± 0.026

0.347 ± 0.024

Nsig

49.6 ± 6.0

34.4 ± 4.4

W → lν

1.59

Z → ll

0.00010 ± 0.00004

diboson

0.08

± 0.08

0.11 ± 0.08

¯
tt

0.08

± 0.08

0

QCD

0

+0.17
−0

0.01 +0.02
−0.01

Total Nbg

1.75

+0.24
−0.18

1.57 ± 0.15

Nobs

2

σBL
sig

± 0.13

1.36 ± 0.13
0.095 ± 0.005

2

Table 3.1: Expected and observed numbers of events in 1.04 fb−1 in ref. [54] from the various
background sources in each decay channel for MT > 891 GeV, the region used to search for
a W with a mass of 1500 GeV. In the table, the signal selection efficiency, sig , and the
prediction for the number of signal events, Nsig , obtained with this efficiency. The last two
columns are the expected number of background events, Nbg , and the number of events
observed in data, Nobs . The uncertainties are statistical.

GeV, the hard cut on MT is required to be

MT > 891 GeV,

for M
= 1500 GeV.
W

(3.37)

The rate for signal and backgrounds are listed in the Table 3.1. We notice that after all the
cuts, the significance is large in the simulations. For the MT cut imposed for other masses
of W , we refer to the ATLAS paper [54] for details.
117

3.3.3

Upper Limits on W masses at the LHC

At the current LHC, since there is no excess on searching for the heavy resonance in the
charged lepton and missing transverse energy final states, one would like to set the limits
on this heavy resonance in this final states. Based on the expected and observed numbers
of events after all the cuts, the Beyasian or frequentist analysis is used to set the limits at
the 95% confidence level (CL). Here we will take the W signal with 1500 GeV mass at the
LHC as an example to show how to calculate the upper limit. From the Table 3.1, we can
read out the signal selection efficiency sig after all the cuts, the simulated numbers of signal
Nsig , the simulated numbers of background Nbg , and the observed numbers Nobs . Then
the expected number of events is

Nexp = sig Lint σB + Nbg ,

(3.38)

where σ is the production cross section, B is the decay branching ratio, and Lint is the
integrated luminosity of the data sample. Using Poisson statistics, the likelihood to observe
Nobs events is
( sig Lint σB + Nbg )Nobs −(
sig Lint σB+Nbg ) .
e
L(Nobs |σB) =
Nobs !

(3.39)

Taking a flat prior probability density, the posterior probability can be obtained. From the
appendix, we know that the 95% CL upper limit σ95%CL , is given by
(σB)95%CL
0

P(σB|Nobs )d(σB) = 95%.
118

(3.40)

If one don’t take the uncertainties of signal and backgrounds into account, the upper limit
has a simple formula to calculate for a single channel,

1−

Γ(Nobs + 1, (σB)95%CL sig Lint )
= 95%.
Γ(Nobs + 1, Nbg )

(3.41)

Using the numbers in the Table 3.1, we obtain the upper limit at the 95% CL is

(σB)up = 8.79184 fb
(σB)up = 12.968 fb

in the electron channel ,
in the muon channel ,

(3.42)
(3.43)

which are the same as the results in the ATLAS paper [54]. In principle, one can calculate
the combined limit using the equations in appendix. Here we will only use an approximate
formula in the Gaussian limit:

1

µ2 =
up

.

(3.44)

(σB)up = 7.277 fb,

(3.45)

N 1
i µ2
iup

The combined limit is

which is still close to the results in the ATLAS paper [54].

3.4

Discovery Potential and Significance

In the early LHC7 experiment, the combined constraints from current direct searches and
indirect EWPTs play the crucial role to specify the unexplored parameter space. Given the
119

allowed parameter space discussed in the previous chapter, we are able to predict the time
scale to probe the W and Z at 5σ discovery evidence. We are able to provide the following
information:
• The integrated luminosities, with which the LHC can discover the W and/or Z for
certain masses in different G(221) models.
• The region of the parameter space that could be accessed for different luminosities and
energies in the LHC run.
• The possibility to identify different models in our classification once the W and/or Z
are discovered.
To be specific, we consider two different scenarios: an early run with

√
s = 7 TeV and

√
an integrated luminosity of 5.61 fb−1 ; a long run with s = 14 TeV and O(103 ) fb−1
integrated luminosity is expected finally.
To get the expected luminosity contour, one has to calculate the signal and background
cross sections at LHC7 and LHC14 for each point in the parameter space of the models.
In principle, the complete Monte Carlo simulations for the signal and background including
efficiency analysis in the G(221) models have to be used to obtain the needed luminosity
for the discovery or exclusion at 7 TeV and 14 TeV. However, in the Drell-Yan production
process, all the model-independent effects, including the kinematic cuts, can be factorized out
from the model-dependent part, which only depends on the gauge couplings and branching
ratios. Therefore, the simulation on one benchmark model, such as the sequential W and Z
model, can provide the needed luminosity information for the other models. At the LHC7,
the complete simulation on the signal and backgrounds including detection efficiency have
120

been done in Refs. [54, 55]. At the LHC14, the ATLAS TDR [58] have done the detailed
studies on the discovery potentials for the sequential W and Z model. The luminosity
needed for other new physics models can be obtained by properly scaling the luminosity
obtained for the sequential model.
Here we summarize the event analysis procedures at the current LHC and in the ATLAS
TDR. At the LHC7, we adopt the ATLAS simulation and analysis with integrated luminosity
at about 1 fb−1 . Both electron and muon channels are considered in both W and Z
searches. For the W searches, the missing energy in both channels are required to be above
the threshold energy of 25 GeV. Furthermore, the cut on the transverse mass of the lepton
and missing energy system varies as the W mass increases. For more detailed information,
please refer to Refs. [54, 55]. In the ATLAS TDR, for the sequential W , the simulation on
the lepton plus missing transverse energy signal at high mass region is performed. We list
the event selection and cut-based analysis as follows:
• Events are required to have exactly one reconstructed lepton with pT > 50 GeV within
|η| < 0.25, and isolated from jets with ∆R j = 0.5;
• The lepton reconstruction is smeared by σ(1/pT ) = 0.011/pT ⊗ 0.00017, while the jet
resolution is taken as σ(ET ) = 0.45 ×

ET ⊗ 5%;

mis
• Missing transverse energy ET > 50 GeV;
¯
• To reduce the di-jet and tt backgrounds, a lepton fraction is required to be

pT /

pT +

;
121

ET

> 0.5

(3.46)

• Transverse mass mT =

2pT E mis (1 − cos ∆φ) > 0.7 × M , where ∆φ is the angle
T
W

between the momentum of the lepton and the missing momentum.

For the sequential Z , we list the event selection and analysis on the di-lepton final states as
follows:

• Events are required to have exactly two reconstructed same-flavor opposite-charged
leptons with at least one lepton pT > 30 GeV, within |η| < 0.25;

• Di-lepton invariant mass window |m

−M |<4×Γ .
Z
Z

Next we explore the LHC sensitivity to W and Z bosons. We can quantify the sensitivity
to new physics discovery or set exclusion limits on it based on statistics. Specifically, for the
case of discovery we would like to know the statistical significance (S) for discovery, which
characterizes the inconsistency of the experiment data with a background-only hypothesis.
If there is no discovery at a given luminosity, we set exclusion limits on new physics. In the
counting experiments, suppose one has an experiment that counts n events, modeled as a
Poisson distribution with mean s + b, where s is the expected signal rate, b is the expected
background rate. The probability of measuring n events is therefore

P (n|s, b) =

(s + b)n −(s+b)
e
.
n!

(3.47)

Using a profile likelihood ratio as the test statistic, the expected significance is obtained as
follows [117]
S=

2 ((s + b) ln(1 + s/b) − s) .
122

(3.48)

For sufficiently large b we can expand the logarithm in s/b and obtain the widely-used
significance formula
s
S = √ (1 + O(s/b)) .
b

(3.49)

In addition to establishing discovery by rejecting the background hypothesis, we can consider
the signal hypothesis as well. It is common to use confidence level (CL) α and the related
p-value to quantify the level of incompatibility of data with a signal hypothesis. The profile
likelihood ratio qµ is used as the test statistic [117]. For a sufficiently large data sample the
probability density of qµ takes on a well defined χ2 distribution with mean µ and variance
ˆ
σ for one degree of freedom. Given the p-value for each number of signal events s, we can
ˆ
obtain the upper limit sup on the number of signal events,

sup = µ + σ Φ−1 (1 − α) ,
ˆ ˆ

(3.50)

√
where the mean and variance of the χ2 distribution are µ = n − b, and σ = b for large
ˆ
ˆ
data sample, and Φ is the cumulative distribution of the standard Gaussian with zero mean
and unit variance. For the expected upper limit, in which the data count is taken as the
background sum, the upper limit at confidence level α = 95% is
√
√
sup = Φ−1 (0.05) · b = 1.64 × b .

(3.51)

So for a sufficiently large data sample the equivalent significance Z for excluding a signal
hypothesis is given by
sup
Z = √ = 1.64.
b

123

(3.52)

For instance, when expressing the significance for 5σ discovery with the exclusion upper limit
at the 95% CL, a factor S/Z = 5/1.64

3 needs to be applied.

Denoting by σs (σb ) the inclusive cross section of the signal (background), s ( b ) the
cut acceptance of the signal (background), and L the integrated luminosity, the number of
signal (background) events can be written as

s = σs s L ,

(3.53)

b = σb b L .

(3.54)

For a sufficiently large data sample, both S and Z have a scaling behavior on the integrated
luminosity,
S

Z

√
σ
s
√ = √ s s × L.
σb b
b

(3.55)

Figures 3.8, 3.9, 3.10, 3.11, 3.12, 3.13 displays the 5σ discovery potential (fb−1 ) for
LHC7 via W leptonic decay channel, and current combined constraints are within solid
black contour. The LRD(T) and MLR models can be further constrained when the integrated luminosity for LHC7 reaches its maximum 5.6 fb−1 . However, the other models
need much more luminosity, which even exceeds the total integrated luminosity (5.6 fb−1 )
at LHC7. Therefore, the W leptonic decay channel cannot make further contributions
to discovering these G(221) models, except in some small region in LRD(T) and MLR. In
Figs. 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, it shows that the EWPTs constraints are stronger than those
from the Tevatron and the LHC7, except LRD(T) and MLR. This means that compared to
EWPTs, the LHC7 direct search via the W leptonic decay channel for the new physics
models with G(221) gauge group structure can put further constraint only on LRD(T) and
124

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

LRD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MW’ [TeV]

4

5

10-1
104
103
102

LRT

10
1
1

2
3
MW’ [TeV]

4

5

10-1

Figure 3.8: 5σ Discovery potential (fb−1 ) in the LRD and LRT models for different luminosity at LHC7 via W leptonic decay channel. The current combined constraints are within
solid black contour. The dashed black lines in LRD represent MLR models.

125

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

LPD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MW’ [TeV]

4

5

10-1
104
103
102

LPT

10
1
1

2
3
MW’ [TeV]

4

5

10-1

Figure 3.9: 5σ Discovery potential (fb−1 ) in the LPD and LPT models for different luminosity at LHC7 via W leptonic decay channel. The current combined constraints are within
solid black contour.

126

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

HPD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MW’ [TeV]

4

5

10-1
104
103
102

HPT

10
1
1

2
3
MW’ [TeV]

4

5

10-1

Figure 3.10: 5σ Discovery potential (fb−1 ) in the HPD and HPT models for different luminosity at LHC7 via W leptonic decay channel. The current combined constraints are within
solid black contour.

127

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

FPD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MW’ [TeV]

4

5

10-1
104
103
102

FPT

10
1
1

2
3
MW’ [TeV]

4

5

10-1

Figure 3.11: 5σ Discovery potential (fb−1 ) in the FPD and FPT models for different luminosity at LHC7 via W leptonic decay channel. The current combined constraints are within
solid black contour.

128

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102

SQD

10
1
1

2
3
MW’ [TeV]

4

5

10-1

Figure 3.12: 5σ Discovery potential (fb−1 ) in the SQD models for different luminosity at
LHC7 via W leptonic decay channel. The current combined constraints are within solid
black contour. The dashed black lines in SQD represent MSQ models.

129

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

TFD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MW’ [TeV]

4

5

10-1
104
103
102

UUD

10
1
1

2
3
MW’ [TeV]

4

5

10-1

Figure 3.13: 5σ Discovery potential (fb−1 ) in the TFD and UUD models for different luminosity at LHC7 via W leptonic decay channel. The current combined constraints are within
solid black contour.

130

MLR. For the other models, the direct search at LHC7 for s-channel W production with
leptonic decay cannot compete with seeking for deviation from SM predictions via EWPTs.
Figures 3.14, 3.15, 3.16, 3.17, 3.18, 3.19 shows the 5 σ discovery potential (fb−1 ) shows
the 5σ discovery potential (fb−1 ) at the LHC7 via the Z leptonic decay channel, and the
current combined constraints are within solid black contour. We can see that for LRD(T),
SQD, TFD, UUD, MLR and MSQ, further discovery via the Z leptonic decay channel needs
more than 100 fb−1 , which is definitely far beyond the total integrated luminosity before
LHC switches away from 7 TeV. However, some corner of the parameter space of LPD(T),
HPD(T) and FPD(T) can be further tested when LHC7 reaches 5.6 fb−1 . Especially, for
HPD(T) and FPD(T), there are small regions where Z can be discovered with a few fb−1
luminosity or these parameters can be excluded with less than one fb−1 luminosity. At
the LHC7, the Z leptonic decay channel is more efficient than EWPTs on constraining
the parameter regions in the LPD(T), HPD(T) and FPD(T). For LRD(T), LPD(T) and
HPD(T), EWPTs are more sensitive to the large cφ region, where LHC7 cannot compete
with EWPTs. For SQD, TFD and UUD, both W and Z leptonic decay channel at LHC7
cannot help further, because the constraint from EWPTs for UUD is much stronger than
Tevatron or LHC7 data, as shown in Figs. 2.5, 2.6, 2.7, 2.8, 2.9, 2.10.
Figures 3.20, 3.21, 3.22, 3.23, 3.24, 3.25 presents the 5σ discovery potential (fb−1 ) at
the LHC14 via the W leptonic decay channel, and current constraints are within the solid
black contour. After the LHC14 collects 10 fb−1 , a sizable region of parameter space will
be further tested, except for all the phobic models, LPD(T), HPD(T) and FPD(T). For the
phobic models, very large integrated luminosity is needed to have 5σ discovery because of
the small total cross section in the W leptonic decay channel, which is either suppressed by
131

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

LRD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MZ’ [TeV]

4

5

10-1
104
103
102

LRT

10
1
1

2
3
MZ’ [TeV]

4

5

10-1

Figure 3.14: 5σ Discovery potential (fb−1 ) in the LRD and LRT models for different luminosity at LHC7 via Z leptonic decay channel. The current combined constraints are within
solid black contour. The dashed black lines in LRD represent MLR models.

132

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

LPD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MZ’ [TeV]

4

5

10-1
104
103
102

LPT

10
1
1

2
3
MZ’ [TeV]

4

5

10-1

Figure 3.15: 5σ Discovery potential (fb−1 ) in the LPD and LPT models for different luminosity at LHC7 via Z leptonic decay channel. The current combined constraints are within
solid black contour.

133

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

HPD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MZ’ [TeV]

4

5

10-1
104
103
102

HPT

10
1
1

2
3
MZ’ [TeV]

4

5

10-1

Figure 3.16: 5σ Discovery potential (fb−1 ) in the HPD and HPT models for different luminosity at LHC7 via Z leptonic decay channel. The current combined constraints are within
solid black contour.

134

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

FPD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MZ’ [TeV]

4

5

10-1
104
103
102

FPT

10
1
1

2
3
MZ’ [TeV]

4

5

10-1

Figure 3.17: 5σ Discovery potential (fb−1 ) in the FPD and FPT models for different luminosity at LHC7 via Z leptonic decay channel. The current combined constraints are within
solid black contour.

135

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102

SQD

10
1
1

2
3
MZ’ [TeV]

4

5

10-1

Figure 3.18: 5σ Discovery potential (fb−1 ) in the SQD models for different luminosity at
LHC7 via Z leptonic decay channel. The current combined constraints are within solid
black contour. The dashed black lines in SQD represent MSQ models.

136

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103

TFD

10
1

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

102

2
3
MZ’ [TeV]

4

5

10-1
104
103
102

UUD

10
1
1

2
3
MZ’ [TeV]

4

5

10-1

Figure 3.19: 5σ Discovery potential (fb−1 ) in the TFD and UUD models for different luminosity at LHC7 via Z leptonic decay channel. The current combined constraints are within
solid black contour.

137

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

LRD

2
3
MW’ [TeV]

4

5

10-3
104
103
102
10

LRT

1
10-1
10-2
1

2
3
MW’ [TeV]

4

5

10-3

Figure 3.20: 5σ Discovery potential (fb−1 ) in the LRD and LRT models for different luminosity at LHC14 via W leptonic decay channel. The current combined constraints are
within solid black contour. The dashed black lines in LRD represent MLR models.

138

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

LPD

2
3
MW’ [TeV]

4

5

10-3
104
103
102
10

LPT

1
10-1
10-2
1

2
3
MW’ [TeV]

4

5

10-3

Figure 3.21: 5σ Discovery potential (fb−1 ) in the LPD and LPT models for different luminosity at LHC14 via W leptonic decay channel. The current combined constraints are
within solid black contour.

139

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

HPD

2
3
MW’ [TeV]

4

5

10-3
104
103
102
10

HPT

1
10-1
10-2
1

2
3
MW’ [TeV]

4

5

10-3

Figure 3.22: 5σ Discovery potential (fb−1 ) in the HPD and HPT models for different luminosity at LHC14 via W leptonic decay channel. The current combined constraints are
within solid black contour.

140

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

FPD

2
3
MW’ [TeV]

4

5

10-3
104
103
102
10

FPT

1
10-1
10-2
1

2
3
MW’ [TeV]

4

5

10-3

Figure 3.23: 5σ Discovery potential (fb−1 ) in the FPD and FPT models for different luminosity at LHC14 via W leptonic decay channel. The current combined constraints are
within solid black contour.

141

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
10

SQD

1
10-1
10-2
1

2
3
MW’ [TeV]

4

5

10-3

Figure 3.24: 5σ Discovery potential (fb−1 ) in the SQD models for different luminosity at
LHC14 via W leptonic decay channel. The current combined constraints are within solid
black contour. The dashed black lines in SQD represent MSQ models.

142

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

TFD

2
3
MW’ [TeV]

4

5

10-3
104
103
102
10

UUD

1
10-1
10-2
1

2
3
MW’ [TeV]

4

5

10-3

Figure 3.25: 5σ Discovery potential (fb−1 ) in the TFD and UUD models for different luminosity at LHC14 via W leptonic decay channel. The current combined constraints are
within solid black contour.

143

the production rate of the W , such as HPD(T) and FPD(T), or suppressed by the decay
branching ratio, such as LPD(T) and FPD(T). With a 10 fb−1 luminosity, for LRD(T), the
discovery potential for W mass can reach more than 3 TeV, and the W mass discovery for
MLR can reach more than 4 TeV. Furthermore, for the large cφ region in LRD(T), LHC14
search via W leptonic decay channel can easily probe the large M
region with several
W
fb−1 . In BP-II, the current constraints already pushed the W to the large mass region.
However, with a 10 fb−1 integrated luminosity, for SQD, TFD and UUD models, most of the
allowed region below 5 TeV W mass can be further tested. For relatively small cφ in SQD
and TFD, a few fb−1 luminosity can even probe W boson beyond 5 TeV. When the LHC is
upgraded to 14 TeV, SQD, TFD and UUD can be further tested, exploring the region where
current constraints cannot reach. This shows that the capability of LHC14 is far beyond
LHC7. However, even LHC14 cannot test all the phobic models, such as LPD(T), HPD(T)
and FPD(T), via only W leptonic decay channel.

Figures 3.26, 3.27, 3.28, 3.29, 3.30, 3.31 shows the 5 σ discovery potential (fb−1 ) for
the LHC14 via the Z leptonic decay channel, and current combined constraints are within
solid black contour. For the models other than LRD(T), UUD and MLR, the LHC14 can
already test the parameter space effectively with the integrated luminosity less than 1 fb−1 .
However, for the FPD(T), SQD and TFD models, EWPTs are more sensitive to the large
cφ region. Also if the luminosity can reach 10 fb−1 , we can test a large parameter space
region, where we can either discover new physics based on these models or constrain the
parameters in the relevant region. For LRD(T), UUD and MLR, when integrated luminosity
is accumulated to more than 100 fb−1 , LHC14 data can have sizable parameter space further
tested up to even beyond 5 TeV MZ . For LRD(T) Z leptonic decay channel is less effective
144

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

LRD

2
3
MZ’ [TeV]

4

5

10-3
104
103
102
10

LRT

1
10-1
10-2
1

2
3
MZ’ [TeV]

4

5

10-3

Figure 3.26: 5σ Discovery potential (fb−1 ) in the LRD and LRT models for different luminosity at LHC14 via Z leptonic decay channel. The current combined constraints are
within solid black contour. The dashed black lines in LRD represent MLR models.

145

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

LPD

2
3
MZ’ [TeV]

4

5

10-3
104
103
102
10

LPT

1
10-1
10-2
1

2
3
MZ’ [TeV]

4

5

10-3

Figure 3.27: 5σ Discovery potential (fb−1 ) in the LPD and LPT models for different luminosity at LHC14 via Z leptonic decay channel. The current combined constraints are
within solid black contour.

146

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

HPD

2
3
MZ’ [TeV]

4

5

10-3
104
103
102
10

HPT

1
10-1
10-2
1

2
3
MZ’ [TeV]

4

5

10-3

Figure 3.28: 5σ Discovery potential (fb−1 ) in the HPD and HPT models for different luminosity at LHC14 via Z leptonic decay channel. The current combined constraints are
within solid black contour.

147

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

FPD

2
3
MZ’ [TeV]

4

5

10-3
104
103
102
10

FPT

1
10-1
10-2
1

2
3
MZ’ [TeV]

4

5

10-3

Figure 3.29: 5σ Discovery potential (fb−1 ) in the FPD and FPT models for different luminosity at LHC14 via Z leptonic decay channel. The current combined constraints are
within solid black contour.

148

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
10

SQD

1
10-1
10-2
1

2
3
MZ’ [TeV]

4

5

10-3

Figure 3.30: 5σ Discovery potential (fb−1 ) in the SQD models for different luminosity at
LHC14 via Z leptonic decay channel. The current combined constraints are within solid
black contour. The dashed black lines in SQD represent MSQ models.

149

cosφ

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

104
103
102
1
10-1
10-2

cosφ

1

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

10

TFD

2
3
MZ’ [TeV]

4

5

10-3
104
103
102
10

UUD

1
10-1
10-2
1

2
3
MZ’ [TeV]

4

5

10-3

Figure 3.31: 5σ Discovery potential (fb−1 ) in the TFD and UUD models for different luminosity at LHC14 via Z leptonic decay channel. The current combined constraints are
within solid black contour.

150

than W channel. However, for the phobic models, such as LPD(T), FPD(T) and HPD(T),
there is no O(1/x) suppression on the couplings of Z to fermions, unlike the couplings
of W to fermions. So Z leptonic decay channel is much more effective than W for the
investigation based on the LHC14 data. Especially, for the small cφ region, a few pb−1
luminosity can probe very large M . In the phobic models, observing a Z alone cannot
Z
rule out the possibility of a non-Abelian gauge extension of new physics.
In BP-II, both Z and W leptonic decay channel are suitable to explore the allowed
parameter space of the models. Since the mass of W and Z are degenerate in BP-II,
discovering degenerate W and Z in the leptonic decay channels at the same time will
be the distinct feature compared to the models in BP-I. Compared to the LHC7 discovery potentials in Figs. 3.8, 3.9, 3.10, 3.11, 3.12, 3.13 and 3.14, 3.15, 3.16, 3.17, 3.18, 3.19.
Figs. 3.20, 3.21, 3.22, 3.23, 3.24, 3.25 and 3.26, 3.27, 3.28, 3.29, 3.30, 3.31 show that for LHC
the upgrade of the CM energy from 7 TeV to 14 TeV is much more efficient than accumulation
of luminosity. For instance, for FPD(T) the Z leptonic decay channel at LHC14 with less
than 1 fb−1 can explore some region of parameter space, while LHC7 needs more than 104
fb−1 luminosity to achieve the similar sensitivity. For all these G(221) models, LHC14 can
exceed the capability of current combined constraints and has promising discovery potential.
If the heavy gauge bosons W and/or Z are not discovered, the potential for discovery
can be converted to the 95% CL exclusion limits on the heavy gauge bosons W and/or Z
using the relations Z = S/3 as discussed above. Equivalently, the luminosity for exclusion
limits is about one order of magnitude lower than the discovery luminosity. Therefore, as
shown in Figs. 3.8, 3.9, 3.10, 3.11, 3.12, 3.13 supposing no signals found, via W leptonic
decay channel, W mass in LRD(T) can be further excluded by about 100 GeV after the
151

LHC7 collects 5.61 fb−1 luminosity. Figs. 3.14, 3.15, 3.16, 3.17, 3.18, 3.19 shows that via
Z leptonic decay channel, one can expect slightly further exclusion on LPD(T), HPD(T)
and FPD(T) at LHC7. At the LHC14, as show in Figs. 3.20, 3.21, 3.22, 3.23, 3.24, 3.25
and 3.26, 3.27, 3.28, 3.29, 3.30, 3.31, exclusion region can extend very fast when luminosity
is accumulated. For instance, via W leptonic decay channel, 1 fb−1 can exclude most of the
parameter region for LRD(T), SQD, TFD and UUD, and 10 fb−1 can completely remove the
possibility of M less than 5 TeV in these models if there is no any sign of W production.
Z
For LPD(T), HPD(T) and FPD(T), Z leptonic decay channel at the LHC14 can be used to
exclude most of the parameter space region with only 1 fb−1 luminosity. Then data with 10
fb−1 luminosity at LHC14 may leave LPD(T) and HPD(T) and FPD(T) only a corner of
parameter space at large cφ to survive. The shapes of the exclusion contours are the same
as these at the discovery contours.

3.5

Summary

In this paper we have discussed the potential for discovering, or setting limits on, the extra
heavy gauge bosons W and/or Z using two different scenarios at the LHC: an early run with
√

√
s = 7 TeV and total integrated luminosity of 5.61 fb−1 ; a long run with s = 14 TeV and

100 fb−1 integrated luminosity. The EWPTs, Tevatron and LHC data have been used to set
bounds on the allowed parameter space. We showed that direct searches give tighter bounds
than EWPTs in BP-I. Although constraints from LHC data surpasses Tevatron and EWPTs
constraints in LRD, LRT models, in other models the parameter space depends non-trivially
on the present bounds, especially during the early LHC runs. The unexplored parameter
space will become accessible for 5σ discovery at different time scales. In LRD(T) it is more
152

Models

Current M
Limit
W

M
LHC14 Reach
W

LRD (LRT)

1.72 (1.76) TeV

3.2 - 5 TeV

LPD (LPT)

0.55 (0.55) TeV

No improvement

HPD (HPT)

0.46 (0.35) TeV

0.55 TeV

FPD (FPT)

0.5 (0.4) TeV

No improvement

SQD

1.25 TeV

3.5 - 5 TeV

TFD

1.7 TeV

2 - 5 TeV

UUD

3.1 TeV

4 - 5 TeV

Table 3.2: The current lowest limits and discovery reaches at the LHC14 with 100 fb−1
luminosity on W masses.

Models

Current M Limit
Z

M LHC14 Reach
Z

LRD (LRT)

2.25 (3.2) TeV

2.8 - 5 TeV

LPD (LPT)

1.8 (1.8) TeV

3.5 - 5 TeV

HPD (HPT)

1.7 (1.7) TeV

3 - 5 TeV

FPD (FPT)

1.75 (1.75) TeV

1.75 - 5 TeV

SQD

1.25 TeV

1.5 - 5 TeV

TFD

1.7 TeV

2 - 5 TeV

UUD

3.1 TeV

3.3 - 5 TeV

Table 3.3: The current lowest limits and discovery reaches at the LHC14 with 100 fb−1
luminosity on W and Z masses.

153

efficient to use W leptonic decay channel for discovery or exclusion than Z leptonic decay
channel. In the phobic models, it is challenging to discover a W decaying into leptonic mode.
Hence, observing a Z alone cannot rule out the possibility of NP models with non-Abelian
gauge extension of the standard model. In BP-II models, both Z and W leptonic decay
channel are suitable to explore the allowed parameter space. Discovering degenerate W and
Z in the leptonic decay channels at the same time will be the distinct feature in BP-II. In
Tabs. 3.2 and 3.3, we summarize the current constraints and LHC14 reaches with 100 fb−1
luminosity on the W and Z masses in various models. If one needs to identify new physics
models more precisely, one has to combine different discovery channels, such as top quark
pair, single top quark production for the heavy resonances, or study angular distributions,
or other properly defined asymmetries, in the most promising regions of parameter space
of the models considered. For example, the LPD(T), HPD(T), and FPD(T) models can be
further explored by examining the single-top production, the associate production of W
and W (or Z) bosons, and the production of weak gauge boson pairs from electroweak gauge
boson fusion processes.

154

Chapter 4
PROPERTIES OF HEAVY
CHARGED GAUGE BOSONS

4.1

Introduction

The leptonic channel is the golden channel to discover heavy charged gauge bosons at hadron
colliders. The most sensitive observable to discover this heavy resonance is the transverse
invariant mass of the lepton and missing energy system. After discovery, using the smeared
peak behavior, it is possible to determine its basic properties, such as the mass and width.
At the same time, the angular distribution of the final state lepton can be used to determine
its spin, although it is difficult to do the angular analysis for the charged gauge boson due to
the missing neutrino in the final state. To study its properties in detail, it is more efficient
to look at other decay channel, such as the top quark channel.
The top quark plays a special role in the standard model, and offers a window into
possible new physics beyond the SM. In many models, because the top quark is naturally
155

related to EWSB, new physics is very sensitive to the top quark observables. The observables
such as angular, spin correlations and polarization of the top quark can be used to determine
the particle properties and its chiral structures. Since t-channel single top production with
W exchange are much more suppressed than s-channel single top process if W masses are
larger than the top mass, one can reconstruct the invariant mass from the resonance peaks
of the final state top and bottoms. Therefore, for a lepto-phobic W , the single top channel
will be the best channel to do discovery and further identification. Furthermore, in the top
quark channel, because the final state lepton of the decaying top quark is sensitive to the
chiral structure of the charged gauge bosons, it is possible to probe the chiral structure of
the charged gauge bosons using the top quark polarization. We will study the heavy charged
gauge bosons in the single top final states.

The forward-backward asymmetry indicates the chiral structures in the process. Both
CDF and DØ at the FermiLab Tevatron observed the large forward-backward asymmetry of
the top quark in the top pair final states, deviated by about two standard deviation (2σ) from
the standard model expectations. Many new physics models have been proposed to explain
the enhanced asymmetry. One possible model suggests that this large asymmetry might
come from t-channel flavor-changing W exchange in the top pair production. Measuring
the forward-backward asymmetry will give us the information on the chiral structure of the
W -boson.
156

4.2

s-channel Production of W : Resonances in Single
Top Channel

4.2.1

New Physics Models

In the typical W models, the leptonic final state will be the primary channel. However,
because the W searches in the leptonic final states at the LHC are pushing to higher and
higher W mass region, it is also worth to search W in the quark channel, especially single
top quark channel in the low mass region. For example, in some special models in which the
W is lepto-phobic, the third generation quark final state will be the best channel to look
at. We list the following possible models which prefer the single top final state,
• In some models, the top quark is involved in the electroweak breaking. The this
case, the top quark couplings of new gauge bosons will be larger than the light quark
couplings. In the top-flavor model, the W prefer to couple with the third generation
quarks.
• Motivated from the W jj anomaly at the LHC, it is possible that heavy gauge boson
can be lepto-phobic. From the di-jet final states, the resonance of di-jet can not tell
us whether it is a W or neutral gauge bosons, such as Z or axi-gluon. In this case,
the single top process will be the discovery channel for the lepto-phobic W .
• Motivated from top pair forward-backward asymmetry, it is possible to have the flavorviolating couplings between the first generation quarks and the third generation quarks,
¯
such as t − u − Z , and b − d − Z . So through d¯ initial state, there is a s channel
¯
b
single top by decaying to t¯ final state. On the other hand, through ud initial state,
u
157

there is a t channel single top by decaying to tb final state.

In this thesis, we focus on a lepto-phobic W model. To keep this analysis model independent, the most general effective Lagrangian is written as

g2
L = i√
2

u c t
¯ ¯ ¯



¯
 d 
 
 
γ µ (fL PL + fR PR )  s  Wµ+ + h.c. ,
 ¯
 
¯
b

(4.1)

where g2 = e/ sin θW is the weak coupling, while PL/R are the usual chirality projection operators. For simplicity, we consider only the case with a purely left-handed current
(fL = 1, and fR = 0), but our study can be extended easily to other cases. The triple
gauge interaction of the W and SM gauge bosons is included although such a non-abelian
interaction is suppressed for large W mass (m ) by W -W mixing effects, which are of
W
order O(m2 /m2 ).
W
W
This W decays to quarks with decay width

Γ

¯
W →q q

2
9g2 mW
48π

2
2
fL + fR .

(4.2)

2
2
For fL + fR = 1, Γ
∼ m /40, indicating that the W is narrow.
W
W
The parameter constraints are listed here. Previous studies of W s decaying to tops have
focused on the LHC at center of mass energies of 14 TeV [71]. We examine the potential
of the early LHC to identify W s during the early LHC as well as the experimental reach.
Many natural models of new physics beyond the SM have relatively light W bosons. They
are needed to cancel quadratic divergences induced by SM W bosons on the higgs potential.
158

However, the closest exclusion limits come from leptophobic W s which decay to dijets [70, 69]

280 < m
< 1500 GeV.
W

(4.3)

and direct searches from single top production.

4.2.2

Cross Section and Event Simulation

u

t

u

W
¯
d

d
W

¯
b

b

(a)

t
(b)

Figure 4.1: Feynman diagrams of the s-channel and t-channel single top production with
exchanging a W .

The s- and t-channel signal processes are

p p → t b + h.c.

pp → tq

or → t q,

(4.4)

Feynman diagrams are shown in the Fig. 4.1. From the current constraints on the W mass
in the top quark channel, one notes that the allowed W mass is larger than the threshold
energy of the top and bottom system. Therefore, the s-channel contribution will dominate,
while the t-channel contribution is small. Considering the semi-leptonic decay of the top
159

σ(pp → W’ → b b l ν) [pb]

10

f L = 1, f R = 0
1

10-1

10-2

10-3

1000

1500

2000
MW’ [GeV]

Figure 4.2: Total cross section of W for fL = 1, fR = 0 at the LHC.

160

quark, the final state of this channel is

pp → W → t¯ tb) → blν¯
b(¯
b.

(4.5)

The total cross section for W production and subsequent decay to a top quark and bottom
quark at the LHC are computed with Madgraph [107], which is shown in Fig. 4.2. We
use the CTEQ6L1 set of parton distribution functions [74] and set the factorization and
renormalization scales to the W mass.
Signal and background events are generated with Madgraph [107] and are normalized to
the corresponding LO cross sections. We choose the benchmark point as 1500 GeV mass,
g2
= 7.21 GeV. The
and coupling fL = √ and fR = 0. The total decay width is Γ
W
2
dominant backgrounds to the final state of lepton and two jets are from top quark pair
production and W boson production in association with jets. For top pair production we
¯
include the lepton+jets final state, tt → blν ¯
bjj. Smaller backgrounds are from single top
quark production in association with a W boson (t + W ) or with jets (t + jets, t-channel
and s-channel) and from diboson+jet (W V ) production.
We use the anti-kt algorithm in the FastJet [75] package to cluster quarks and gluons
into final state jets with parameter R = 0.4. Detector resolution effects are simulated by
smearing jet and leptonic energies according to a Gaussian:

δE
=
E

A
E/GeV

⊕B,

(4.6)

where δE is the energy resolution, A is a sampling term, B is a constant term, ⊕ represents
E
addition in quadrature, and all energies are measured in GeV. For leptons we take A = 5%
161

and B = 0.55%, while for jets we take A = 100% and B = 5%, chosen to represent the
ATLAS and CMS detector performance [76, 77]. We do not smear ET . We model b-tagging
/
as a flat 60% probability to tag b-quark jets and a 0.5% probability to mistag non-b-quark
jets (including charm quarks).
At the analysis level, all the signal and background events are required to pass the basic
selection cuts listed here:

At least two jets with

p

j
≥ 25 GeV,
T

ηj ≤ 2.5

pT ≥ 25 GeV,

Exactly one lepton with
Missing energy

∆Rjj,j > 0.4,

≤ 2.5,

ET > 25 GeV,

Separation with

η

∆R

> 0.2.

(4.7)

The kinematic distribution of the W signal and the various backgrounds after these cuts
are shown in Figs. 4.3 and 4.4.
To isolate the W signal and suppress the SM backgrounds, a set of final cuts is applied
on the jet PT and on HT , listed as follows,
j
pT1st ≥ 200 GeV,
j
p 2st ≥ 80 GeV,
T
HT ≥ 600 GeV,

(4.8)

where HT is the scalar sum of the system pT . These cuts effectively suppress most of the
SM backgrounds while passing much of the W signal.
In order to further improve the sensitivity of the analysis, the reconstruction of the W
162

Normalized Weight

signal
top + W
top + jets
W + jets
WW + WZ
top pair

0.2

(a) Leading jet

0.1

0
0

200

400

600

800 1000
PT [GeV]

Figure 4.3: The normalized PT distributions of the leading jet for signal and backgrounds.

163

Normalized Weight

0.25
signal
top + W
top + jets
W + jets
WW + WZ
top pair

0.2
0.15

(b) Second jet

0.1
0.05
0
0

200

400

600

800 1000
PT [GeV]

Figure 4.4: The normalized PT distributions of the subleading jet for signal and backgrounds.

164

Normalized Weight

signal
top + W
top + jets
W + jets
WW + WZ
top pair

0.2
0.15

System HT

0.1
0.05
0
0

500

1000

1500
2000
HT [GeV]

Figure 4.5: The normalized HT distributions of the system for signal and backgrounds.

165

and its invariant mass is required. For this reconstruction it is necessary to first obtain the
2
neutrino momentum. In the top quark reconstruction, we demand m2 = MW first. The
lν
longitudinal momentum of the neutrino is formally expressed as

pνL =

1
2 p2
eT

A peL ± Ee

2 /
A2 − 4 peT E2
T

,

(4.9)

2 /2
2
/
where A = MW + 2 peT · ET . If A2 − 4 peT ET ≥ 0. If there are two solutions, we will pick
up the solution with central rapidity. If the solution is complex, we will pick up the real part
of this solution. With the neutrino identified properly, we reconstruct the mass of the W as

mrec = m(pν + pl + pjet 1 + pjet 2 )
W

(4.10)

Figure 4.6 shows the reconstructed invariant mass. The signal peak is clearly visible above
the smoothly falling background.

To increase the local significance of this process, after all the optimized cuts one will
select certain bins of the invariant mass distributions. We then impose a window cut on the
invariant mass difference between the reconstructed invariant mass and the theory W mass
under consideration,

mlνjj − MW

< 400 GeV .

Table 4.1 shows the number of events passing each set of cuts, in units of fb.
166

(4.11)

Cross section [fb]

signal
top + W
top + jets
W + jets
WW + WZ
top pair

0.01

jjlν inv. mass

0.005

0
0

500

1000

1500

2000 2500
Mjjlν [GeV]

Figure 4.6: After all the cuts, the reconstructed invariant mass distributions of the system
for W signal and backgrounds.

167

Cross section [fb]

’

WL truth
’
WL cuts

0.002

’

WR truth
’
WR cuts

0.0015
0.001
0.0005
0
-1

-0.5

0

0.5
1
cos(θl)

Figure 4.7: Before and after basic cuts, the cos θ angular distributions of the final state
leptons for left-handed W and right-handed W .

168

σ(fb)

Signal

t + jets

no cuts

58.96

18877

basic cuts

33.38

+ smearing

¯
tt

t+W

WV

W + jets

2861

25840. 9888

4018600

4049.

833.

7206.7 2265

284516

33.83

4067.

815.

6878.6 2193

296371

+ hard cuts

28.65

158.5

11.1

232.56

3.67

7836.2

+ HT cuts

26.45

59.80

5.69

137.6

3.51

3782.

+ b-tagging

22.26

44.79

3.44

115.7

0.19

61.98

+ mass window

21.59

7.80

0.13

3.97

0.03

16.02

Table 4.1: Cross sections for the signal and various background processes at the 7 TeV LHC
in different cut levels are listed.

4.2.3

Lepton Angular Distribution and Chiral Structure

The charged lepton from top-quark decay is maximally correlated with top-quark spin. The
connection between top-quark spin and the charged lepton can be found from the distribution
in θhel , the angle of the lepton in the rest frame of top quark relative to the top-quark
direction of motion in the overall c.m. frame. This is usually named as “helicity” basis. The
angular correlation of the lepton + is given by
1 ± cos θhel
dσ
1
=
,
σ d cos θhel
2

(4.12)

with the (+) choice for right-handed and (−) for left-handed top-quarks; Clearly, the charged
lepton from a right-handed top-quark prefers to move along the top-quark moving direction
while the one from a left-handed top-quark is to against the top-quark moving direction. In
the top-quark rest frame, 75% (25%) of charged lepton from tR (tL ) decay follows top-quark
moving direction.
169

After the discovery of the W , one would like to know its mass, spin, and couplings.
Angular distributions of its decay products can be investigated to definitively determine the
spin and chiral structure of the W to the SM fermions. The chirality of the W coupling
to SM fermions is best measured from the polarization of the top quark. Among the top
quark decay products, the charged lepton from t → b + l + ν is the best analyzer of the top
quark spin. For a left-handed top quark, the charged lepton moves preferentially against the
direction of motion of the top quark, while for a right-handed top quark the charged lepton
moves along the direction of motion of the top quark. The angular correlation of the lepton
is (1 ± cos θl )/2, with the (+) choice for right-handed and (−) for left-handed top quarks,
where θl is the angle of the lepton in the rest frame of top quark relative to the top quark
direction of motion in the center-of-mass (c.m.) frame of the incoming partons. In Fig. 4.7,
we plot the cos θl for fL = 1, fR = 0 (WL ) as well as fL = 0 andfR = 1 (WR ). We
expect a flat angular distribution for the SM background because the top quark and anti-top
quark are not polarized. Therefore, the angular distributions of the lepton can be used to
discriminate W models in which the chirality of the W coupling to SM fermions differs.

4.3

t-channel Production of W : Top Quark Forwardbackward Asymmetry

4.3.1

Forward-backward Asymmetry of Top Quark

The observed forward-backward asymmetry in rapidity At B of top quarks [72, 73] at the
F
Fermilab Tevatron deviates by about two standard deviations (2σ) from standard model
(SM) expectations [79]. After corrections for detector acceptance and resolution, At B in
F
170

the center-of-mass (c.m.) frame is 15.8 ± 7.5% at CDF [72] and is 19.6 ± 6.5% at D0 [73].
In addition to At B , the D0 group also reports a positive forward-backward asymmetry of
F
charged leptons from top quark decays of AF B = (15.2 ± 4.0)% compared with the small
value 2.1 ± 0.1% from simulations of the SM [73, 78]. The definition of AF B is
n − nB
,
AF B = F
n +n
F
B

(4.13)

where nF (nB ) is the number of events with q y > 0 (q y < 0), and q and y are the
sign and rapidity respectively of the charged lepton from the semileptonic decay of a top or
¯
anti-top quark in the tt production.
In this thesis, we investigate the kinematic and dynamic relationship between the two
asymmetries At B and AF B . We study quantitatively the influence of the top-quark boost
F
on the kinematics of the charged lepton, showing how the distribution of leptons in the
laboratory frame is related to the polarization state of the top quark parent. We show
that current data on the ratio of the two asymmetries favor models in which more righthanded than left-handed top quarks are produced. The fact that AF B , At B , and the ratio
F
AF B /At B are larger than the SM predictions indicates that the charged lepton strongly
F
prefers to move in the same direction as the top quark from which it originates. This
result can arise if right-handed top quarks [80, 81] play a significant role in At B or if a
F
non-standard mechanism produces more highly boosted top quarks at the Tevatron, as we
explain below.
Many new physics (NP) models have been proposed to explain the enhancement of At B ,
F
such as flavor-changing Z [82], W [83] 1 , axigluon G [84, 85, 86] models , etc [89, 90].
1 The next-to-leading order quantum chromodynamics corrections to the process of
171

The first two models produce predominantly right-handed top quarks, whereas the axigluon
model generates unpolarized top quarks. It is important to validate these models at the Large
Hadron Collider (LHC) and/or at the Tevatron. For example, the heavy flavor-changing Z
(

mt ) model is disfavored because it predicts too many same-sign top quark pairs at the

LHC [91, 92]. In this thesis, we focus on how consistently the W models can describe both
At B and AF B .
F

4.3.2

Kinematics

The charged lepton in top quark decay is a powerful analyzer of the polarization of the top
quark [93]. In the rest frame of a top quark, the distribution in the polar angle θhel of a
decay lepton + is
1 + λt cos θhel
dΓ
1
=
,
Γ d cos θhel
2

(4.14)

where λt denotes the top quark helicity. Here, λt = + is for a right-handed top quark (tR )
while λt = − for a left-handed top quark (tL ). The angle θhel is measured with respect
¯
to the direction of motion of the top quark in the overall center-of-mass system of the tt
production process. The distributions are shown in Fig. 4.8(a). The charged lepton from a
right-handed top quark decay prefers to move along the top quark direction of motion, while
a lepton from a left-handed top quark moves preferentially against the top quark direction
of motion. In the rest frame of the top quark, 75% (25%) of charged leptons from tR (tL )
decay follow the top quark direction of motion, i.e. cos θhel > 0.
Once the top quark is boosted, the angular distribution of the charged lepton relative to
¯
q q → tt induced by the flavor-changing Z and W are calculated in Ref. [87] and Ref. [88],
¯
respectively, with the result that the NP prediction at the leading order is reliable.
172

Γ-1dΓ/dcosθhel

1

0.9
0.8

(a)

0.7
0.6
0.5

0.4
0.3

tR
tL

0.2
0.1
0
-1

-0.5

0

0.5

1

Γ-1dΓ/dcosθtl

cosθhel
3

2.5
2

(b)

Et=200GeV

tR
tL

1.5
1
0.5
0
-1

-0.5

0

0.5

1

cosθtl
Figure 4.8: (a) cos θhel distribution in the top quark rest frame for both tL and tR . (b)
cos θt distribution in the boosted frame for a top quark with Et = 200 GeV.
173

the direction of motion of the top quark is sensitive to the energy of the top quark Et (or
equivalently its velocity β). We derive
1 − β cos θt + λt cos θt − β
dΓ
=
,
Γd cos θt
2γ 2 1 − β cos θt 3
where β =

(4.15)

2
1 − m2 /Et , γ = Et /mt . As an illustration, we plot in Fig. 4.8(b) the distrit

bution of cos θt of the charged lepton for a fixed energy of the top quark Et = 200 GeV,
where θt is the angle between the charged lepton and the moving direction of its parent top
quark in the laboratory frame. We find that about 60% of + follow (i.e., cos θt > 0 ) the
top quark moving direction for a tL , and almost 100% for a tR for Et = 200 GeV.

The distribution of charged leptons in the laboratory frame depends on the top quark
kinematics, including the top quark energy and its rapidity, and the top quark polarization.
The probability for finding a positive charged lepton in the forward region when it originates
from a top quark with a velocity β, rapidity yt , and polarization λt is defined as
NF
,λ
,
RF t (β, yt) =
N +N
F
B
where NF (NB ) denotes the number of leptons

(4.16)

in the forward (backward) region in the

laboratory, and
yt ≡ ln

Et + pt
z
Et − pt
z

(4.17)

with pt being the z-component of top quark momentum along the proton beam. After
z
174

,λ
lengthy algebra, it can be shown that the ratio R t (β, yt ) is
F

1

1
λt coth2 yt
 +

+
,

2


−2 coth2 y 1/2 4βγ 2 1 + γ −2 coth2 y 3/2

2 1+γ
t
t






y ∈ 0, y max
 t

t





,λ
RF t (β, yt) =





1
λt coth2 yt
1

 −

−
,
2


−2 coth2 y 1/2 4βγ 2 1 + γ −2 coth2 y 3/2

2 1+γ

t
t






y ∈ −y max , 0
 t
t

(4.18)

where
max =
yt

1 1+β
ln
.
2 1−β

(4.19)

To illustrate the effect of the top quark boost, we plot in Fig. 4.9 the fraction RF as a
function of yt . We choose two characteristic top quark energies, Et = 200 GeV (Fig. 4.9
(a)) and 600 GeV (Fig. 4.9 (b)). The former energy represents top quarks produced around
the threshold region, while the latter pertains for highly boosted top quarks. Note that
max = 0.53 for E = 200 GeV. When a top quark moves perpendicular to the beam line,
yt
t
i.e. yt = 0, there is an equal number of leptons in the forward and backward regions, leading
to RF = 0.5, independent of Et and the polarization of the top quark.
We focus on the behavior of RF in the region of yt > 0, since the results for RF (yt < 0)
is just (1 − RF (yt > 0)). For right-handed top quarks tR (black-solid lines in Fig. 4.9), RF
increases rapidly with yt in the region of yt > 0. This is because the charged lepton likes to
follow the direction of motion of the top quark as we already see in Fig. 4.8. The direction
of top quark can be used to well approximate that of the charged lepton in the laboratory
175

RF

1
0.9

tR
tL

0.8
0.7
0.6
0.5
0.4
0.3

(a)

0.2

Et=200GeV

0.1
0

-0.4

-0.2

0

0.2

0.4

y
RF

t

1
0.9

tR
tL

0.8
0.7
0.6
0.5
0.4
0.3

(b)

0.2

Et=600GeV

0.1
0

-1.5

-1 -0.5

0

0.5

1

1.5

y

t

Figure 4.9: The ratio RF as a function of yt for a top quark with fixed energy: (a) Et =
200 GeV and (b) Et = 600 GeV.
176

frame, and such an approximation woks better for a highly energetic top quark. We can also
see that when Et becomes larger, i.e. the top quark is more energetic and the lepton is more
boosted, RF rapidly reaches its maximum value 1.
On the contrary, in the case of tL ’s, the ratio RF does not significantly with yt compared
with tR , owing to the anti-boost effect on the charged lepton. For Et = 200 GeV, the boost
causes charged leptons to distribute nearly uniformly (as seen in Fig. 4.8(b)), as a result, RF
is around 0.5 for all the allowed range of yt , as seen in the red-dotted curve in Fig. 4.9(a).
When the energy of top quark is large enough, the large boost forces most of the charged
leptons from top quark decays to move along the top quark direction of motion, even if they
move against the top quark direction of motion in the top quark rest frame. The boost yields
a large value RF in the region of large yt , as shown by the red-dotted curve in Fig. 4.9(b).
The competing influences leave the tL curve slightly below the tR curve. One can expect
that, in the limit of Et → ∞, the direction of charged lepton is the same as its parent top
quark, independent of polarization of the top quark.
Moreover, in Figs. 4.10 and 4.11, we show how RF varies with top quark transverse
momentum pt and yt . The distributions for right-handed top quarks tR ’s do not vary
T
greatly with pt due to the fact that most of the charged leptons like to follow tR . However,
T
the shapes of the curves for left-handed top quarks, which are the focus in the discussion
below, are very different between the low pt and high pt regions, as is seen in the red-dotted
T
T
lines. For the top quark away from yt = 0 with fixed pt , the boost becomes more significant
T
because the energy of the top quark must be increased compared with yt = 0. Therefore,
more leptons are forced to move along the direction of the top quark in the laboratory frame.
On the other hand, when yt changes, a portion of the decay leptons which are initially in
177

RF

1
0.9
0.8
0.7

tR
tL

0.6
0.5
0.4
0.3

(a)

0.2

pt =10GeV

0.1

T

0
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

y
RF

t

1
0.9
0.8
0.7

tR
tL

0.6
0.5
0.4
0.3

(b)

0.2

pt =50GeV

0.1

T

0
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

y

t

Figure 4.10: The ratio of the charged lepton in the forward and backward region as a function
of the top quark rapidity for top quarks with fixed transverse momentum pT = 10, 50 GeV.
178

RF

1
0.9
0.8
0.7

tR
tL

0.6
0.5
0.4
0.3

(a)

0.2

pt =100GeV

0.1

T

0
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

y
RF

t

1
0.9
0.8
0.7

tR
tL

0.6
0.5
0.4
0.3

(b)

0.2

pt =300GeV

0.1

T

0
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

y

t

Figure 4.11: The ratio of the charged lepton in the forward and backward region as a function
of the top quark rapidity for top quarks with fixed transverse momentum pT = 100, 300 GeV.
179

the forward/backward region (y > 0/y < 0) will then be in the backward/forward region.
In summary, there are two factors which affect RF : the boost and the rearrangement of the
distribution of charged leptons in the forward (y > 0) and backward (y < 0) region. The
former always increases RF while the latter may increase or decrease the RF depending
on how energetic the top quark is at yt = 0. Generally speaking, when the boost is not
significant (low pt and small yt ), RF decreases when yt increases from yt = 0, as we
T
can see in the drop in the red-dotted curves in Fig. 4.10(a) and 4.10(b). When the boost
is big enough, RF always increases with yt . The platform-like behavior around yt

0

in Fig. 4.11(a) arises because the leptons accumulate nearly uniformly around the axis of
√
motion of the top quark when pt = mt / 3 100 GeV.2 Therefore the ratio RF is rather
T
stable as the top quark changes its direction of motion direction around yt = 0.

4.3.3

At B and AF B
F

The observed positive top-quark asymmetry At B indicates more top quarks are produced
F
in the forward region than in the backward region of rapidity. Both tR and tL can generate
a positive lepton asymmetry AF B with positive At B . However, as shown in Fig. 4.9, tL
F
would need a large boost along the proton beam line (i.e. in the large forward rapidity
region) to overcome the fact that most of the charged leptons from its decay move against it
in its rest frame. A right-handed top quark tR can yield a positive AF B even for top quarks
¯
near the tt threshold region. Therefore, the large positive top quark and lepton asymmetries
At B and AF B observed by the D0 collaboration indicate that the top quark polarization
F
and kinematics of top quarks, like yt and Et , may be playing a non-trivial role. In this
2 The critical value, m /√3 is obtained by solving ∂RF |
=0
t
∂yt yt =0
180

section we present a general analysis of the correlation between At B and AF B so that one
F
can have a better understanding about the numerical results derived from NP models to be
shown in Sec. 4.3.4.

The top quark asymmetry At B can be expressed as a sum of contributions from the SM
F
and NP as:
NP
SM
NP
SM
(NF + NF ) − (NB + NB )
t
AF B =
(N NP + N SM ) + (N NP + N SM )
F
F
B
B
t, NP
NP + At, SM × (1 − RNP ),
= AF B × R
FB

(4.20)

where
NP
NP
SM
SM
NP
Ntot
t, NP NF − NB
t, SM NF − NB
NP =
, AF B =
, R
,
AF B =
SM
NP
N SM + N SM
N NP + N NP
Ntot + Ntot
F
B
F
B

(4.21)

with N SM and N NP being the numbers of events in which the top quark moves with
F (B)
F (B)
SM(NP)
yt > 0(yt < 0) in the SM and induced by NP, respectively, and Ntot
is the total
number of events predicted in the SM (induced by NP). The NLO QCD contribution to the
t,SM
¯
production process q q → tt could generate a value AF B ∼ 5% at the Tevatron, which is
¯
much less than the central value of experimental data.

To simplify the discussion of the correlation between At B and AF B , we assume in this
F
section that At B is generated completely by NP, but all SM contributions (including the
F
NLO QCD effects) are retained in the numerical calculations we present.

The contributions to At B from different polarizations of top quarks can be separated
F
181

as:
t , NP
t , NP
× RNP ,
At B ≈ ρt A L
+ ρt A R
F
R FB
L FB
where

(4.22)


λ
λ
N t −N t
λ , NP  F
B 
,
A t
=
FB
λt
λt
N +N
B NP
F

(4.23)



ρλ =
t

N λt , NP
.
NP
Ntot

λ , NP
Here, A t
denotes the forward-backward asymmetry of the top quark with polarization
FB
¯
λt generated only by NP, while ρλ is the fraction of top quarks with polarization λt in tt
t
events induced by NP. One advantage of decomposing At B into contributions of different top
F
quark polarizations is to monitor the chirality of the couplings of NP particles to top quarks.
Another advantage is to make the connection between AF B and At B more transparent.
F

As discussed in Sec. 4.3.2 the ratio RF depends on the top quark kinematics (β, yt and
λt ). To compute the total probability of finding a charged lepton in the forward region, one
must convolute the top quark production cross section with RF on an event-by-event basis,
i.e.
1
,λ
¯
N tt ⊗ R t =
F
Ntot
Ntot
1

,λ
¯
N tt (β, yt, λt )R t (β, yt),
F

(4.24)

¯
¯
where N tt labels the tt production rate for a top quark with specific kinematics (β, yt , λt )
and Ntot stands for total number of events. The lepton asymmetry AF B generated by a
top quark with polarization λt can be therefore expressed as
182

,λ
A t
FB

=
NP

=

=

λ
,λ
λ
,λ
λ
,λ
λ
,λ
(NF t ⊗ RF t + NBt ⊗ RF t ) − (NF t ⊗ RB t + NBt ⊗ RB t )
λ
λ
NF t + NBt
NP
λ
,λ
λ
,λ
N t ⊗ 2R t − 1 + N t ⊗ 2R t − 1
F
F
B
F
λ
λ
N t +N t
F
B
NP
λ
λ
,λ
(N t − N t ) ⊗ 2R t − 1
F
B
F
.
(4.25)
λt
λt
N +N
F
B
NP

Here,
NB
,λ
,λ
R t (β, yt) ≡
= 1 − R t (β, yt),
B
F
N +N
F
B

(4.26)

,λ
,λ
and we use the following relation between R t and R t in our derivation,
F
B
,λ
,λ
R t (yt ) = R t (−yt ) .
B
F

(4.27)

λ
λ
The quantities N t and N t in the convolutions in Eq. (4.25) should be understood as the
B
F
¯
¯
distributions N tt (β, yt, λt )Θ (yt ) and N tt (β, yt, λt )Θ (−yt ), respectively, where Θ (x) is the
λ
λ
Heaviside step function. The quantity N t − N t should be understood as
F
B
¯
¯
N tt (β, yt , λt ) − N tt (β, −yt , λt ) Θ (yt ) .
,λ
Because R t in Eq. (4.25) cannot exceed 1, we have AF B
F

(4.28)

,λ
At B . When R t is close
F
F

¯
to a constant RC , e.g. RC ∼ 1/2 around the tt threshold (Et ∼ 200GeV) for left-handed
183

top quark or RC ∼ 1 for a highly boosted top quark, Eq. (4.25) can be simplified as

λ
λ
N t −N t
λ , NP
,λt
B 
× 2RC − 1 = AFtB
AF B
× 2RC − 1 .
=  F
λt
λt
NP
N +N
F
B NP


(4.29)

Equation (4.29) and Fig. 4.9 show that:
,t
¯
• A L ∼ 0 when the tt pair is produced around the threshold region;
FB
,tL
• AF B

,tR
AF B ≈ At B in the large mtt region.
¯
F

Although Eq. (4.29) is approximate, it helps in understanding the NP prediction obtained
from a complete numerical calculation.

4.3.4

Flavor-changing W Model

In this subsection we focus on a flavor-changing W model [83]. We examine how this models
can accommodate the values of both At B and AF B measured by the D0 collaboration.
F
¯
A different class of NP models to explain the tt forward-backward asymmetry is based on
t-channel kinematics. Such models involve large flavor-changing interactions. A model with
a non-universal massive neutral vector boson Z [82] is one of the possibilities. However, it
is disfavored because it implies a large rate for same-sign top quark production at the 7 TeV
LHC [91], not supported by data [92].
We consider a flavor-changing W which couples an incident d-quark to the produced
t-quark [83],
¯
L = g2 gR dγ µ PR tWµ + h.c. ,

(4.30)

¯
where g2 is the weak coupling. In the W model, in addition to the SM process q q → g → tt,
¯
184

AlFB (%)

20
18
16
14
12
10
8
(200GeV,400GeV)

6

(400GeV,600GeV)

4

(600GeV,800GeV)

2

(800GeV,1000GeV)

0
10 12

14

16

18

20

22

24

26

28

30

At (%)
FB
Figure 4.12: Correlation between AF B and At B in the W models. The point correspondF
ing to the D0 data is also shown. The numbers within the parentheses label the lower and
upper limits of the mass of the W .

185

fraction of events

0.25

0.2

0.15

0.1

0.05

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
AlFB/A t
FB

Figure 4.13: The statistics for the ratio of predicted AF B to At B in the W models.
F

186

¯
the tt pair can also be produced via a t-channel process with a W mediator. Apart from a
ˆ
g 2 g 2 sδac δbd
, the helicity amplitude M t (λq , λq , λt , λt ) is
common factor 2 R
¯
¯
W
2
ˆ
4 t−m
W
Mt (+ − −−) =
W

2
2 + rW

Mt (+ − −+) =
W

2
2(1 − β) + rW (1 + β) (1 − cos θ),

1 − β 2 sin θ,

2
Mt (+ − +−) = − 2(1 + β) + rW (1 − β) (1 + cos θ),
W
2
Mt (+ − ++) = − 2 + rW
W

1 − β 2 sin θ ,

(4.31)

where rW = mt /mW .

4.3.5

Correlations on AF B and At B
F

In the region β

1, the nonzero helicity amplitudes are

2
Mt (+ − −+) ∼ 2rW (1 − cos θ),
W
Mt (+ − +−) ∼ 4(1 + cos θ) .
W

(4.32)

2
In order to produce top quarks in the forward region, one needs 2rW < 4, which is always
true for the region of W masses (heavier than the top quark) considered in this paper. At
¯
the Tevatron the β distribution of the top quark in tt production peaks around 0.6, and
therefore most of the top quarks are not significantly boosted. We can also easily see that
more right-handed top quarks are produced compared to left-handed ones in the W model,
ρt > ρt . Since the t-channel propagator contributes a minus sign, the total forwardR
L
backward asymmetry results from a competition between the square of the purely NP term
187

4
and the interference term of NP with the SM. The former is proportional to gR and the latter
2
to gR . We plot the correlation between At B and AF B for the W model in Fig. 4.13. The
F
strong correlation is fit well by

AF B

Moreover, for a relatively light W (

0.75 × At B − 2.1% .
F

(4.33)

600) GeV, both At B and AF B can be consistent
F

with the D0 data within 1 σ.

For the W model, Fig. 4.13 shows the statistics for the ratio of the predicted AF B to
At B , based on the scattered points in Fig. 4.12. The total number of events is normalized to
F
1. he data from D0 shows AF B /At B about 78 ± 33%. The ratio in the SM is close to 40%.
F
The comparison to the D0 point shown in Fig. 4.12 indicates that top quark events with
a large proportion of right-handed top quarks are favored. Constraints on flavor-changing
currents in the W model allow only right-handed couplings to the top quark, consistent with
the D0 AF B results. Improved statistics would help, as well as a measurement of AF B by
the CDF collaboration. For W model, since the events that contribute to At B are almost
F
locate region of 2RF − 1 > 0, it is very transparent from Eq. (4.25) that AF B /At B can be
F
as large as 0.75, see Eq. (4.33). For SM, one may expect that AF B /At B > 0.5. However,
F
the predicted values are about and 0.4 for SM, respectively. Therefore, one should be more
careful to take in to account the boost effect due to different frames, i.e.
AF B (lab) At B (lab)
AF B (lab)
F
=
.
At (c.m.)
At (lab) At (c.m.)
FB
FB
FB

(4.34)

It is known that the boost effect will reduce At B in laboratory frame related to the c.m.
F
188

frame [95]. For SM, the reduction is about 30%. One can then easily understand that
the predicted values AF B (lab)/At B (c.m.) < 0.5 even AF B (lab)/At B (lab) > 0.5. We
F
F
therefore encourage D0 collaboration to measure the At B in the laboratory frame as well,
F
in order to have a more transparent comparison with NP prediction, and we expect the
value of AF B /At B in the laboratory frame will be larger than 78% which is the result
F
of current data measured from two different frame. Combined with the results from CDF
measurements, if large AF B and AF B /At B are confirmed, it would be a strong evidence
F
for the new physics signal and the chirality of coupling between new particle(s) with SM
particles.

4.4

Summary

The deviation of the top quark forward-backward asymmetry At B from its SM prediction
F
may indicate the presence of new physics. Based simply on the large value of At B , the
F
charged lepton forward-backward asymmetry AF B should also be expected to be larger
than the SM expectation. Indeed, the D0 collaboration reports AF B = 15.2%, about 3σ
away above the SM value. In this paper, we study the kinematic and dynamic aspects of
the relationship between the asymmetries At B and AF B based on the spin correlation
F
between charged leptons and the top quark with different polarization states. Owing to the
spin correlation in top quark decay, AF B and At B are strongly positively correlated for
F
right-handed top quarks. However, for left-handed top quarks, the nature of the correlation
depends on how boosted the top quark is. For large enough top quark energy, left-handed
top quarks will also generate a large charged-lepton asymmetry, similar to that for righthanded quarks. However, if the top quark is not boosted (Et
189

200 GeV), AF B from

left-handed top quarks will be less than At B /2 for a positive At B . Since most of the
F
F
¯
tt events are produced in the threshold region, one may use the large positive values of
At B and AF B measured at D0 to conclude that production of left-handed top quarks is
F
disfavored. Confirmation of the D0 result and greater statistics are desirable. There is great
value in making measurements of both At B and AF B because their correlation can be
F
related through top quark polarization to the underlying dynamics of top quark production.
We focused on a flavor-changing W model which produced dominantly right-handed top
quarks. To determine free parameters, we required that these new physics models fit At B
F
¯
as well as the tt total cross section at the Tevatron at 1σ level of accuracy. As we have
shown, there is a strong correlation between At B and AF B in both models. The best fit
F
to the relationship is AF B 0.75 × At B − 2.1%, within 2σ of the D0 result. To generate
F
AF B satisfying the data to better than 1σ accuracy, a light W (lighter than 600 GeV) is
favored.

190

Chapter 5
EXOTIC HEAVY CHARGED
GAUGE BOSONS

5.1

Introduction

In this chapter we study a different W model, named a “top-philic” model, in which the
W couples only to third-generation quarks (top and bottom quarks) and is produced only
in association with a top quark. It decays only into a top quark and a bottom quark pair.
¯
The collider signature of the events is a tt pair plus one b-jet. We explore the discovery
potential of the top-philic W boson at the LHC at a center of mass energy of 14 TeV with
an integrated luminosity of 100 fb−1 . We compute the inclusive cross section, simulate the
signal and backgrounds, and investigate a set of optimal cuts. Our study shows that the
prospects are promising to discover the top-philic W in tW associated production despite
the presence of SM backgrounds that exceed our signal by three or four orders of magnitude.
¯
The key is to identify as a b jet the extra jet produced in association with tt. A 1 TeV W
191

with the same coupling strength as the SM W -t-b interaction could be discovered with a
statistical significance of 5 standard deviations at the LHC at 14 TeV.
Motivated by the observation of large parity violation in top quark pair-production at
the Tevatron [72], several authors have recently proposed a W boson with a flavor changing
d-t-W interaction [109, 110, 86, 111, 112]. This W boson is also produced in association
with a top quark, but it differs from the top-philic W we discuss in that it decays into a
¯
top quark and a non-b quark, yielding a final state of tt plus a non-b jet [113]. This final
¯
state suffers from a huge ttj background that cannot be mitigated by b-tagging on the jet
produced in association with the top quark pair. As a result, a large coupling strength would
be needed for discovery of the flavor changing W at the LHC.

5.2

The model

A top-philic W can arise from a new non-abelian gauge symmetry which breaks generation
universality [30, 31, 32]. A summary may be found in Ref. [24]. In this study we adopt an
effective Lagrangian approach rather than focusing on specific NP models. The effective,
renormalizable interaction of the W to the SM third generation fermions is

g2
¯
L = i √ tγ µ (fL PL + fR PR )b Wµ+ + h.c. ,
2

(5.1)

where g2 = e/ sin θW is the weak coupling, while PL/R are the usual chirality projection
operators. For simplicity, we consider only the case with a purely left-handed current (fL =
1, and fR = 0), but our study can be extended easily to other cases. The triple gauge
interaction of the W and SM gauge bosons is not included because such a non-abelian
192

interaction is suppressed for large W mass (m ) by W -W mixing effects, which are of
W
order O(m2 /m2 ).
W
W
The W decays entirely to a top quark and bottom quark pair with decay width

Γ

W →t¯
b

2
3g2 m
W
48π

2
2
fL + fR .

(5.2)

2
2
For fL + fR = 1, Γ
∼ m /100, indicating that the W is quite narrow.
W
W
The top-philic W boson is produced predominately through a gluon-bottom-quark fusion
process, as depicted in Fig. 5.1. The W decays into a top and bottom quark pair, and the
¯
overall final state is then ttb. Since this final state has not been used to search for a W
at the Tevatron and LHC, none of the current collider limits constrain our top-philic W
model. It is possible the top-philic W boson could be as light as a few hundred GeV.
¯
¯
The recent CDF measurement of the ratio of the cross sections for tt + 0 jets to tt + n jets
is consistent with the SM expectation [114]. Top-philic W production will contribute to the
¯
tt + n jets rate. However, our numerical calculation shows that tW production is too small
¯
to be of concern, e.g. σ(tW − + tW + ) ∼ 3 fb for m
= 200 GeV, fL = 1, and fR = 0.
W
Moreover, the cross section drops rapidly with m . We conclude that the top-philic W
W
¯
model is consistent with tt current measurements at the Tevatron.
¯
In Fig. 5.2, we display the leading-order inclusive cross section for tW − and tW +
¯
production, σ(tW − + tW + ), as a function of the W mass (m ) at 14 TeV with a purely
W
¯
left-handed W -t-b coupling, i.e. fL = 1 and fR = 0. Note that σ(tW + ) = σ(tW − )
owing to equality of the parton distribution functions for initial state b- and ¯
b-quarks. The
CTEQ6L parton distribution functions are used in our calculation with the renormalization
and factorization scales chosen as mW . The production cross section is at the picobarn level
193

g

g

t
t
W

b

¯
t

(a)
g
W

¯
b
(c)

+

b
W

b

−

¯
t

(b)
g

¯
t
¯
t

b

b

−

t

¯
b

t
¯
b

¯
t

¯
b

t
W

(d)

+

¯
b

Figure 5.1: Feynman diagrams for the associated production of a W and a top quark: (a,
¯
b) W − t, and (c, d) W + t.

for a W with a few hundred GeV mass and at the femtobarn level for a multiple TeV-scale
W .

194

’-

σ(tW + tW’+) [fb]

104

f L = 1, f R = 0
103

102

10
400

600

800

1000
MW’ [GeV]

¯
Figure 5.2: The inclusive cross section of the tW − and tW + associated production at a
14 TeV LHC with fL = 1 and fR = 0.

195

5.3

Collider phenomenology

¯
Our signal consists of both tW − and tW + production channels because the two channels
¯
give rise to the same tt plus one b-jet collider signature after the W decay. The b-jet could
originate from a b or a ¯ as one cannot now distinguish b- and ¯
b
b-jets experimentally. At
the event reconstruction level the b-jet together with one top quark would yield a heavy W
¯
resonance. The main SM background is from production of a tt pair plus one b jet. We
also take into account the possibility that a light quark jet fakes a b jet. The signal and
background events are generated with MadGraph5/MadEvent [107].
In order to trigger on the signal event, we demand a leptonic decay of the top quark
¯ bjj. The signal processes are
t → b + ν and hadronic decay of the antitop t → ¯
¯
pp → tW − → ttb → bW +¯ − b → bb¯ + jjν,
bW
b
¯
¯b
pp → tW + → tt¯ → bW +¯ −¯ → b¯¯ + jjν.
bW b
bb

(5.3)

The topology of our signal is characterized by one isolated positive charged lepton, five high
miss
energy jets, and a large missing transverse momentum (ET ) from the missing neutrino.
Both electrons and muons are used in our analysis.
We separate the SM backgrounds according to the flavor of the jet produced in association
¯
with the tt pair:

¯
¯
ttj : pp → ttj → bW +¯ − j → b¯
bW
bjjj + ν,

(5.4)

¯
¯
ttb : pp → ttb → bW +¯ − b → b¯
bW
bbjj + ν.

(5.5)

196

m
W
400
¯
ttb
¯
ttj
500
¯
ttb
¯
ttj
600
¯
ttb
¯
ttj
700
¯
ttb
¯
ttj
800
¯
ttb
¯
ttj
900
¯
ttb
¯
ttj
1000
¯
ttb
¯
ttj

No cut

basic

32920
1.9×105
3.13×107
15115
1.9×105
3.13×107
7361
1.9×105
3.13×107
3843
1.9×105
3.13×107
2110
1.9×105
3.13×107
1215
1.9×105
3.13×107
720
1.9×105
3.13×107

6929
23849
3×106
3324
23849
3×106
1666
23849
3×106
874
23849
3×106
490
23849
3×106
290
23849
3×106
172
23849
3×106

optimal b-tagging ∆M cut
5240
2712
306062
2621
2709
306057
1300
2524
288098
638
1781
212153
405
1060
130122
187
594
74342
106
337
42423

3018
1537
6984
1513
1529
6895
754
1429
6214
369
1026
4441
197
620
2346
107
353
1052
62
199
505

2166
297
967
1120
449
577
565
437
385
282
303
304
154
189
214
85
110
62
50
64
35

cut
6.6 %
0.15 %
3.1×10−3 %
7.4 %
0.23 %
1.8 ×10−3 %
7.7 %
0.23 %
1.2 ×10−3 %
7.4 %
0.16 %
9.7 ×10−4 %
7.3 %
0.10 %
6.8 ×10−4 %
6.9 %
0.058 %
2.0 ×10−4 %
7.0 %
0.034 %
5.4 ×10−5 %

Table 5.1: The numbers of signal and background events at 14 TeV with an integrated
luminosity of 100 fb−1 before and after cuts, with fL = 1, for seven values of m
(GeV).
W
The top quark decay branching ratio 2/27 is included in the “no cut” column, and the b
tagging efficiency is included in the fifth column. The cut acceptances cut are also listed.

197

¯
The “extra” jet in association with the tt originates from a light-flavor quark or gluon in the
first case and from a b or ¯ in the second case. As shown below, the two backgrounds are
b
suppressed by different kinematic cuts. In the generation of background events, we demand
the transverse momentum (pT ) of the extra jet to be harder than 10 GeV to avoid soft
and collinear divergences from QCD radiation. After kinematic cuts, the contributions from
other SM backgrounds, e.g. W + W − jjj, are quite small and are not included in our analysis.
For an integrated luminosity of 100 fb−1 , the numbers of signal and background events
at the event generator level are shown in the second column of Table 5.1. The top quark
¯
decay branching ratio Br(tt → b¯ + νjj) = 2/27 is included in the numbers. We choose six
b
benchmark points for the mass m . We set the W -t-b couplings at fL = 1 and fR = 0.
W
The rates for other values of fL can be obtained from simple scaling

2
σ = fL × σ(fL = 1).

5.3.1

(5.6)

Selection cuts

At the analysis level, all the signal and background events are required to pass the basic
selection cuts listed here:

p

j
≥ 25 GeV,
T

pT ≥ 25 GeV,
∆Rjj,j ,

ηj ≤ 2.5
η

≤ 2.5,

> 0.4, ET > 25 GeV
198

(5.7)

where pT denotes the transverse momentum, ET is the missing transverse momentum from
the invisible neutrino in the final state, and ∆R is the separation in the azimuthal angle
(φ)-pseudorapidity (η) plane between the objects k and l

∆Rkl ≡

ηk − ηl 2 + φk − φl 2 .

(5.8)

We smear the final state hadronic and leptonic energy according to a fairly standard Gaussiantype detector resolution given by

δE
=
E

A
E/GeV

⊕ B,

(5.9)

where A = 5(100)% and B = 0.55(5)% for leptons (jets).
As shown in the third column of Table 5.1, roughly 1/3 of the signal events pass the
basic analysis cuts. At this stage, the SM backgrounds are dominant over the signal. A set
of optimized cuts, based on the kinematic differences between the signal and backgrounds,
is needed to extract the small signal.
There are five jets in the final state. Jets from a heavy W boson decay tend to have a
harder pT than jets in the backgrounds. We order the jets by their values of pT . Jet charge
would also be a possibility for labeling jets, but the charge of jets is not well measured
experimentally. Figure 5.3(a) displays the normalized pT distribution of the jet with largest
pT for a 1 TeV W . The signal and background curves are normalized by their individual
cross sections. The signal distribution (black solid curve) peaks around 450 GeV while the
backgrounds peak around 60-80 GeV. The leading jet in the signal is mainly the b-jet from
W → tb decay. It shares energy with its top quark partner; therefore its pT is about
199

1 / σ dσ / d PT [bin / 10 GeV]

0.09
ttb signal
ttb background
ttj background

0.08
0.07

(a) First jet

0.06
0.05
0.04
0.03
0.02
0.01

1 / σ dσ / d PT [bin / 10 GeV]

0
0

200

400

0.14

600
800
PT [GeV]

ttb signal
ttb background
ttj background

0.12
0.1

(b) Second jet

0.08
0.06
0.04
0.02
0
0

100

200

300

400
500
PT [GeV]

Figure 5.3: Normalized pT distribution of (a) the leading jet, (b) the second leading jet
distributions. The black-solid curves represent the signal (m
= 1 TeV), the red-dashed
W
¯b background, while the blue-dotted curves the ttj background.
¯
curves the tt
200

1 / σ dσ / d PT [bin / 10 GeV]

0.25
ttb signal
ttb background
ttj background

0.2

(a) Third jet

0.15
0.1

0.05
0
0

100

200

300
400
PT [GeV]

1 / σ dσ / d HT [bin / 20 GeV]

0.08
ttb signal
ttb background
ttj background

0.07
0.06

(b) HT

0.05
0.04
0.03
0.02
0.01
0
0

500

1000

1500
2000
HT [GeV]

Figure 5.4: Normalized pT distribution of (a) the third leading jet, as well as (b) the normalized HT distribution. The black-solid curves represent the signal (m
= 1 TeV), the
W
¯b background, while the blue-dotted curves the ttj background.
¯
red-dashed curves the tt
201

m /2. On the other hand, the leading jet in the backgrounds is predominately from top
W
quark decay. Its pT spectrum peaks around mt /3 ∼ 60 GeV. These distinct pT spectra
motivate a hard cut on the leading jet pT .
The pT spectrum of the background is independent of mW whereas the pT spectrum
of the leading jet in the signal is sensitive to m . Absent prior knowledge of m , a first
W
W
step in a search might be to introduce a mass independent cut to suppress backgrounds,
such as to require pT > 120 GeV. This cut could then be increased or decreased to probe
for a signal, as we expect experimental collaborations will do to search for heavy resonances.
Since the signal strength is much smaller than the background, we perform Monte Carlo
simulations to find the best cut for each m . This best cut is provided by the simple
W
parameterization
p1st ≥
T

50.0 +

m
W
5

GeV,

(5.10)

which works well for 400 GeV < m
< 1.0 TeV. We think of these m
dependent cuts
W
W
as different cut thresholds. Our mW dependent cuts are optimized for discovery, and the
numbers shown in the fourth column (labeled “optimal”) in Table 5.1 should be viewed as
optimized results for each m .
W
Figures 5.3(b) and 5.4(c) show the pT spectra of the second and the third leading pT
jets. Similar to the leading jet, the 2nd and 3rd leading jets in the signal are harder than
those in the backgrounds. We impose kinematic cuts on the 2nd and 3rd jets as follows:

p2nd ≥
T
p3rd ≥
T

m
W
10
m
20.0 + W
50
20.0 +

202

GeV,
GeV.

(5.11)

Another useful variable is HT , the scalar sum of the pT ’s of all the visible particles in
the final state,
+
H T = pT +
j

j
pT .

(5.12)

Figure 5.4(d) shows the HT distributions for the signal and backgrounds. Involving a massive W in the final state, the signal distribution peaks above 1 TeV while the background
¯
distributions peak near the mass threshold of a tt pair (∼ 400 GeV). This difference enables
us to impose a hard cut on HT to further suppress the SM background:

HT >

m
m
− W
W
10

GeV,

(5.13)

The fourth column of Table 5.1 shows the number of signal and background events after
¯
the optimized cuts listed in Eqs. (5.10-5.13). The ttb background is suppressed significantly,
¯
but the ttj background still overwhelms the signal. However, as we now show, if b-tagging
¯
¯
can be applied to the extra jet (the jet produced in association with the tt pair), the ttj
background can be suppressed efficiently. This improvement arises because the extra jet
¯
in the signal originates from the b quark in the W decay while the extra jet in the ttj
background is from a non-b quark.

5.3.2

χ2 -template and extra-jet tagging

Superficially, the only difference one sees among the final states in Eqs. (5.3-5.5) is that the
¯
¯
signal and the ttb background produce final states with 3 b jets, whereas the ttj background
¯
has only 2 b jets. The key to suppressing the ttj background is to identify the extra b jet in
the final state. To do this, we first exploit the difference in pT between the extra jet and
203

the other jets, and then we require b tagging to identify it as a b jet.
The extra jet in the signal comes from the heavy W decay and tends to have large pT .
The extra jet in the SM backgrounds, mainly from QCD radiation, tends to have a much
softer pT . However, a complication is that top quarks in the signal events are boosted and
jets from top quark decay have large pT . One of the jets from top quark decay could play
the role of the leading jet. Our simulation shows that the extra jet (from heavy W decay)
serves as the leading pT jet in 62% to 84 % of the cases for m
ranging from 400 GeV to
W
1000 GeV. In view of small signal rate for a heavy W , a more efficient method is needed to
identify the extra-jet.
In this study we use a χ2 -template method based on the W boson and top quark masses
to select the extra jet. For each event we pick the combination which minimizes the following
χ2 :
2 (m − m )2 (m − m )2
t
t
jlν
jjj
2 = (mW − mjj ) +
+
.
χ
∆m2
∆m2
∆m2
t
t
W

(5.14)

There is two-fold ambiguity in the reconstruction of the longitudinal momentum of the
neutrino from top quark decay. Making use of the W -boson on-shell condition, m2 = m2 ,
W
lν
we can determine the longitudinal momentum of the neutrino (pνL ) as

pνL =

1
2p2
eT

A peL ± Ee

A2 − 4 p 2 E2
eT T

,

(5.15)

where A = m2 + 2 peT · ET . If A2 − 4p2 E2 ≥ 0, the value of pνL that best yields the
W
eT T
known top mass is selected via m2 = m2 . Once detector resolution is taken into account,
t
lνb
this ideal situation need not hold. In this case, the value of pνL is chosen which yields the
204

minimum χ2 . The reconstruction efficiencies ( ) for a 1 TeV W compared with Monte Carlo
truth are found to be:

extra = 99.8%,
tlep = 98.9%,
thad = 92.3%.

(5.16)

Such high efficiencies at the parton level arise mainly from the fact that the jets are highly
boosted. Since there are combinatorial ambiguities in the final state, the efficiency for
reconstruction of a top quark decaying leptonically (tlep ) is higher than for a top quark
decaying hadronically (thad ).
Once the extra jet is identified by this kinematic method, one can require it to be a b-jet,
¯
reducing the ttj background by about a half, as is shown in the fifth column of Table 5.1. To
retain as many signal events as possible, we require only one jet to be b-tagged. A tagging
efficiency of 60% is used in our analysis. We take into account a mistag rate for a light non-b
quark (including the charm quark) to mimic a b jet, with mistag efficiency j→b = 0.5%.
For the Monte-Carlo truth events of the signal and backgrounds, we expect that 60% of the
¯
¯
signal and ttb background events pass the b-tagging, while 0.5% of the ttj background events
¯
pass the b-tagging. Recall that the ttb background is suppressed by the hard pT and HT
¯
cuts. The b-tagging will further suppress the ttj background events with an efficiency 0.5%,
if one can perfectly identify the extra-jet out of the five jets in the final state. However, the
extra-jet identification with the χ2 template method is not perfect. The jet identified as
the extra-jet has three sources: the true extra-jet, b-quarks from top (antitop) quark decay,
and the light-non-b quark from W − -boson decay. Multiplying the extra-jet fraction with
205

the corresponding jet-tagging efficiency, we show below that one obtains a net jet-tagging
¯
efficiency of 1.2 % for the ttj background, cf. Eq. (5.20), about twice as large as the case of
perfect extra-jet identification (0.5 %).
In Table 5.2 we show the tagging efficiency b−tag of the extra jet after the χ2 -template
fit. It depends on the reconstruction efficiencies for the extra jet: correct denotes the correct
fraction from the χ2 -fit, wrong−b is the fraction of b jets from top quark decay that fake
the extra jet, while wrong−light is the fraction of light jets from top quark decay that fake
the extra jet. As an example, consider the b-tagging efficiency in the signal process with
1 TeV W mass. Since there are five jets in the final state, it is possible that after event
reconstruction the extra jet is a b-jet from the top quark or anti-top quark decay (which
we label “wrong−b”), or a light-flavor jet from hadronic top quark decay (which we label
“wrong − light”), or a b-jet from the W decay (which we label “correct”). Note that the
b-tagging is applied to the extra jet (which we call “extra-j-tagging”, to avoid confusion
with the original b-tagging), but not to the truth b-jet from the W decay. Taking the
reconstruction efficiencies into account, we evaluate the net b-tagging efficiency of the extra
jet extra−j−tag as

extra−j−tag = ( correct + wrong−b ) × 0.6 + wrong−light × 0.005 ,

(5.17)

¯
for the signal process. A similar analysis gives us the same formula for the ttb background.
For a 1 TeV W with hard cuts, we find the extra jet is a b-jet with 99.9% probability, and
a light jet with 0.4% probability. Therefore, the b-tagging efficiency for the signal is

0.999 × 0.6 + 0.004 × 0.005 = 0.60.
206

(5.18)

¯
For the ttj background, the correct jet is a light-flavor jet from the real radiation associated
with top pair production. The formula changes to

extra−j−tag = ( correct + wrong−light ) × 0.005 + wrong−b × 0.6

(5.19)

¯
¯
for the ttj background. For the ttj background, in the case of a 1 TeV W with hard cuts,
we find extra jet is a light jet with 98.8% probability, and a b-jet with 1.18% probability.
¯
Therefore, the b-tagging efficiency for the ttj background is

0.988 × 0.005 + 0.012 × 0.6 = 0.012.

(5.20)

We use Eqs. (5.17-5.20) to explain the numbers in the b-tagging column in Table 5.2. Ta¯
ble 5.2 shows that about 60% of the signal and ttb background events pass b-tagging even
with the imperfect reconstruction of the extra jet. However, the extra-jet tagging efficiency
¯
for the ttj background is always larger than the b-tagging efficiency of 0.005 because it always
is possible to mistag a b-jet when the extra-jet tagging is done.

5.3.3

Mass window ∆M cut

After full event reconstruction, one can compute the W mass formed from the extra jet and
¯
¯
the reconstructed t- or t-quark. Since our signal events consist of both tW − and tW + , one
half of the signal events exhibit a peak in the invariant mass spectrum of the extra jet and t
quark (denoted as mtj ) while the other half have a peak in the invariant mass of the extra jet
¯
and t quark (denoted as mtj ). Figure 5.5 shows the reconstructed mtj and mtj distributions
¯
¯
¯
¯
for the signal (red), ttj (blue) and ttb (green) backgrounds. The signal distribution shows
207

m
W
400
¯
ttb
¯
ttj
500
¯
ttb
¯
ttj
600
¯
ttb
¯
ttj
700
¯
ttb
¯
ttj
800
¯
ttb
¯
ttj
900
¯
ttb
¯
ttj
1000
¯
ttb
¯
ttj

correct

wrong−b

wrong−light

extra−j−tag

98.15 %
98.34 %
96.67 %
98.53 %
98.34 %
96.7 %
99.32 %
98.34 %
96.75 %
99.4 %
98.6 %
97.15 %
99.66 %
98.93 %
97.6 %
99.87 %
99.24 %
98.17 %
99.79 %
99.5 %
98.65 %

1.63 %
1.5 %
2.96 %
1.35 %
1.5 %
2.92 %
0.59 %
1.48 %
2.88 %
0.51 %
1.27 %
2.55 %
0.31 %
1.0 %
2.17 %
0.12 %
0.69 %
1.63 %
0.14 %
0.43 %
1.18 %

0.22 %
0.15 %
0.37 %
0.12 %
0.16 %
0.37 %
0.08 %
0.15 %
0.36 %
0.09 %
0.12 %
0.29 %
0.03 %
0.07 %
0.23 %
0.01%
0.06 %
0.2 %
0.06 %
0.07 %
0.17 %

59.9 %
59.9 %
2.26 %
59.9 %
59.9 %
2.23 %
59.9 %
59.9 %
2.22 %
59.9 %
59.9 %
2.02 %
60 %
60 %
1.79 %
60 %
60 %
1.46 %
60 %
60 %
1.2 %

Table 5.2: The efficiency for extra jet reconstruction with the χ2 -template method. The net
b-tagging efficiencies ( extra−j−tag ), calculated with Eqs. (5.17) and (5.19), are shown in
the last column.

208

Number of events / 10 GeV

18
16
14

ttb signal
ttb background
ttj background

(a)

12
10
8
6
4
2

Number of events / 10 GeV

0
0

500

1000

1500
Mtj [GeV]

18
16
14

ttb signal
ttb background
ttj background

(b)

12
10
8
6
4
2
0
0

500

1000

1500
Mtj [GeV]

Figure 5.5: (a) Reconstructed invariant mass distribution of the leptonic decaying top and
extra jet; (b) reconstructed hadronic decaying top and extra jet invariant mass distribution.
209

a sharp peak at the input value of m . The pin shape reflects the narrow width of the
W
top-philic W boson, e.g. the W width is about 8 GeV for a 1 TeV W . The long tail
into the small mass region comes from the one-half wrong combination. The peaks of the
background distributions around 800 GeV are caused by the combination of hard kinematic
cuts and jet identification (with combinatorial factors included).
Once mW is known, we can impose cuts on mtj or mtj to further suppress backgrounds.
¯
¯
We first demand large invariant masses for both tj and tj,

mtj > 250 +

mW
,
4

mtj > 250 +
¯

mW
,
4

(5.21)

and that one of the following two mass window cuts be satisfied,

mtj − m
W

<

mW
, or
10

mtj − m
¯
W

<

mW
.
10

(5.22)

The mass window suppress both SM backgrounds by a factor of 10 while it keeps most of
the signal.

5.3.4

Discovery potential

The SM backgrounds are suppressed efficiently such that less than 1 background event survives after cuts with an integrated luminosity of 100 fb−1 . For a 1 TeV W with the same
coupling strength as the SM W -t-b interaction, we obtain a 5 standard deviations (σ) sta√
tistical significance, defined as S/ B where S and B denotes the number of signal and
background events, respectively. For a lighter W , the significance is larger for fixed coupling strength. The 3 σ and 5 σ discovery curves are plotted in Fig. 5.6. The region above
210

the 5 σ curve is good for discovery.

211

fL

2
5 σ with 100 fb-1
3 σ with 100 fb-1

1.8
1.6
1.4
1.2

S>5σ

1
0.8
0.6
0.4
0.2
0
400

600

800

1000
MW’ [GeV]

Figure 5.6: The discovery potential for the top-philic W 14 TeV with an integrated luminosity of 100 fb−1 .

212

5.4

W -t-b coupling and t-polarization

After the discovery of this W boson, one would like to know its mass, spin, and couplings.
The invariant mass or transverse momentum distributions of its decay products can be used
to determine its mass. Angular distributions of its decay products can be investigated to
confirm its spin and the chiral structure of the W couplings to SM fermions.

The

chirality of the W coupling to SM fermions is best measured from the polarization of the
top quark [71, 94]. Among the top quark decay products, the charged lepton from t → blν
is the best analyzer of the top quark spin. For a left-handed top quark, the charged lepton
moves preferentially against the direction of motion of the top quark, while for a right-handed
top quark the charged lepton moves along the direction of motion of the top quark. The
angular correlation of the lepton is

1
(1 ± cos θl ), with the (+) choice for right-handed and
2

(−) for left-handed top quarks, where θl is the angle of the lepton in the rest frame of top
quark relative to the top quark direction of motion in the center-of-mass (cm) frame of the
incoming partons. In Figs. 5.7 and 5.8 we plot the cos θl distribution for fL = 1, fR = 0
and fL = 0, fR = 1 couplings. The curves clearly show the main characteristic features of
1
the (1 ± cos θl ) behaviors for purely right- and left-handed polarized top quarks from W
2
decay, even after kinematic cuts are imposed. We note that due to the pT and ∆R cuts, the
distributions are distorted and drop significantly in the region cos θl ∼ −1 for fL = 1 and
fR = 0, and cos θl ∼ 1 for fR = 1 and fL = 0. We expect a flat angular distribution for
the SM background because the top quark and anti-top quark are not polarized. Therefore,
the angular distributions of the lepton can be used to discriminate top-philic W models in
which the chirality of the W coupling to SM fermions differs.

213

Number of events

4
Left-handed W’
Right-handed W’

3.5
3

signal

2.5
2
1.5
1
0.5
0
-1

-0.5

0

0.5

1
cosθl

Figure 5.7: The angular distributions of the final lepton cos θl for the left-handed and the
right-handed W .

214

Number of events

8
7.5

ttb background
ttj background

7

backgrounds

6.5
6
5.5
5
4.5
4
3.5
-1

-0.5

0

0.5

1
cosθl

Figure 5.8: The angular distributions of the final lepton cos θl for the SM background.

215

5.5

Summary

In this chapter we examined the LHC phenomenology of a top-philic W model. In the model
the W boson is produced in association with a top-quark and it decays into a top quark and
¯
bottom quark pair, yielding a collider signature of tt plus one b-jet. We exploited the different
kinematic features of the signal and backgrounds to suppress the large standard model
¯
¯
backgrounds from ttj and ttb production. Examining the distributions of the signal and
¯
backgrounds, we found that hard pT cuts and cuts on HT can suppress the ttb background.
After full event reconstruction, we showed that tagging the extra b-jet can further suppress
¯
the ttj background. We showed that discovery of a top-philic W with SM-like coupling
strengths is promising at 14 TeV with L = 100 f b−1 . A resonance peak in the top quark
and b-jet invariant mass distribution is a distinct signature of W discovery. Top quark
polarization can be used to measure the chiral structure of the W -t-b coupling. Top quark
pair and hard b-jet final states are worth examining even in a model-independent way. This
final state is a new unexploited channel at the LHC.

216

Chapter 6
CONCLUSION
This thesis presents a study of the phenomenology of heavy charged gauge bosons. We
studied the heavy gauge bosons within the framework of an effective field theory with the
gauge symmetries SU(2)1 × SU(2)2 × U(1)X models (the so-called G(221) models). Such a
formalism provides a unified view of the possible symmetry breaking patterns to recover the
standard model as well as the gauge couplings of the fermions. The G(221) models can be
separated into two classes of breaking patterns: breaking pattern I (BP-I), and the breaking
pattern II (BP-II). Examples of models in BP-I are the left-right (LR), leptophobic (LP),
hadrophobic (HP) and fermiophobic (FP), while examples in BP-II are the sequential (SQ),
the ununified (UU) and non-universal (NU) models.
We studied the parameter constraints on the G(221) models and placed bounds on the
masses of the heavy gauge bosons. For the electroweak precision constraints, we performed
a global-fit analysis on a set of 37 electroweak observables, including Z pole data, the mass
and the width of the W ± boson, and various low-energy observables. Moreover, the direct
searches of the W and Z bosons at the Fermilab Tevatron and CERN LHC further con217

strained the G(221) model parameter space. We summarized here the most important results
from the indirect and direct constraints. For the left-right model, the direct search data at
the LHC with 7 TeV energy gives a stronger constraint on W and Z masses than both the
electroweak and Tevatron constraints. It excludes the region where the W mass is smaller
than 1.7 TeV, and where the Z mass is smaller than 2.3 TeV. For the sequential model,
although W and Z with degenerate masses below 500 GeV are allowed by the EWPTs at
large cφ , the limits from Tevatron and LHC exclude the region where W and Z masses
are smaller than 1.5 TeV. For all other models, the electroweak precision data still gives the
strongest constraint on the W and Z masses, because of the non-universal flavor structure
in these models. In breaking pattern I, with all combined constraints, all the phobic models
can still have relatively light W with a mass of 500 GeV mass, but with a heavier Z of
mass 1.5 TeV. For the non-universal models, such as top-flavor and un-unified models, the
electroweak indirect constraints are tighter than Tevatron and LHC7 direct constraints, and
push the new gauge boson mass up to more than 2 TeV in top-flavor model and 3 TeV in
un-unified model.

The Drell-Yan production, which has the simplest topology and very clean leptonic final
states, is the most promising channel for early discovery of W and Z bosons. In the general
SU(2)1 × SU(2)2 × U(1)X models, we calculated the Drell-Yan production cross section at
the next-to-leading order. Our study revealed that the discovery potential of W and Z
bosons is highly limited by the indirect constraints. With a full scan of the parameter space,
we are able to predict the time scale to probe/exclude the W and Z by accessing more
parameter region as the luminosity goes up at the LHC. In the left-right model it is more
efficient to use the W leptonic decay channel for discovery or exclusion than the Z leptonic
218

decay channel. In the phobic models, observing a Z alone cannot rule out the possibility
of a non-abelian gauge extension of new physics. In BP-II models, both the Z and the W
leptonic decay channel are suitable to explore the allowed parameter space. Discovering a
degenerate W and Z in the leptonic decay channels at the same time will be the distinct
feature in BP-II.
The top quark plays a special role in the standard model, and offers a window into
possible new physics beyond the SM. In many models, because the top quark is naturally
related to EWSB, new physics is very sensitive to the top quark observables. The observables
such as angular, spin correlations and polarization of the top quark can be used to determine
the particle properties and its chiral structures. We studied the heavy charged gauge bosons
in the single top final states. We showed that it is possible to probe the chiral structure of
the charged gauge bosons using the top quark polarization. As another example, since both
CDF and D0 confirmed a large forward-backward asymmetry in the top pair production,
we studied how a flavor-violating W model might explain this asymmetry. We investigated
the correlation between the top quark At B and the final-state lepton Al B in the flavorF
F
violating W models. We studied the kinematic and dynamic aspects of the relationship
between At B and Al B , arising from the spin correlation between the charged lepton and
F
F
the top quark with different polarization states. Our study revealed a strong correlation
between these two asymmetries, and we conclude that a model which produces more righthanded than left-handed top quarks is favored by the present data.
We also looked for unexpected new signatures for the exotic W discovery. We studied a
”top-philic” model, in which the W couples only to third-generation quarks and is produced
only in association with a top quark. The final state, top quark pair plus a hard b-jet, is
219

a new unexploited channel at the LHC. We performed a detailed study on the top quark
pair plus a hard b-jet final states, and expect that our analysis can be used in a model¯
independent study. To discriminate signal from the large background ttj, we studied the
distributions of the signal and backgrounds and found that several kinematic features, such
¯
as HT and b-tagging, can be used to suppress the large ttj backgrounds. We showed that
discovery of a top-philic W with SM-like coupling strengths is promising at 14 TeV with
L = 100 f b−1 . A resonance peak in the top quark plus b-jet invariant mass distribution is
a distinct signature of W discovery. Furthermore, top quark polarization can be used to
measure the chiral structure of the W -t-b coupling. The Top quark pair plus hard b-jet final
state is a new unexploited channel at the LHC, which is worth continued examination in a
model-independent way.
As a final outlook, we would like to mention several theoretical considerations left for
future investigation. At the LHC, if there is enough luminosity, the vector boson pair production, and the vector boson and Higgs associated production channel will be the primary
channels to study. The reason is that the vector boson channel is essential to explore the
spontaneous symmetry breaking at the TeV scale. The vector boson channels will probe a
different part of the theory about resonances, whether the resonance is a gauge boson or
just a vector boson, whether the resonance is relevant to spontaneous symmetry breaking
or not. In addition, there are also other kinds of W -bosons, such as T-parity odd W , and
Kaluza-Klein W resonances which can be studied in future.
With the start of the LHC, a new era of discovery in particle physics has just begun. It
is the time to study the collider phenomenology of new physics beyond the standard model.
Although in this thesis we focus on the charged gauge boson resonances, similar strategies can
220

be performed to study the collider phenomenologies for other kinds of new heavy particles.
We look forward to how the LHC will revolutionize particle physics by opening the TeV
energy region to direct experimental exploration.

221

APPENDICES

222

Appendix A
Statistics for Upper limit and
Significance
In the high energy experiments, a very important aim is to search for new physics beyond
the standard model. Statistical data analysis will be used either to set limits on new physics
or to claim a new discovery. To distinguish a possible new physics signal s from the existing
standard model background b, hypothesis testing in statistical method plays an important
role. When setting limits, the model with signal plus background plays the role of null
hypothesis, which is tested against the background-only hypothesis H1 . When discovering
a new signal process, one defines the null hypothesis as background, which is tested against
the alternative signal hypothesis.
The Bayesian and the classical frequentist, are the two statistical approaches commonly
used in high energy physics to do hypothesis testing. The difference between frequentist and
Bayesian is:
• in frequentist statistics, data are repeatable random sample, while underlying param223

eters remains constant during this random process (parameters are fixed);
• in Bayesian statistics, parameters are unknown random variables, while data are observed from the realistic experiments (data are fixed).
They are different points of view: the frequentist statistician does many repeated trials given
an underlying true theory, while the Bayesian tries to get the degree of belief of an incomplete
theory given the existing measurements. For more details, we refer readers to refs. [115]. In
the following, we will apply the hypothesis testing in the counting experiments to illustrate
the two approaches. Suppose one has an experiment that counts n events, modeled as a
Poisson distribution with mean s + b, where s is the expected signal rate, b is the expected
background rate. The probability of measuring n events is therefore

P (n|s, b) =

A.1

(s + b)n −(s+b)
e
.
n!

(A.1)

Upper Limit

Both Bayesian and the classical frequentist methods use confidence level (CL) to quantify
the level of incompatibility of data with a signal hypothesis. Usually one requires a 95% CL
for excluding a signal hypothesis.

A.1.1

Bayesian Approach

In the signal hypothesis, one would like to set limit on the expected cross section σ of a
signal process. Consider there are N channels (or bins), and for channel i, one observes ni
events, assuming the expected background rate is bi , and the acceptance is i with integrated
224

luminosity Li .
In the absence of systematic errors from subsidiary measurements, the joint likelihood
for combined N channels can be written as the product of individual likelihood
N
L(n|σ, b, ) =
i=1

(σLi i + bi )n! −(σL +b )
i i i ,
e
n!

(A.2)

where the vectors indicate the variables are arrays carrying index i.
To quantify the degree of belief on this signal hypothesis, the posterior probability is defined to be related with the confidence level. Using Bayes’ theorem, the posterior probability
density can be obtained, given the likelihood and the prior probability density π(σ) for the
signal cross section. The posterior probability density is

P(σ|n, b, ) =

1
L(n|σ, b, )π(σ),
N

(A.3)

where the normalization factor N integrated all the possible σ. Typically the function P(σ)
is commonly taken to be flat for σ ≥ 0 and zero otherwise.
The limit is now obtained by integrating this posterior probability density for σ until we
achieve the 95% of the total integral from zero to infinity. The 95% CL upper limit σ95%CL ,
is given by
σ95%CL
P(σ |n, b, )dσ = 95%.
0

(A.4)

In the case of single channel with constant prior probability density, the final expression is

1−

Γ(n + 1, σσ
L)
95%CL
= 95%.
Γ(n + 1, b)
225

(A.5)

To include the systematic errors (nuisance parameters), one would consider the prior
probability density from the subsidiary measurements π(σ, b, ). Then one marginalizes the
nuisance parameters by integrating over them. The posterior probability density can be
rewritten as
P(σ|n) =

1
N

L(n|σ, b, )π(σ, b, )dbd .

(A.6)

The 95% CL upper limit σ95%CL , is given by
σ95%CL
P(σ |n)dσ = 95%.
0

(A.7)

To get the expected limit, one would treat n as b in above integration.

If the number of observed events is large, the Poisson likelihood asymptotically tends to
a Gaussian with mean n and standard deviation

(s + b)n −(s+b)
e
P (n|s, b) =
n!

√

n:

→

−
e

(s + b − n)2
2n
.

(A.8)

In practice, the approximation is already good for n ≥ 10. The integration
s95%CL
0

s95%CL
P (n|s, b)ds
P(s|n, b) = 0 ∞
0 P (n|s, b)ds

(A.9)

can be calculated analytically
√
√
ERF (s95%CL + b − n)/ 2n + ERF[(n − b)/ 2n]
√
= 95%.
1 + ERF[(n − b)/ 2n]
226

(A.10)

The expected limit can be obtained through
√
ERF (s95%CL )/ 2b = 95%.

(A.11)

The final result is

s95%CL = 1.96 ×

√

√

b

→

b
,
σ95%CL = 1.96 ×
L

(A.12)

where one can infer a scaling behavior

1
σ95%CL ∼ √ .
L

(A.13)

For the combined upper limits of N channels, the combined likelihood can be written as

N (σ L + b )n
i i
i i −(σ i Li +bi )
e
P (n|s, b) =
ni !
i=1

N
→

−
e

(σ i Li + bi − ni )2
2ni
.

i=1
(A.14)

From the properties of product of Gaussian distribution, the Gaussian likelihood can be
rewritten as

ni − bi 2
σ−
i Li
(σ − µ)2
N −
−
2
2ni /( i Li )
2κ2 .
=e
P (n|σ, , b) =
e
i=1

227

(A.15)

where

µ =

κ2 =

N ni − bi
i 2n /( L )
i i i ,
1
N
i 2n /( L )2
i i i
1
.
2
N ( i Li )
i
ni

(A.16)

(A.17)

we can get the combined upper limit

s95%CL = 1.96 × κ.

(A.18)

From the above, we can obtain the following relation

2
σ95%CL =

A.1.2

1
1
N
i σ2
95%CL i

.

(A.19)

Frequentist Approach

The frequentist approach requires the selection of a test statistic q as measure of discrepancy
between the data and the hypothesis, with higher values of q corresponding to increasing
disagreement. As a signal-vs-background discriminator, there are many kinds of choices of
the test statistic:

• a simple choice will be the possible observed number of events n during the trial
measurement;

• By the Neyman-Pearson lemma, the likelihood ratio Q is the most powerful discrimi228

nator, defined as
qµ = −2 ln

L(n|µs + b)
,
L(n|b)

(A.20)

where µ is a strength parameter defined such that µ = 0 is the background-only
hypothesis and µ = 1 is the hypothesis of background plus signal. This test statistic
was used by LEP and the Tevatron. In references, there are various modifications of
this method: Feldman-Cousins, CLs , Power-Constrained Limits, etc.
• At the LHC, the profile-likelihood test statistic qµ is used. The profile likelihood ratio
is defined as
ˆ
L(µ, ˆ
b)
λ(µ) =
.
L(ˆ, ˆ
µ b)

(A.21)

ˆ
ˆ
Here L(µ, ˆ is the likelihood maximized ˆ for the specified µ i.e., it is the conditional
b)
b
maximum-likelihood (ML) estimator of b (and thus is a function of µ). L(ˆ, ˆ is the
µ b)
maximized (unconditional) likelihood function, i.e., µ and ˆ are their ML estimators.
ˆ
b
The profile likelihood ratio is thus

tµ = −2 ln λ(µ).

(A.22)

In the absence of systematic errors, for the Poisson likelihood with given observable n

L(µ, b; n) =

(µs + b)n
n!,
e−(µs+b)

(A.23)

the profile likelihood with given n is

tµ = −2 ln λ(µ) = 2 (µs + b − n) + n ln
229

n
.
µs + b

(A.24)

For large µs + b, the test statistic can be expanded around x = (n − µs − b)/(µs + b)
as
tµ =

(n − µs − b)2
µs + b

1−

x x2
+
+···
3
6

.

(A.25)

Here the distribution for n is approximately a Gaussian distribution with mean µs + b
and variance σ0 =

√
µs + b. So we can rewrite
(µ − µ0 )2
,
2
σ0

tµ

where µ0 =

(A.26)

√
n−b
and σ0 = µs + b/s. For the multi-channel case, the test statistic
s

is
N
tµ = −2 ln λ(µ) =

2 (µs + bi − ni ) + ni ln
i=1

ni
.
µs + bi

(A.27)

Having defined the test statistic, next one constructs the probability density f (tµ |µ)
of the chosen test statistic qµ under the signal hypothesis through many repeated pseudoobservations. Using the probability density f (tµ |µ) one can quantify the level of disagreement by defining the p-value

pµ =

∞
tµ,obs

f (tµ |µ) dtµ ,

(A.28)

where tµ,obs is the value of the statistic tµ observed from the data. Such a p-value is
denoted as CLs+b , which relate to confidence level as 1 − CLs+b . The probability densities
are different for different test statistics, such that in the simplest choice of test statistic n
the probability density is a simple Poisson distribution.
230

At the LHC, the default test statistic is the profile likelihood ratio, because this method
has the simplest way to incorporate systematics into hypothesis, and its known asymptotic
properties. We will focus on this profile likelihood method and calculate the upper limit.

In order to find the p-value, the sampling distribution of the test statistic f (qµ |µ) need
to be obtained. We also need the distribution f (qµ |µ ) with µ = µ to find how this is
distributed (the median) if the data correspond to a strength parameter different from the
one being tested. Here we present an approximate distribution for the profile likelihood
ratio, valid in the large sample limit. In the large sample limit, one can model µ, which is
ˆ
the value of µ that maximizes the likelihood, as a Gaussian distributed variable. If µ follows
ˆ
a Gaussian distribution with a mean µ and standard deviation σ, we can approximate the
test statistic as
−2 ln λ(µ) =

√
(µ − µ)2
ˆ
+ O(1/ N ) .
σ2

(A.29)

and determine its distributions of the test statistics. The standard deviation σ of µ is
ˆ
obtained from the covariance matrix of the estimators for all the parameters,

−1
Vij = −E

∂ 2 ln L
∂θi ∂θj

,

(A.30)

where here the θi represent both µ as well as the nuisance parameters. If all nuisance
parameters are fixed, µ will be the value
ˆ

E[µ] = E[

n−b
n−b
]=
,
s
s
231

(A.31)

which maximizes the likelihood. For the variance, one can take θ0 = µ, so
−1
V00 = −E

nµ2
∂ 2 ln L
=E
∂µ∂µ
(µs + b)2

=

s2
,
µs + b

(A.32)

where L is a Poisson distribution, and E[n] = µs + b. Then we obtain

σ 2 = V00 =

µs + b
.
s2

(A.33)

If µ is Gaussian distributed, one can show the statistic tµ follows a non-central χ2
ˆ
distribution with one degree of freedom,

f (tµ ; Λ) =

1
2

1
1
√
exp −
2
tµ 2π

tµ +

√

Λ

2

+ exp −

1
2

tµ −

√

Λ

2

, (A.34)

where the parameter Λ is
Λ=

(µ − µ )2
.
σ2

(A.35)

For the special case µ = µ one has Λ = 0 and −2 ln λ(µ) approaches a chi-square distribution
for one degree of freedom.

For purposes of computing limits, we introduce a modification to the profile likelihood


 −2 ln λ(µ) µ ≤ µ ,
ˆ


ratio definition as
qµ =

(A.36)



0

µ>µ,
ˆ

The reason for setting qµ = 0 for µ > µ is we want to determine the smallest µ such that
ˆ
there is a fixed small p-value to find data as compatible with that value of µ or less. Similar
232

to the above, the pdf f (qµ |µ ) can be written as

f (qµ |µ ) = Φ

µ −µ
σ

δ(qµ ) +

√

1 1
1
1
√ √ exp −
2 2π qµ
2

qµ −

µ−µ
σ

2


 ,

(A.37)

In the special case µ = µ is a half-chi-square distribution:

1
1
1 1
f (qµ |µ) = δ(qµ ) + √ √ e−qµ /2 .
2
2 2π qµ

(A.38)

The cumulative distribution is

F (qµ |µ ) = Φ

√

qµ −

µ−µ
σ

,

(A.39)

and the corresponding special case µ = µ

F (qµ |µ) = Φ

√

qµ .

(A.40)

The p-value can be calculated as

pµ = 1 − F (qµobs |µ) = 1 − Φ

qµobs ,

(A.41)

and the upper limit is
µup = µ + σΦ−1 (1 − α) .
ˆ

233

(A.42)

where α is the upper limit of pµ . At a confidence level of 95%, p-value is 1 − α = 5%, so one
has
µup = µ + σΦ−1 (0.05) = µ + 1.64σ .
ˆ
ˆ

(A.43)

To determine µ and σ one further take the Gaussian approximation. For large µs + b, the
ˆ
distribution of n is Gaussian distributed with mean µ and σ. For given n, µ is the maximized
ˆ
ˆ
√
√
µs + b
ˆ
n
n−b
and σ can be approximately treated as σ =
=
. For
value of µ, with µ =
ˆ
s
s
b
√
the expected upper limit µ = 0 and σ = b/s, so we get
ˆ
√
b
.
µup = 1.64
s

(A.44)

We then derive the combined upper limit. The test statistic for N multi-channel processes
is
N
(µi − µi )2
ˆ
=
χ2 ,
i
2
σi
i=1
i=1
N

qµ =

(A.45)

where µ is the is a strength parameter, or saying the ratio between Higgs cross section and
reference standard model Higgs cross section. Since the product of Gaussian distribution
has such a property:
ˆ
N µi
i σ2
i ,
N 1
i σ2
i
1
,
N 1
i σ2
i

µ =
ˆ

σ2 =

234

(A.46)

(A.47)

the test statistic of N multi-channel can be rewritten based on µ and σ 2 as
ˆ

qµ =

(µ − µ)2
ˆ
.
σ2

(A.48)

At a confidence level of 95%, p-value is 1 − α = 5%, so one has
µup = µ + σΦ−1 (0.05) = µ + 1.64σ .
ˆ
ˆ

For the expected upper limit µ = 0 and σ =
ˆ

√

(A.49)

b/s, so we get the combined limit

√
µup = 1.64

b
= 1.64 × σ ,
s

(A.50)

bi
= 1.64 × σi .
si

(A.51)

and for each channel
µiup = 1.64
From Eq. A.46, one can get

1

µ2 =
up

N 1
i µ2
iup

.

(A.52)

The result is the same as the one using Bayesian approach.
To include the systematic errors, one can define the complete profile likelihood ratio with
background errors. When one does the maximization of the likelihood, one need to solve the
ˆ
ML estimators ˆ and ˆ There is another method to include systematic errors, in which all the
b
b.
errors are included as nuisance parameters. In this case, one need to do the marginalization
for the likelihood, and then take the ratio of the profile likelihood.
235

A.2

Discovery Significance

In the case of observing an excess of events, the statistical significance of an observed signal can be quantified by means of a p-value or its equivalent Gaussian significance in the
frequentist approach, and Bayes Factor in the Bayesian approach.

A.2.1

Bayesian Approach

In the Bayesian approach, the Bayesian could define a interval that reflects the degree-ofbelief for each hypothesis. The ratio of these intervals is defined as Bayes factor,

B=

likelihood with s = 0, marginalized over b
,
likelihood with µ = 0 marginalized over s and b

(A.53)

which is quite similar to the definition of the profile likelihood ratio, with Bayesian marginalization replacing maximization. Unlike the frequentist approach, one doesn’t need to know
the probability density of all possible outcomes of the measurement for a given hypothesis. However, the disadvantages with this Bayesian approach are that one has to assume
prior distributions for each hypothesis, and one is only allowed to make relative confidence
statements about two hypotheses. Due to the above disadvantages, the Bayes factor only
has limited usage on new physics discovery in high energy physics. We will focus on the
discovery in the frequentist approach.
236

A.2.2

Frequentist Approach

For the discovery, the p-value is defined as the probability of getting an observation as or
less compatible as seen in data for the background-only hypothesis, in the following

p0 =

∞
t0,obs

f (t0 |b) dt0 ,

(A.54)

where t0,obs is the value of the statistic t0 observed from the data. The p-value can be
converted into an equivalent significance Z, defined such that a Gaussian distributed variable
via
Z = Φ−1 (1 − p) .

(A.55)

For a signal process, one commonly regards the rejection of the background hypothesis with
a significance of at least Z = 5 as an appropriate level to constitute a discovery. This
corresponds to p = 2.87 × 10−7 . For purposes of excluding a signal hypothesis, a threshold
p-value of 0.05 (i.e., 95% confidence level) is often used, which corresponds to Z = 1.64.
We will focus on the test statistic q0 used at the LHC [116]. From the modified definition
of q0 , we therefore have

 2 2
 µ /σ
ˆ


µ≥0,
ˆ
(A.56)

q0 =


0


µ<0,
ˆ

where µ follows a Gaussian distribution with mean µ and standard deviation σ. One will
ˆ
get a similar probability density with µ = 0. For the special case of µ = 0, the probability
density reduces to
1 1
1
1
f (q0 |0) = δ(q0 ) + √ √ e−q0 /2 ,
2
2 2π q0
237

(A.57)

which is a mixture of a delta function at zero and a chi-square distribution for one degree of
freedom. The corresponding cumulative distribution is

F (q0 |µ ) = Φ

√

µ
q0 −
σ

.

(A.58)

and the special case µ = 0
F (q0 |0) = Φ

√

q0 .

(A.59)

The p-value of the µ = 0 hypothesis is

p0 = 1 − F (q0 |0) ,

(A.60)

√
Z0 = Φ−1 (1 − p0 ) = q0 .

(A.61)

and therefore the significance is

In the counting experiment, if we regard b as known, the data consist only of n and thus
the likelihood function is
(µs + b)n −(µs+b)
e
,
L(µ) =
n!

(A.62)

The test statistic for discovery q0 can be written


 −2 ln L(0) µ ≥ 0,

ˆ

L(ˆ)
µ
q0 =


0

µ<0,
ˆ

(A.63)

where µ = n − b. For sufficiently large b we can use the asymptotic formula (A.61) for the
ˆ
238

significance,
Z0 =

√







2 n ln

q0 =


0


n
+b−n
b

µ ≥ 0,
ˆ
(A.64)
µ < 0.
ˆ

To approximate the expected (median) significance assuming the nominal signal hypothesis
(µ = 1) we replace n by the Asimov value s + b to obtain

med[Z0 |s + b] =

q0,A =

2 ((s + b) ln(1 + s/b) − s) .

(A.65)

For large b, the test statistic can be expanded around x = (n − b)/(b) as

q0

(n − b)2
.
b

(A.66)

Here the distribution for n is approximately a Gaussian distribution with mean b and variance
√
σ0 = b. So we can rewrite

n−b
 √

µ ≥ 0,
ˆ

√
b
Z0 = q0 ∼


0

µ < 0.
ˆ

(A.67)

Expanding the logarithm in s/b one finds the expected (median) significance assuming signal
rate
s
med[Z0 |s + b] = √ (1 + O(s/b)) .
b

(A.68)

√
Although Z0 ≈ s/ b has been widely used for cases where s + b is large, one sees here that
this final approximation is strictly valid only for s

239

b.

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