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GLOBAL PRECISION ANALYSIS OF SU(2) S SU(2) e U(1) MODELS
iBy

Kai Ruven Sclnnitz

A THESIS

Submitted to
Michigan State University
in partial fulfillment Of the requirements
for the degree Of

MASTER OF SCIENCE
Physics and Astronomy

2009

ABSTRACT
GLOBAL PRECISION ANALYSIS OF SU(2) st SU(2) e U(1) MODELS
By

Kai Ruven Schmitz

G (221) models extend the electroweak gauge group Of the Standard Model by an
additional 5 U (2) which results in the presence of three new heavy gauge bosons Z ’
and W'i with masses at the TeV scale. In this thesis a global fit analysis Of the most
prominent G (221) models -——— the left-right (LR), leptophobic (LP), hadrophobic (HP)
and fermiophobic (FP) models as well as the ummified (UU) and non-universal (NU)
models ~— is presented. Utilizing a modified version Of the Fortran plotting package
GAPP the G (221) models are fitted to a set Of 37 electroweak observables including a
multitude Of Z pole observables, the mass and the width of the Wi boson, the mass
Of the top quark and various low-energy observables. The experimental precision with
which the electroweak observables have been measured allows to put strong bounds
on the parameters Of the G(221) models and to constrain the masses Of the Z’ and
the W ’ i. As a confirmation Of the power Of the Standard Model the scale Of the new
physics in the G(221) models is generally found to be very high. For each G(221)
model under study the most important observables that drive the minimization Of
x2 can be identified. Among the most. relevant observables are the hadronic cross

+ annihilation and the weak vector charge Of cesium-133. To illustrate

section in 6—6
which values of the G(221) parameters are consistent with the experimental data
plots Of the parameter space are presented that indicate the viable regions at 95%
CL. Likewise plots Of the Z ’ and W’i masses demonstrate which masses Of the new
heavy gauge bosons are already ruled out by the data and which are still possible. In
a closing remark the constraints from the Z W+W“ vertex on the G’ (22 1) parameters

are considered. AS it turns out the bounds on the Z W+W_ coupling do not affect.

the results of the global fit analysis.

ACKNOWLEDGMENT

In last August I came tO Michigan State to learn more about elementary particles
and the fundamental forces of nature. I had not any attended lecture on high-energy
physics in my luggage, just my thirst for knowledge and the perception that I would
miss a great and intriguing field in physics if I did not take the Opportunity and study
particle physics at MSU. Now, almost one year later, I find myself becoming a high-
energy physicist that is fascinated by the topics he is working on. All this would not
have been possible without the help and support of many people that stood by my
side in the last months. I would like to take this opportunity tO express my gratitude
to all Of them.

In the first place, I wish to thank my advisor, Prof. Chien—Peng Yuan. When I
first stepped into his Office in last August looking for someone I could work for as a
Research Assistant he Offered me to join his group right away. Brimming with ideas
and passionate about the projects he proposed to me, he made me sense from the
very first what a blessing it would be to have him as my advisor. Since then he has
proven to be an excellent guide on my way through the landscape Of particle physics.
Thanks to his mentoring I learned much more in the last year than I had ever dared
to hope.

I would also like to thank Prof. Wayne Repko and Prof. Carl Schmidt who have
made themselves available as members Of my advisory committee. Not only do I
appreciate their willingness to read my thesis and to be on my committee, I am also
very thankful to them for the deep insight into Quantum Field Theory and Quantum
Electrodynamics that I could gain in their classes. Had I not taken their courses I
would not have been prepared for writing this thesis.

One Of the pillars Of this project was the Fortran package GAPP that has been
written by Professor Jens Erler. I would like to express my deepest gratitude to

Professor Erler for placing the newest version of the GAPP code at my disposal and

iii

for helping me so patiently while I was making my first steps with GAPP. Without
his support it would have taken me much longer to get started with my thesis.

My stay at Michigan State only became possible due tO the cooperation program Of
the German National Merit Foundation with the MSU Physics Department. I deeply
acknowledge the support that. I received from both institutions. On the American
Side it was Prof. Wolfgang Bauer that helped making my year abroad a success. The
driving forces on the German side were Dr. Astrid Irrgang, Dr. Inga Scharf, Dr.

Julia Schuetze and Annette Schwarzenberg. My special thanks go to all Of them.

Furthermore, I am especially grateful to Ken Hsieh who assisted me in carrying
out this project in countless ways. He taught me so much about the Standard Model
and electroweak physics in general, was always there for me when I needed assistance
and never became tired of answering my questions. In the last months he was not
only an excellent teacher and collaborator but became also a good friend. My thanks
go out to Jianghao Yu who especially assisted me at the first stage Of my project.
Without his help I would not have been able tO start working on this thesis. Over

the last year he has become much more to me than just an Office mate.

My Master studies would not have been the same without my friends that I made
at MSU during the last months. I can consider myself fortunate that I met Kirstie
Sieloff and Martin Schuetz. The adventures that we had will always be on my mind.
I am also particularly thankful to Sarah Heim, Christine Sung and Martin Kell for
all the good times we had together. Stefano Di Chiara, Mohammad Hussein and
Baradhwaj Panayancheri-Coleppa made the Office we worked in together to a pleasant
place. I would like to thank them for creating such a nice atmosphere. Furthermore,
my thanks go out to my friends in Old Europe that did not forget. me while I was
studying in the New World. I am looking forward to see you again soon.

Finally, my Sincere gratitude goes to my parents, my brother and my sisters who

supported me from home as best as they could and last but definitely not least to my

iv

girlfriend and best friend, Monia Glaeske. I only made the decision to study abroad
because I knew that our relationship would be able to endure one year Of separation.
She gave me strength and confidence throughout this year and in the end it was her

encouragement that made this thesis possible.

TABLE OF CONTENTS

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

List Of Figures ................................

Introduction
1.1 Motivation Of This Study .........................
1.2 The SM and Its Extension by a Second 8 U (2) .............

New Physics Models

2.1 Classification ...............................
2.1.1 Symmetry Breaking Pattern ...................
2.1.2 Higgs Representation at the First Breaking Stage .......
2.1.3 Assignment Of the Fermion Charges ...............

2.2 Masses Of the Gauge Bosons .......................
2.2.1 Higgs Representations ......................
2.2.2 Gauge Couplings and Mixing Angles ..............
2.2.3 Mass Matrices and Mixing Of the Gauge Bosons ........
2.2.4 Physical Boson States and Masses ................

2.3 Gauge Interactions Of the Fermions ...................
2.3.1 Fundamental Fermion—Boson Interactions ............
2.3.2 Effective Lagrangian at the Electroweak Scale .........
2.3.3 Effective Four-Fermion Interactions ...............

Global Fit Analysis with GAPP
3.1 Parametrization ..............................

3.1.1 Fundamental Model Parameters .................

3.1.2 Relations tO the Standard Parameters ..............
3.2 Electroweak Observables .........................
3.2.1 Overview of the Included Observables ..............
3.2.2 New Physics Corrections .....................
3.3 Numerical Analysis ............................

3.3.1 Introduction to GAPP ......................
3.3.2 Modification Of the Code .....................
3.3.3 Fitting and Scanning the Models ................

Results

4.1 Fits to the Electroweak Data .......................
4.1.1 Best Fit Values ..........................
4.1.2 Higgs Mass Dependence .....................

vi

viii

49
50
50
54
61
61
79
90
91
93
99

102
103

4.2 Allowed Regions in Parameter Space .................. 110

4.2.1 General Features ......................... 112

4.2.2 Observables Driving the Plots .................. 114

4.3 Concluding Remarks ........................... 121
4.3.1 Constraints Fl‘om Triple Gauge Boson Couplings ........ 121

4.3.2 Future Prospects ......................... 126
Experimental Input 129
Al Reference Observables a, G’ F and M Z ................. 129
A2 Electroweak Observables ......................... 130

B MS—Bar Mass of the TOp Quark 133
C Bounds on the G(221) Models 137
CI Parameter and Mass Plots ........................ 137
C2 Pull Distributions ............................. 147
Coupling Coefficients 153
D1 Couplings in .%W_ ............................. 153
D.1.1 Fermion Couplings to the Z Boson ............... 153

D.1.2 Couplings Of the New Physics Currents ............. 158

D2 Couplings in 24f ............................. 170
D.2.1 Couplings Of the Neutral Fermion Currents ........... 170

D.2.2 Couplings of the Charged Fermion Currents .......... 173
Bibliography ..................................................... 189

vii

2.1

2.2

2.3

2.4

2.6

2.7

2.8

2.9

2.10

3.1

3.2

3.3

3.4

LIST OF TABLES

Charges of the fermicms under the G (221) gauge group ........
Representations of the Higgs fields (I) and H ..............
Lagrangians Of the Higgs fields (I) and H ................
Fundamental gauge couplings and mixing angles ............
Entries Of the fundamental boson mass matrices ............
Masses Of the physical gauge bosons ...................

Building blocks of the fermion Lagrangian 2, .............

Fermion couplings to the heavy Z ' boson in the current K 0’“ .....

Fermion couplings to the heavy W ’ "J: boson in the current K 33’“

. . 2
Relations between the Lagrangians ’2flfi1nd ,

Z’wrn alld 34f ......

Results for the coefficients C5,, C 5, and Cd ...............

Shifts 69f ( f) and 691% f) in the left- and right-handed couplings gf (f)
and gg( f) Of the fermions tO the Z boson ...............

Shifts 69% (f) and 69% (f) in the vector and axial couplings g€( f) and
gg (f) Of the fermions tO the Z boson ..................

Shifts rig?“ f ) in the left-handed coupling gill f ) of the fermions to
the Wi boson ...............................

Overview of the modified GAPP files ..................

viii

13

16

20

26

33

36

38

40

41

60

82

83

86

3.6 Maximum allowed deviations AX2 from Xfiiin. at 95% CL for different

numbers of free model parameters. ................... 100
4.1 Best fit values Of 5:, tag), .933, 111 H and m ................ 105
4.2 Bounds 011 the Higgs mass M H ..................... 109
4.3 Numerical evaluation of A011,“. /0had.,Sl\=I ................ 116
4.4 Numerical evaluation Of AA F 3(b) /A F 3531(1)) ............. 116
4.5 Numerical evaluation of AAIW /]lfw,31\,1 ................. 117

, 2 . 2

4.6 Numerical evaluation of A (9Z9) / (g'fi‘éM) ............. 117
4.7 Numerical evaluation of AQW (133Cs) /Qllr',SM (133Cs) ........ 118
4.8 Overview Of the observables driving the parameter plots ........ 120
Al Experimental values for the high—energy Observables .......... 131
A2 Experimental values for the low-energy observables .......... 132
D1 5,? (f) in the LR-D, LP-D, HP-D and FP—D model. .......... 154
D2 571% (f) in the LR-T, LP-T, HP-T and FP-T model ............ 154
D3 5% (f) in the UU-D and NU-D model ................... 154
D4 5% f) in the LR-D model. ........................ 155
D5 5% f) in the LP-D model. ........................ 155
D6 §IZ?( f ) in the HP—D model ......................... 156
D7 §g( f ) in the FP-D model. ........................ 156
D8 @112“ f ) in the LR—T model. ........................ 156
D9 glzg f) in the LP-T model. ........................ 156
D.10 512“ f ) in the HP—T model ......................... 157

ix

D.11 ggfl) in the FP-T model. ........................ 157

D.12 fiéfl) in the UU-D and NU-D model ................... 157
D.13 03;? (f131, f2,J') in the LR—D model .................... 160
D.14 6‘wa (f1’i, f2,J-) in the LP-D model .................... 161
DIS 0%? (f1JJ-, fQ’J') in the HP-D model .................... 162
D.16 03119017,. fgaj) in the FP-D model .................... 163
D.17 Cg? (f1,J°,f2,J-) in the LR—T model .................... 164
D.18 Cwa. ( f1,J-, fgyJ) in the LP-T model .................... 165
D. 19 Cwa (f1 1, f2 J) in the HP- T model .................... 166
D. 20 0'wa (f1 ,7, f2 J) in the FP- T model .................... 167
D. 21 C211? (f1JJ, f J) in the UU-D model .................... 168
D. 22 67ng (f1J- fgj) in the NU-D model .................... 169
D. 23 0%? (f1,- fgj) in the LR—D, LP—D, HP-D and FP-D model ...... 170
D 24 age? ( f1,, f3 J) in the LR T LP T HP T and PP T model ..... 171
D 25 cg? (f1 ,- f3,J-) in the UU D model .................... 171
D. 26 CS“? (f1 1, f3 J) in the NU-D model .................... 172
D. 27 C5? (f1 J, f2J) in the LR D model .................... 174
D28 051;? (fly, f2,J-) in the LP-D model .................... 175
D. 29 CJJC (f1 1, ng) in the HP- D model .................... 176
D30 0%? (fly, fed) in the FP-D model .................... 177
D31 0,139 (fLi, fed) in the LR-T model .................... 178

X

D32 C Etc (f1,J-, f23J) in the LP—T 1110(ch .................... 179

D33 CE}: (f1 ’1, fQJJ') in the HP-T model .................... 180
D34 Cyfc (f1J-, fgaj) in the FP-T model .................... 181
D35 (3'ny (f1,i, fgjj) in the UU-D model .................... 182
D.36 Cffc (f1,J-, fQJj) in the N U-D model ........... i ......... 183
D37 of}? (fly, 153,1) in the LR-D model .................... 184
D38 CEJC’ (f1,iaf3,j) in the LP-D model .................... 184
D39 Cffc(f1,i, 1(3),) in the HP-D model .................... 185
D40 CE'JC (f1JJ-, f3JJ) in the FP-D model .................... 185
D.41 C4CJC (f1,J-, f3,J-) in the LR—T model .................... 186
D42 CEJC (fly, f3,» in the LP-T model .................... 186
D43 CfJC(f1J-, f3J.) in the HP-T model .................... 187
D44 Off ( f1,i, f3,J-) in the FP-T model .................... 187
D45 CEJC (f1’i, 151,-) in the UU-D and NU-D model. ............ 188

xi

2.1

2.2

2.3

3.1

4.1

4.2

4.3

4.4

4.5

8.1

C.1

C2

C3

C4

LIST OF FIGURES

Images in this thesis are presented in color

Classification of the G (221) models under study ............ 9
Mixing Of the gauge bosons according to the first breaking pattern . . 28

Mixing Of the gauge bosons according to the second breaking pattern 29

Overview of the model, standard and fit parameters .......... 53
X2 in dependence of the Higgs mass AI H ................ 107
X2 in dependence of the top mass 771,; .................. 111
Influences Of some key observables on the parameter bounds ...... 119
Bounds on the (BP-I,D) models from the Z W+W‘ vertex ....... 125
Bounds on the the LP-D model with the anticipated values for QW (p)

and Qw(e) being included into the analysis .............. 128
Mass renormalization of the top quark ................. 134
Bounds on the LR—D model ........................ 138
Bounds on the LP-D model ........................ 139
Bounds 011 the HP—D model ........................ 140
Bounds on the FP-D model ........................ 141

xii

C.5 Bounds on the LR—T model ........................

C.6 Bounds on the LP-T model ........................

C.7 Bounds 011 the HP-T model ........................

C.8 Bounds on the FP-T model ........................

C.9 Bounds 011 the UU-D and the NU—D model. ..............

C.10 Pull distributions in the LR. models ....................

C.11 P1111 distributions in the LP models ....................

C.12 Pull distributions in the HP models. ..................

C.13 Pull distributions in the F P models ....................

C.14 Pull distributions in the UU-D and N U-D models ............

xiii

142

143

144

148

149

151

152

Chapter 1

Introduction

1.1 Motivation Of This Study

What holds the world together in its inmost folds? The Standard Model (SM) Of
particle physics is the best answer physics can give to that question at present. Being
considered as the most successful theory in the history Of physics, the SM is able to
describe and predict the behavior of elementary particles under the influence of the
electromagnetic, the weak and the strong force with unique precision, all the way to

the scale Of the nucleon (10"15 111).

However, it is clear that the SM is not an ultimate theory. It neglects gravity in
the description of the subatomic world and therefore does not encompass all forces
of nature. In that respect it does not meet the expectations towards a theory Of
everything from the outset. Furthermore it requires the masses and the mixing Of the
fermions and the mass Of the Higgs boson as external input parameters instead of
providing an unified explanation of how these quantities originate from more funda-
mental parameters. It lacks the answers to many fundamental questions, such as the
nature Of dark matter and dark energy or the origin Of the baryon asymmetry in the

universe. On a more technical level it faces difficulties such as fine-tuning [1] or the

1

violation Of unitarity [2] at high energy scales in the absence Of a light Higgs scalar.

In view Of all these deficiencies it is one Of the main tasks Of modern high—energy
physics to investigate theories beyond the SM. Theoretical work has to be done on
two fronts: On the one hand it is necessary to construct new physics (NP) models
and to examine in which directions the SM could be extended. On the other hand the
phenomenology Of new physics models has to be studied to be able to correctly inter-
pret the data Of experiments that aim at measuring effects of new physics. Presently,
this second task is as important as never before. The particle physics community is
about to enter a new era ~— in the near future the Large Hadron Collider (LHC) at
CERN will collect precise data on the TeV scale and it is widely expected that it
will provide evidence for physics beyond the SM. Now, at the eve Of the LHC, it is
therefore Of special importance to study the phenomenology of new physics models.
It is the goal of this thesis to contribute to that effort.

We will examine the compatibility of a certain class Of models, the so-called G (221)
models, with the most recent precision data 011 a number Of electroweak Observables.
111 a global fit analysis we will investigate the constraints on the parameters Of these
models in order to find out in which regions in parameter space the respective G (221)

models are consistent with the data.

1.2 The SM and Its Extension by a Second S U (2)

Before we concentrate our attention to the physics beyond the SM we briefly summa-
rize the main characteristics of the SM that will be relevant for our analysis.

The SM is a gauge theory. Its ansatz for the gauge group Of the electroweak
sector is the S U (2) L <8) U (1)1» in which the S U (2) L entails weak interactions of left-
handed fermion currents and the U (1)1/ acts on fermions that carry hypercharge

Y. The fermion content of the SM is accommodated in three generations Of leptons

2

and quarks. Mathematically, they are incorporated in the SM as representations Of
the electroweak gauge group: Left-handed fermion states form doublets under the
S U (2) L, right-handed fermions are represented by S U (2) L-singlets. The hypercharge
Y is constructed such that it adds with the third component T13! of the weak isospin

to give the electric charge, Q 2 T2 + Y.

If the electroweak gauge symmetry were unbroken in nature the SM would feature
four massless vector bosons acting as mediators of the electroweak force. However, we
know from the experiment and the fact that. the only long-range interactions in nature
are those Of electromagnetism and gravity that three electroweak gauge bosons are
massive. An elegant explanation for that Observation is provided by the Higgs mech-
anism that interprets the masses of the gauge bosons as a consequence of symmetry
breaking triggered by a scalar particle, the Higgs field. The SM assumes the Simplest
case and represents the Higgs boson by a S U (2) L-doublet. In the Higgs mechanism
the Higgs boson spontaneously acquires a non-vanishing vacuum expectation value
(VEV) which breaks the gauge symmetry of the Lagrangian and results in the oc-
currence of boson mass terms. In the course Of spontaneous symmetry breaking the
ftmdamental gauge bosons mix with each other to form the mass eigenstates that we
see in the experiment: The neutral, massive Z boson, the charged and massive W+

boson as well its anti-particle, the W " boson, and the neutral and massless photon

A.

Many models beyond the SM presume the existence of further gauge bosons that
account for new forms of particle interactions at high energy scales. The introduction
of new gauge bosons in the theory corresponds to the extentions of the electroweak
gauge group by another symmetry group. An additional U (1) results, for instance,
in the appearance Of a second massive uncharged boson, the Z’. Extensions of the
SM with a SU(2) <8) U (1)1 (29 U(1)2 gauge group in the electroweak sector —— or with

a G (211) gauge structure as we may say equivalently . »— have been studied in large

3

detail [3]. One of the next natural steps after adding a U (1) is the extension of the
SM gauge group by a second S U (2) In these G (221) models the electroweak gauge

group is given as the:

G(221): SU(2)1®SU(2)2®U(1)X

in which, depending 011 the specific G(221) model, the two S U (2)s can either accom—
modate left- or right-handed fermion doublets and the U (1) X introduces a new form
of hypercharge X. The inclusion of a second S U (2) results in the presence Of three
new gauge bosons, the Z ' , the W ’ + and the W’ ‘1 As these hypothetical gauge bosons
have escaped detection so far, they are assumed to be very massive. In this thesis we
will study the constraints on a class of G (221) models. One of our key questions will
be which masses for the new heavy gauge bosons are still allowed and consistent with

the data.

The explicit G(221) models that we will consider in this work are: The left-
right model (LR) [4, 5, 6], the leptophobic model (LP), the hadrophobic model (HP)
and the fermiophobic model (FP) [7, 8, 9], as well as the ununified model (UU)
[10, 11] and the non—universal model (NU) [12]. The LP, HP and UU models are
incomplete which manifests itself in the anomalous non—conservation of chiral fermion
currents. For purposes of completeness, we will, however, include them into our
analysis nonetheless. Especially the LP and the HP models are worth being discussed

as they represent, in a sense, intermediate steps in the transition from the LR to the

FP model.

In fitting models with a G (221) gauge structure to the electroweak data we follow
up the work of many earlier theoretical and phenomenological analyses. In the lit-
erature a number of studies can be found that perform global fits to various G(221)

models in the same spirit as our work. To be aware of the footing this thesis stands

4

on we may give a brief overview of these studies now: Polak and Zralek investigated
the symmetric version of the LR model in which the new charged vector bosons, the
W'+ and the W", couple with the same strength to the fermions as the charged SM
bosons, the W + and the W“. In Refs. [13] and [14] they Obtained constraints on
the LR parameters from the Z pole observables and from the low-energy data respec-
tively. The non-symmetric case was considered by Chay et al. [15]. They put bounds
011 the Z’ mass and the two mixing angles by combining the precision electroweak
data from LEP-I and the experimental data on low-energy neutral-current processes.
The tree-level and one—loop calculations in the FP model were carried out by Donini
et al. [16]. They showed that precision electroweak data and flavour physics provide
stringent constraints on the FP parameter space. Chivukula et al. [17] used the data

from precision electroweak measurements to put. strong bounds on the UU model.

However, no study encompassing all C (221) models at once has been presented so
far. The goal of this thesis will be to close that gap. While the studies in the liter-
ature differ in terms Of their focus and teclmiqes, we will present one comprehensive
consistent analysis for all C(221) models. In particular, we will pursue an effective
Lagrangian approach that will equip us with very universal and flexible expressions
for the different fundamental quantities in the G ( 221) models. Proceeding in this way
allows us to address the various G (221) models on an equal footing and ensures that.

the respective results are contrastable.

We ask ourselves: Which bounds do the experimental data place on G (221) mod-
els? we will give an answer to that question in three steps: In the second chapter we
will investigate the intrinsic properties of the G(221) models under study. This part
of our work mainly aims at providing us with the analytic expressions for the gauge
boson masses and the fermion currents. In the third chapter we will use the results of
our calculations to derive the new physics corrections to the electroweak observables

to which we fit our models. Subsequently, we will give a short introduction to the

Fortran plotting package GAPP [18] that we utilize in a modified form to perform our
numerical analysis. The fourth chapter is devoted to the presentation and discussion

of our results.

Chapter 2

New Physics Models

Tobe able to fit the G(221) models to electroweak precision data we need to know how
these models are constructed and what their respective properties are. In this chapter
we will try to develop an understanding of the models under study by addressing two
key points: The masses and the mixing of the gauge bosons and the gauge interactions
of the fermions.

The gauge bosons acquire their masses while the fundamental G (221) gauge group
is broken down to the U (1)0111. This breaking is successively accomplished by two
Higgs fields (I) and H that Spontaneously acquire non-zero VEVS at different energy
scales. Our strategy to extract the gauge boson mass matrices from the Higgs con-
tributions $71, and .20}; to the ftmdamental Lagrangian .Z’ is the following: First,
we discuss which representations for the Higgs fields comply with the mechanisms
by which the fundamental G'(221) gauge group can be broken. Once we know the
charges Of (I) and H under the G(221) symmetry groups we can write down $11, and
2H explicitly. We then have to clarify what is meant by the model parameters ap—
pearing in these two Lagrangians. When we have fully understood the structure of $11)
and 3H we can finally concentrate our attention to the breaking of the fundamental

symmetries and the generation of the boson masses.

7

The second contribution to .2” that we are interested in is the interaction of the
fermion with the gauge boson sector. As, from the perspective of new physics, the
electroweak Observables appear as low—energy data this is best done in an effective
field theory approach. Successively, we will integrate out the massive gauge bosons
until we end up with the effective Lagrangian below the electroweak scale and hence
the effective four-fermion interactions.

However, before we begin with any calculation we may categorize the G(221)
models under study. This will give structure to the analysis that we are going to

perform and thereby Simplify later considerations significantly.

2. 1 Classification

The G (221) models under study are the LR, LP, HP, F P, UU, and NU model. Three
criteria will help us to classify these models: The choice of the breaking pattern,
the representation of the Higgs field (I) and the charge assignments Of the fermions.
Fig. 2.1 on the following page gives an overview of the hierarchy among all the classes
into which our G(221) models can be grouped. The following discussion basically

serves as a comment on that diagram.

2.1.1 Symmetry Breaking Pattern

The gauge group of all C (221) models in the electroweak sector is the S U (2)1 (8
S U (2)2 (8 U (1) X- If this symmetry were unbroken it would lead to the presence of
seven massless gauge bosons in nature. However, experiments tell us that there is
only one massless force carrier belonging to the electroweak interaction, the photon.
As already discussed in the introduction all other bosons must acquire masses through
the effect of spontaneous syrmnetry breaking.

G(221) models go beyond the SM by extending its gauge group, the S U (2) L <8

8

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,3
e.
b
V)
e] A 9
.5 2 3
a n'. d
9. e. g
1‘ ®
pug g?
‘e’m s
o? co
“’3
52
“‘8
SE
a3 :5” E:
”.2 e e
si A PF
25 :1 A
:5 °~ “
:3 £9. 5".
® 1-3
N -
a E'.‘
s 5
to
8E e.
55 fit
a a a ,1
2'5” —€*: “-
g 1'5... (:1
:3 a. m I
mfiats S
.g 9'
.4 a:
I I .. DJ
8 8 n e

 

 

 

 

 

 

Figure 2.1: Classification of the G (221) models under study ~—~— the ten G (221) models
considered in this work fall into three distinct classes that differ from each other in
terms of the mechanism by which the fundamental G (221) gauge group is broken and
the choice for the Higgs representation (I) at the first breaking stage. Two breaking
patterns are available; the Higgs field (I) can either be represented by a doublet or
triplet. Referring to these three classes of G'(221) models we will speak Of the (BP-
I,D), (BP—I,T) and (BP-II,D) models.

9

U (1)y, by an additional S U (2). However, the SM has proven to be a very successful
theory at low energies. At a first stage the breaking mechanism of G (221) models
therefore has to reproduce the symmetry group of the SM. From there on, the break-
ing proceeds, of course, as in the SM: At a second breaking stage the SM breaking
SU(2)1; (8 U (1)1/ -+ U (1)9111 is mimicked. It is thus clear that the general G(221)

breaking pattern must have the following form:

SU(2)1®SU(2)2®U(1)X i SU(2)L8U(1)Y i U(1)..,,,

We have already introduced the two Higgs field (I) and H that are responsible for
the two symmetry breakings in the introduction to this chapter. Now we define (I)
to be the Higgs field that is responsible for the first breaking and H to be the field
responsible for the second breaking. We expect (I) to acquire a VEV at the TeV scale
furnishing the new gauge bosons W’i and Z’ with very heavy masses. H plays the
same role as the familiar Higgs doublet in the SM. It gets a VEV of roughly 250 GeV
resulting in masses Of the wt and the Z boson as seen in experiment. Note, that
the expected hierarchy of the two VEVS is, in principle, a model assumption.

There are two ways to break the SU(2)1 ® SU(2)2 ® U(1)X down to the U(1)em
splitting the set of all G (221) models naturally into two groups. In the following dis-
cussion we will refer to both mechanisms as the first and the second breaking pattern,
breaking pattern one and two or just BP-I and BP-II when an abbreviation is needed.
The choice of one of either breaking patterns has a substantial phenomenological im-
pact and in fact, it is the most important criterion for the classification of the G (221)

models. We now present both patterns in detail:

First breaking pattern: The first breaking mechanism identifies the first S U (2)
as the left-handed S U (2) L that we know from the SM and the second S U (.2)

10

as its right-handed counterpart. We would like the first symmetry breaking to
provide us with the SM gauge group and as the S U (2) L is already present right
from the begirming --— in form of the S U (2)1 ~ we only have one choice for
what to do at the first breaking stage: The S U (2)2 and the U(1) X have to be

broken to the SM hypercharge group U(1)1/2

SU(2)1 —~> SU(2)1. ; SU<2>28U<1>X 3’4 571),.

Then everything is set up correctly for the second breaking stage at which H can
break the SU(2)1, 69 U(1)y to the U(1)em. Fig. 2.2 on page 28 illustrates this
first breaking mechanism in a schematic diagram. From the above mentioned

models the first four, i.e. the LR, the LP, the HP and the FP model, belong to

the first breaking pattern.

Second breaking pattern: The second breaking pattern arrives at the S U (2) L of
the SM by breaking the direct product. Of the two S U (2)s to the diagonal sub-
group. As for the remaining group, now the U (1) X: there is again only one

choice left. It has to be identified with the U (1)y right from the beginning:

SU<2>lesU<2>2 3» SU(2)1. ; U(1)X —> way

At the second breaking stage we again encounter S U (2) L®U (1)y i U (1)0111.
Fig. 2.3 on page 29 visualizes this breaking mechanism. The two G (221) models
in which the symmetry breaking proceeds according to pattern two are the UU

and the NU model.

11

2.1.2 Higgs Representation at the First Breaking Stage

There are no uniquely defined represeiItatitms for (I) that have to be employed when
either breaking the SU(2)2 and the U(1)X to the U(1)1/ or the SU(2)1 ® SU(2)2 to
the S U (2) L- In fact, we are free to decide between different multiplets for either kind
of breaking pattern. In this work we will consider the simplest scenarios and restrict
ourselves to: S U (2)2-doublet (D) and S U (2)2—triplet (T) respresentations in the case
of breaking pattern one and a bi-doublet (D) representation for breaking pattern
two. A bi—triplet would be possible as well for the second breaking mechanism. But
as the only expected difference to the bi-doublet case would be a rescaling of some
parameters we will neglect that possibility. As Opposed to (I) the representations Of
H are fixed. The condition that the second synnnetry breaking has to resemble the
SM breaking mechanism respectively leaves only one choice for H in either breaking
pattern.

In total we will thus deal with ten different models: four models (LR-D, LP-D,
HP-D, F P-D) in which the gauge symmetries are broken according to pattern one
and (I) is represented by a SU(2)2-doublet, four models (LR-T, LP-T, HP-T, FP-T)
with the same breaking mechanism but utilizing a S U (2)2-triplet representation of (1)
instead and two models (UU-D, NU-D) that follow breaking pattern two and use a
bi-doublet for (I).

Since we are only interested in classifying the G (221) models these cormnents on
the Higgs representations should suffice for the moment. In subsection 2.2.1, when

we begin to derive the boson masses, we will go into the details.

2.1.3 Assignment of the Fermion Charges

Separating the C(221) models by breaking pattern and Higgs representation at. the

first breaking stage leads to three subsets of models: (BP-I,D), (BF-LT) and (BP-

12

U(1)X

 

 

 

 

Model SU(2)1 SU(2).2 Quarks: Leptons:
LP” (3:) ’ (:2) (32) . (:2) 1/6 —1/2
LP (311:) ’ (:2) (3:) 1/6 YSM
HP (14:) ’ (:f.) (:2) YSM -1/2
FF (:2) ’ (:f) YSM YSM
UN (3:) (:fi) YSM YSM
NU (:2) 15t,211d ‘ (:2) 15431111 (iii) 3rd ’ (:2) 31d YSM YSM

 

 

 

 

 

 

 

 

 

Table 2.1: Charges of the fermions under the C(221) gauge group —— the displayed
iSO—doublets are eigenstates Of the weak interaction. Unless otherwise specified, they
represent all three generations. YSM stands for the usual values Of the electroweak
hypercharge in the SM: YSM (UL) = YSM (61.) = -%t YSM (612) = “*1, YSM (UL) =

YSM (dL) = 11; YSM (UR) = i2; and YSM (d3) = -:lz~

13

II,D) in shorthand. Within each of these three groups the respective models differ

from each other in terms of the fermion charges.

Historically, the LR model represents the original G (221) model. It assumes that
the right-handed fermions transform as doublets under the S U (2)2, just as the left.-
handed fermions transform as doublets tmder the S U (2) 1, viz., the weak S U (2) of
the SM. In the LR model all right—handed fermions, thus, interact with the new
heavy charged W'i bosons. If we let the W’ i gauge bosons talk to quarks or leptons
exclusively we arrive at the LP and HP models. If we only invoke the presence Of a
second S U (2) in the electroweak gauge group, but assume that the W'i gauge bosons
do not interact with any SM fermion, we get the FP model. All four models belong
to the first breaking pattern.

Regarding the second breaking pattern we can either separate the SM fermion
representations by flavor or by generation which provides us with the UU and the
(family) NU model respectively. The UU model assigns the quark doublets to the
first, the lepton doublets to the second 3 U (2); the NU model Singles out the third
fermion generation reserving the S U (2)2 exclusively for it. In that respect the NU
model is the only G (221) model under study in which the charge assignements do not

apply cross-generationally —~~- the only model that breaks family symmetry.

Tab. 2.1 gives an overview of the fermion charges under the fundamental G (221)
gauge groups for all considered models. In order to save space we only include the
fermions Of the first generation in Tab. 2.1. Unless otherwise specified, these are,
however, understood to collectively represent all three fermion generations. Moreover,
as indicated in Tab. 2.1 some of our G'(221) models not only extend the particle

content of the SM by additional gauge bosons but also by new right-handed neutrinos.

Our set of new physics models comprises all phenomenologically different G (221)
models. Further G(221) models that we do not include into our analysis could be

constructed from the models of the second breaking pattern by exchanging the fermion

14

representations between the two S U ( 2)s. As such a permutation would, however, only
correspond to a redefinition of some model parameters we will not consider these

models in our study.

2.2 Masses Of the Gauge Bosons

The gauge bosons get their masses and mix with each other in the course of spon-
taneous symmetry breaking that is triggered by the two Higgs fields (I) and H. To
calculate the masses of the physical gauge bosons we first have to discuss the contri-

butions 2(1) and EH from (I) and H to the total Lagrangian .2.

2.2.1 Higgs Representations

In all C (221) models the Lagrangians ftp and 3H have the following form:

.541, ~ D[(D,,<I>)*(D#<1))] ; .211 ~ T1 [(01,119 (1311-11)]

The coefficients of $11, and EH get fixed by the choice of the Higgs representa-
tions and the trace symbol only has an effect when the corresponding product Of the
covariant derivatives is non-scalar —— which is the case if a Higgs field is represented
by a triplet or a bi-doublet. Constructing the Lagrangians for the two Higgs fields we
have to use the covariant derivative DJ” rather than the ordinary partial derivative
81, in order to ensure local gauge invariance.

The goal Of our analysis in this subsection is to explicitly write down the covariant
derivatives DH<I> and DHH in terms of the gauge bosons and the gauge couplings.
These derivatives depend on the charges Of the respective Higgs fields under the
G (221) gauge groups. Tab. 2.2 on the next page lists the Higgs representations our

models are constructed with. The second column of Tab. 2.2 gives the quantum

1St stage Repr. Multiplet and VEV

 

 

 

 

 

 

 

 

 

BP—I (D) ‘1’ N (112%) (I) 2 (if) i ((1)) : F (“(9)

BP-I (T) <1) ~ (1,3,1) <I> = 3 (£20 VIZ-45+) i (‘1’) I A (ti; 8)
BP-II <1) «4210) ‘1’ I (:3 if) i ((1)) z i (3 2)

2nd stage Repr. Multiplet and VEV

BP-I (D) H «42120) H = (if; 11:12:) ; (H) 2 £2— (Cg 3:)

BP-I (T) H N (2120) H 2 (I? [(7123) ; (H) 2 % (Cg 8:)
BP-II H ~ (1.2. 5) H : (iii) ; (H) 2 E (g)

 

 

 

 

 

 

 

Table 2.2: Representations of the Higgs fields (I) and H for the three classes of con-
sidered models — at the first stage of breaking pattern one (I) can either be chosen to
be a doublet or a triplet under the S U (2)2. In the second breaking pattern (D is rep-
resented by a bi-doublet. The second breaking stage mimics the symmetry breaking
of the SM; the respective representations of H are therefore fixed.

16

numbers of (I) and H for all three considered classes of G ( 221) models in the following

form at :

Higgs field N (T1, T2, X)

Here T1 and T2 denote the main mtantum numbers of the S U (2)1 and SU(2)2
isospins and X is the charge under the U (1) X. The actual multiplets that represent <1)
and H as well as their VEVs are listed in the third column of Tab. 2.2. We generically
denote the non-zero VEVS at the first and second breaking stages by {I and i} resp. In
the case of breaking pattern one we put a small subscript on {1. to distinguish between
the doublet (D) and the triplet (T) representations for (I). Since the same Higgs H
is used in both (BP-I,D) and (BP-I,T) models such a discrimination is, however, not
necessary at the second stage. By contrast to pattern two the first breaking pattern
introduces a further degree of freedom through the VEVs of the Higgs fields. In the

first breaking pattern (H) actually features two different VEVs, it and 12’:

We obtain the expression given in Tab. 2.2 when we relate the VEV 0 and the

angle 8 to F: and 123’ as follows:

~ ~ ~/
It: “CB ; H

II
(‘N
CD

to:

The tilde (~) over the angle fl and the VEVS Ft, Fz’ and it indicates that these param-
eters are intrinsic model parameters. We will elucidate the details in Subsec. 2.2.2.

c5. and 83 are, of course, abbreviations for cos (L3) and sin ([3). In the following we

will abbreviate the trigonometric functions of any arbitrary angle a in this 111anner:

17

3a E sin(a) ; ca. E cos(a) ; ta E tan(a)

Two criteria apply to the selection of those multiplet components that acquire
non-zero VEVs: On the one hand we require the Lagrangian to remain invariant
under U (llem transformations while the fumiamental gauge symmetries are broken;
that is, we have to assign the VEVs to the respective Higgs fields such that the electric
charge Q is a good quantum number in the end. On the other hand, as complex or
purely imaginary VEVs would lead to, e.g., unphysical mass terms, the Higgs VEVs

must be real.

In order to fulfill the first condition we define the electric charge Q to be the sum

of T3, T3 and X:

Qzfi+fi+x an

with T13 and TS being the third components of the isospin vectors T1 and T2
and assign the non-zero VEVs to multiplet components for which this sum takes
the value zero. The Higgs VEVs then do not carry electric charge and the U (1)0111
remains unbroken. In Tab. 2.2 the electric charges of the Higgs fields are indicated

by superscripts.

It is not surprising that the relation in Eq. (2.1) represents the proper definition
of Q. In models that employ the first breaking pattern T13 is equivalent to TIP: and T;
and X add up to Y. If breaking pattern two is used X and Y are the same quantum
number and the sum of T? and T23 equals T2. Eq. (2.1) thus does nothing else than

mimicking the familiar SM identity:

Q2T£+Y=Ti3+(T23+X) = (T§+T23)+X

18

Furnishing only the real parts of the respective multiplet components with non-
zero VEVs always introduces factors of % when going from a Higgs field to its VEV.
To give an example: The ho that belongs to the H field of the second breaking pattern

is given as:

1
0 0 - 0 , . 0 0
h = fl (hr + z - hi) where hr, hi. 6 R

where :15 plays the role of a normalization factor. (H) is supposed to be real and so

only h? gets a VEV and h? vanishes in the vacuum:

0—_L~ 2'. :i
(h)——\/§(u+ (J) x/2

With all these general remarks being made, the charges of the Higgs fields (I) and
H given in Tab. 2.2 now allow us to explicitly write down the covariant derivatives
DM<I> and DpH. Tab. 2.3 on the following page presents the Lagrangians .241, and
.‘ZH in terms of gauge couplings and vector bosons for all three considered classes of
G (221) models. The overall prefactors take care of the proper normalization of fig)
and EH. They ensure that in both Lagrangians, if expandend in the components of

their Higgs multiplets, all terms have a prefactor of 1:

5a = Z (Dim-fl (0/1196...) ; 2H = 2 (Dual (0%)

2' z
The gauge couplings of the S U (2)1, the S U (2);; and the U (1) X that enter into
3.1, and 53H are denoted by {11, fig and g X resp. We will concentrate our attention to
them in the next subsection. The gauge bosons belonging to the three fundamental

symmetry groups are given as:

SU(2)1: W11,W2,W13 ; SU(2)2: W21,W_?,W23 ; U(1)X: BX

19

1St stage

Lagrangian

 

 

 

 

 

 

 

 

 

 

 

BP-I (D) .221, (D#,<I>)l (We)
(afle’r + iggqfl‘rg - 111/{fl + rage)“ - B”) x
x (amp — 'iggTZI’q) . WZW — lime - 85;)
BP-I (T) 3.1, 2 - Tr[ (Dfld))l (13%)]
2-Tr[(8#<l>l+i§2[<1>l,T§] W20,“ + i§X<Dl . B”) x
x (amp — igg [7119. e] .wgw —— @ch - 352)]
BP-II .241, T1-[ (Dp,<b)l (13%)]
Tr[(a,.<1>’r + 15716171“ - r1713“ — 1g2T§<I>T - 111/2”,”) x
x (3M1) — iglrfrp - ch’“ + 1572M; - ng
2nd stage Lagrangian
BP-I 3H é -T1-[ (1),.11)’f (191111)]
§ -n[(a,)HT +i§1HTTf - Biff,” 433273111 - W3“) x
x (au'H — 2'91er . Wf’“ + iggHTQd - wzdm
BP-II 2H (1),,11)l (DHH)

(6qu + 252 leg . L13“ +1§X§HT - B”) x
x (8"H — i§2T2bH - Wy- — igX§H . 135:.)

 

 

Table 2.3: Lagrangians of the Higgs fields (I) and H - - depending on their charges
under the G' (221) gauge group the Higgs fields (I) and H are accounted for by different
contributions to the total Lagrangian. Once the Higgs fields acquire their VEVs the
gauge symmetry of these Lagrangians gets broken and six linear combinations of the

seven fundamental gauge bosons become massive.

20

 

Since the Lagrangians $4, and 3H are invariant under the full G (221) gauge
group, these seven gauge bosons are still massless. The terms in which they appear
in .241, and 3H represent. the gauge interactions of the Higgs fields (P and H —-— mass
and mass mixing terms do not appear until we break the G (221) symmetry. In our
language of Higgs Lagrangians the mechanism of spontaneous symmetry breaking
(SSB) is accounted for by the substitution of the Higgs fields (I) and H with their

VEVs ((1)) and (H):

SSBZ .241) -—* 3”,) ; 3H —--> $(H) (2.2)

We will examine the effects of these substitutions in Subsec. 2.2.3. Now we focus

on the gauge couplings.

2.2.2 Gauge Couplings and Mixing Angles

The treatment of the parameters that are involved in our analysis requires special
care: In the next chapter it will be our goal to calculate the new physics corrections
of our G(221) models to the electroweak observables and to add these corrections to
the GAPP code. GAPP, however, is designed to fit the SM to the electroweak data
and therefore employs the usual parameters of the SM. The old physics parameters
in our G (221) models differ from these SM parameters, as they receive contributions
from new physics, and so we will have to develop a dictionary of relations that will
help us translate our parameters to those of the SM, that is to those used by GAPP.

In [19] Burgess et al. present a detailed study of constraints on new physics
derived in a model-independent effective Lagrangian approach. They give a detailed
discussion of the relation between new physics and SM parameters and in this work
we will basically follow their procedure. Burgess et al. distinguish between time

different ideas: Model parameters, standard parameters and reference observables.

21

In the following we will make use of all three concepts which is why we now want to

shortly comment on each of them:

Model parameters are the parameters by which the fundamental Lagrangian of our
G(221) model is parametrized as well as all other quantities that are derived
from them. This definition also applies to the reparametrized Lagrangian in
which the VEVs of the Higgs fields are used rather than the fields themselves.
Burgess equips these parameters with a tilde; in our previous discussion we
already adopted this notation when we introduced the Higgs VEVs ft and 1"),
the angle 6 and the couplings of the three fimdamental gauge groups g1, {12
and fix. The rest of this subsection will deal with the important secondary

quantities that can be constructed from these couplings.

Standard parameters are parameters whose analytical relations to the experimen-
tal input take exactly the same form as in the SM. In other words: They
represent the actual parameters of the SM transfered to and incorporated into
our G(221) models. In order to distinguish them from our model parameters
we will furnish them with the index SM. It is the standard parameters that are
used by GAPP and as mentioned above one of our goals that we have to accom-
plish before we can start fitting will be to translate our model parameters that
have equivalents in the SM to these standard parameters. In fact, the relevant
standard parameters will be the fine structure constant, 018M, the VEV of the
electroweak symmetry breaking, ng, and the electroweak mixing angle 853M

We will come back to the standard parameters in Subsec. 3.1.2.

Reference observables are observables that have been measured with high preci-
sion and that can therefore be employed to deduce the numerical values of the
standard parameters. GAPP uses the fine structure constant a and Fermi’s

constant G F as reference input. Optionally, the mass M Z of the Z boson can

22

be fixed. we note that a is identical to (ISM — the two notations for the fine
structure constant just either emphasize its role as an observable or as a model
parameter. Once we have expressed our new physics corrections in terms of the
standard parameters the reference observables will serve as the ultimate link

between our theoretical analysis and the experimental data.

The breaking of a more fundamental initial gauge group to a smaller subgroup
has two effects: First, the new gauge group will be associated with its own gauge
couplings that are related to the couplings of the fundamental gauge group. The
actual form of the breaking pattern has to tell us how these relations exactly look
like. And second, the mechanism of symmetry breaking will lead to the mixing of
the gauge bosons belonging to the fundamental symmetry group. It will therefore be
convenient to change the basis of the gauge bosons after the symmetry breaking by
performing a rotation about a certain angle.

Both the coupling constants of the new gauge groups as well as the mixing angles of
the gauge bosons can be constructed from the gauge couplings of the initial symmetry
group, in our case fil, fig and fiX. In the following we show how this is respectively

done at the first and the second breaking stage of our G (221) models.

First breaking stage The first breaking of pattern one, SU(2)g @- U (1) X —>
U(1)y, mimics the SM breaking SU(2)L <8) U(1)y —9 U(1)em. Hence, we define the
mixing angle a; between the bosons of the the S U (2)g and the U ( 1) X similarly to the
electroweak mixing angle 08M — the tangent of (13 is given as the ratio of the U (1)
coupling, fiX, to the SU(2) coupling, fig. As for breaking pattern two we can either
set the tangent of d to fig /fi1 or the inverse of that. Breaking the two S U(2)s to the
diagonal subgroup, SU(2)L, we have to treat both groups on an equal footing and
it must therefore not matter which choice we make. We decide for the first option

which leads us to:

23

I

l ; BP-II: t

BP-l: t6; ~ (2.3)
‘ g2

<5

belie:
H N:

After the first breaking stage we arrive at the S U(2) L ® U (1) X of the SM for
either kind of breaking pattern. Analogously to the standard couplings 9L,SM and
9Y,SM we denote the gauge couplings of this first subgroup of the fundamental G (221)
gauge group by fiL and fiy. As the first. breaking pattern directly identifies the first
S U (2) with the S U (2) L the corresponding coupling, fi1, has to be equivalent to fiL.

111 breaking pattern two fi L follows from the mixing of both fundamental S U (2)s:

1
9i.

l ...

1313-1; nggl ; BP-II: + (2.4)

‘Qz
tow H

2
1

be

If we combine Eqs. (2.3) and (2.4) we can rewrite fil and fig in terms of the weak

isospin coupling constant fi L and the first. stage mixing angle 9'9:

~ 9L ~ 9L ~ ~ ~
BP-II: 91 = —- ; 92 = —- © 9L : Sggl = 04392 (2-5)

0
6» 1

Similar arguments apply to the definition of fiy. In breaking pattern two the
U (1) X is not touched during the first symmetry breaking. fi X therefore is equivalent

to fiy. The first breaking pattern mixes the S U ( 2)g and the U (1)X at the first stage:

1 ~ .
9X

BP-I: —1—

fig:
tom "“

2
Y

Q2

The combination of Eqs. (2.3) and (2.6) provides us with relations similar to those

in Eq. (2.5), new for the first. instead of for the second breaking pattern:

BP-I: 92 = -—S~ ; gX -—- —— e 91/ =— 80792 = ngx (2-7)
é . ,

Second breaking stage The second breaking stage resembles the electroweak sym-
metry breaking of the SM. For both breaking patterns the analogue of the Weinberg
angle 98M in our G(221) models, é the electric charge 6? and the fine structure con—
stant 6: are defined as:

1 1 (E2

+ ;d—
47r

III
IH
Ill

(b:
(Q2 I
t is
C!)

rial +-‘

2
Y

(Q1

2
L

is:

Compare also with Eq. (3.6). Just as in the SM, we then have the following rela-
tions between the gauge couplings and the electric charge as well as the electroweak

mixing angle:

@Y =

Clem’

According to the definitions of fi L and fiy in Eqs. (2.4) and (2.6) the electric charge

takes the following form in either breaking pattern:

1_1+1+1__1+1+1
é? 9% a a 9% £23 9%.

GAPP only knows indirectly about coupling constants. Instead of the SM gauge
couplings 9L,SM and 91/,SM the fine structure constant (ISM and the electroweak
mixing angle 63M are implemented in the code. When we have come to calculate
the corrections to the precision observables we will also discard all gauge couplings
trading them for Er, 63 and d. The corresponding relations between fil, fig and fi X on
the one side and d, g and a; on the other side follow from Eqs. (2.5), (2.7) and (2.8).
We present the results in Tab. 2.4 that summarizes all of the important relations

discussed in this section.

BP 91AM) gg/errd) fix/(Md) at? @172 1,, t,

 

 

c? c? 91 gg +9X fix/62 6Y/6L

__ _ _ _ _ ~_2 ~ ~ ~ .. ..
II 1&1 561C621 6611 91 +92 9X 92/91 9Y/9L

 

 

 

 

 

 

 

 

 

 

 

 

Table 2. 4. Fundamental gauge couplings and mixing angles —— overview of the rela-
tions between fil, fig and fix on the one side and the parameters 61, 6 and 6 as well
as the couplings constants g L and fiy and mixing angles <15 and 6 on the other side.

2.2.3 Mass Matrices and Mixing of the Gauge Bosons

As we now have understood all quantities that enter the two Higgs Lagrangians $11,
and EH we can return to Eq. (2.2) and finally perform the symmetry breaking. The
first stage of symmetry breaking generates masses for the two heavy gauge bosons

W’i and 2’:

_1’7’2 £17.11 ”‘72 371+ * /—,,1

The hats () over the boson symbols indicate that these gauge bosons are not yet
mass eigenstates and thus not physical boson states. During the second symmetry
breaking two bosons I/I/Zt and 2 will acquire masses and — that is the point ——
mix with the W'i and the 2’. In order to find the physical gauge bosons it will
be necessary to diagonalize the mass matrices in the neutral (2’, Z) and charged
(W’i, Wt) gauge boson sector. Z’, Z, W ' i and Wi will then be represented by
some linear combinations of the hatted gauge bosons. The tilde ever [TI/é, and HEW
reflects the fact that these masses are only internal model parameters which do not
represent masses of physical states. The masses of the physical gauge bosons will be
constructed from these tilde masses.

At the second stage of symmetry breaking H gets its VEV furnishing two further

26

gauge bosons, IVi and Z, with masses:

3e) +$<H>

Ill

1 ~. . . 1 N A, . A _~ . .
51”;qu + 1; (113+ 1311;) ZLZW + 6111§2,Z#Z”“'(2.9)

W’ W’
+ 61W 2 (vi,}L W"’“ + W; W’W‘)

+ 5173.111; WW + (112. + A1172. )wyw-w

W Ii"

The next goal 011 our way to the masses of the physical gauge bosons is to de-
termine the mass parameters in Eq. (2.9) and to discuss how the hatted bosons are
constructed in terms of the fundamental gauge bosons W113, W123, W13,2 and BX. We
will collect our results for the mass parameters in Tab. 2.5 on page 33.

In a first trivial step we transform the fundamental bosons that belong to the two

S U(2)s such that they become eigenstates of the electric charge operator:

1 71 ' 72 7-— 1 1 ' 2
WK, = E (VI/1,2 — 111-1,2) ; 141,92 = :5 (W1,2 + sz)

If we take the bosons W13, 14"?2 and B X to be the fundamental basis of the
G (221) gauge bosons the effect of the twofold synnnetry breaking triggered by (I) and

H can be summarized as follows:

SU(2)1 ® SU(2)g ® U(1)X Photon EVV bosons NP bosons
Wi, W3, Wi, Wg, BX —+ A wi, z w’i, z’
massless massless massless massless massive massive

Figs. 2.2 and 2.3 show diagrannnatktally the steps that lead from the initial funda-
mental gauge bosons to the physical states for both breaking patterns. We will now
successively examine both stages of symmetry breaking and the diagonalization of the

mass matrices. The following discussion can essentially be regarded as a comment on

Figs. 2.2 and 2.3.

27

 

><
3 ‘3. m.
b 5:;
‘3’ A
N H
® H
a; V
e {:1 b
’6? :3
SF”)
0)
><
A
H
v
b l :39
>< I s
on he ‘NHN
0
®
in:
”a 6‘ a 7’
N "’ .a m .q, ‘N
A m 0
N
v
be-
CQ (:0... cm.) {3°
r» ~——>§ Q
a;
(X)
'H “H 'H
N T \
1+§o——-><:§o-——><§
R
N
SH
‘0 +1... -H 41
+§o———><§o——><§

 

Breaking Pattern I

heavy

Z!

light

E”
3.311
k
4.9
.51
£944

First stage Second stage Mass eigenvalues

28

Figure 2.2: Mixing of the gauge bosons due to spontaneous symmetry breaking ac-
cording the first breaking pattern — the text colors indicate whether the respective
bosons have already acquired masses (red symbols) or whether they are still massless
(green symbols). For all pairs of bosons that mix at a certain stage of symmetry
breaking the respective mixing coefficients are given on either side of the correspond—
ing arrows. Trigonometric functions written on the left-hand / right-hand side of an
arrow belong to the left / right boson at the preceding stage of synnnetry breaking.
See Tab. 2.2 for the definition of the VEVs 11 and i3 and the angle 6.

 

 

SU(2)1, ® U(1)Y
U(1)em
Breaking Pattern II

><
A
H
v
b
>< 5"
m o——> m m sco—-—> <1:
a;
‘8 a»
aq, 8
—-> < .
N a? is: N N
A
N
V
DO— .,, E
to ”F, c? .99
-> ”3 <No———><N N
U
‘3’ t
+1... 9:3" a 14 g 14
mpg ' .§.——».g g;
:4 6°
N
v
b a
03 -H,4 a? -H :1 «H
*E re Ԥ ' ' (T E
03

First stage Second stage Mass eigenvalues

Figure 2.3: Mixing of the gauge bosons due to spontaneous symmetry breaking ac-
cording to the second breaking pattern "~— the text colors indicate whether the re-
spective bosons have already acquired masses (red symbols) or whether they are still
massless (green symbols). For all pairs of bosons that mix at a certain stage of
symmetry breaking the respective mixing coefficients are given on either side of the
corresponding arrows. Trigonometric functions written 011 the left-hand / right-hand
side of an arrow belong to the left / right boson at the preceding stage of symmetry
breaking. See Tab. 2.2 for the definition of the VEVs i2 and {v and the angle 13.

29

First Breaking Pattern At the first breaking stage the charged S U (2)2 bosons
W2i acquire masses whereas their S U (2)1 equivalents Wli remain massless. These
two sets of charged gauge bosons do not mix and so the transition from Wit and W?
to Wi and W’i is nothing else than a renaming. In the neutral gauge boson sector
2' emerges as a certain linear combination of W? and B X- The orthogonal linear
combination to 2’ represents the neutral vector boson By of the hypercharge group
U (1)y. We diagonalize the neutral mass matrix by rotating the (W753 , B X) boson

basis about the angle 6.

W31: 1 0 Wft 2’ c~

W’i 0 1 W; By 3;) c~. BX

The coefficients of the boson products ZA'LZAW and Ii’f'iifl’fi‘ in the Lagrangian
fig» provide us with the masses M3, and 1,1713”. As can be seen from Tab. 2.5 these
masses depend on the choice for the representation of (I). In fact, it is this difference in
the numerical prefactors of the masses WT; and A731], that accounts for the different

phenomenology of the (BP-I,D) and the (BP-I,T) models.

Since the Lagrangian 3’11) does not involve the bosons of the S U (2)1 the neutral
W? stays unaffected during the first symmetry breaking. Once the S U (2)2 <81 U (1) X
symmetry is broken to the U (1)y we identify the W? as the neutral boson of the
weak isospin group, WE. The bosons that remain massless after the first breaking

stage therefore directly correspond to the fundamental gauge bosons of the SM.

At the second stage of symmetry breaking we encounter the usual SM breaking
mechanism: The W3 and the By combine resulting in the massive Z boson and the
massless photon, A. We construct Z and A by rotating the (W3, By) basis about

the angle 6:

30

3
WL

O
Q):
|

Cl)
%1

CI:

CD:

69* BY

The charged bosons W'i belonging to the S U (2) L acquire masses as well. We
obtain M; and 117%,, as prefactors of 2,12“ and I/i"lj"W"’“ in the Lagrangian $011
Moreover, the presence of the new physics bosons 2' and W’i results in mass mixing
terms 6117i; 2’ and éfivzi/W” Finally, the second symmetry breaking leads to shifts
AME, and Afifi/l in the masses Mg, and 117%, that are proportional to the VEV
6 of the low-energy Higgs field H. The results for all these mass parameters are
presented in Tab. 2.5. We notice that the angle 6 that we introduced in Subsec. 2.2.1
as the mixing between the two VEVs R and 175’ appears in Tab. 2.5 exclusively in the

W ‘W’ mass mixing parameter 6M3, From now on we will, hence, regard ,6 not

virl'
only as the R—R’ but also, if not mainly, as the W 41/ ’ mixing angle.

Second Breaking Pattern Breaking pattern two involves, of course, the same
gauge bosons as the first breaking pattern but constructs the bosons after the first
breaking stage in a different way. As opposed to breaking pattern one Wit and
W2i do not represent mass eigenstates once (I) has acquired its VEV. W:t and W’i
are now introduced as linear combinations of WIi and W; after the charged mass
matrices has been diagonalized by a rotation about the angle 6. The massive 2" does
not receive contributions from BX in the second breaking pattern but is constructed
from W13 and W3. The neutral boson of the weak isospin, W3, follows as the linear

combination of W? and W23 that is orthogonal to 2’:

W’i c —.s~ W1:t 2’ C7 "' W?

,3
W2

31

By is identical to BX; the second stage breaking proceeds as in breaking pattern
one. We determine the mass parameters in Eq. (2.9) in the same way as for the first
pattern and collect them in Tab. 2.5. As an interesting consequence of the particular
structure of the second breaking pattern we notice that the 2' and the W ' i are
degenerate in terms of their masses and that they receive the same mass corrections

in the course of the second synnnetry breaking: see also Eq. (2.9):

5*,
N2 1 ~2 ~2 ~2 ~2 , A“. .2 e~2 2 ~, 2
ill/[2, = Z ( l + 92) U 2 Alli” , A1112, 27— 1‘92?) 2 A111“,

This behavior is expected since the second breaking pattern breaks the S U (2)1 <8)
S U (2)2 t0 the diagonal subgroup not distinguishing between charged and uncharged
gauge bosons. The mass generation for the 2" and the 17V”: proceeds in exactly the

88.1116 way.

2.2.4 Physical Boson States and Masses

We now know all parameters in the mass Lagrangian of the gauge bosons after the
second breaking stage, 201$:er E 3“,) + $(H1’ for all three classes of G (221) models.
The considerations that will lead us from 219;, to the masses of the physical bosons
apply generally and so we now return to our BP-independent analysis: .3432ng can be
written in its most compact form when we introduce 1722, and %,.W, as notations

for the mass matrices of the neutral and charged gauge bosons:

A A

(2) 1 - . ~ Z . . ~ Wi
finass : 5 (Z Zl) J/{ZZI A, + (Vi/i VV’i) "flii’W’ .
Z

IV’i

with flZAZ/ and J/lww, being of the form:

32

(BP-I,D)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

22. r2. ~22 12.
M Z I Z’ M W MW,
1 ~2 ~2 2 1 ~2 ~2 ~2 1~2 2 1~2~2
7.1 (91. + 91/) U 71 (92 + 9X) “D 19L” 21.92%
72.. "2. 172. . “72.
611122, 111112, 6. [W W, AMW,
6403)“- c213- 323-- -
- T; 919ng1)? T(' >931)? - L.f—)91.92'02 21193122
(BF-LT)
~2 ”T2 ’72. 7V2.
M 2 112, MW MW,
1 ~2 ~2 2 ~2 ~2 ~2 1~ 1~ ~
21 (9L + 91/) ”U (92 + 9X) “T 19%}? 2931121“
’72. . 7’2. 7’2. . ~2.
511122, AMZ, (SAIWW, AMW,
12(3)- ~ ~ 2 02(3))2 2 308).. ~ .2 1~2 2
— 4g 9L929XU T92“ ‘T9L92'U 21927}
(BF-II,D)
~2 ”7'2 772 ”7’2
A/IZ A/IZA, JI‘IIi; AIL?"

1 ~2 ~2 .2 1 ~2 ~2 2 1~2 ,2 1 ~2 ~2 2
21(9L + 9Y) '0 1(91+92)u 19L“ 21(91+92)“
”7’2 2 5’2 ~,2
61kIZZ‘, AA! I 61 I‘i/r"ifl ijlvi/I

s2(c§)~ ~ ~ 2 320-2 2 1 ~2 ~2~ 2 1 ~.. ~2 2
- 4g ' 91.9291/v T921) -;1 92 - ngLv 21' (92 - 91) v

 

 

 

 

 

 

 

 

Table 2.5: Entries of the fundamental boson mass matrices — the first symmetry
breaking generates masses Mg, and MW’ for the 2’ and the W’ i. In the course of

the second symmetry breaking the Z and the Wi bosons acquire masses M; and
117%,. Additionally, the Z’ and W'i masses get respectively shifted by Ali/7%, and

~2 . . ' ' [7’2 , . A72 . . _
AMW’ and mass Imxmg terms (”[22, and 6MWW’ occur, cf Eq. (2.9).

33

1. 112 517222, , N 117,2, 617,2, W,

.. __ Z . .. _
" ZZ’ — ~ 2 ‘11 111w" —

6M; 1172,+Z112172,6172 1172 , + 11172

w W W’

The masses of the physical gauge bosons are given as the eigenvalues of .6122,
and flWW” the physical bosons themselves as the corresponding eigenvectors. Since

we expect AI; and AIEAV’ to be. very large,

~2 ~2 ~2 ~2 . ~2 ’72
M Z’ >> M ,AMZ, ,6 112-2 , 1~1I12i , >> MW ’AMW” 6MWW, ,
it is appropriate not to calculate the exact eigenvectors and eigenvalues of .1722,

and WWW” but to restrict ourselves to a series expansion in powers of Mg; and

[LIV-v2, resp. We diagonalize the mass matrices by performing rotations about the

‘ )' "~i o. - /~y Av A -
angles to Z Z’ and WW W"

 

6172 ., M72 ,
5’22” 2 A72 E372 ; “Dvi’v‘v” : 372 vfjp
Z’ Z w’ W

Up to linear order in M27? or M63 the physical bosons Z, Z ’ , Wi and W'i are

then given as:

_ A A, . I I _ A, '~, A A A
fit: _ 7:1: ~ 7H: _ Ii _ Af’fil: 1"». . A721:

The eigenvalues corresponding to these eigenvectm’s, that is, the physical masses

AIZ, M 2,, [WW and AI2/,, take the following values:

34

6174 ~ 6A74

A12 =AY2— ——Z—Z' ; A12 :1112+ ——Z——Z’ (212
Z A72, ’ Z' 2’ A72, )
6A74Z 61174
.2 _732 _ wn’ , ,2... ~2 WW’ .
MW _ .112, 222 , .112 V,—_— 11122244r 222 (2.1.5)
”I W,

These general relations represent important results in their own right. However,
we are also interested in more explicit expressions that tell us how the masses of the
six massive gauge bosons depend 011 the Higgs VEVs, the mixing angles etc. for each
of the considered classes of G (221) models. We obtain such expressions in three steps:
First, we insert our results for the mass parameters as given in Tab. 2.5 into the above
relations. The expressions for ME, M?“ Ma; and A12” we get this way still involve
the gauge couplings of the various symmetry groups. In order to prepare the global
fit analysis with GAPP we subsequently rewrite the coupling constants according to
Tab. 2.4 in terms of the fine. structure constant (3 and the gauge boson mixing angles.

In a third step we introduce the parameter 5: as the ratio of the squared VEVs 112

and 272,
112) {1.2 112
(BP-I,D): 1: a 22— ; (BP IT) 1; 2 72—2— ; (BP II ,:§:D) a 22— , (2.14)

As for ME and M a we consider

and expand the physical boson masses in 2.

terms up to first order in 22.111 case of the heavy gauge bosons we will only keep
terms proportional to 5:. The parameter it allows us to quantify the relation between
the scale of the new physics and the electroweak scale. It will play a crucial role
in our later analysis: As we will see the corrections to the SM observables in our

C(221) models will scale with and de pending on whi( h i range in parameter spa< e

35

2 —2 2 .2 —2 2 2 ,2 ,2 .2
MZ/(cé MO) [LIZ/(LE MO) MW/M0 MW,/MO

 

 

- —-i 4 2 72* _.l 2- ,2 T2”
(BP LD) 1 33C¢ t98¢ :1: 1 1321‘ tgsqs .1?
- __1_~ 4' 2~j2~‘ ___1_~_ 2~ 2:2"
(BP LT) 1 4$C¢ 4t65¢ .7. 1 233826 ”68¢ :1:
BP-II,D 1— is": 5% 1— $34 3:26:25:
( ) 5" ¢ (P 1” <25 (15 (:5

 

 

 

 

 

 

 

 

 

Table 2.6: Masses of the physical gauge bosons in terms of the model parameters i",
q’), 0 etc. M0 = (flab/(4.93) has the same form as the SM mass of the Wi boson,

but is defined in terms of G (221) model parameters.

is allowed by the experimental data it becomes more likely or less likely that a certain
G(221) model can be probed in collider experiments. For that reason the expansion

extracts the leading new physics contributions to the boson masses and sorts

Htlr—I

in

1

out higher order terms that we are not interested in. In that respect expanding in E

has the same effect as the expansions in M}? or [Mpg/2, that we performed above.

The results for ME, Mg“ M 3V and MIQV’ that we obtain after having gone through
all three steps are presented in Tab. 2.6. In this table we introduce the mass M0 that

allows us to write down the boson masses in a nice and compact form. .MO is defined,

 

 

~2~2 ~~2
8 ”U 7TCI‘U

Mg 5 = , (2.15)
453. 53

such that it resembles the SM expression for the mass of the Wi boson. However,
the definition of MO employs model and not standard parameters and so A10 only

corresponds to the SM W i mass in the limit 55 —> oo.

36

2.3 Gauge Interactions of the Fermions

The fermions that are incorporated into our G(221) models couple to the vector
bosons through gauge interactions. In the third chapter, when we will have come to
calculate the corrections to the electroweak precision data, these interactions will rep-
resent the heart of our analysis ‘-—— the theoretical description of the various precision
observables to which we will fit the G (221) models either requires the fundamental
fermion currents or the effective currents that arise in the low-energy theory. In this
section we will thus first discuss the direct interactions between fermions and gauge
bosons and then present how one arrives at the effective theory by successively inte-
grating out the heavy gauge bosons. The latter part will first lead us to an effective
SM-like Lagrangian at the electroweak scale. After removing the electroweak gauge
bosons from the theory we will end up with the effective four-fermion interaction
below the electroweak scale. The three Lagrangians that we will obtain in this way
will represent very powerful and flexible tools that will allow us to perform many

calculations for all C(221) models at once.

2.3.1 Fundamental Fermion-Boson Interactions

Our G (221) models accommodate the fermions f in iso—multiplets 2p each of which is

represented by a corresponding term $1,“, in the total Lagrangian .2”:

31;) Z “DA/H Du 115’

Two things are necessary to ensure that .2?) is invariant under local gauge trans-
formations: First, we have to employ the covariant derivative Dpzb instead of the
partial derivative amp. Second, the term —2/_)M¢z[) reflecting the fermions’ masses is
not included into $1!) but generated through spontaneous symmetry breaking in the

Yukawa sector.

37

BP S U (2)1 Doublet S U (2)2 Doublet Charged under the U (1)X

 

I éLzz’L’r“ T1“ Wfi ,fi’JL QQYIT’RfiflTQbVV‘g/le fixi’r“ (XLPL + XRPR) BX,/fl/J

H §1¢L7“'Tf1Wfiud’L him/“T5 W25, ”14% hi?“ (YLPL + YRPR)BY,y1/J

 

 

 

 

 

 

Table 2.7: Building blocks of the fermion Lagrangian 3,1, that account for the gauge
interactions of the fermion multiplet w with the vector bosons —~ 2,1, is composed as
a sum of these blocks according to the chosen breaking pattern and the charges of it
under the G (221) gauge group.

Just as we expanded the covariant derivatives of the Higgs fields, DHCIJ and DflH,
in terms of the gauge couplings and bosons in the previous section we will now expand
Due. Generally, Dmb can be split into a kinetic part represented by Buzb and terms
leading to the gauge interactions. To give an example: In the LR-D and LR—T model
the covariant derivative of a lepton multiplet dig and the corresponding Lagrangian

$1193 read as follows:

I'. ._ U” ' f. ‘ I" H" b /'b v
DMW — 811W - szTf' V" f,pPLifi/"€,L - z.612T2VL gupnlfian,
- iéx (XLPL + XRPR) Bxfl’e
. T , ~ 7 r , .I ~ 7 b b
W = 1%?pr + mam“ T? W fl#,PL71«"€,L + 92W,R’Y“T2 Wz,uPR¢13,R

+ forth?" (XLPL + XRPR.) BX,;.W

The gauge interactions differ from fermion to fermion, though, in dependence of
the respective charges under the G (221) symmetry groups. In particular the fermion
interactions with the S U ( 2)2 gauge bosons W51,” have different chiral structure for
both breaking patterns. In Tab. 2.7 we list the pieces that can enter the gauge
interaction part of the Lagrangian 20¢. ‘

The expressions given in Tab. 2.7 stand for the fermion interactions with the weak

38

eigenstates ”711,2, W122, ”"132 and B X- We are, however, interested in the fermion
currents that couple to the physical gauge bosons. It will be these couplings that will
go into the calculation of the corrections to the electroweak observables. In a first
step towards this goal we calculate the fermion couplings to the gauge bosons that
we obtain after the second stage of symmetr)r breaking, Z, 2’, W'i, W’i and A. We
take the contribution joint. to the total Lagrangian E that accounts for all fermion

gauge interactions,

joint. 5:- : ($15; — 'iIEV’HaMé’) 7
w

transform from the fundamental gauge basis to the basis of the boson charge
eigenstates and perform subsequently the rotations that we discussed in Subsec. 2.2.3.

The result of this calculation is $9)

mt , the fermion—boson interaction Lagrangian after

. . . 2
the second breaklng stage. We may wrlte 'Ziht.) as:

31%) = ZHJO’“ + W; JW + W; r,“ (2.16)

+ 2;,KW + Wl’erW + W;— KW

+ AHLO’I"

with Jo, J i, K 0, Ki and LO denoting the fermion currents coupling to the
respective gauge bosons. J0, Ji and LO correspond to the usual SM currents, K 0
and K i represent new fermion currents that emerge due to the new physics in our

G(221) models. For the contribution of a particular fermion f to J0 and L0 we find:

39

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

§(fl, u, 2’) §(z7, 1/, Z’)
LR -%8;,§XPL + (%ng§2 - éSg§XWR %’5q;§XPL + (%Cq~,§2 + %8,,3§X)PR
LP 7196,9th + (see -— %8g§X)P , $8,352.th
HP ~8g§x(z1;PL + 31%) ésqglePL + (écqgéz + $9,393an
FP -Sg§.X'(%PL + gl’n) %SQ3§XPL
UU %Cg§1PL —%8¢3§2PL
NU d to (5511191, %cd~)§1PL
(13", 2“ )
NE 4.363572% —%3¢3§2PL
(3r )
g(&, d, z") §(é, e, 2’)
LR —%;5(§§XPL — ($65,572 + %8g§X)PR %5(;,£7XPL — @0692 — %;8,,3§X)PR
LP —¥(1;S¢3§XPL — (éCégg + %8égx)PR 3¢3§X(%PL + PR)
HP -Sg§X((1;PL — :13PR) %3d7,§XPL — (écgh -(1;8q3§X)PR
FP ‘5q3§X((13‘PL - zips) 3¢3§X(%PL + PR.)
UU eagle. éséea
NU d —%C§5§1PL ‘%C93£71PL
(lat7 2n
NE ) %sq~5§~]2PL $3 (59211
(3r )

 

 

Table 2.8: Fermion couplings to the heavy Z ' boson in the current K 0"“ — in the
first breaking pattern K 0’“ contains both left- and right-handed parts; in the second
breaking pattern it is purely left-handed. Cf. Eqs. (2.16) and (2.21).

40

 

 

 

 

 

 

LR. 313$?ng 7%.‘72PR
LP JfiggPR 0
HP 0 $512139.
FF 0 0
U‘U JECéglpL —\—}—§s&§2PL
NU (15‘, 2nd) fiCéQIPL "fngéiPL
NU (3rd) _\—}§3¢3§2PL —%8q‘5§2PL

 

 

 

 

 

 

 

Table 2.9: Fermion couplings to the heavy W’i boson in the current K i’“ -— in the
first breaking pattern K 1,11. is purely right-handed. It therefore introduces charged
fermion interactions with a V+A structure standing in contrast to the charged V-A
interactions of the SM. As the contributions from K i“ are always suppressed by %

they can, however, be neglected in many cases. In the second breaking pattern K i’“
is purely left—handed. Cf. Eqs. (2.16), (2.22) and (2.23).

41

J?“ = 3:79 - f7“ [(TEU) — sgee<f>) PL + (-8§Qe(f)) PR] f (2.17)

11‘)“ = éQe(f) - m (2.18)

The currents J i belonging to a specific generation of quarks (u, d) or leptons (V, 8)

take the following form:

 

J+’“ = 914— -d'. 11P u ; J+’“ = g—L— -é ~“P 2.19
(1 \/-2- L7 L L 1 \/2 L/ LVL ( )
— 9L _ —. , 9L _

J "t = — -u *th d ; J (1 = '1/ “(IF 6 2.20
q V? L”) L L 1 \/2 LI L L ( )

The currents coupling to the light gauge bosons and the photon thus have exactly
the same structure as the SM fermion currents. Both sets of currents only differ
in the definition of the parameters they employ. In order to be able to present the
new physics currents in a compact form we introduce the functions §(f, f, 2') and

§(f, f, VV’i) that allow 11s to write K0 and Ki as:

K?” 2 NW“. f, 2’)f ; (2.21)
K?“ E d7“§(d, u, W'H')u ; K142” E (Ea/“fie, V, IAVH‘)1/ (2.22)
Kg,“ 5 m“§(a,d.1i’")d ; Ky“ E 17)“§(17,e,l/l/'_)e (2.23)

In Tab. 2.8 on page 40 we sunnnarize the results that we get for the fermion
couplings to the new heavy Z ’ boson. Tab. 2.9 on the previous page lists the couplings

to the new heavy “flit boson.

The derivation of the fermion currents that couple to the gauge bosons Z, Z’,

42

Wit, W’i and A completes our discussion of those contributions to the fundamental
Lagrangian A? that we are interested in. After the second stage of symmetry breaking

. . . . . . 2
all information relevant to the further analysrs 1s contained 1n the sum of .Z-Elazgs and

3(2) which we may denote by $911.:

int. fun

$15.33.. E 33.33.. + «it?

= $22,211 + g. (117;, + 21217;) mm + 5117222112”
+ III/3.12; WW + (Eff; + A173,”) ng+W’-M

+ 61173,“; (W;Lif’—~# + W; WWI)

+ ZHJ0*“ + W;.J+vl1 + W; JW

+ ZLK‘W + WfiK‘L’“ + Vii—KW“

0. .

Since the fermion coupling to the photon will play no role in the further discussion

we omit it from now on.

2.3.2 Effective Lagrangian at the Electroweak Scale

At energies below the masses M Z’ and M W, the new heavy gauge bosons Z ’ and W ' i

are too heavy to be produced. In experiments probing observables at the electroweak

scale the Z ' and the W'i are solely noticeable though their virtual interactions. As

a consequence, the theoretical description of low-energy processes only accounts for

them in form of their propagators, 85.1,}(q2) and Sfif,(q2)z
1

1w 2 N _ . w 2 N
52'! (q ) (12 _ [ll/1%, a Sui/(q )

As the momentum transfer in these processes, q, is much smaller than 1112/ and

43

 

 

 

(1St step) Fundamental theory Effective EVV theory
Integrate out Z', IV’i: 54121de —+ few,

(2nd step) Effective EVV theory Four-fermion interactions
Integrate out Z, W 2t: .21.,“ ——> 34f

 

 

 

 

 

 

Table 2.10: Relations between the Lagrangians $313 (1., flew, and 34f —- the elec-
troweak theory described by fifew can be derived from the fundamental Lagrangian

3813 d. by integrating out the heavy gauge bosons Z' and W’i. The subsequent re-
moval of Z and W35 from .%W, provides us with the effective four-fermion Lagrangian

24,.

MW, it is justified and fittingly to expand the propagators Sgt/(f) and Sig/,(q2) in

inverse powers of the large masses M , and MEI/r Doing so will lead us to an effective

2
Z
theory at the energy scale of electroweak interactions.

However, we can already take care of the expansions in 115,2 and M {V2, in the
classical Euler-Lagrange equations of motions for Z’ and W’i. In this equivalent
approach we do not have to worry about the treatment of Z ’ and W'i as full-fledged

boson fields comparable to Z, Wi and A just to render most of our work dispensable

p.11
W’

for the large masses of the Z’ and the W’i right from the beginning by expanding

in the end when expanding the propagators S glf(q2) and S (q2). We better account
the solutions of the equations of motion. If we insert the expressions we get this way
for Z’ and W’i into the Lagrangian $313 d. we will obtain the effective theory at the
electroweak scale as well. The Lagrangian of the effective electroweak theory, few”

will then only contain fermion currents, the Z and the Wi.

Similar arguments apply to processes that take place at even lower energies. If

the momentum transfer in a low-energy experiment is smaller than the masses of

44

the Z and the wit we can also integrate out these bosons. This will lead us to the
effective four-fermion interaction Lagrangian 24f. Our program for this and the next
subsection can thus be summarized as shown in Tab. 2.10.

We begin with integrating out the Z' and the Iii/i. In a first step we have to
transform Emmi. from the basis of the hatted gauge bosons, Z, Z’ , Wi and W’i, to
the basis of the physical bosons, Z, Z’, Wi and I-V’i. Eqs. (2.10) and (2.11) tell us
how to do that. Subsequently, we formulate the equations of motions for the Z’ and

the W ’i:

(2) (2) (2) (2)
30m _ 251111131. 2 . (9a (Wannai ‘9ng4; = 0
‘ I , I ‘ r/
6 (8QZL) <92; 8 (BQWM ) 8%“
Since Kfimd. does not contain any derivatives of gauge bosons these two equations

reduce to the condition that the derivatives of Zfimd with respect to Z’ and W’i

vanish. Up to first order in ME? and My}? we find:

 

(raga) K0
)u [If . , A12,
(2) i
m _ fl? _ _ KH- _1__
6W’i _ 0 :> M It “ ”,V 2 “'4
p“ Alli,” 1% WI

Plugging these expressions into .Kfund yields the effective Lagrangian at the elec-

troweak scale:

_ 1 '2 ' 2 7+ r— .
few. = 511.122,,241 + MWWH a .u

0 - 7 ~. 7— “, r I.
+ Z,,J..,;¢f + a ;J:;.,{‘ + VI ,, Jew.“ + $35. (2.24)

few. exhibits three structurally different parts: Ordinary mass terms for the light

physical bosons Z and Wi, effective fermion currents chfi and Jélfd‘ that couple to

45

the Z and the W:t and an effective four-fermion interaction 3&5, that emerges due
to the self-interaction of the new physics currents K 0 and K i. The masses Mg and

Ma, and the currents Jgfi and Jail“ are related to the parameters of the fundamental

 

Lagrangian 'gffifd as follows:
117 . “’2. .
M2 _ W2 _ d [ZZ’ . M2 _ 7‘72 _ 0 UWW’
‘ Z”‘ Z “’2 ’ W""W ~2
M « M .
2’ a”
61112.. ., 5M2. . ,
Jae = J0?“ —— #112?“ ; .13.)“ = 13541 — ~—;—%M1—Ki’“ (2.25)
MZ, MW,

As expected, the results for Mg and [VEV in .208“), agree with the expressions
in Eqs. (2.12) and (2.13). we find that the low-energy effects of the new heavy
gauge bosons Z ' and W ’ i are reflected in shifts in the masses and currents that are
proportional to the inverse of the heavy masses Mg, and Ma”. The Lagrangian .213va

takes care of the exchange of virtual Z ' and 1.1/Ii bosons in four—fermion interactions:

KK __ 1 r0 0,;1. 1 + —,,u
30,, _ —m—I\;K — 377K; K (2.26)
2! viii

2.3.3 Effective Four-Fermion Interactions

Processes at energies below the electroweak scale only involve the Z and the Wi as
virtual off-shell particles. Observables are best described in an effective four-fermion
interaction framework in which the Z and the W i are integrated out. To arrive at
the corresponding Lagrangian, 24f, we repeat the steps of the previous subsection.

First, we formulate the equations of motions for the Z and the IVi:

836W . 839W . aflWV . 80%) W .

————————— =0 ; a———————————_:
a (302;) 82,, 08(8011f) await

 

3a

46

which are again equivalent to the condition that the derivatives of the Lagrangian
gem with respect to the Z and the ”"3: vanish. Up to linear order in the inverse of

the heavy masses we then get:

’72. ,. ‘74 .
Z__igflfl___1_. J0_§_IEZ_Z1KO __:)_A_I_Z_ZLJ0+§ _1_
‘1' _ M 2 _ 1171’? “‘ 332. ” M2. M4 “’ ~4.
Z Z 2' Z’ Z 2’

i *2. . ~4. .
W$:__J§‘LE—_—_;. Ji__(_>.A_[WflKi __§%KMJi+fi __1__
u ,. 2 ~,2 u ”#2 u ”#2 ~4 H ~4

AI”? AIviX' l’lIVi/l AIfiIIMI/i/ MW,

Inserting these results into few provides us with the effective four-fermion inter-

action Lagrangian:

-,~2 4
1 2011'] « ' 6611A »
—___ .1ng _ ”—1-222132041 + __,.. 2:2,: JOJO’“

its
m

 

2.37% .112, ME, ME “
~2 N4
1 6A! A .. JNIA A,
— ——-—.... fir,” — ———...EVW’ (J+K"“' + J‘K‘L’“) + ————.... ‘K‘i‘.’ fir,”
M 2 H M 3, ” “ M2. ,MQ. "
W W IV W
1 0 .0 1 + —.,
— ~2 KuK # — TKMK *1 (2.27)
2MZ, MW,

Neglecting terms that are of second or higher order in ME; or 671;? this result for
the four-fermion Lagrangian 24f may equivalently be written in terms of the effective

i . .. . .
currents may and J9“)? and the K K Interactions:

 

1 1 1 ( 0. 1 —. . XX
2’ W’ Z ‘ W

The discussion of 34f completes our analysis of the fundamental properties and

47

features of the G (221) models under consideration. In the last two sections we derived
and examined the three Lagrangians .Yéizd, few, and 34f each of which describes
the old and new physics that emerge in the G (221) models at a different energy scale.
We now have the masses of the physical gauge bosons and the gauge interactions of

the fermions at our disposal and are ready to proceed with the preparation our global

fit analysis.

48

Chapter 3

Global Fit Analysis with GAPP

Given the precision that has been reached in measuring the electroweak observables
it is a matter of fact that the SM accounts for most of the physics that governs the
electroweak interaction and that new physics effects can play the role of corrections
to the predictions of the SM. In this chapter we will calculate these corrections to the
SM in our G(221) models and examine which regions in the new physics parameter

space are still allowed for by the experimental precision data.

Our strategy to obtain constraints on the new physics contributions in the G (221)
models is the following: First, we will carefully define the parameters in terms of which
the corrections to the SM expressions are going to be parametrized. In this discussion
we will link the model to the standard parameters and show how the connection to
the reference observables is eventually realized. Once we have established a working
base of input and fit parameters we will be able to turn towards the calculation of the
new physics corrections. After an overview of the observables that will be included
into the fits we will revisit the effective Lagrangians few, and 2 4 f and derive the
set of those operators in terms of which the electroweak observables are defined. This
crucial step will enable us to write down the corrections to any observable that we

are interested in.

49

After all analytical work is done we will turn towards the numerical part of our
global fit analysis and alter the code of the plotting package GAPP. In some examples
we will present the structure of the GAPP files and illustrate the general procedure
by which we implement our results. This modification of the GAPP code will finally
allow us to test our G (221) models by comparing their predictions to the electroweak
data. For each model we will scan over a grid in parameter space and identify the
regions permitted by the data. In this context we will have to explain how. the grid
must be set up and to develop the algorithm that will provide 11s with the contours

in parameter space.

3. 1 Parametrization

The goal of our fit analysis is to examine the bounds on new physics effects. For
that reason we now have to review the fundamental model parameters of our G (221)
models and clearly separate new from old physics parameters. We will fix those
combinations of the fundamental model parameters that have equivalents in the SM
by means of experimental data. Parameter combinations that do not correspond to

parameters of the SM will serve as free parameters during the numerical fits.

3.1.1 Fundamental Model Parameters

The gauge sector of the SM has "SM 2 3 model parameters: the coupling constants
9L,SM and 9Y,SM of the two gauge groups SU (2) L and U (1)Y and the VEV ”SM
of electroweak symmetry breaking. ()ur G(221) models either feature fi(BP-I) =
6 or fi(BP-II) = 5 model parameters depending on the mechanism by which the
fundamental gauge group is broken. For both breaking patterns the extension of the
SM gauge group by a second SU(2) results in an additional coupling constant. In

breaking pattern one the VEV of the Higgs field (I) exhibits two degrees of freedom,

50

in breaking pattern two it exhibits one. According to our discussion in the second
chapter the fundamental parameters of all three considered classes of G (221) models

are given as:

(BF-LT): {@1,§2,§X.fir»3=’5‘}

(BF-II,D): {Weak-.1117}

Compare especially with Tab. 2.2 on page 16, Fig. 2.2 on page 28 and Fig. 2.3 on
page 29. Three combinations of the G (221) model parameters have analogs in the SM
and must thus be fixed by the reference observables. Fitting the models belonging to
the first breaking pattern we are hence left with three free parameters. In the case of

the second breaking pattern we will deal with two fit parameters.

One possibility to distinguish new from old physics parameters would be to con—
sult the relations between the three gauge couplings 91, $32 and g X and the two SM
couplings 9L,SM and 9Y,SM and to fix combinations of these three couplings accord-
ingly. However, gauge couplings represent rather less intuitive parameters since they
are only indirectly related to physical quantities and GAPP does not use them. In-
stead of 9L,SM and 9Y,SM it employs the Weinberg angle 63M and the fine structure
constant 013M. We follow the example of GAPP and trade the couplings @1, fig and § X
for our model fine structure constant 6: and the mixing angles a; and (if. The relations
that tell us how this has to be done in principal are listed in Tab. 2.4. Subsequently,
we exchange the angle 6 for the sine squared, 3%. It is this parameter rather than 5
itself that will frequently appear in our calculations.

Moreover, we expect the ratio squared of the two scales of symmetry breaking,

ii and 17, to be the best measure for the effect of new physics 011 the electroweak

51

observables. This dimensionless parameter that we introduced as 5: in Eq. (2.14) sets
the mass scale of the new heavy gauge bosons (see Tab. 2.6) which is the key criterion
for the impact of new physics. In a second step we thus trade 11 for if: such that we

end up with:

~

BP—I: {51, ,s

@453

Among these model parameters we recognize three parameters that we already
know from the SM: (1, 27 and 3%. We will fix these parameters by the experimental

input. The remaining two or three parameters will take the role of the fit parame-

Z
0

parameters 018M, USM and 838M which we carry over from the SM:

ters. We remove 62, 27 and s from our calculations by relating them to the standard .

2 2
e . 1 7m: . v .
_ _ M . _ . 2 _ SM M 2
471' fizz S C
'31“ "98M 93M

We obtain numerical values for QSM, USM and 838M making use of the precise
measurements of the fine structure constant oz, Feri‘ni’s constant G F and the pole
mass M Z of the Z boson. In conclusion, the way we organize the model parameters

can be summarized as depicted in Fig. 3.1:

The new physics effects in our G (221) models all scale with the masses of the new
heavy gauge bosons and if we were to take them to infinity our model parameters
should be identical to the parameters of the SM. Deviations in Cr, 27 and 3% from 08M,
”SM and 838M thus are expected to appear at first order in 3,1.- In other words: We

already know that the relations we are looking for must. be of the following form:

52

 

 

 

 

 

 

 

 

 

 

 

i2 8
Q) I—I1
4: .D
Q) A g
a 2 FM H
2 g 2
:8 g , .D
.. O)
E 2 5 0
CG ‘0 C:
"C d W'" a)
Cl w—2 a...)
C3 - ci-i
4.“: Q)
m I a:
A W
H
‘Q. A 09
PM "9‘ “9‘ 453

15 w—z w—v
H
is .. d
we H L: Q.

I

cu 9.. :3
m m [.14

 

 

 

 

 

 

 

1%; PM E.)
re re 53
is is” CD
«a "J: E
i9“ i5 3
id id 9..
w-I wa '_'
a)
.. '8
"i" "T‘
o.. 0.. 2
CG m

 

 

 

Figure 3.1: Overview of the model, standard and fit parameters ~~ model parameters
that do not have an equivalent in the SM (new physics parameters) will be used as fit
parameters in our global fit analysis. Parameters with analogs in the SM (old physics
parameters) have to be related to the standard parameters and subsequently fixed by
the reference observables.

53

i2
L 1
v :17SM [1+ 7C7} + (Z (3)] (3 3)
II:
I
2 __ 2 _ ~ _
863 _895M [1+ £C6+§ 532)] (3 4)

Once we have expressed the coefficients C5,, C9 and C9~ in terms of SM and fit

parameters these relations in combination with Eq. (3.1) will enable us to fix the

2.

values of Ex, 27 and 56

. In the following subsection we derive Ca, Ca and Cg one after

another.

3.1.2 Relations to the Standard Parameters

Fine structure constant (3: The first case turns out to be trivial. The electric
charge 6 is defined as the coupling constant of the fermionic gauge interactions with

the photon A.

WW 2 e . easier/1.. (3.5)

A comparison of this definition with our result for L38“ in Eq. (2.18) lets us
conclude that there is no difference at all between all the introduced versions of the

fine structure constant:

2 6 = 65M => 5' = Oz = 058M => Cc} = 0 (3.6)

Cb?

VEV of the Electroweak Symmetry Breaking 27: We derive the coefficient Ci}
by relating the two VEVs 2'} and ”SM to Fernii’s constant CF. As C F is determined

from experiment its numerical value is fixed. It is the same in any model and can

54

thus serve as a link between 23 and USM- The definition of ”SM in terms of C F is given
in Eq. (3.1). To obtain the corresponding relation between 27 and G F we first have to
understand the origin of Eq. (3.1): G F is model independently defined as prefactor

in the analytical expression for the lifetime 7'” of the muon )2, compare with [20]:

 

 

 

 

where:

 

 

F()1—81:+83r3 —12.r.21n:r
156815_ 518 2 8395 67 4 8 2
C = —— — 1 2
2 5184 8163“ H 7207r + 67r n
2 m 1
—1 -—i u
(I (mu) 2 (1 — 37111 (E)3 + a;

As we are only interested in the leading order shift in USM the higher order correc-

1 are of no interest for us. For the moment it is sufficient to only consider

tions to 7';
the tree-level expression for the muon lifetime; ’7'; 1 = (G%m2)/(192773). The explicit
computation of Ty in the SM is based on the effective theory below the electroweak
scale which is governed by the Lagrangian $3M,4f‘ As for the decay of the muon,
we are only interested in the contribution 3&1,“ to ESL-1,4 f that takes care of the

interaction of charged currents:

(h
3” LSi4f2A12—JSLI,)1.JSLI
W ,SM

The muon if decays into an electron e‘, a muon neutrino up and an electron

anti-neutrino 176: n" —+ e- + V), + 176. The corresponding fermion currents are

CI!
0"!

consequently given as:

_ 9L SLi _
J ’, = -e’)' 'P 1/ ; J ,, i“ : —-’——— -1/, “P
SLi fl L e SL1 fl #7 L“

 

Inserting these cm'rents into Egg 4f and going through the algebra yields the SM

expression for the muon lifetime:

r

4 .
_1 9L,SM mil

SL1" 4 ' . 3
H 32MW’SM i925

 

This result allows us to identify CF. Employing the relation Ala/38M = :11 9381.11ng

we recover Eq. (3.1):

0F : Q . 32% I __19__
8 MWsLi fil’SM
We observe that G F is determined by the prefactors of the fermion couplings to the
charged bosons W'+ and W _ in the effective four-fermion Lagrangian. This insight.
sets up our strategy to relate i} to CF: We will take that part of 34f that accounts
for charged interactions, $41,. , compute the effective couplings to the charged gauge

bosons and relate the results to T; 1and G F as in the SM. $5113 has been calculated
in Subsec. 2.3.3:

       

 

6M? 6174.
59;?- = —- ———,.1. JJJ’W— ——’VVVV (J;K 11%.]; K+’“)+ ———'VVVV’ J;J M
M. 1172., L72. ,L72.
W W W
_ 1
L72,

Depending on their chiral structure the new pl‘iysics currents K+’“ and K ‘1”
might or might not enter the calculation of T; 1 in our G (221) models. The reason lies

in the fact that it is the square of G F that enters the expression for the muon lifetime.

56

Repeating the SM calculation we have to square the fermion couplings to the charged
gauge bosons. The contributions from the currents 1&7?“ and K ‘W are, however,
suppressed by the mass HEW of the new heavy ll/"i gauge boson. If K 4”“ and K7“
are right-handed —- being the case for the models of the first breaking pattern —« the
J K and K K operators do not interfere with the left-handed J J operators in the limit
of massless fermions. The non—zero terms involving the currents K +’“ and K “3‘" that
we are left with are all of order Maj and hence negligible. By contrast, in the models
belonging to the second breaking pattern the new physics currents are left-handed
resulting in non—vanishing interference terms. In the following we discuss both cases

separately.

We derived the currents J +3“ and J ‘3“ in Subsec. 2.3.1 and found that they have
the same structure as their SM analogs; see Eqs. (2.19) and (2.20). To calculate G F
for the first breaking pattern we therefore only have to replace g L,SM by Q L and take

Vi

into account that the shift in the l mass introduces a second J J operator in the

effective Lagrangian:

 

~2 ’74. .
BP-I- G ____\/§. 9L 1+———6M‘VW'
‘ F 8 .132. L32. A72.

W W" W

Making use of the expressions for g L and the l/Vi mass parameters in terms of the
fundamental model parameters that we collected in the last chapter we can rewrite

this result as follows:

1 53% 1 333
V (BF-LT): CF: 1+

BP-I,D: G 2: 1 —~— ; ~
( ) F @132 .L‘ @172 217

 

 

 

The dependence 011 the representation of the Higgs field (I) is induced by the mass

M 3V, that takes different values for different (I) representations. Keeping terms up to

57

first order in % the comparison with the SM result finally yields:

533 335
2:1: 4r

In other words: The coefficient C17, takes the value Cg. = $533 if (I) is remesented
1

by a doublet the value C5. = 1.93 5, if the triplet representation is chosen for (I). In the
/

models of the second breaking pattern the currents KV’” and K_’f‘ may be written

 

as:
. Six _ V _ , §K -
K+’“ = —— -€7#P I/ ; K ,,u =2 — - I/yf‘P 6
fl L fl L
where g corresponds to the entries in Tab. 2.9 with the factor J— and the
K «5
projection operators taken out. Now all operators in $111} contribute to G F and we
obtain:
~ "f 4 7'2 S
V2 9% WWW V? ‘5MWW’ ~ ~ V32 9%
8 M2. M2. M2, 8 M2 M2, 8 M2.”
W W’ W W’ W W
~2 72. ’74. . ‘72. . ~ ~2
: Q.iL_1+fi/_ %1_2%M£I:+!K
8 "2 T2. 74‘ 217:2 .62 ”2
A I W 11 W’ A W W L 9L

In the UU-D and the NU-D model all corrections due to new physics cancel each

other and Fermi’s constant C F reduces to the same expression as in the SM:

1

BP-II: GF = J?”
z:

 

=> ’5’ = vSM => C5 =

[\D

2.
0
angle 3% by equating the SM expression for the mass of the Z boson M Z in Eq. (3.1)

Electroweak Mixing Angle s We obtain the shift in the electroweak mixing

58

with our C(221) results that are listed in Tab. 2.6. With AID being defined as in

Eq. (2.15) we can write for 11122:

 

4 4
”2 C7 ~.~2 cT
(BP-I,D): Mg 2 LL) 1__~¢2 : ”€121 1__ 3))

 

2 2 2
Cg 89-65
4 4
112 0* M1272 6*
(BF-LT): 11; = i—‘l 1——33— — ,, 1— if
C2. 4513 5‘: .. 413
0 a a
4 4
M2 5: M752 3*
(BF-II,D): 111% = 79— 1—70 = 2 2 1——.‘”
C~ 11 8~C~ :1:
6 6 6

The results of our previous discussion of d and 7? allow us to remove all model

1

parameters from 11.1% expect for the mixing angle 6. Up to linear order in 5:,- we obtain:

2 r'
WOSKI’USRI 1 4 2
_ ; A» 2 — __ _ _ ( - _ )
(BP LB) 12 3% [0% 1 :2: c¢ s ]

. 2 -
ESMVSM

1
Sc: if
50 0 -

(BF-LT): 1.1% =

2 .-
WOSM’USM—

1

1
- - 1,2 I __ _ 4
(1313 II,D). NZ 5% _ 535(1)]

 

The comparison with Eq. (3.1) then yields:

. F 1
- 2. 2. z 2 2 ._ _ 4.. 2-
(BP-I,D). 366.0 SOSMCQSM ‘1 j: (c¢ 323)] (3 7)
.22_2 2 l_1}4_}2
(BF-LT). Sgcg _ 'SOSMCQSM L1 :2 (4ch 3213)] (3 8)
_ 1
_ - 2 2. z 2 ,2 _ _, 4.
(BP II,D). 5000 SQSMCOSM -1 533(1)] (3 9)

 

C5,. Cg, Cé

 

 

_ 1 2‘. — . V 1 I 4“. — 2~
_ 1 ‘2~ _ ‘ ’ . 1 4 _ 1 2~

(BF-II,D) 0 0 -f (93M) ' 8

 

 

 

 

 

 

 

 

Table 3.1: Results for the coefficients C5,, C73 and C6 that parametrize the shifts

in the model parameters fr, 1"; and 6 resp. at first order in 2% — compare also with

Eqs. (3.2), (3.3) and (3.4). f (63M) is given as f (63M) '5 chM/ (038M — 538M).

. 9 .
Replacmg C; by 1 — 55 we can solve these three equations for 3%:

 

 

 

 

1 a
2 _ 2 SM 4 2
(BP"I’D) Sci _ SVSM 1 — E ' 2 _ 2 (ch _ 82*)
_ 98M 95M
— 1 Cg 1 1
_ 2:2 __. SM _4__2-
(BP LT) 86 395M 1 5i: 2 _ 32 (404) 2826)
_ 93M 93M
- 2_2 _l. %M 4
(BP II,D) 36 _ 598M 1 i 2 _ 82 3(1)]
_ 95M 98M

These findings for the electroweak mixing angle complete our analysis of the model
parameters. Tab. 3.1 summarizes our results for the coefficients C ~ , Cf) and C5 that we
found in this subsection. Together with Eqs. (3.2), (3.3) and (3.4) and the definition

of the SM parameters in Eq. (3.1) this table enables us to fix the values of d, 27 and

2.
6

to parameterize the operators that we need for the calculation of the new physics

3 . We are now prepared to return to the effective Lagrangians $93, and 24f and

corrections in terms of the fit parameters 5:, a3 and B. First of all we will, however,

discuss the considered electroweak observables in more detail.

60

3.2 Electroweak Observables

The corrections to the SM expressions for the electroweak observables constitute the
foundation of our global fit analysis. Before we turn to the explicit algebra we,
however, first have to introduce and define all the observables that we are going to
consider. After we have developed an understanding of all the quantities that will

enter our fits we will proceed with the actual calculations.

3.2.1 Overview of the Included Observables

The electroweak precision observables to which we will fit the G (221) models fall into
two classes: Observables defined in terms of operators that appear in the effective
Lagrangian .%W, at the electroweak scale and observables that are related to operators
in the four-fermion Lagrangian $4]: below the electroweak scale. Both classes can
be further subdivided into certain sets of observables. We now briefly characterize
these groups mentioning in each case which and how many observables they contain
and how much experimental data is available respectively. In total we will fit 37

observables; 46 experimental values are at our disposal.
Observables derived from few;

0 Z pole data: Partial decay widths T Z ( f D of the Z boson, fermion left-right
asymmetries A L R( f ) and various other observables that can be constructed
from these quantities. 21 observables, 25 experimental values (LEP and SLAC

data) .

o Wi pole data: Mass [WW and total width PW of the W'i boson. Two observ-

ables, four experimental values (LEP and TeVatron data).

0 Mass mt 0f the top quark: t: One. observable, one experimental value (total

TeVatron average).

61

Observables derived from 24f:

o Neutrino-nucleon scattering.‘ Left- and right—handed neutral current quark cou-
. T . .
plmgs 9'15!” and 9'sz and ratlos of neutral-to-charged current cross sections RV

and R17. Five observables, eight experimental values.

0 Neutrino-electron scattering: Vector and axial neutral current electron cou-

plings 9%”; and 9518. Two observables, two experimental values.

0 Parity-violating processes: Weak charges of cesium, thallium and the electron,
188 20.5 . , . , . .

QW ( Cs), QW ( T1) and Qw(e). Two hnear combinations C1 and C2 of

the quark vector couplings CM and Cld. Five observables, five experimental

values.

0 Lifetime T7 of the 7' lepton: One observable, one experimental value (world

average) .

This selection of observables differs slightly from the set of observables that is
used by the default 2009 version of the GAPP code [21]. Originally, GAPP does not
consider the width I‘w of the W i boson but additionally includes the value of the
anomalous magnetic magnetic of the muon %(gp — 2), the measurement of the unitary
of the first row in the CKM-matrix and data related to the b —+ 37 decay. fig” — 2)
and the b —> 37 amplitude only receive new physics corrections at the loop-level.
Since we are only interested in tree-level effects induced by new physics we do not
include the corresponding observables into our analysis. In this study we consider the
new physics tree-level corrections, if a? is large, to be of the same order of magnitude
as SM loop effects — new physics loop terms are therefore negligible. Furthermore,
we take out the experimental constraints on the CKM unitarity as we do not consider

new physics in the flavor sector in this analysis.

62

In the following we will allude to all these groups of observables separately. We
will introduce the respective quantities in terms of which the individual observables
are defined and set up everything such that we are prepared to calculate the new

physics corrections in Subsec. 3.2.2.

High-energy observables derived from few,

Z pole data: At tree-level the partial width I‘ Z ( f f‘) of the decay of a Z boson

into a fermion pair ff is given as:

 

z(ff)= — "C3 ) 1:22;” ([géml2 + [9:20am (3.10)

" 6 C6
By writing the parameters appearing in this expression without tilde or SM index
we indicate that this relation holds independently of the model employed. When we
have come to calculate the new physics corrections in Subsec. 3.2.2 we will evaluate
this general expression for I‘ Z (f B and all the other definitions presented here in the
SM as well as in our G(221) models. g€( f ) and 9624 (f) denote the vector and the
axial couplings of the fermion f to the Z boson. They are related to the left— and

right-handed couplings 9% (f) and 912?“) as follows

gém s ; (gém + 91%)) ; 9.20:) a g (9122(1) — 92(1))

and parametrize the coupling of the Z boson to the neutral fermion current J?”

6

0 H
wa-Z

6:61? (gt/(111,. + 9A(f)71n'5) .12” (3.11)

‘1 (9% be. + gases) fZV'

 

seq;

nc( f) in Eq. (3.10) stands for the color multiplicity of the fermion f. The charged

63

leptons 6 as well as the corresponding neutrinos Vg are represented by color singlets,

the quarks q come in tree colors ——— red, green, blue:

710(021 1 nc(’/€)=1 i 'nc(Q)=3

M Z is the experimental value for the mass of the Z boson. If we sum over the
partial widths F Z ( f f) of all fermion pairs to which the Z can decay, that is, all pairs
expect for the top pair tf, we obtain the total width 1" Z of the Z peak. Summing only

over quarks q in the final state provides us with the hadronic decay width P Z (had):

F2 5 :FZ (ff) ; F2 (had-) 5 212016)
f?ét qyét.

The individual partial widths I‘ Z ( f f‘) for decays into fermion pairs f f , the total
width F Z and the hadronic width I‘ Z (had) are the ingredients for a wealth of sec-
ondary observables. For instance, the total hadronic cross section chad. representing

a fundamental QCD quantity that is accessible experimentally can be expressed in

terms of I‘Z (eé), FZ and F2 (had):

127r

E —— ~ 1" (ca—6+) F (had)
,. 2 2 Z Z

a had.

The partial widths for decays into charged leptons F Z (66) and decays into quarks
I‘ Z (qr‘j) are used to define the hadron—to-lepton ratios R(€) and the hadronic branching

ratios R(q) respectively:

 

I‘Z (had)
R t’ E ——-——-—-—— : 66 . .,
F _
R(q) " ——Z (W) ; q E {ud,c,s. b}

: FZ (had)

64

As for the light quarks, u, d and 8, experimental data is available for the ratio of

R(s) to the total branching into light quarks:

_» R(8)
MS) 2 Ha) + R(d) + 12(5)

 

Coming back to the left- and right-handed fermion couplings 9121 f ) and gg( f)
to the Z boson we can write the polarization or left-right asymmetry A L R( f ) of a

fermion f as follows:

[92(1)]2 —- [914(1)]2
[92 (1)12 + [9.2.0912

 

ALRU) 5 (3-12)
The combination of the quark branching ratios R(q) and the left-right asymmetries

A L R(q) yields the hadronic left-right asynnnetry Q L R3

QLR E Z R(9)ALR(Q)" Z R(Q)ALR(q)

q=d,s,b QIU,C

A second class of asynnnetries, the forward-backward asynnnetries AFBl f), emerges
from the convolution of the A L R( f ) asymmetries with the polarization asymmetry

A L 3(6) of the electron. The hadronic charge asymmetry Q p B is defined accordingly:

AFBUEZALRreMLRm ; QFBEEALWQLR

Having introduced these last two quantities we have covered all relevant definitions
pertaining the Z pole data. We will include the following 21 observables into our

global fit analysis:

 

 

R(C) 1 RU?) 1 ALR(€) 1 ALRUL) 1 ALetT) 1 ALR(3) ,
ALR(C) 1 ALR(b) 1 4113(6) 1 AFB(#) 1 AFB(7') 1 AFBlS) 1
AFB(C) 1 AFbe) 1 QFB

 

 

 

 

 

 

l/l’":t pole data Due to their opposite charges the two charged electroweak bosons,
W+ and W‘, couple to different fermion pairs. Respectively, the following decays

are allowed:

W+ _.1 6+1») ; W+ _. WI,- (8.13)
W" —1 £17) ; W" ——1 11,11,- (3.14)

with 2' = 1, 2 and j = 1, 2, 3 representing generation indices. Since the top quark
t is too heavy to be produced in W i decay z' = 3 is excluded. The respective decay
widths of the W+ and the W‘ are, of course, identical. In order to avoid writing
down every expression twice we will consider the properties of the WJr only in the

following discussion. All results derived for the W+ will apply to the W “ as well.

In analogy to Eq. (3.10) the tree-level expression for the partial width of a VV+

decaying into a lepton—neutrino pair €+Vg reads as:

 

n. (F) ,1 2 2
mica); :8g-Mw([g}}<e] + [91-11 (51])

The left- and right-handed couplings g2( f ) and 92% f ) of the fermions to the

W i are defined similarly as the corresponding couplings to the Z boson:

66

J+6 W+ 11 E 71—6171,,9 (ng (5)131, + 911101911) WWW-

LLWV’“ 3 «5017's (9fi (“PL + giri (0P1?) Ul’l”+’“

 

In the SM the W i does not couple to right-handed fermion currents. The only
reason why we include 9211f) into our discussion is that it might play a role in our
C (221) models later 011. However, we learnt from our calculation of the muon lifetime
in Subsec. 3.1.2 that. right-handed contributions are always suppressed by the mass
117W, in the C(221) models. As it is [9W ( f )]2 that. enters the partial decay width
PW (6+ I/g) these contributions will be negligible. Calculating the Wi decay width

explicitly up to order 6 (MIT?) we will therefore always use the following relation:

 

IV . 2 1'
-MW [9L (5)] (8.1.1)

The partial width for the decay into a quark pair PW (u1z-cfj) is slightly more com-
plicated than PW (6+ I/g). To take into account the mixing of the strong eigenstates of
the down-type quarks in the case of weak interactions we have to include the entries

of the CKM—matrix V into I‘W (uddij):

net-ta)

_ , 2

Owing to the unitarity of V these coefficients become irrelevant once we only

consider the combined widths PM; (11.1):

 

2
Zj:‘l/1j12=l => P11 11.1) EZF11( (uidj) 271462;) ’lfwg[L V11(,')] (3.16)

67

The arguments E and uz- of gin f ) in Eqs. (3.15) and (3.16) are not intended to
indicate a dependence of 92% f ) 011 the quantum numbers of the lepton K or the up-
type quark uz- but are supposed to reflect the fact that 92% f) may vary between
fermions belonging to different S U (2)s in our G (221) models.

If we finally sum up all leptonic and hadronic decay widths we obtain the total

width rW of the W: boson:

FW E 2: PW (”fl/r) + 2 PW (W)
e z'

This is one of the two Wi pole observables to which we will fit our G (221) models;

the other being the mass [MW of the W'i boson:

 

 

Fl/V , A’I W

 

 

 

 

 

 

Mass of the top quark: Besides the mass of the W:t boson we will also include

the pole mass mt of the top quark into our fits.

 

 

 

 

 

mt

 

 

111 the G(221) models that we consider mt, however, does not receive corrections
due to new physics. To see why that is we must have a closer look at the origin
of mt within the theory: In the SM as well as in our G(221) models the masses
of the fermions are generated in the Yukawa sector through spontaneous symmetry
breaking. The fermions couple to the Higgs bosons —— once the Higgs fields acquire
their VEVs the Yukawa interactions turn into fermionic mass terms. The generated
masses are then given in terms of the Higgs VEVs and the initial Yukawa couplings I

Gf. In the case of our G(221) models we may write for mt:

mt 2 Ct - H.111?) (3.17)

68

where the functional form of f (i: {2) depends 011 the details of the respective model.
Eq. (3.17) shows us that the value of mt can always be set to any desired value just by
choosing the Yukawa coupling Ct accordingly. In fact, Ct is an additional fundamental
parameter of our G(221) models. Due to its trivial relation to mt the problem of
constraining Ct can, however, be completely separated from the remaining analysis.
In this work we will choose Ct such that the G(221) prediction of mt corresponds to

the SM value:

mt E mt,SM

Note that it is the on—shell mass mt of the top quark that we will use as an
observable. Fitting the G (221) models to the data in Subsec. 3.3 we will, by contrast,
employ the 11% mass mt as a free fit parameter. In appendix B we briefly outline the

relation between these two definitions of the top quark mass.

Low-energy observables derived from 34f

Neutrino-Nucleon Scattering Deep inelastic scattering (DIS) experiments allow
to probe the coupling of neutrinos I/ to nucleons inside an atomic nucleus N. For
measuring the electroweak mixing angle, it is advantageous to choose an isoscalar
target. As neutrinos are capable of exchanging both Z and W i bosons with the up
and the down quarks that constitute the nucleons neutral (NC) as well as charged
(CC) current interactions occur in V-DIS experiments. In the case of, for instance,
incident muon neutrinos V“ and muon anti-neutrinos Du the following reactions take

place:

69

NC: I/flN ——+ l/fl,X ; DAN —> DMX

CC: l/flN ——> ILL—X ; DflN —> p.+X

where X denotes an arbitrary hadronic final state. These weak interaction pro-
cesses are governed by the effective four-fermion Lagrangian 34f below the elec-

troweak scale. If we assume that only the usual left-handed SM neutrinos play a role

in v-DIS experiments the contribution ZEC’VN to $41: that accounts for the neutral

current neutrino interactions with the up and the down quarks is given by:

NC,l/N _ CF
34f = fl” 11(11—75) VZq7“l€L(CI )(1-25)+€R(Q)(1+"/5)lq
q: u, (1
As the chiral fermion structures that we encounter in this Lagrangian will appear

over and over again in the analysis that is still to come we 110w introduce the follow-

ing notation for left— and right-handed as well as vector and axial fermionic spinor

products:
f17”(1 - 75)}‘2 E (ilk)? : f1““(1+75)f2=— (f2)R
f1?’“f2 E (f1f2)(12 ; f1?” 7: 5:f2 — (f1f2)’:1
. . . , . NC,1/N . , .. .
Wlth these abbreVIatlons we can wrlte .204 f 111 a more compact f011n.
NC, 1/N_ GF _ _ a .
34f 75(WlL,” Z l8L(q)(<1‘q)’£+ 512(4) (qq)R] (3.18)

q=u,d

7O

The Lagrangian Effigy“ which is the counterpart of KEC’VN being responsi-

ble for charged current interactions has the following structure in the case of muon

neutrino experiments:

cc, N CF _ - _ — . cc, N
as . 7, [ml/asses + erasures] was

with $133521qu accounting for effects beyond the SM. The actual physics of neutrino-
hadron scattering is contained in the coefficients 5 L (q) and 5 R (q) of the effective
four-fermion operators in the neutral current Lagrangian. In Subsec. 3.2.2 we will
calculate the shifts of all V—DIS observables just be determining the new physics cor-
rections to 5 L (q) and 5 R (q). At tree level the left— and right-handed neutral current
quark couplings g’iN and 9'sz can be expressed in terms of the coefficients 5 L (q) and
5 R (q) as follows:

_ [V
ggN 25% (magi (d) ; g’fz 5%,(u.)+s2R(d)

Accordingly, the differences between the respective coefficients can be used to

define the quantities (SEN and (SEN:

T _ ~IV _
63;" = 5% (u) _ 5% (d) ; (1* 1 = 532 (u) — 8212 (d)
- - VN ‘l/N ° 3 _ .. ' _ . - -. VN uN
Comblmng 6 L and OR With the left and rlght handed couplings g L and QR

allows us to construct the observable FLVN :

Isl/N E 0% - ($121")2 + 0%. (gng + of, (dz/N)2 + cg. (53,411")2 (3.19)

Ail/N can be understood as a measure for the effective uuqq coupling in V—DIS

processes [22]. It has been measured experimentally by the CCFR collaboration at

71

Fermilab; cf. Ref. [23]. In Eq. (3.19) C9 , Cfi, C2 and 0%, represent weight factors
that depend on the specific experimental setup; according to GAPP Ci and 0% are
of order 0(1), C2 and 0% of order 6’ (10’2).

However, the actual observables of interest in the context of V-DIS experiments
are the ratios RV and R9 of the neutral-to—charged current cross sections which we
denote by UN Nand 0318 in the case of neutrinos and by UV N Cand 0,918 in the case of
anti-neutrinos. By construction many theoretical uncertainties cancel in RV and RD.

For that reason it is these quantities that many V—DIS experiments are interested in.

111 the lowest-order approximation we may write:

 

 

051$ VN 2 VN 2 Ogle 2 1 l/N 2 ‘
RV E 030.. — —(gL )+ +(gR ) r ; Rp E CC E (gLN ) + ; (9R ) (3.20)
UVN JPN
r E egg /01S31(V3 denotes the ratio of the charged current cross sections (ICE and

001$ and can be measured directly Ru and RD complete the set of neutrino-nucleon

scattering observables that we will use in our fit analysis.

'2 2
(933”) . (g?) , W . RV , Hz?

The expressions for RV and R1; in Eq. (3.20) involve the ratio 7“ that we do not

 

 

 

 

 

 

 

 

have a handle on. However, this does not represent a problem. In practice, RV and

R17 are usually written as linear combinations of 5 L (q) and E R (q):

The advantage of this notation is that it clearly separates experimental from

theoretical influences on R, and Rp. While the coefficients (5, (I, 0L, R(q) and (‘1 L, R(q)

72

are fixed by the conditions under which the experiment is carried out the theoretical
details are entirely incorporated into 5 L (q) and ER (q). Higher-order corrections to
RV and R9 only apply to E L (q) and 5R (q); the coefficients are always the same. The
values of 6, 5, a L, R(q) and ii L, R(q) for the various V-DIS experiments are implemented

in the GAPP code. Our task will be to derive a L (q) and ER (q) in our G (221) models.

Neutrino-Electron Scattering Not only the scattering of neutrinos off nucleons
but also off electrons can be probed in low-energy measurements. The most pre-
cise data on neutrino—electron scattering comes from the CHARM II [24] experiment
at CERN that utilized muon neutrinos and anti-neutrinos. I11 the theoretical de-
scription of the CHARM II 111easurements we only need to consider the Lagrangian

$30,111: that takes care of neutral current interactions. 330”", is identical to the

- NC VN . . .
Lagrang1an 24 f ’ that accounts for neutral current neutrino-hadron mteractlons,

see Eq. (3.18), except for the fact that it involves electrons instead of quarks:

1 V8 G _ _ _ . ,
ESQ = ——\7_§ (VI/)L‘u (5L (6) (66),: + ER (6) (66);;2] (3.21)

In the case of scattering of electron neutrinos I/e off electrons also the charged

interaction Lagrangian 3430’1/6 has to be included.

_ _ _ _ ‘ cc.
if E 75 [warms + emcee] WWI»;

However, the physically relevant information is again entirely contained in the
coefficients 8 L (e) and 5 R (e) of the four-fermion operators of the neutral current
Lagrangian. Just as in the hadronic case these two coefficients are used to define
the effective four-fermion couplings. Instead of employing left- and right—handed
couplings one usually formulates the neutrino—electron interaction in terms of vector

and axial couplings gi’f and 9f:

73

ail/8 E€R(€)+€L(€) ; 928 5512(6) -€L(€)
0gfq.C,I/e ' _ ,.' 3.
4f can then equ1valently be w11tten as.
NC,I/e _ CF _ 1/ _ 11 1/3 _ 11
34f “ ”:50”le lgvefeeh + 9A (€€)Al

The observables that are typically measured in the experiment are the total cross

sections 03160 and 0,9160 or their ratio 03160 /0,1—>16C. In the limit of large incident neutrino

energies. E, >> me, the cross sections are given as:

 

02 77leE1/ '- 2 1 2-
0320 = —————F (917 + 9.?) + r (956 - 91f)
27r _ 3 j
02 m E ' 1 -
N e V / 2 2
0176C = F27, —— (91/6 - 91f) + g (95? + 95f)

 

 

If the scattering of electron neutrinos V5 is studied the contribution from the
charged current interactions must be considered as well. Effectively, the corresponding
cross sections 01,8 and 0,78 are obtained by substituting 9"}? A + 1 for gfi’f A in 056C and
0,2160. We do not have to care about these details as the experimental results are
usually boiled down to the fundamental couplings 956 and gfie. In our fit analysis
with GAPP these two couplings will be the only observables related to neutrino-

electron scattering:

 

 

V8

91/ a g 11:16

 

 

 

 

 

 

Parity Violating Processes: The interaction of charged leptons with other charged
fermions is dominated by the Coulomb force. For the most part it is described by the
QED Lagrangian gQED : 1}},qu see Eq. (3.5), in which the fermion current [138“

has a vector structure. Taking the product of L38” with the vector field A“ results

74

in a scalar Lagrangian XQED- QED processes are therefore invariant under parity
inversion. However, the electroweak force as we know it from the SM has a V—A
structure resulting in maximal parity violation. New physics effects may introduce
V+A interactions softening the extent of parity violation. But as they will always be
suppressed at low energies parity violation is an intrinsic property of the electroweak

force below the scale of new physics.

Since the detection of parity violation in the mid 1950’s [25] many experiments
were devoted to the investigation of parity-violating interactions in electroweak pro-
cesses. The related observables that we will consider in our analysis originate from
three different measurements: The observation of atomic parity violation (APV), the
study of left—right asymmetries in Moller scattering [26] and the analysis of deep in-
elastic electron scattering on nuclear targets. APV and electron-hadron scattering
experiments probe parity—violating interactions between electrons and the quarks in
atomic nuclei. In the case of APV it is the atomic electrons that interact with the
nucleons in the core. e-DIS experiments feature free electrons beams. The Moller
scattering experiments examine electron-electron instead of electron-hadron interac-

tions.

To account for the parity violation in these experiments one introduces the weak
vector charge QW. In the description of APV experiments the electroweak physics is
accommodated in the weak charge QW (’4 Z) of the isotope under study where Z and
A denote atomic charge and mass number respectively. 111 this work we will consider
the weak Charges of cesium-133 and thallium-205. Moller scattering experiments allow

to extract the weak charge of the electron QW(e).

To understand what is meant by QW (AZ) we first have to define the weak vector
charge Qw(q) at the quark level. I11 a first step we introduce the Lagrangian ESQ“!
that incorporates the parity-violating contributions to the effective quark-electron

interactions below the electroweak scale:

 

NC, _ GF _ _ -_ _
24f 64 = _ fl EC]: [C1q(ee)A,#/(qq)(‘, + C2q(ee)V,# (qq)':(4] (3.22)

The coefficients C1 f and Cgf in this Lagrangian play a similar role as e L (q) and
5 R (q) in filial/N or 956 and 9:38 in $4179”? We will focus on them when it comes
to calculating the new physics corrections in Subsec. 3.2.2. Since Efic’eq mixes
vectorial with axial fermion products, that is, parity-odd with parity—even terms,
it transforms as a pseudoscalar under parity transformations -— hence the parity
violation in the quark-electron interactions. The general idea behind QW(q) is to
mimic the parametrization of the QED vector current Lg’” in terms of the electric

charge Qe(q). If we define QW(q) as:

QW(C1) E 2 ' Clq

we can rewrite the SM tree-level expression for the electroweak neutral current

Jg’“ as follows:

8

a.,.,..z# 2 [away - [Qw,s1v1(q)(€7Q)V,u 1 (am 2“

59Si1093t1

The comparison of this form of the neutral current with the one given in Eq. (3.11)
allows us to relate Qw,3M(q) to the vector and axial couplings gg SM(q) and gg SM(q):
91%,SM(q)

QW,SIVI((I) = Z
lgA,SM(q)l

By convention Qw,3M(q) is normalized such that the prefactor of the axial part

in JO’é‘M has an absolute value of 1. The sign of the axial component is given by the

sign of gfi SM(q). As we now know how the weak charges of the 11p and the down

76

h. j.
in .

‘l t

quark are defined we can write down the weak charges of composite particles. The

weak charges of the nucleons, the proton p and the neutron n, are given as:

QWUD) E 2Qu/(u) + QWfd) ; QWUI) E wa‘U-l + QQWW)

Equipped with these basic charges we are able to calculate the weak charge

QW (AZ) of an atomic nucleus consisting of Z protons and N = A — Z neutrons:

QW (AZ) E Z ' QWUD) + N ' wa’l) = 2 ' [(Z + A) ' Clu + (M — Z) ' Cid]

To take care of parity-violating interactions in electron-electron scattering pro—

- . NC.ee
cesses we 111troduce the pseudoscalar LagrangIan .5,” 4 f ' :

NC. G17 _ _
34f '66 = V; ~01.3 (ee)A,#(ee)(fr (3.23)

In analogy to Qw(q) the weak charge of the electron is essentially given by the

coupling constant C18:

Qer) E 2 ° C1e

Finally, it is possible to extract certain linear combinations of the coupling coef-
ficients Clu and Old from polarized electron-hadron scattering data. In our global
fit analysis we will use the values for the linear combinations C1 and C2 that were

determined experimentally by Young et al. [27]:

€129~C1u+4-C1d ; CQE—4'C1u‘i'9‘01d

In summary, the included observables related to parity—violating processes present

77

themselves as follows:

 

 

can/(1330s) . Qw (20511) . ewe) , cl . C2

 

 

 

 

 

 

Lifetime of the tau lepton: Lastly, we can extract the lifetime T7- of the tau 7'

from the effective four-fermion Lagrangian 34f.

 

 

 

 

 

7'7'

 

 

The derivation of T7- in the effective theory below the electroweak scale follows
exactly the same steps as the computation of the lifetime Tu of the muon; cf. our
discussion of Ty, Fermi’s constant C F and the electroweak VEV z”) in Subsec. 3.1.2.
There is, however, one detail that we have to pay attention to in repeating the cal-
culation of 7'”: The 7' lepton might couple differently to the currents K i’“ than the
fermions to which it decays. In the UU-D model this caveat applies to the hadronic
decay modes of the T; in the N U-D model —— as the 7' belonging to the third fermion
generation only decays to first and second generation fermions —— the final state cou-
plings always differ from the initial state couplings. Deriving "r,J we did not have to
worry about this subtleness since we were only dealing with purely leptonic decays
within the first two fermion generations, ,u.‘ ——> e‘ + V“ + 176.

Although these differences are of theoretical interest as they illustrate the different
features of the respective models they are of no practical importance to our analysis.
Just as in Subsec. 3.1.2 it turns out that the fermion couplings to the new physics
currents K 35’“ do not contribute to the tau lifetime 'r—r once we discard all terms that

are of order 6’ ($4). In the lowest-order approximation we thus find for 7'7-2

 

5
_1 2 mr
TT CF 192%3

Given the heavy mass of the T we also include the leading-order correction to the

tree-level result into our expression for 77:

78

 

m5 m2 1/
——1 __ 2 . 'T;_ ‘ T

3.2.2 New Physics Corrections

The aim of this work is to examine the leading-order effects of new physics in the
G(221) models. Calculating the new physics corrections to the SM predictions we
will therefore only work at tree-level. For a given electroweak observable O that we
want to include into our fit analysis the GAPP code already knows the SM tree-
level expression Gaff. The task that is left to us is to calculate the corresponding
expression 012??? in our C(221) models. Consequently, the dominating effects of new

physics, AOtree, are reflected in the deviation of (91%;? from Ogrlfff:

tree __ tree tree
A0 = ONP — OSM (3.25)

In this and the next subsection we will calculate AOtrCC for all fundamental observ-
ables, that is, all observables that cannot be constructed from other basic observables.
The computation of the secondary observables will then be taken care of by GAPP.
We will organize our discussion in the same way as our overview of the included ob—
servables in Subsec. 3.2.1. First, we will revisit the Lagrangian 29w, Subsequently,

we will concentrate our attention to the low-energy data.

Corrections to the high-energy observables

Z pole data: All Z pole observables can be formulated either in terms of the partial
decay widths F Z ( f f) or the polarization asymmetries A L R( f ) It therefore suffices
to only calculate the corrections to these two quantities. All other Z pole observables
will then be covered automatically. We obtain AI‘ Z( f ) and AA L R( f ) by comparing

the SM expressions F Z,SM( f f ) and A L R,SM( f ) with their equivalents in the C(221)

79

models. According to Eqs. (3.10) and (3.12) we may write:

 

 

_ , C M. . 2 2
I“Z,Sl\"1(ff) = n éf) ' 92 z? ([95,3rdfll + [gismffl] )
'QSAIC9St4
Z 2 ' Z 2
. , — , f
ALR,SM(f) = [gL’SMfl] 'gR’Sm )] (326)

[gismml 2 + :91Z7,Sl\1(f)] 2

'C A15 '~ 2 ~ 2
mm = n f,” 2f; (_g€(f)] + [giml )

 

 

~ ~

60

_ [gflnf— [2am]?
Amm -_ fee] Hague]

 

(3.27)

The fermionic couplings to the Z boson play a key role for both observables. In a

first step towards AF Zf f) and AA L R( f ) we therefore focus on these couplings. The

SM gives the following expressions for 912,,SM( f), gg SM( f), 95 SM( f ) and 9A; SM( f ):

gismm = T530) — 858,,Qan ; game) = —s§SMQe<f> (3.28)
1 1
912/,SMU) E 5T2”) “ SgSMQeff) ; grasmff) E —’2‘T13;(f) (329)

111 our G (221) models two effects lead to deviations from the SM couplings. First
of all, g? (f), 51% f), §5( f) and fig (f) depend on the model parameters rather than
on the SM parameters. We gave a detailed discussion of the shifts in the respective
parameters in Sec. 3.1. Anyway, if this were the only difference the G (221) couplings
would still have the same form as the SM expressions; compare with the result for
the electroweak neutral current J?” in Eq. (2.17). The second effect that we have
to consider is the mixing of the Z boson with the new heavy Z’ in the electroweak

r . . 0.
theory. At the electroweak scale the Z couples effectlvely to the fernnon current Jew”, ,

80

F .
1V5]

t,

 

see Eq. (2.25), that also involves the new physics current K03“. The shifts in the
model parameters as well as the explicit form of K 0’“ are model-dependent which
is why we cannot state universal results for the fermion couplings to the Z boson
that apply likewise to all C (221) models. What the couplings in the different models,
however, do have in common is that they all reduce to the SM expressions in the limit

fit—>00:

a? (f) a 933W) + 69% (f) : age) 2 933w) + age) (3.30)
gm 2 game) + 696m ; age) a gismm + agfic) (3.31)

The deviations @121 (f), 69}? f ), 6g€( f) and (591% (f ) can be expanded in inverse
powers of 5:3 with the lowest-order terms being proportional to 3%. As hitherto we will
only keep these contributions and neglect higher orders. Tab. 3.2 summarizes our
results for égf(f), dggU); in Tab. 3.3 we present our results for 696”) and 69%(f).
To get an impression of the values behind the left- and right handed couplings 9% (f)
and 912% (f) for the different fermions f in the different C (221) models we numerically
evaluate the expressions that we derived in this subsection; see Tabs. D.1 to D.11 in
the appendix. The experimental input values that were employed to generate these

tables are given in Subsec A1.

The deviations 69% (f) and 6 gIZz( f ) in the left— and right-handed couplings enable
us to calculate the corrections AA L R( f ) to the polarization asymmetries. 6 g‘Z/( f ) and
691% (f) allow us to write down the shifts AFZ( f) in the partial decay widths. With

the aid of Eqs. (3.7), (3.8) and (3.9) we obtain the following results for AFZ( f):

81

£595 (f)

 

 

 

 

 

 

 

 

 

 

 

 

 

(BF-I’D) Hes-1) (cg—s; )Qe(f) +s ZCQI Tan—earn
(BF-LT) f (em) (Etc; — $3313) Qe(f) + £8ch [sz — 62.0)]
(13mm) f can - siege) + 8:, [cyan — ng§<!>]

T691220)
(BP-LD) f (esm) (c3, — 3,3) Qe(f) + 32c? [Tg‘m — can] + chgm
(BP-Irr) f (em) (a; — as; )Qe(f)+2113:C:3 [Tgm — can] + agar»
(BF-II,D) f (9m) - ngeU)

 

 

 

Table 3. 2. Shifts 6 g L ( f ) and 6 g R( f ) 1n the left- and right- handed couplings g L ( f ) and
g R( f ) of the fermions to the Z boson —— compare also with Tabs. D. 1 to D. 11 111 Sub-

sec.D.1.1. Thefunt n 6 ,. . . c 6 ,. = (c2 — ).
C 10 f ( SM) iS em 11 IS f ( SM) 898M CQSM/ 98M 893M

82

 

(BF-ID) f (63M) (cg, — 335,) Qe(f)

$33363, [1330) + Tm) — 2am] —+— target)
(BP-LT) f (65M) (gt—cg, — $333) Qe(f)

+3158;ch mm + T2‘(f) — 2am] + Serge)

(BF-II,D) f (em) . siege) + as? [cyan — Sims]

 

 

 

 

 

 

 

 

¢

mic)
(BP-I,D) 55%,) [T23( f) — Tg(f)] + $0ng f)
(BF-LT) ésgcg [T23( f) — T21 f)] + ,15c4T23( f)

(BF-II,D) —%s

 

 

 

 

 

 

Table 3.3: Shifts dg‘Z/( f) and (591% (f) in the vector and axial couplings 95( f ) and
9% (f) of the fermions to the Z boson. The function f (65M) is given as f (65M) E

.9 c2 (02 _32 ).
%M %M %M

2
9sr1

83

"c AI ’ ,
(BF-I’D): Arzm = "I“. Z? {293,sm(f)595(f)+29§,sn(f)69§(f)

3 .2
q95tic9SM
1 Z 2 Z 2 4 2 .
+ :i: ([er’sxiffll + [9A,SM(f)] > (cg—325,) (3.32)
(

TC AI. , - , -
(BF-LT): Arzm = I.” 2 ZS {296,3M<f)ag€(f)+29%,SM(f)og§(f)

S ,.
BSAI 93rd

.5
2 2
+ 11? (lggaSMml + [933340)] ) Ge; —— $333) } (3.33)
(

 

 

 

6 AI .
(BP’H’D): NZ”) 2 n 3f) ‘ 2 i3 {QQIermUMgflfl + 29181,,“ mgfi ( f)
SQSAI’QSAI
l 2 g 2
+ :1: ([965le4 + [g/ZISAMfl] ) 833} (3.34)

In Eqs. (3.32), (3.33) and (3.34) we present the corrections AF Zf f ) to the partial
decay widths in a compact form. To obtain the actual expressions that we will imple-
ment in the GAPP code we still need to replace the couplings 95,SM( f) and 9§,SM( f )
and the deviations 69% f) and 6g§( f) by the terms that are given in Eq. (3.29) and
Tab. 3.3 respectively. As these expressions will turn out to be rather cumbersome and
as they will not yield further insight into the physics behind the corrections AFZ( f)
we do not present them here. This argument applies to all new physics corrections
that we are going to discuss in this and the next subsection. In each case we will
only Show as many steps of the respective calculations as necessary to illustrate our
procedure. However, notice that we will still employ the corrections AF Z( f ) as an
example for the modification of the GAPP code in Subsec 3.3.2.

Making use of Eqs. (3.26), (3.27), (3.30), (3.31) we obtain the following expression

for the corrections AA L R( f ) to the left-rigl'it asynnnetries:

9123,3M(f)59§(f) _ Highvlfflégfff.)
2 2 2
([93531ffll + [933M(f)] )

84

 

AALRff) = 491%.s1xi(f)91Z?.s1\1(f )

 

Our results for AF Z( f ) and AA L R( f ) is everything we need to implement the
shifts in the Z pole observables in the GAPP code. We can now proceed with the

width FW and the mass A! W of the W’i boson.

Wt pole data: According to Eqs. (3.15) and (3.16) we need the mass MW of the
wit first before we can calculate its width. In the SM the tree-level expression for

[My is given as:

\/ ”dz/SM

Ill-[1425M = CHSM - [W Z = 89
SM
To obtain AMW we simply have to consult our results for M 3V listed in Tab. 2.6,
express Mg in terms of the standard and fit parameters, take the square root and

substract AJWSM- These steps result in:

 

 

 

2
1 Ce .
.. : 1. f 11’ {,7 I , Z. — . Shl ( 4. —— 2..)
(BP LD) A] [W /1 111,51“ 2.2? C2 _ 82 Cd) 323
93rd 93rd
1 0% 1 1
_ I l/ , :__ SM _4__2~
(BP LT) All/[W /1 [WSM 2f 62 _ 82 (46¢ 252l3
9StI 93t4
2
1 SGShd 4
(BF-II,D): AA’IW/A’IWfiM : ‘2? ' Sq;

2 2
C — 8
93M 98M

In the SM the left—handed coupling gEVSM( f ) of a fermion f to the W:t boson is
just given by the gauge coupling 9L,SM of the left-handed SU(2)L. The SM partial

decay widths FW,SM( f ) can therefore be Written as:

2
9L,SM
Pvt/,SMU) = nc(f) ' mil/1mm

In the G (221) models that belong to the first breaking pattern the right-handed

85

 

 

(BP-I,D)

(BF—LT)

(DP-II,D)

 

 

 

 

 

 

Table 3.4: Shifts 6921/”) in the left-handed coupling 92( f) of the fermions to the
[it _ . . ‘ . .‘ r. _ ‘ ‘ . E 2 _ 2 .
VI boson. The function f (95M) 1s given as f (03M) CgSM/ (COSM 808M)

couplings g}; (f) are suppressed by :15 which is why we do not have to consider them

in the Wi width. The left—handed couplings unchangeably correspond to the S U (2) L

gauge coupling QL- As for the second breaking pattern the fermion coupling is purely

left—handed and the mixing between the Wit and the W’i introduces a shift in 9})“ f ).

Our result for the charged fermion current Li}? in Eq. (2.25) leads us to:

BP-I: 92%!) = 9L

BP-II: g?”

(f) 2 a (1+ :53, [c3T1<f) — s3T2<f>])

In a last step we take into account the deviation of £7 L from its SM analog g L,SM

due to the shift in the electroweak mixing angle. Doing so we obtain the corrections

(Mi/W f) to the left-handed fermion couplings g?“ f), see Tab. 3.4.

W

With the results for 6g?

9L (f) 2 gi‘fsmm + eye) a 9L,s.\«1 + 69%)

(f) at hand we can write down the shifts AFW( f ) in the

86

partial Wi widths FW( f):

ncff)

AFWU) Z 48% (29L,SMCSQI:V(f)*MW’351\’I + 9124,31“AA ’IW)

 

Corrections to the Low-Energy Observables

Neutrino Scattering and Parity Violation All of our observables that are ex-
tracted from either V-DIS, Meller scattering, e-DIS, or APV experiments can be
traced back to the couplings in the effective four-fermion Lagrangian $41!. In the case
of neutrino—hadron scattering the left— and right handed quark couplings E L (q) and
53 (q) are the quantities of interest, see Eq. (3.18), to calculate the vector and axial
electron couplings 9"”; and gffle in neutrino-electron scattering we need the coefficients
5 L (e) and E R ( 6), see Eq. (3.21), and to obtain the weak charges of atomic nuclei,
QW (AZ) , and the electron, Qw(e), we have to know the couplings CM and 016, see
Eqs. (3.22) and (3.23). Because of these similarities we can address the corrections
to most of the low-energy observables in one go. Only the calculation of the shift
in the T lifetime has to be taken care of separately; we will discuss A77- in the next

subsection.

Before we turn to the new physics corrections we still have to hand in the SM
expressions for the effective four-fermion couplings that are involved in our analysis.
One finds for the couplings 5 L,SM( f ) and ER,SM( f) of a charged fermion f —— an up

or down quark or an electron in our case - to the neutrino:

5L,SM(f)Eggfiswdl/Mfitiff) ; ER,SM(f)=29f,31\-1(V)91fz,sm(f) ; f€{u,d,e}

The couplings gliSM( f) and 912?.SM( f) are given explicitly in Eq. (3.28). With T Z( V) =

87

% and Q€(I/) = O we obtain:

€L,SM(f)=gf,SM(f)=T£(f)—S§SMQ(f) ; eR,SM(f)=gIZ,,W(f)338,320)

The corresponding couplings C 161,31“ and C1e,SM in electron-quark and electron-

electron interactions are very similar to these results. We find:

Z Z .
C1f,SM E 9A,SI\.1(€)9V,31\1(f) , f E {11,6123}
In accordance with Eq. (3.29) and using T3 = —.~£; we rewrite C1 f,SM as:

Cifsn = T20) — 2838,31Qe(f)

In the next step we calculate the corrections to these expressions. Most of the
work has already been done. We derived the effective four—fermion Lagrangian 34f in
Subsec. 2.3.3. The fermion currents and the boson masses that constitute .204}: were
the subject of the discussion throughout the entire second chapter. Now we reap
the fruits of our labor. Instead of only calculating these couplings in terms of which
our observables are defined we perform a general analysis and compute all effective
four-fermion couplings. First, we write the neutral and charged current components

of 34f as follows:

C _ _
3413‘: E —:/—g f; Zcffc (f1,i:f2,j) (f1f1)i,#(f2f2)§t
1: 2 W

G _ _
$430 .=_ 7% f; 205,0 (f1,.-,f3,j) (f1f2).,,..(f3f4)5-‘
1, 3 W

The fermion sums run over up and down quarks, neutrinos and electrons, f E

{22,01,146} and z" and j denote the chirality of the respective fermions, 2', j E {L, R}.

88

 

The fermion pairs (f1, f2) and (f3, f4) in .Z’CC f re present iso-doubk ts under either of

the two 5 U (2)s in our models.

Similarly, we can rewrite the effective four—fermion interactions flfiK in the La-
grangian at the electroweak scale KM; We separate ZQK into neutral and charged

current contributions, compare with Eq. (2.26),
KK _ NC CC
x‘w. : ZFW’. + ’ZEW'.

and define the coupling coefficients ngC? (f] ,i, fgj) and GEE ( f1,z’» f3”) such that

$611? and 26%? turn into:

C — — . . .
$3119 5 — F ‘52 E :CEE( (f1,ivf2,j) (f1f1),3,,(f2f2)§' ; 1,] = LR
\fivfi f2 2']

«90.2%? E —07= F2 ZCS1C( (f1,z',f3,j) (f1f2),,u(f-3f4)§ ; M = LR
f11f3 iaj

The couplings in .2031? and .54ch do not directly appear in the definitions of our
low-energy observables, but contribute indirectly to them as they are integrated in
the low- -energy couplings C4fC(f1,,f2,J-) and C419 (f12,f3 3). Besides that they are
also important in their own right as they represent major consequences of the new
physics in the G(221) models at the electroweak scale. We therefore do not restrict
our analysis to the four-fermion interactions in 34f, but also examine the couplings
in @1qu-

We Obtain (7%qu (f1,2’af2,j) 052E (f1,2’vf3,j)C Nfc,(f1 iaf2,j) alldC CfC(f1 ,2':f3,j)
from the Lagrangians few, and 34f in Eqs. (2.24)4 and (2.27) by inserting our results
for the effective fermion currents and boson masses and trading the model parameters
that we are not going to fit for the standard parameters. Inserting the experimen-

tal values for the reference observables provides us with numerical results which we

89

 

present in the appendix see Tabs. D.13 to D.22 for Giff (fly, fg’j), Tabs. D.23 to
D26 for CE“? (fLi, f“), Tabs. D27 to p.36 for Cfic (fly, fzj) and Tabs. 13.37 to
p.45 for Off (fLi, m).

Finally, we remark that the modification of the GAPP code requires the analytical
expressions for the four—fermion couplings. As the explicit results are rather lengthy

we, however, do not present them here.

Lifetime of the Tau Calculating the corrections to the T lifetime turns out to be
trivial: The only quantity in our ex1:)1'essi0n for 7‘7 that receives a shift is the mass
MW of the Wi boson; compare with Eq. (3.24). we can immediately write down

the leading-order shift AT;1 as:

r 1 r
Lin—120%. m} .3 __m3 /). (3.43”)

 

 

Notice that the corrections to 77— only emerge from the subleadjng term in Eq. (3.24).
AT;1 is consequently suppressed by the ratio of the mass of the ’T to the SM mass
of the W'i boson which is why we expect the shift in the T lifetime to play only a

minor role in our global fit analysis.

3.3 Numerical Analysis

With the calculation of the corrections to our 37 observables we have completed the
analytical part of our study. Now we are ready to determine the bounds on our
new physics parameters numerically with GAPP. In this section we will give a short
introduction to GAPP, say how the code has to modified in order to accommodate
the G(221) model and discuss how we actually run it. For the moment we focus on
the technical details of our fit analysis ~ the results that we obtain are presented in

the next chapter.

90

3.3.1 Introduction to GAPP

GAP P, short for Global A71.(1,l;(/s2s of Paxrticlc Properties, is a Fortran package developed
by J. Erler that allows to perform precision tests of the SM and to determine its
f1n1damental parameters. For this work J. Erler kindly provided us with the most
recent GAPP version which is up-to-date as of 2009. GAPP is written in such a
way that extensions of the SM can be easily a(tcommodated in the code: The default
version of GAPP already comes with the option to examine various models beyond
the SM that feature a Z ’ as a new heavy gauge boson; in this work we utilize GAPP
to test. our C(221) models.

At its core GAPP ('-.a.lculates the deviation of the theoretical predictions for the

various precision observables from the experimental data in terms of chi-square, X23

1 _ . 2
:72? : ——. (are — are)

. .0-
2 2 1

Here @E'XP‘ stands for the central value, 0, for the total uncertainty of the experi-
mental result; OSXP' = (22%” :l: 02'- 0,- subsumes the experimental errors as well as the
theoretical uncertainties that affect the interpretation of the experimental data. The
individual contributions to x2 from the different observables, also called the pulls, are
denoted with 732-.

Confronting a given theoretical model with experimental data x2 is a measure for
the agreement between theory and experiment —- the larger the value of x2 the less
likely is it that the physics underlying the experiment is described by the considered
model. In other words: If X2 takes a too large value one can conclude that the testes
model is ruled out by the experiment. ()11 the other hand, if one assumes that a given
theory represents the correct description of the experiment, x2 can be employed to
determine or constrain the values of the parameters in the examined model. In this

case X2 is regarded as a function of the model parameters; those parameter values for

91

 

which x2 takes a minimum value are. considered to be the best estimate for the true

values.

The calculation and 111ini1nization of x2 is the actual purpose of GAPP. In order
to find the smallest possible X2 value it employs the minimization program MIN UIT
[28] that is included in the CERN program library. MINIUT can either be initialized
by external data or it can directly be driven by Fortran subroutine calls. The current
version of GAPP runs MIN UIT in the data-driven mode. This means for us that the
fit parameters have to be defined in an external data file. In this file, called smf it . dat
in the default GAPP version, each parameter is assigned a number, a name, a starting
value and a starting step size. Additionally, one is able to set bounds on the ranges
in which the respective fit parameters are allowed to vary during the minimization.
In the same file MINIUT is given all the commands that specify which actions it is
supposed to perform. In our analysis we will either use GAPP to calculate X2 for
a given set of parameter values or to find the minimum value of X2. Accordingly,
we will either just give a simple return command to MIN UIT or feed it with the

connnands minimize, improve and seek.

MINUIT always requires a Fortran subroutine that calculates the value of the
function of interest. In the case of GAPP this subroutine is called fcn and located in
the file chi2.f. Before X2 is calculated f cn defines certain constants, initializes the
quantum numbers of the fermions and sets flags that trigger the inclusion of various
higher-order corrections. Subsequently, it calls the Fortran function ch12 that is
contained in the same file and that takes care of the calculation of X2- ch12 stores
the experimental values and errors, calls the different subroutines that compute the
theoretical predictions, calculates the pulls and finally determines X2- Once ch12 has
returned the X2 value to fcn any final computations are processed. Depending on
how the flags were set by the user the likelihood L ~ exp (—X2/ 2) corresponding to

the calculated X2 value might be determined or the. results of the computation might

92

File Subroutines Observables

 

 

Z pole data lep100.f zonle FZ (ff), ALR(f)

Wi pole. data sin2th.f sin2thetaw MW

wwidth. f wwprod I‘W (f)

UN scattering dis . f nuh 5L (q), 53 (q)

we scattering nue.f nue g’f‘, gfff
PV processes pnc . f apv C lq

w moller QW(e)
T lifetime taulife.f taulifetime T771

 

 

 

 

 

 

 

 

Table 3.5: Overview of the modified GAPP files - ~ the subroutines in these files
compute the SM quantities for which we have calculated the G(221) corrections in
Subsec 3.2.2. Once we have i111plemented the new physics shifts into these files the val-
ues for the electroweak obse1vables calculated by GAPP will automatically represent
the predictions of our G (221) models.

be written to an output. file.

In Subsec. 3.2.2 we calculated the new physics corrections to a variety of fun-
damental quantities 'with which all of our observables can be constructed. These
quantities are calculated by GAPP in seven different Fortran files; see Tab. 3.5. Set-

ting up the GAPP code such that we can fit our G' (221) models we will have to modify

these files. The calculation of the observables will then be taken care of by GAPP.

3.3.2 Modification of the Code

In order to implement our G (221) models into the GAPP code three modifications
are basically necessary: The new physics fit parameters have to be defined in the

input file that drives the fit, the additional quantum numbers of the fermions have

93

 

to be provided to the function f cn and the corrections to the observables have to
be added to the SM expressions in the respective Fortran files. In the following we
will comment on each of these 1no<lifications. As the inclusion of the new quantum

numbers represents the simplest step we. allude to it first.

Additional Quantum Numbers

In the models belonging to the first. breaking pattern we have to introduce two new
quantum numbers: The third component T23( f ) of the S U (2)2 isospin T2( f ) and the
charge X (f) under the U (1) X gauge group. As X( f) may be depend 011 the chirality
of the fermion f it effectively represents two quantum numbers: X L( f ) and X R( f ),
the charges of the left- and right—handed versions of f. The models of the second
breaking pattern do without the introduction of new quantum numbers: T13( f ) and
T;( f) correspond to the weak isospin quantum number T2( f); X (f) is identical to
the weak hypercharge Y( f)

we take care of the different charge assignments by creating for each model a
Fortran file (sm.f, 1r-d.f, hp-d.f, nu-d.f) in which we specify the numerical
values of the new quantum numbers T§( f ), XL( f) and X R( f ) In the SM, the UU-D

and the NU-D model these quantum numbers are set to zero for all fermions. Within

the function fcn we import these files using the Fortran command include.

Fit Parameters

In Subsec. 2.3.1 we came to the conclusion that the new physics corrections to the
electroweak observables are best. parametrized in terms of 5:, (13 and, in the case of
breaking pattern one, 5’. As this is certainly true, we will, however, not give these
three parameters as direct input. to MIN UIT. In analyzes like ours one usually chooses
fit parameters that are defined on the. whole real axis. Additionally, we consider it

natural to define the fit parameter such that they reflect. the structure of the fitted

94

analytical expressions as well as possible. Instead of if, (.75 and [3 we therefore let

~ 9
GAPP scan over 111.2", 1‘: and s; F’,:

.~I ’7 ~ ~ 2 2 ~
{1.9. .13} ——+ {111 1:, Q1»): 82$}

The advantage ()f 111:2: and t3.) over 5." and (b is that both functions range over the

2~
23

every expression that depends on ,3 it, however, appears in the form 333. Every other

whole real axis. 3 takes, of course, only values between 0 and 1. Essentially in

choice would therefore be unnatural.

h-"Ioreover, t3) represents a convenient compromise between two functions, c3; and
533, that are particularly important in the two breaking patterns that we are consid-
ering. Due to the different definition of the mixing angle (13 in both breaking patterns
it is either 633 or 8% that appears all over in the analytical expressions describing
the respective C(221) models; see for example the % contributions to the Z mass in

Tab. 2.5 that are either proportional to 6:15 or 54. It is this difference in the expressions

(23

for M Z that leads to the different shifts in 3% in the first and second breaking pattern;
compare with the calculation in Subsec. 3.1.2. In the end, all these distinctions can
be traced back to the fundamental relation of the S U (2)2 coupling 572 to the mixing

angle (ii:

A.

 

BPJ:-——jz———=c: ; BPJL ——JQ——-=s~
~2 ~2 ‘9 ~2 ~2 Q5
92 + 9X 91 + 92
2..

and 52’~ as fit parameters we include them in the GAPP

<15 23

input file. To prevent GAPP from scanning unphysical parameter values or values

Having decided for 111 1'3, t

that we are not interested in we set. bound 011 the ranges over which our parameters

are allowed to vary:

BP-I: 111i6[0.0(),10.0()] ; 1:6[0.01,1()0.00] ; s§B€[0.00,1.00] (3.35)

BP-II: 111.i:€[().00.10.00] ; t%€[0.03,30.00] (3.36)

If hri‘ becomes very large the new physics in our G (221) models decouples from
the energy scale of the SM and thereby from the physics that is essential to the
electroweak observables. At large. values of 1112': we will therefore not be able to
discriminate between our G(221) models which is why we set an upper bound of
111 5311121.): = 10 011 hri. The bormds 011 t: originate from the intrinsic structure of our
G (221) models and the condition that perturbativity must not be violated: Depending
011 the breaking pattern tq; is either defined in terms of g X and {12 or Q1 and fig, see
Eq. (2.3). In both cases the gauge couplings carmot take arbitrary values as they
are related to the electroweak couplings Q and fiy, compare with Eqs. (2.4) and
(2.6), which are constrained by the experiment. Using the experimental data listed in
Appendix A.1 we find that 911,554 and 9933M take approximately the following values
in the SM:

6 e .
gL,SM : :— 3 0.626 ; gY,SM : ;— Ft: 0.346
'6311 68M

These numbers represent lower bounds 011 531, 572 and Q X1 Q2 respectively:

813-1: @092 > gysm ; BP-II: éraé2 > 9L,SM

Our G' (221) models are perturbative quantum field theories in which the typical
expansion parameters are given as al E §¥/47r, 02 _=_ 93/“ and a X E §§/47r. To
ensure the validity of perturbation theory all three parameters must be smaller than

1. This places an upper constraints on 571, {)2 and fix.

96

 

 

531,512,9X < V47r

Combining both arguments and putting the numerical values into Eqs. (2.3), (2.4)
and (2.6) we are able to roughly estimate the bounds on ti; see Eqs. (3.35) and (3.36).
We expect that for high 513 values changes in t: will be irrelevant to the calculation
of the observables. Setting limits 011 the allowed range of t: therefore does not only
take care of the mentioned theoretical constraints but also avoids the risk of GAPP
getting lost during the minimization of X2 while it scans over always higher t3; values.

Fitting the G (221) models to the electroweak observables we will also allow the
WIS mass of the top quark mt, see appendix B, and the mass of the SM Higgs boson
M H to float. In doing so we will see how much of an effect the new physics in our
C(221) model has on these crucial SM parameters. Especially we are interested in
the question whether the considered extensions to the SM are consistent with larger
masses of the Higgs or whether they constrain M H to similar values as the SM. In

order to find the minimum value of X2 in the respective models we will thus let GAPP

vary five parameters.

 

 

 

 

1115: , t3; , s2; , flit , AIIH

 

 

 

 

When it comes to scanning the parameter space of the new physics parameters

we will fix mt and Al H at their respective best fit values.

New Physics Corrections

In Subsec. 3.2.2 we calculated the new physics corrections AOtree to the electroweak
observables O at tree-level. After providing GAPP with the new quantum numbers of
the fermions and the definition of the fit parameters we 110w are ready to implement
the results of our calculations into the GAPP code. The crucial question in this

context is where to put our expressions for AOtm". In the case of some observables

97

 

the GAPP code is structured in the best possible way from our perspective: First
GAPP computes the tree—level expression OtS’rfie, higher-order corrections 08181 =
Oéfimp + Ogfi®+ are calculated subsequently and successively added to the tree-

level result.

_ tree l-loop 2— loop+
08M “ OSM + OSM + Osiu

When then just have to add AOl‘m)‘ to 0;??? in order to get the theoretical pre-

diction ONp of our new physics models.

ONP z (0g? + Act”) + Oéfi’ol’ + (9513;0th . (3.37)

However, in other cases the pure tree-level SM expressions are not accessible in the
code. GAPP might start the calculation of OSM using quantities that already contain
higher-order terms right from the beginning. OSM would then be initialized by some
approximation that is constructed from tree—levelas well as loop terms 0(Rdree’HO).

Moreover, GAPP might also include higher—order contributions multiplicatively in-

stead of just adding them to (93M:

t 1-loo t .,
08M : OSESF ° (1+ 031“ p/Osrlfie) ' (1+ ...)

111 any case, no matter 110w GAPP calculates OSM: we stick to the procedure
illustrated in Eq. (3.37). As early as possible after their initialization we modify the
individual SM quantities in the GAPP code. Doing so will introduce mixing terms
in the calculation of some observables that we actually do not want to include into
our analysis. For instance we will get products of our new physics corrections AOtree
with SM loop terms. However, all terms that we introduce unintentionally are small

and can be neglected:

98

 

 

tree 1-loop N tree 2-loop ~ ~

According to the general comments made in this subsection we now implement
our results for AOtree in the respective Fortran files, see Tab. 3.5. The shifts in
the electroweak observables differ from model to model. We take care of that by
introducing a switch variable modtype in the Fortran code that allows to respectively
include only those new physics corrections that correspond to the model that is being
fitted. Implementing the corrections in that way gives a modular structure to our
modification. Once the code is set up for one specific class of G(221) models it is
easy to add the corrections for all other models. The extension of GAPP by further
models that are similar to our G ( 221) models should be accomplishable without much

effort .

3.3.3 Fitting and Scanning the Models

Finally, GAPP is configured in such a way that we can run it and examine the
compatibility of our G (221) models with the electroweak precision data. Two tasks
are on our agenda: First, we will let GAPP minimize X2 for each model. These model
fits will tell us which values for the new physics parameters and which masses for the
top quark and the Higgs boson are most preferred by the data. Subsequently, we will
scan the parameter space of the new physics parameters in the respective models.
Based on these model scans will be able to identify the parameter values that are still

consistent with the data.

Minimization of X2: For each model we determine the smallest possible value
xiiin. of x2. In order to find X‘fmn. we let GAPP vary five parameters: Infi“, t:, 833,
772,; and M H- This first step is trivial as it just requires a simple call to MINUIT. The

99

 

 

71:1 71:2 7123

 

 

AX2(950.:..n) . 3.84146 5.99146 7.81473

 

 

 

 

 

 

 

 

Table 3.6: Maximum allowed deviations Ax? from Xfiiin. at 95% CL for one, two

or three free model parameters ~— at these values of X2 the cumulative distribution
function F (X2, 72) of the X2 (.listribution with 11 degrees of freedom takes the value

F (x2. 71) : 0.95.

results of this first numerical analysis are presented in Subsec. 4.1.1. See especially

Tabs. 4.1 and 4.2 and Figs. 4.1 and 4.2.

Constraints on the new physics parameters: 111 our second analysis we fix mt
and M H at their respective best fit values and focus exclusively on the bounds on the

new physics parameters. The criterion by which we decide whether a certain set of

‘) . . . . . . . .
‘~, 32 ~ 1s Viable and consistent w1th the data 18 the dev1at10n A)? in

<15 25
x2 from the respective minimum value X2 . . If 2 is lar er than 2 = x2 - + AX2
mm. X g Xref. mm.

values for In if, t

for some parameter values we conclude that these values are ruled out by the data;

all parameter values that yield a x2 smaller than xfilin. + AX2 are still fe(sible.

X2 < Xfinin. + sz :> Parameter values are allowed.

X2 > Xfiiin. + AX2 => Parameter values are ruled out.

The choice of AX? : AX2 (CL, 72.) depends on the desired confidence level CL and
the number n of free model parameters. In this work we would like to describe the
properties of the parameter space at a 95% CL. The models belonging to the first
breaking pattern feature three, the models of the second breaking pattern two new

physics parameters. In Subsec. 4.1.2 we will examine the 11le dependence of X2- 111

100

 

 

that context we will need AX? corresponding to one free parameter. We calculate
AX2 (95%, 1), AX2 (95%, 2) and AX2 (95%, 3) employing the cumulative distribution
function F (X2, 71) of the X2 distribution with 71 degrees of freedom and present the
results in Tab. 3.6.

The points of interest in the 1.1aramcter space are those where X2 falls below the
threshold of Xr2nin. + AX2 or where it becomes larger than xiii“. + AX2. Together
these points form the bormdaries for the allowed regions in parameter space which
represent the goal of our analysis. To find these points we let GAPP scan over a grid
in parameter space and calculate X2 at each point. If X2 is larger (smaller) than Xgef.
at a given grid point and smaller (larger) than Xgef at the following grid point we
lineally interpolate between the involved parameter values to find the values on the

boundary between the allowed and forbidden regions.

101

 

 

Chapter 4

Results

Our global fit analysis provides 11s with a wealth of information about the G(221)
models under study. After we have given a detailed discussion of our numerical
approach in the last. chapter we now discuss our results and draw conclusions about
the underlying physics.

2 in the

First we will present the values of the fit parameters that minimize X
individual models. These best fit values will give us an idea of the scale of the new
physics in the G (221) models and they will tell us which masses 111 H of the Higgs
boson are respectively most preferred by the data. As we will see M H tends to
take smaller values than in the SM for the models of the first breaking pattern and
roughly the same value as in the SM for models of the second breaking pattern. To
get bounds on M H for all ten G (221) models as well as for the SM we will plot X2
as a fimction of 111 H- Doing so will allows us to read off those M H values for which
AX2 is smaller than AX2(95%, 1). To get a better understanding of the X2 plots we
will also examine the pulls P,- of the electroweak observables for M H either fixed at
a very small or a very large value. The corresponding pull distributions will help us

identify the observables that cmmtrain MH.

In the second step we will turn to our parameter scans and present the bounds

102

 

011 the new physics parameters. Subsequently, we will translate the boundaries in
parameter space to bounds on the masses of the new heavy gauge bosons. These
results will show us which gauge boson masses are still consistent with the data and
whether one could hope to detect the Z ' and / or the W’i at the LHC. Again we will
examine the pull distributions to identify those observables which drive the parameter
plots. In a last section we will calculate the explicit numerical expressions for these
important observables in the respective models and try to reconstruct the plots of the

bounds in parameter space.

4.1 Fits to the Electroweak Data

4.1.1 Best Fit Values

In Subsec. 3.3.3 we described how we minimized X2 for the ten G (221) models under

2 ,
25,, MH and

consideration as well as for the SM by varying the values of ln ft, ti, 3
frat. The results of that analysis are 110w presented in Tab. 4.1.
Inspecting Tab. 4.1 we make several interesting observations: First of all, we notice
that the values of Xr2nin. for the G (221) models are of the same order as for the SM.
This finding tells us 011 the one hand that none of the G(221) modles is ruled out
by the data — all models yield reasonable Xiiiin. values that are comparable to the
one of the SM. 011 the other hand we also see that our results in Tab. 4.1 prove
once more how excellently the experimental data is described by the SM. Our G (221)
models can barely improve the SM value of Xfiiin} only in four models, the LP-D,
LP—T, F P-D and the FP-T model, we obtain a slightly smaller value. The other four
models that belong to the first breaking pattern yield approximately the same value
as the SM; the UU-D and N U-D values of X1211in_ are slightly higher. As we will see

later it is not just by chance that the models of breaking pattern one split into two

groups. The analysis of the pulls of the electroweak observables in Subsec. 4.2.2 will

103

 

 

reveal that the fits of the LP and the FP model and the fits of the LR and the HP
are respectively driven by the same sets of observables.

Given the best fit values for i" it is, however, obvious why the minimum X2 values
are so close to each other. The. scale of the new physics in the C(221) models is
throughout very high. In all models if is pushed to very large values resulting in
a substantial suppression of the new physics corrections. It is a testament to the
power of the SM that the experimental data apparently favors small up to negligible
contributions from new physics. In none of the considered models the best fit value
for i" is smaller than 160. In the case. of the NU—D model we even reach the bound that.
we set on lnrit -—— we demanded that. lnfi‘. must not take values larger than Inf 2 10.
With it = 22026 in the NU-D model we exactly reach that limit. This explains why
we put a long dash ( — » ) into the corresponding entry in Tab. 4.1. As the NU-D
model apparantly favors neglible new physics corrections the best fit value for t: is
meaningless as well.

In surmnary, we. conclude: The smaller the deviation from the SM the better
in agreement with the experiment. are the predictions of the G(221) models. This
insight will help us in the further interpretat1011 of our results. Especially, when we
have come to discuss the bounds 011 the new physics parameters in Sec. 4.2 we will
take the SM as the best description of the experimental data ~ the fact that some
regions in parameter space are ruled out by the data can then be explained with the
new physics corrections being too large in these regions.

Moreover, we find that the best fit value for :2? in any (BF-LT) model is always
smaller than in the corresponding (BP—I,D) model. This relation is expected since
choosing a triplet instead of a doublet representation for H in any model of the first

breaking pattern leads to suppressing prefactors of the c4~ and 52 terms in the new

a 2.3
physics corrections: see, e.g., Tab. 3.1. 111 (BF-LT) models the (3:; contributions are

always four times smaller than in the (BP—I,D) models. The 8%,; terms receive a

104

 

 

 

 

 

 

M“ We“ £11196“

SM 41.95 — 4 102.8 93.24

LR-D 42.40 2028 99.99 0.9998 102.8 72.33 L
LP-D 41.00 1055 100.0 0.5499 102.7 08.94
HP-D 42.24 041.9 100.0 0.3348 102.7 70.88 ‘
FP-D 41.09 812.9 04.05 0.4312 102.7 07.50

LR—T 42.44 997.1 99.95 0.9992 102.8 72.17

LP-T 41.00 203.8 100.0 0.2750 102.7 08.94

HP-T 42.24 100.5 100.0 0.1074 102.7 70.88

FP-T 41.09 203.1 04.72 0.2153 102.7 07.48

UU-D 43.10 318.0 0.03010 -— 102.8 94.00

NU-D 43.34 —— —— —— 102.8 93.48

 

 

 

 

 

 

 

 

 

 

 

Table 4.1: Best fit values of i“, 1‘3), 333, M H and flat for all ten G (221) models as well
. A 2 ~

2

and (3.36). In the case of the N U-D model 5: reaches the maximum allowed value;

the corresponding best fit ti; value is thus meaningless.

as for the SM —— the bounds that were set on In 5:, 1%, s are given in Eqs. (3.35)

105

 

prefactor of %. This effect will also be. evident. in the plots of the bormds on the new
physics parameters: 111 Subsec. will see that the boundaries that separate allowed
from forbidden regions in parameter space are always shifted to lower :i? values if H
is chosen to be a triplet.

111 most models t“; reaches either the lower or the upper bound that we set 011 that.
parameter. In the models of the first. breaking pattern a high t: value seems to be
preferred; in the models belonging to the second breaking pattern t2 tends to take
smaller values. Both observations lead 11s to the same conclusion: The experimental
data can be best explained if the coupling 02 of the second S U (2)2 in the G (221) gauge
group is taken to be small —— which is just another way of saying how successful the

ansatz of the S U (2) L 8 U(1)y gauge. group in the SM is.

4.1.2 Higgs Mass Dependence

Another insight that we gain from Tab. 4.1 is that the G(221) models prefer a Higgs
mass M H in the same. range as the SM. The best fit values for M H in the models
of the first breaking pattern are smaller than the SM value by roughly 20 GeV. Our
results for the second breaking pattern are almost identical to the SM value. Again
we notice that the models belonging to breaking pattern one fall into two groups:
The Higgs mass values of the LR and the HP models on the one hand and the results
for the LP and the FP models on the other hand are respectively comparable to each
other.

To get a better impression of how X2 depends on the mass of the Higgs boson we
calculate X2 for values of M H between 30 and 300 GeV in all G (221) models and in
the SM. Doing so we fix all other fit parameters at their best fit values such that the
Higgs mass remains as the only free parameter. The results of that step are shown in
Fig. 4.1. Since the Higgs mass always appears logaritlnnically in loop contributions

to the electroweak observables, a quadratic dependence 011 111(AIIH) is expected if we

106

 

 

[lllllllllllllllllli

  
 

AX
1,111

15

10

 

 

0|
llllllllllll’l’l’lllll

 

Figure 4.1: X2 in dependence of the Higgs mass M H for all ten G(221) models and
the SM — the X2 curves of the (BP-I,D) models differ so little form the curves of the
corresponding (BP-I,T) models that they are mostly covered by them. See Tab. 4.2
for the allowed ranges of the Higgs mass that we deduce from this plot.

107

 

expand X2 around its minimum value. For small deviations from Xfiiin. the plots in
Fig. 4.1 certainly confirm this expectatitm.

Fig. 4.1 cannot only tell us the X2 value for a given Higgs mass but also help 11s to
answer the reverse question: Which masses M H correspond to a certain value of X2?

According to our considerations in Subsec. 3.3.3 all values of M H that correspond to

2 _

ref. ..._ Xian. + AX? (95%, 1) are. consistent with the experimental

a X2 smaller than X
data at 95% CL. We determine these allowed ranges of the Higgs mass for all G (221)

models as well as for the SM and present the results in Tab. 4.2.

We find that in none of the considered models M H can be smaller than 38 GeV or
much larger than 150 GeV. To find out which observables constrain the Higgs mass we
perform two further fits with M H being fixed at M H = 25 GeV and M H = 250 GeV
respectively. During both finds we let GAPP vary Inf, 1%, 833 and mt to minimize
X2. Proceeding in this way removes the dependence of the pulls on the new physics
parameters and the top mass -_ we isolate the contributions from the Higgs mass. The
pull distributions after our fits point onto three observables that significantly deviate
from the measured values: For M H = 25 GeV the forward-backward asymmetry of
the bottom quark A F 3(0) (observable N9 17) contributes with a large pull to X2. If
AI H is set to A! H = 250 GeV, the theoretical predictions for the left-right asynnnetry
of the electron A L 3(6) (observable N9 21) and the W i mass MW (observable N9 29)

are far off the experimental results.

The physically relevant observation is that the measurement of the Wi mass
constrains the allowed range for the Higgs mass. Such a correlation is expected as
M H enters the expression for MW at the loop—level, in the SM as well as in our G (221)
models. In Fig. 4.1 we observe that the allowed M H ranges of the BP-I models are
shifted towards lower M H values compared to the curves belonging to the SM and
the BP-II models. In other words: The BP-I models prefer smaller Higgs masses than

the SM or the BP-II models; compare with Tab. 4.2. A closer look at the new physics

108

11107101 .41ij (412.0,) [GeV] 3111012111...) [GeV] .11}?’(X§Gf.) [GeV]

 

 

 

 

 

 

 

SM 55.37 93.24 148.12
LR-D 41.91 72.33 117.76
LP—D 39.62 68.94 112.91
HP-D 40.94 70.88 115.64
FP-D 38.62 67.50 110.85
LR-T 41.80 72.17 117.47
LP-T 39.62 68.94 112.91
HP-T 40.94 70.88 115.64
FP—T 38.63 67.48 110.89
UU-D 56.17 94.60 150.30
NU-D 55.50 93.48 148.51

 

 

 

 

 

 

 

Table 4.2: Bounds on the Higgs mass M H *- MSW and M}? are those masses of the
Higgs boson for which X2 takes the value Xfef. = X12nin. + AX2(95%,1). Therefore
MEW represents a lower and MED an upper bound on the Higgs masses that are

consistent with the data. Note that the values of M H for which the X2 curves in
Fig. 4.1 reach their respective minima are identical to the best fit values given in
Tab. 4.1.

109

corrections A111 W to the Ill-"i mass reveals why that is: In Tab. 4.5 in Subsec. 4.2.2 we
present numerical expressions for the shift AM W — plugging the best fit values for C:

and 82 ~ or 5% respectively into our results for Alt/[W we notice that the rib-dependent

26

contributions are almost negligible: Consequently, AMW is practically zero in the
UU-D and NU-D models. In the models of the first breaking pattern the shift does
not vanish; it is clearly dominated by the 3;? term. The best fit M H values in the
BP-I models therefore differ from the SM value in order to compensate the non-zero

new physics contributions A111“; to MW. ()n the other hand, due to the negligible

new physics shift in the BP—II models, the best fit values for the Higgs mass in these

 

Inodels are basically the same, as in the SM.

The large pulls for A F 3(1)) and [4113(6) are. less meaningfull in the context of the
AI H dependence of X23 In the best fits of all of our models A F 3(1)) is the observable
With the largest contribution to X2 anyway, see Tab. Al. The pull of observable
N3 21, A L 3(8), is prone to been blown up by the exceptionally small experimental
error,

In a last step we address the question of how X2 behaves if we do not vary A! H
but, the top mass m. For all models under study we calculate X2 for a set of fixed
T711: values and show the result in Fig. 4.2. As the best fit values for mt in Tab. 4.1
are already almost the same for all models we expect rm to be constrained to a very
llar row range. Fig. 4.2 exactly confirms this expectation: If we demand that AX2 is

Smaller than AX2 (95%, 1) the top mass fizt cannot be smaller than N 160 GeV and

larger than N 165 GeV in any model.

4-2 Allowed Regions in Parameter Space

F‘ ,.
133$- C.1 to C9 in the appendix show the main results of our study: The bounds 011

t1
19’ 118w physics parameters for all G (221) models and — for all models of the first

110

 

100

  
   

----LR-D ---—LR-T

-------- HP-D HP-T

----LP-D '---LP-T

—FP-D «PP-T

- ---- uu—o -------- NU—D

 

 

 

Figure 4.2: X2 in dependence of the top mass mt for all ten G(221) models and the
SM — the X2 curves of the (BP-I,D) models differ so little form the curves of the
corresponding (BF-LT) models that they are mostly covered by them.

111

breaking pattern — the bounds on the masses M Z’ and A! W’ of the new heavy gauge
bosons. We obtain the plots of the gauge boson masses by taking the parameter values
on the contours in parameter space and plugging them into the expressions for M Z’
and MW’ that we derived in Subsec. 2.2.4, see Tab. 2.6. In that respect the mass plots
are nothing else than direct translations of the parameter bounds into constraints on
the masses of the Z ’ and the I’V’i. In the UU-D and the N U-D model the masses of
the Z ' and the W ’ i boson are degenerate which is why we do not include mass plots
for these models. They would just show straight lines in the .MZI—MW/ plane. we
find that the minimum masses of the Z ' and the IV”: that would still be consistent
with the data are respectively given as 2.49 GeV and 3.66 GeV in these two models.

In the following subsection we will discuss the general properties of the parameter
and mass plots. Subsequently, we will identify the observables that drive the plots

and try to quantitatively understzmd how the plots come to their specific shapes.

4.2. 1 General Features

As discussed in Subsec. 3.3.2 (,5 either enters the new physics corrections to the elec-

troweak observables in form of the 0 <13 or the 303. For that reason we decide to plot

the bounds on the parameters in the Eli—co; or 53-5613 plane respectively. The parame-

ter space of the models belonging to breaking pattern one in which 835’ introduces a

further degree of freedom is actually three—dimensional. In fact, the boundaries be—
tween the allowed and the forbidden parameter values are given by two—dimensional
surfaces in these models. However, we stick to a two-dimensional representation and

color-code the values of .32 ~ on the parameter contours.

25

If we fix 3% at different values we obtain different parameter bounds in the (ii—cg;

plane. In other words: Looking at different slices of the two-dimensional boundary
surface along the 833 axis changes the beimds on i: and ng- The allowed regions shown

in Figs. 01 to 08 represent the maximum allowed regions that we obtain combining

112

 

the projections of all slices in respectively one plot. Proceeding in this way represents

a conservative approach: We exclude as few parameter values as possible. Only when

2 ~
23

ruled out by the data.

‘) . ~ .
no value for 3 leads to a x2 < Xfef we say that a given set of :1: and ca; values is

For each model we include two different parameter contours into our plots: One
contour that has been calculated with M H and flu being fixed at the best fit values
AIEP and 7723?“) of the respective model and one contour calculated with M H and 171,
being set to the values MEM = 93.24 GeV and 772%“ = 162.8 GeV that we obtained
fitting the SM. With the Higgs and the top mass given by MEI) and ml“) the contours
are more relaxed —~ M H and fit are harmonized with the specific properties of the

G (221) models and larger regions in parameter space can be opened up.

In all plots we indicate the bounds that we set on the parameter 1&2 for our nu-

a5
merical analysis, see Eqs. (3.35) and (3.36), by dotted lines. In the parameter plots

showing the irked; or the iii—sq; plane these bounds simply result in straight horizontal

0’)

lines that cut off regions where c ~ or 303 takes too high or low values. As for the mod-
els of the first breaking pattern, these boundaries in the 51—0 (5 plane are independent

of the value of 32 ~. If we translate these parameter bounds into the AIZr—MW/ plane

26

we can, in principle, choose between different 52~ values. Depending on our choice

213

we would get slightly different constraints on the gauge boson masses. As it turns out

the effects of the 5%, contributions to [W Z’ and .MW/ are, however, very small. They

are suppressed by 1‘1/2 and calculating the bounds on M Z’ and A/IW/ that follow
from the constraints on Ca; we can just neglect them. The dotted lines that are shown

in the mass plots correspond to a 52 ~ value of 32 ~ 2 0.

23 23

The fact that 52 ~ becomes larger when we follow the parameter contours to higher

25

values of C3) can be explained with the different signs of the c3; and 533 terms in the
shifts of the electroweak observables, compare with Tab. 3.1. As discussed in our anal-

ysis of the best fit values, see Subsec. 4.1.1, the experimental data apparently requires

113

the contributions from new physics to be small. In order to keep the corrections low
35, has to increase when we reach higher values of egg.

Furthermore, we observe that the parameter plots of the (BF-LT) models are all
shifted to lower :2: values with respect to the corresponding (BP-I,D) models. This
can again be attributed to the changes in the prefactors of the c: and 536 terms that
occur when the doublet. representation of (I) is replaced by a triplet representation.
The prefactors of c4~ and 52 ~ also explain why 52/. tends to take. smaller values in the

a5 2fi 23

upper Ca; regions if (I) is a triplet: Multiplying c; by 211 and 333 by % effectively enhances

the effect of the 533 terms by a factor of 2. To keep the new physics contributions
small we now have to choose smaller values for 5% ~.

An obvious feature of the parameter plots for the first breaking pattern is that
they again fall into the same two groups: The plots for the LR—D, LR—T, HP-D and
HP-T models and the plots for the LP-D, LP-T, F P-D and F P-T models respectively
resemble each other. Furthermore, the models of breaking pattern two seem to join
either of these groups: The UU-D parameter contours have the same shape as those of
the LR and HP models. Our results for the NU-D model look similar to what we get
for the LP and F P models. These similarities can be explained with the respective
observables that yield the largest to contributions to X2 in the individual models.

In the next subsection we will identify those observables and try to quantitatively

reconstruct our parameter plots.

4.2.2 Observables Driving the Plots

For each G(221) model under study we set the new physics parameters to some
exemplary values in the forbidden regions in parameter space and plot the resulting
pull distributions, see Figs. C.10 to (3.14. These plots directly point onto those
observables due to which the respective regions are ruled out. We, however, note that

the pull distributions shown in the appendix represent rather auxiliary material than

114

actual results of our analysis. They just allow us to identify the important observables
~— once we know which observables we have to look at our further discussion will be

based 011 the explicit expressions for those observables.

In the first place, the pull distributions confirm that models with similar contours
in parameter space are driven by the same observables: In the LR and HP models
as well as in the UU—D model the hadronic cross section Uhad. in e"e+ annihilation
(observable N9 3) clearly has the largest pull. Further large contributions come from
the forward-backward asynnnetry of the bottom quark A 17 3(1)) (observable N9 17)

and the “"3: mass ll-IW (observable N9 29).

AS expected also the LP and F P models are driven by the same observables. At
low values of cg; the weak vector charge QW (133Cs) of cesium-133 (observable N9 61)
apparently plays an important role. We present the corresponding pull distribution

for the FP models in Fig. C .13. For the same value of Cd”) the pulls of the LP models

95 is set to higher values QW (133Cs) looses its

- 2
Influence and the left-handed neutrino—nucleon coupling (gzN) (observable N9 48)

would basically look the same. If e

becornes the driving force behind the plots. We show an example pull distribution
for the LP model in Fig. C.11. Again, at the same (,6 value the result for the F P

IIIOdel would be similar. For even larger values of 03 we would see that also in the LP

and PP models the pulls of A F.3(b) and mass MW can go up. In the NU-D model
uN 2 -
(9 L ) is the most important observable.

T0 understand the origin of the pulls we compute the relative new physics cor-
rections to Chad.» AFB(b)v MW’ (QZNY and QW (133Cs) in all G(221) models and
insert the values of the experimental reference observables a, CF and M Z, see Sub—
SeC' A- 1 - The results of our calculation are shown in Tabs. 4.3 to 4.7. We new list
the most important conclusions that we draw from these results. First, we focus on
the InOdels of breaking pattern one. The upshot of our argumentation is pictorially

311m - . .
mar 120d in the sketches shown 111 F1g. 4.3.

115

 

iAUhad. /0had.,SM

 

LR-D —1.13-c; — 0.142 . c: + 0.0432 - s2
LP-D 0.346 - c: -— 0.142 - 0:; + 0.0432 - 32-
HP-D -—1.33 - (:35 —- 0.142 - c3.) + 0.0432 - s2~

FP-D 0.0035 - c: — 0.142 - c: + 0.0432 - .33

,3
Um) 4,889 . 32. _ 0.0132 . s4.
(.6 d9
NUD 0.533 . 32. _ 0.0132 - 347
Q9 (D

 

 

 

 

 

 

 

Table 4.3: Numerical evaluation of A0113.(1./011m1.,81\1 — the expressions for the (BP-
I,T) models follow from the (BF-ID) results by multiplying c: and (3:3 by 211 and 83,;

by %.

i‘AAFB(b)/ AFB.s.\1(b)

 

 

_‘ ,2 -~.4_‘ .~.2

LR—D 30.0 Cd) + 67.6 Ce 20.6 32
LP-D —40.1 -c% + 67.6174, — 20.0 - 52
o e 2

HP-D —30.9 - c: + 67.6 - 0:3 — 20.6 - .92 ~
FP-D —47.0 - a: + 07.0 - c: — 20.0 - 32 -

LTU-D 0.101 . .52. + 6.29 - 34~_
o <9

NU-D 14.2 - s

 

 

 

 

 

 

Table 4.4: Numerical evaluation of AA F 3(6) /.4 F B,SM(b) ~~— the expressions for the
(BP'LT) models follow from the (BF-ID) results by multiplying c: and 0:3 by 211— and
2
,3 ~ . 1

2.3 by 2-

116

i‘AAIIi'/A[‘I;SI\I

 

 

_ ( 4_ ( 82~
(BPI,D) 071.) cc; 0.-713 23

_ -. 4.- . - . 2-
(BPI,T) 0.130 cqb 0.300 523
UU-D 0.219 - s:

NU—D 0.219 - s:

 

 

 

 

 

 

Table 4.5: Numerical (waluation of AllIw/A'IW’SM

~ 6 V") 62:20)

‘Jk'A.

 

 

(1313—11))
(BF-LT)
UU-D

N U-D

 

0.0875+1.91cg+0.839 —2.84-sg

C4 u:
CC? 73

0.()219+0.478 C: +0210 C~-—1.42 82
(P 2/3

0.839 - s4.
(.2

r _ 5' ‘ .2 . 4
2.08 0.383 S¢+0.839 8G5

 

 

 

 

 

Table 4. 6:

2
Nume ri( r1 evalu rtion of A g’m / WV 2
‘ ‘ 9L SM

117

EAQW (133CS) /QL£SM (133C5)

 

LR-D —0.855 - c

LP—D 3.35 -— 1.95 . c" —- 0.855 - c: — 0.145 - 5

‘10 to
CD:

 

 

 

 

_ _ rr . 4 _ . r . 2~
HP D 0.800 C45 0.140 826
FP—D 2.95 — 1.95 - c2. — 0.855 . c4. — 0.145 - 52,~
(0 (,9 2,13
UU-D —0.855 - s:
NU-D 0.400 + 0.594 - 3E — 0.855 - 54¢,

 

 

Table 4.7: Numerical evaluation of AQW (133Cs) / Qw, SM (133Cs) —~ the expressions

for the (BF-LT) models follow from the (BP-I,D) results by multiplying 0:; and 0:;

by :11 and 53,3, by 4.

, 2
0 The corrections to A F 3(6), MW and (921)“) all prefer smaller values of C¢~>°

If c~ chosen too large the new physics shifts increase and the respective pulls

<15
2
are blown up. A (gzN) and AJWW are especially sensitive to high Car) values:
2
In A (gZN) the c: and the 0:; terms have the same sign so that they camiot

cancel each other; All/I W only involves a c: contribution. A F 3(6) has the largest
effect on the parameter bounds in the LR and HP models as for these models

the coefficient of C: has a smaller absolute value which leads to bad cancellation.

0 The reason for the large impact of QW (133Cs) on the LP and F P bounds lies
in the fact that the corrections AQW (133Cs) involve an absolute term in these
models. Only for large 08 values the negative c: and 0:; terms can compete with
that absolute contribution. The consequences are that the low-c 03 region is ruled
01113 in the LP and PP models and that the parameter contours start at higher

57 Values than in the LR. and HP models. At higher ca) values, once QW (133Cs)

118

 

cos($)

0.4

    
  

 

 

lllllllllllllllllllijlllllllllllllillllLlllllll

Competing cos4 (0;) and
— sin2(2,6) terms.

 

 

illlllllllllllllllllllllllll
A:

X

Figure 4.3: Sketches illustrating the influences of some key observables on the param-
eter bounds for the models of the first breaking pattern » , the parameter contours
in the (BP-I,T) models are shifted to lower 5: values compared to the corresponding
BP'LD) models. Omitting the tick labels on the :2 axis we ensure that the sketches
apply ‘30 (BP-I,D) and (BP-I,T) models alike. The UU-D parameter contour is driven

2
by “bad. as well. In the NU—D model (gZN) is the most important observable.

119

2
Ohad. A F 3(1)) (gZN) QW(133Cs) Set of other obs.

 

 

LR, HP 6) ®
LP, FP ® 6)

 

 

UU 6) ®
NU 6) ®

 

 

 

 

 

 

 

 

 

Table 4.8: Overview of the observables driving the parameter plots — the most and
second most important observables are respectively marked with the symbols G) and
Q). In the UU and NU models only one observable significantly contributes to X2-

does not represent a strong constrait any more, the parameter contours are

2
dominated by the observables A F 3(6), [WW and (gZN) .

o The 52 ~ and 0‘3 contributions always have opposite sign. As discussed earlier this

26 <25

leads to the increase of 333 when 66 becomes larger. The parameter plots of the
FP and LP models suggest that — depending on the exact interplay between
.335, and 603 —— the 3:? terms are be able to overcome the of; contributions
such that the parameter boundaries are pulled back towards lower i: values.
Note, however, that the expressions given in Tabs. 4.3 to 4.7 cannot explain
the branching between the LP and FF contours. To account for that effect we

certainly would have to extend our discussion to other observables as well.

0 In the case of the LR and HP models the 333 terms do not have a chance:
The c: and c: contributions to Aahad. have the same sign and the 5% term
is suppressed by a small prefactor. The pull of Chad. therefore represents a

hindrance for the LR. and HP models that is impregnable for large Cg}:

After these comments on the models of the first breaking pattern it is easy to

120

 

 

understand the shape of the pau'ameter contours in the second breaking pattern: In
the UU-D model all corrections that we present in Tab. 4.3 to 4.7 favor small 365
values. Especially the fact that the 3:7 and 5:, terms in Aahad. have the same sign
leads to the exclusion of the high-s <13 region. For that reason the UU—D plot looks
similar to the plots of the LR and HP models. The contour of the NU-D model is
, 2 2
mainly influenced by the ("("n‘reetion to (92A) . Since A (921V) is small if sq; takes
some intermediate value. we observe a bump in the NU-D contour towards lower if
values for Sci) values around N 0.65.

In conclusion, we summarize our observations as follows: The shapes of the con-
tours for LR, HP and UU models are driven by 011ml.- For the LR and HP models,
A F B(b) also plays an important role. In the LP and PP models, at low C(5- values,

. . 2
QW (13'5Cs) is the most important observable. At higher c 973 the coupling (gZN) is
responsible for the shape of the parameter contours. (To some degree, A F 3(1)) has
the same effect on the LP and PP contours as on the LR and HP contours, though
subdominant compared to the. other observables. The same applies to (gZN)2 for
the LR and HP models.) The NU contour is mainly driven by the pull of (gZN)2.

Tab. 4.8 on the previous page presents these results in a tabular form.

4.3 Concluding Remarks

4.3.1 Constraints From Triple Gauge Boson Couplings

The aim of our analysis is to confront the G (221) models with all important precision
data that is available for electroweak observables. 111 this last section we may thus
include the precise LEP 2 measurement of the Z W+VV" triple gauge boson coupling
into our discussion, The 7W+W' boson coupling has been measured at LEP 2 as

we . . . 7 ‘ . . .
H: We, however, do not consider tins coupling here: Due to QED gauge mvarlance

121

 

the yW+W_ coupling is the same in the. G (221) models as in the SM. It does not shift
which is why it cannot help us to constrain the new physics parameters. In addition to
that, it has been 111easured with less precision than the Z W +W" coupling, anyway.

Employing the Hagiwzu‘a parametrization [29] we can write the Z W+VV— vertex

factor QZWW as a function of the parameter ng2

Z
9 Z W W 2 .9 Z W W' (91 )

which takes the value unity in the SM. The experin'lental value for glz extracted from

the LEP 2 data reads as [30]:

 

9% = 1.001 3% 0.027 a: 0.013

where the uncertainties are the 10 statistical and systematic uncertainties, respec-
tively. The total experimental uncertainty A912 in 9‘? follows from adding these two

uncertainties in quadrature:

glz = 1.001 a; Ag? ; A913 2 0.030 (4.1)

This result is obtained from the analysis of 6-8+ ——> W+W_ events. One deduces
the scattering amplitude Aexp, from the experimental data and determines g1Z in a

single—parameter fit in which all other couplings are kept fixed to their SM values:
_. Z 23
onp- = AsM + 91ASM + A’ém (4-2)

The three amplitudes ~4ng AgM and AIS/M denote the respective contributions

from the s-channel 7 exchange, s—channel Z exchange and t-channel 1/ exchange in the

€Te+

fl W+W " transition. We can now use the result for 912 in Eq. (4.1) to further

122

 

 

 

 

constrain our new physics parameters. In the G(221) models the total amplitude

Aexp. IS given RSI

A...” = .4131. + Afip + AK”) (4.3)

where the G (221) amplitudes can respectively be written as the sum of the corre-

sponding SM amplitude and a new physics correction that is proportional to %:

45/106613) ; a E {7", Z,1/}

HtlH

111 order to find the regions in parameter space that are excluded due to the
Z W+W _ constraints we have to calculate the three amplitudes contributing to Aexp.
in both the G (221) models and in the SM. This is best be done employing the helicity
amplitude method [31]. 111 Ref. [29] Hagiwara et al. present general expressions
for the relevant amplitudes for all possible combinations of (Bi and W3: helicities
which allow us to quickly compute the G ( 221) and SM helicity amplitudes — we only
have to either plug our G(221) results for the various couplings and masses or the
corresponding SlVI expressions into the expressions given by Hagiwara et al.

For the explicit helicity amplitudes we refer to Ref. [2.9]. Here, we only note the
key features of the dependencies on the scattering angle 6 (not to be confused with
the Weinberg angle). For a given configuration of incoming and outgoing helicities,
all three amplitudes .47, AZ and A” are proportional to the Wigner d—functions. For

the s-channel amplitudes, we have:
ALAZ oc dfig’w‘Wl

AS for the t-channel 1/ cx<*-.hange, we have an additional 6 dependence from the

”‘Propagat.or:

 

C ) AmAA

V — d 6
A (X (B 1 + ,132 — 2,13cos6 AJ ( )’

123

 

 

 

 

where B and C depend on the helicity configuration, but are independent of 6. To
finally constrain our new physics parameters, we peform a partial—wave analysis. For

any of the three involved amplitudes we project out the dfig’AAW) component:

- 1
AG E f 1 d(cos 6) Au - dfig‘AAW) ; a E {7, Z, V}

By equating the two expressions for Aexp, in Eqs. (4.2) and (4.3) and projecting
out the dfig’AAW) cmnponent we finally obtain the constraint on the new physics

parameters —- :if, (f) and B can only take values for which the following relation holds:

A.

~ ~Z ~ I , , ”Z ~
Ag”) + AN? + Aiqp I ASM + (1.001 3: A912) 'ASIV'I + .1451“

According to this condition we calculate the bounds on 5:, qfi and ,6 for all possible
helicity configurations. Once we have done that, we combine our results and determine

the maximum regions in parameter space that are ruled out due to the experimental

value of A912 .

Our calculations show that the experimental data on the Z W+W _ vertex gener-
ally does not put stronger bounds on the new physics parameters than the electroweak
precision observables. To illustrate which regions in parameter space are typically ex-
cluded by the data on the Z W+ W _ coupling we show our results for the (BP-I,D)
models in Fig. 4.4. As the only fermion quantum numbers that we have to take into
account are those of the electron we are able to respectively combine our results for

the LR—D and HP-D and the LP-D and FP-D models in common plots. For both

2 ~ values

23

pairs of (BP-I,D) models we present the parameter bounds for the extreme .3
Of 32 1 : 0 2 ~ I:

26 and 52/3 1.

In the case of the LR—D and HP-D models the Z I/V+W_ constraints do not affect

the r eslllts of our global fit analysis at all: For low 0 <13 values 533' is small on the LR-D

124

 

 

003(6)

003(6)

LR—D, HP—D: sin2(2,6)=0
11.(:).1.f '_ '. ,lf‘" T 1“" r "'1. .' ..',."'l“'" '11

0.8 1 1
0.6 « ,,
0.4 . ’
0.2

 

 

0.0

‘--‘l -LA- , 11

l
0 2() 4O 6() 80100

x

LP—D, FP—D: sin2(2Z3)=0
1.0 LL,“ 1 'v "1": , V‘r~1~1-, frirr 'T 1

 

 

“WAI- l—l-J-_L

6O 80 100

cos(g~b)

c0305)

LR—D, HP-D: sin2(2'[3)=1
1.0 1 x _.-.~,,.., 1
0.8 1’ ‘
0.6
0.4
0.2 ,
,1

0.0:..2.
0 20 40 60 80 100

 

 

1”L“L_‘llj~i~ll--1..L.1111¥

~

x

LP-—-D, FP—D: sin2(2fi)=1
1.0 1 “Mr T as... ,

0.8
0.6
0.. ;
0.2

0.0 lyr-j
0 20 40 60 80 100

~

x

 

 

1 .4. “LL-1- .; , l. ..L._J-..L_.-L. 1 _1__1-.1_-1__A-_-.| I

Figure 4.4: Bounds on the (BP-I,D) models from the Z W+W“ vertex .- the shaded
regions are excluded, the blank regions consistent with the measurement of the

ZW+W-

coupling.

 

 

and HP-D parameter contours - in which case the Z W +W " data is only able to
rule out points in parameter space with 5:. S, 20. In the LP-D model our boundary
between allowed and forbidden parameter values is located at (i? values larger than
250. The Z W+VV " constraints do not reach up to that high :15 values. Only in the
FP-D the Z W +11"— contour comes at least close to the one that we found earlier in
our global fit analysis. If ff: is chosen small and 533 set to 1 the contours meet each
other at 60 = 1. However, as the F P-D curve has a much smaller slope than the

Z W+VV ' contour the Z W+WH data does not exclude parameter values that were

not already ruled out by our global fit analysis.

In conclusion, we can say that at present the constraints from the Z W+W" vertex
cannot compete with the bounds set by the electroweak precision data. Our global
fit analysis therefore represent the complete answer to the question that we posed in
the introduction: Which bounds do the experimental data place on G(221) models?
All information on the bounds is contained in the plots of the parameter space and

the masses of the new heavy gauge bosons that we show in the appendix.

4.3.2 Future Prospects

The Q-weak [32] collaboration at Jefferson Lab intends to determine the weak charge
of the proton, Qw(p), by measuring the parity violating asymmetry in elastic e— p
scattering at Q2 = 0.03 GeVZ. Meanwhile the e2ePV collaboration [33], also at
Jefferson Lab, proposes a Meller scattering experiment at Q2 = 0.0056 GeV2 that
would allow to infer the weak charge of the electron, Qw(e), with ultra—high precision.
The experimental results that are anticipated by both collaborations are already

implemented into the GAPP code:

Qw(p) = 0.0715 2t 0.0029 ; Qty-(6) = —0.046900 :1: 0.001079

120

 

 

Given the accurancy that both experiments are aiming at we feel tempted to
investigate how much of an effect it would be if we added these two precise values to
our global fit analysis. In a last step we thus repeat our scans of the C(221) parameter
space with QW (p) and the 02er value for Qn/(e) included as additional constraints.
While the bounds 011 the LR, HP, UU-D and NU—D models are only slightly affected
the results for the LP and FF models change drastically. As an example we present
the i~cé parameter contour for the LP-D model in Fig. 4.5. At low values of ca) the
boundary between allowed and forbidden values is pushed to much larger 50 values.
To find out which observable is responsible for that shift we again pick a point in
the excluded region and examine the distribution of the pulls. It turns out that the
anticipated value for the weak charge of the electron causes the dramatic change of
the LP and PP contmn's. The effect of QW(e) on the LP and PP models is thus
comparable to the one of QW(133Cs) with the only difference being that the region
exluded by Qty/(e) extends to much higher 5: values, compare also with Fig. 4.3. As a
consequence of these further constraints from QW (p) and the e2ePV value for Qw(e)
the allowed region in the MZI—MW/ plane shrinks as well, see Fig. 4.5. The minimum
allowed mass for the VV’i increases, for instance, from m 0.6 TeV to z 1.2 TeV.

This last analysis therefore shows us that our study can only be understood as a
snapshot that tries to capture the current constraints on the G(221) models. Future
experiments such as the measurements of weak charges at low energies might be
able to yield significant corrections to the picture that we have drawn in this thesis.
However, this prospect shall not discourage us. It is exactly the interplay between
increasingly more precise experin‘rents and phenomenological studies like ours that

harbors the chance of detecting new physics beyond the SM.

127

 

 

m...” .. ..............................................................................................................................
r in 0 0.00<sin2(2i§)50.25
oa “ U 0.25<sin2(2i§)so.50
— A 0.50<sin2(2i§)so.75
: <> 0.75<sin2(26)s1.oo
0.6 ——
3 — 0.
7,; _ G~ allowed (95% CL)
8 _ an
NP NP
0.4L— "'ML’Jnf)
L—
SM 8
r —ML {777; M)
#
0.2 —
oolllJlllllllLlllllllLlJLlLl lllj
' 400 soo 600 700 800 900 1000
f
LP-D
s
— 0 0.00<sin2(2§)50.25
4 —— U 0.25<sin2(2f3’)so.50
~ A 0.50<sin2(2i§)so.75
* _, ___ (NP) (NP)
— <> 0.75<sin2(2|3)s1.00 Mn :7”:
A 3 —
(SM) (SM)
% '— MH In:
t; 1—
§ E
E 2__
l—
h
1 .—-—.
—1 l-v-ln-j “““ f1141 ..... l ..... 1i llllll 1 IIIII 1 ..... l 1 1 1 1 l 1 I 1 1 l 1 1 r
°o 1 2 3 4 5
Mz(TeV)

Figure 4.5: Bounds on the the LP-D model with the anticipated values for QW(p)

 

 

 
 
 
 
 
 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

   
   

 

 

 

 

 

and Qw(e) being included into the analysis ~ compare to Fig. CZ.

128

 

 

 

 

Appendix A

Experimental Input

We give the experimental results for the reference observables a, GP and AI Z that are
used by the GAPP code and determine the numerical values of the standard param-
eters 838M and USM. Subsequently, we list the experimental values of all electroweak
observables to which we fit our G (221) models. The experimental data implemented
in the 2009 version of GAPP [21] agrees for the most part with the values given in
the 2008 PDG book; compare with [20].

A.1 Reference Observables a, G p and M Z

The most recent version of the GAPP code uses the following numbers:

a=1/137.03599911 ; 0F:1.16637-10"")GeV‘2 ; MZ=91.1876GeV

The value for (1 represents the low—energy limit of the fine structure constant. Go-
ing to higher and higher energies in the experiment scale-dependent loop corrections
Will cause a to run, that is, to become progressively larger. We have to consider

the running of the fine structure constant when we deduce the numerical value of

129

 

 

 

838M from the Z mass. Expressing Mg in terms of a, 838M and ”SM: see Eq. (3.1),
the involved a does not take the value 1/137.036. Instead it is given as the running
coupling (1(a) evaluated at the Z mass scale. The explicit value of 0 (Mg) depends
on the employed renormalizatn)n scheme. GAPP is based on the modified minimal
substraction (RI—S) scheme. The W value (1 (Mg) for 0: (Mg) is calculated by GAPP

and turns out to be 61(Mg) = 1/127.918. Eq. in Subsec. then actually reads as:

 

2 “fin
56 ‘ C0 ‘
sM sM

We solve this equation for 35 "\I and obtain:

LL,

2 __ 1.... . 2 _ - .
3933141233002 , CQSM —0.706348

The value for Fermi’s constant C F results in a electroweak VEV of:

USM = 246.221 GeV

A.2 Electroweak Observables

We present the 111easured values for the electroweak observables in two tables: Tab. A.1
on the following page lists the experimental results for the Z and Wi pole data as
well as the top mass; Tab. A2 011 page 132 presents the experimental data that is
available for the low-energy observables. Both tables also give the numbers by which

the GAPP code refers to the individual observables, our best fit results for the SM

and the corresponding pulls.

130

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N9 Obs Exp. SM Pull
(2) r2 [GeV] 2.4952 :t 0.0023 2.4956 -0.163
(3) ahad. [nb] 41.541 3: 0.037 41.480 1.658
(4) R (6) 20.804 3: 0.050 20.740 1.278
(5) R ()1) 20.785 :1: 0.038 20.740 1.355
(6) R (T) 20.764 :1: 0.045 20.786 -0.478
(14) 72 (.5) 0.371 i 0.022 0.359 0.535
(16) R (0) 0.1721 :1: 0.0300 0.1722 -0.048
(15) R (0) 0.21629 i 0.00066 0.21580 0.741
( 11) 4 L 12 (6) 0.1498 i 0.0049 0.14787 0.495
(21) 0.15188 i 0.00216 — ” 1.855
(22) 0.1544 2% 0.0060 -_ ” ~ 1.171
(25) 0.162 a: 0.048 — — 0.340
(23) A L R ()1) 0.142 i 0.015 — — -O.358
(10) ALB (7) 0.1439 :1: 0.0043 — —— -O.808
(24) 0.136 i 0.015 — ” — -0.758
(26) A L R (5) 0.895 :t 0.091 0.936 -0447
(20) A L R (c) 0.670 :1: 0.027 0.668 0.078
(19) A L R (b) 0.923 :t 0.020 0.935 -0.588
(7) A F B (6) 0.0145 i 0.0025 0.0163 -0.716
(8) A FB (,1) 0.0169 :1: 0.0013 — ” —— 0.470
(9) A FB (T) 0.0188 i 0.0017 — — 1.477
(18) A F3 (5) 0.0980 j: 0.0110 0.1034 -0493
(18) A F B (c) 0.0707 i 0.0035 0.0738 -0.892
(17) A F B (b) 0.0992 31: 0.0016 0.1033 -2.573
(12) Q F 3 0.0408 :1: 0.0026 0.0423 -O.781
(28) I‘W [GeV] 2.196 i 0.083 2.092 1.257
(80) 2.057 :1: 0.062 —- ” ~— 0559
(27) MW [GeV] 80.376 :1: 0.033 80.378 -0051
(29) 80.432 :1: 0.039 ~— ” _ 1.393
(42) mt [GeV] 172.40 :1: 1.34 172.45 -0035

 

 

 

 

 

 

Table A.1: Experimental values for the Z and Wi pole observables and the top mass

that we employed for our numerical analysis m see the PDG book [20] for references.
The SM values and the corresponding pulls represent our best fit results.

131

 

 

 

 

 

 

 

 

V'° Obs Exp. SM Full
(47) 7.. [fs] 290.93 :1: 0.48 289.08 0.771
(48) (gZN)2 0.3010 :1: 0.0015 0.3089 —1.944
(49) (9)/{NY 0.0308 i 0.0011 0.0300 0.720
(50) HUN 0.5820 :1: 0.0041 0.5830 0252
(51) RV 0.3021 :1: 0.0041 0.3091 -1.705
(52) 0.3096 i 0.0043 — -— 0.119
(53) 171-, 0.403 :1: 0.016 0.386 1.058
(54) 0.384 :1: 0.018 — — -0115
(55) 0.365 a: 0.016 0.881 -1031
(56) 9'6 —0.040 3: 0.015 -0040 -0.016 f
(57) 9316 -—0.507 :1: 0.014 —0.506 —0.044
(58) QW(e) —-0.0408 2: 0.0053 0.0472 1.309
(61) QW (133Cs) —73.16 :1: 0.35 -7315 0032
(62) QW (205Tl) —116.40 a: 3.64 -116.75 0.096

' (63) cl —0.0285 :1: 0.0043 0.0335 1.170
(64) (32 0.342 i 0.063 0.3885 0739

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table A2: Experimental values for the low-energy observables that we employed for
our numerical analysis ——— see the PDG book [20] for references. The numbers N9 in
the first column refer to the numeration of the observables in the GAPP code, the
SM values and the corresponding pulls represent our best fit results.

 

132

Appendix B

MS-Bar Mass of the Top Quark

To renormalize a perturbative quantum field theory one introduces counter terms that
absorb the infinities arising beyond the tree-level. These counter terms are, however,
not unambiguously defined. Different renormalization schemes are available each of
which fixes the counter terms in a different way and each of which, hence, entails

different definitions of important quantities such as masses and coupling strengths.

Connnon schemes include the on—shell scheme in which the particle masses are set
to their physical values 011 the mass shell and the modified minimal subtraction (MS)
scheme which absorbs infinite as well as large logarithmic terms in the renormalization
constants. In this study we require the mass of the top quark in both schemes: The
on-shell mass mt is one of our 37 observables; see Subsec. 3.2.1. The MS mass T‘nt
is one of our five fit parameters; see Subsec. 3.3.2. In this appendix we will briefly
outline the relation between these two definitions of the top quark mass.

All loop insertions into the top quark propagator contribute, in principle, to the
mass renormalization of the top quark. In this discussion we will restrict ourselves to
a couple of example loop effects: We consider diagrams involving a gluon G, a Higgs
boson H, a longitudinal Z boson Z L and a longitudinal LV+ boson W 2' respectively.

The contributions from transverse electroweak gauge bosons are sub-leading which

1 I53

 

 

 

 

Figure B.1: The four considered example diagrams that contribute to the mass renor-
malization of the top quark -1... the Goldstone bosons (b0 and (25+ substitute the longi-
tudinal electroweak gauge. bosons Z L and IV; .

is why we neglect. them here. By virtue of the Goldstone-boson equivalence theorem
(ET) [34, 35. 36] it is possible to replace the longitudinal gauge bosons Z L and W2”
by their corresponding unphysical Higgs bosons 620 and (0+. The ET only requires
that mt > MW, Mg which certainly is the case. Fig. 8.1 presents the four diagrams
under study.

bare

In the on-shell renormalization scheme the bare top mass m is related to the

pole mass mt as follows:

5m 3CD (SrYriUk'
771;)?11‘0 277% (1+ 771—:) 2 "It 1+ T:—— + 7::— + ...)
't ' t t

where (57)/,‘t‘k' subsumes all cmrtributions from the Yukawa couplings of the top quark
to the Higgs bosons H, (250 and 66+, that is, the contributions from diagrams N9 2 to
N9 4 in Fig. 8.1. The dots represent the contributions from all diagrams that we do
not discuss in this appendix. Evaluating diagram N9 1 we obtain an expression for

(SEED/mt:
QCD ,
5m. 30' m2 4
t, 3 w t
:- ———C —A 1 ~-——— —— —— 8.1
mt 477 F ( + n (112 3 ( )
134

 

 

 

 

Here, as stands for the strong coupling constant. At the top mass scale it has a
value of 013(mt) = 0.11 [20]. C F is a. color factor that takes the value C F = 5% in
QCD. A E ”TQ’V -— 73 + 111(477) denotes the regulator that appears in dimensional

regularization. ’y E _—: 0.577... is the E111er-Mascheroni constant. The mass 1‘)arameter

11. is a measure for the energy scale.

Ref. [37] 1_)I'ovi(.les 11s with an expression for 5%‘gk/"lti

 

 

1"— : i%—'—"‘—2— 34—3111 Q +1—4I ...; +2.1 ———§l- +377 (13.2)
mt 10W 8111”,. ,u“ t mt
where
l 9 1
[(5) : dl‘lll[1‘“ + (1 — .r)£ — iC] ; J(§) 2/ d.r1"1n[.r2 +(1— x)€ —- 2'6]
0 0

111 the on-shell scheme the entire shift 677% in the top quark mass is absorbed in
the corresponding counter term. The MS scheme includes, by contrast, the finite real
parts of 6mt in the definition of the mass m. Only the infinite and logarithmic pieces
are cancelled by the renormalization constant. According to Eqs. (8.1) and (B2) the

difference between mt and fi’lt thus amounts to:

 

 

 

303 4 9%, m? M}, M},
: +__C . __ + _ 1—4I +2J — +...
mt 477 F < 3) 16772 8.113. m3 m3

To obtain numerical results for ASFD and AY‘lk' we em loy the followin data: The
t Tnt p . g

experimental value for the pole mass, mt = 172.40 GeV, the strong coupling at the
top mass scale, 073(mt) : 0.11, the SM value 9L.SM : 0.63 for 91,, the weighted mean

of the two measurements of the W i mass, AIW = 80.40 GeV, and the SM best-fit

135

 

 

 

value for the Higgs mass, M H : 93.24 GeV. We find:
A2,?) = —8.05 GeV ; A253?- = 0.93 GeV

The comparison of the SM prediction for the pole mass, mt 2 172.5 GeV, with
the SM best—fit value for the MS mass, flat 2 162.8 GeV, shows that. the contribution

from the neglected diagrams must. add up to roughly 2 —2.6 GeV.

 

136

Appendix C

Bounds on the G(221) Models

In this appendix we present plots for each C (221) model that. indicate which regions
in parameter space are consistent. with the electroweak data and to which masses A! Z’
and ‘MW’ of the new heavy gauge bosons these viable parameter values correspond.
Subsequently, we show pull distributions for each model that were calculated for
exemplary points in the forbidden parameter regions. These pull distributions point

onto the observables that are responsible for the respective parameter bounds.

C.1 Parameter and Mass Plots

For the. models of the first breaking pattern the 1')arameter plots present the fi—cf;5
plane; for the UU-D and the NU-D models the it 50:) plane is shown. The boundary
between allowed and forbidden parameter values is determined twice for each model:

With M H and 771); being fixed at the SM best. fit values in the one case and with M H

and fiat being re-fitted in the new physics models in the second case. In the plots for

the 1110(1913 0f breaking pattern one the information on 533 is color-coded. For each
c <13 the .33 ~ value corresponding to the lowest if is chosen. The dotted lines indicate
1"

the bounds on t; that we discussed in Subsec. 3.3.2.

137

»_' ‘(

 

LR-D

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

   
 
 
  
   

 

 

 

 

 

 

 

 

  

 

 

 

1.0 _m ......................................................................................... c ............. ;.;.._.. ---.-.e ............
g_ . ‘ ‘0‘"‘-—-0‘—’0 ‘0
03 — NP
- - - - M17715: 7
__ MS"): ”ism
0.0
g
(I) ..
8 o 0.00 < sin2(20) s 0.25
°" C! 0.25 < sin2(2§) s 0.50
A 0.50 < sin2(2§) s 0.75
0 2 0 0.75 < sin2(28) s 1.00
0.0—llllJlllJllllllLllllJ_LlLL1llllllJJlllll
200 400 600 000 1000 1200 1400 1000 1000 2000
i
LR-D
5 r
: 0 0.00 < sin2(25) s 0.25
4 "_ D 0.25 < 5111425) 3 0.50
: A 0.50 < $111326) 5 0.75
Z 0 0.75 < sin2(2§) s 1.00
S‘ 3 f
E;
” _ _ _ (NP) (NP)
3 __ _ 730) (SM)
2 — MH ,7”,
1 :_ allowed (95% CL)
— 1 11(1411 ..... 1 ..... l i 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 m 1
°0 1 2 4 5

3
M 2' (TeV)

Figure C.12 Bounds 011 the new physics parameters and the masses of the new heavy
gauge bosons in the LR-D model.

138

 

 

   
    
 
   
  

 

LP-D
1.0 .......................................................................................... 7 ,5; ___________ e .............................
_ 547-4“ 1 " '
I 44%
0.0 :— _ - - Win11”)
_ _ M7310, new

 

 

 

 

 

 

 

 

 

 

 

 

 

 
   
   
   

 

 

 

 

 

 

 

 

 

 

 

g ._
g 1.—
0.4 ___ 0 0.00<sin2(2B')so.25
: :1 0.25 <sin2(2E)s0.50
— A 0.50<sin2(25)s0.75
02 —- ..
_ ° O.75<sin2(25)s1.00
0.0 L l 1 1 l l J L l 1 l l 1
500
LP-D
5
l- ..
4 0 0.00 < sin2(2[3) s 0.25
4 .1 1:1 0.25<sin2(2§)s0.50
_ A 0.50 < sin2(26) s 0.75
_ o _ 2 __
a — H ' '
S
b : MfM,'m: “0
§ _
E 2 _
1 _—
0 — 1 111L1 ..... 1M1 ..... 1 ..... l ..... 1 n m L L 1 1 1 1 l 1 1 1 1 l 1 1 1
o 1 2 3 4 5

 

Figure C2: Bounds on the new physics parameters and the masses of the new heavy
gange bosons in the LP-D model.

139

 

  
   

 

 

 

 

 

 

 

 

 

 

 

 

 
   
   
  
 

 

 

 

 

 

 

 

 

 
 

 

 

 

HP-D
1.0 _ ............................................................................... —:9“;‘:;‘;‘_'_‘_‘;"°';';'I-'—0*" :. .............
0.0 _
___ (NP) (NP)
1 MH .111,
1..
(3”) (SM)
1— MH ’MI
0.0 ~—
E
U)
8 allowed (95% CL) 0 0.00<sin2(2B)s0.25 g
0.4 ~ .
U 0.25<sin2(2[3)50.50
A 0.50<sin2(25)s0.75
0 2 <> 0.75<sin2(2§)s1.00 ' ‘
oo-llllllllllllllilllllllLlllllllllllJllllll
‘200400600000100012001400100010002000
5?
HP-D
5
I 0 0.00<sin2(23')so.25
4 ; 1:1 0.25<sin2(25)so.50
I A 0.50<sin2(2i§)so.75
I 0 0.75<s1n2(2B')s1.00
A 3 __
>
m — ___ (NP) (NP)
t — MH 1”:
§ 7 _ M7811) ”(3M1
_ H , 1
E 2 _
1 ; allowed (95% CL)
L
o — 1 11111 ..... flu-L ........ 1Mi1 1 l 1 L 1 1 l 1 1 1 1 l 1 1 1
0 1 2 3 4 5
M,(TeV)

Figure C.3: Bounds on the new physics parameters and the masses of the new heavy
gauge bosons in the HP-D model.

140

 

 

FP-D

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0 ..........L ............................................................... . ................................................................
*- NANAHAWAMA‘A. ~ -
)- fifl‘ E}. h = = -
BT35] 7 = = -
0.0 P— P) 7 -.
P ’9.
._ _ _ - sz 1’ ”(N
‘— — Mi?) Mi“) ‘5'":
0'6 _ 9“)"
to r- __ ,5“
8 L o 0.00 < sln2(ZB) s 0.25
0.4 _— ~ 5"“.
_ D 0.25 < 31113205 0.50
7 A 050 < “"320 S 0-75 5’ allowed (95% CL)
0 2 _ o 0.75 < sin2(2'6) s 1.00 "
0.0 lilllllllllllllllllllllLLllllllllJllllllllL

 

 

50 100 150 200 250 300 350 400 450
f

 

 

FP-D
5

0 0.00 < sin2(2§) s 0.25

4 Cl 0.25 < sinz(2§) s 0.50

A 0.50 < sin2(2B') s 0.75

0 0.75 < sin2(25) s 1.00

 

 

 

 

__ _ _ MSW. miNP)

—Mff"’.Mfs”’

 

 

 

MW (TeV)

 

allowed (95% CL)

IITTIIIIIITTIIIIIITIIIII

 

 

 

 

Figure (3.4: Bounds on the new physics parameters and the masses of the new heavy
gauge bosons in the F P-D model.

141

LR-T

 

 

  
   
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
  
  
 
  

 

 

 

 

 

 

 

1.0 .......................................................................................................................... . .. ’
03 ’— NP
_ _ _ _ MSW), m: )
“‘ s
L — MS”, 111: ”l
0.0 ~—
§ _
m _ ~
8 — o 0.00 < sin2(20) s 0.25
0" allowed (95% CL) 1:1 0.25 < sin2(2§) s 0.50
A 0.50 < sin2(26) s 0.75
0 2 0 0.75 < sin2(26) s 1.00
0.0 4'. l l l L l l J l l J l 1 l 1 l l 1 L1 1 1 l l l 1 l l I l
50 100 150 200 250 300 350
1?
LR—T
5
1
4 ‘_ 0 0.00 < sin2(25) s 0.25
: :1 0.25 < sin2(26) s 0.50
: A 0.50 < sin2(2'3') s 0.75
g 3 f 0 0.75 < sin2(2'8)s1.00
t :
ES 7
_ P (NP
2 _ - _ - Mg! 1, 7": )
- ._ _ (SM) (3”)
_ M H , 111,
1 —
— allowed (95% CL)
0 _ 1 1 14l11m11l an“ 1 1 . L inns-U1; l 1E1 1 1 #1 m 1
0 1 2 3 4 5
M 2. (TeV)

Figure C5: Bounds on the new physics parameters and the masses of the new heavy

gauge bosons in the LR—T model.

142

 

 

 

 

LP-T
1 .0

0.8

008m

0.4

0.2

LP-T

MW. (TeV)

Figure (3.6: Bounds 011 the new physics parameters and the masses of the new heavy

 

61........4............-....................-.................1..r.z ....................
fi- '

 

(NP)

(NP) 7”.
v 1

H
(SM)
H

"'M

 

 

— M .111?”

 

...............................................

    
  

.....

 

 

 

0 0.00 < saw-(213') s 0.25
1:1 0.25 < sin2(2fi) s 0.50
A 0.50 < sin2(2§) s 0.75

0 0.75 < sin2(26) s 1.00

;.

 

 

 

 

.....................................................................................................................................

I l l l l l l l l l 1 J l

 

 

 

120

 

 

 

0 0.00 < sin2(2B') s 0.25
1: 0.25 < sin2(2B') s 0.50
A 0.50 < sin2(2i§) s 0.75
0 0.75 < sin2(2§) s 1.00

 

 

lllllllllllfllllllllllTT

.........
.....

 

    
 

  
  
 

 

__ _ _ (NP) (NP)
MH .111,

(SM) (SM)
MH ’ m!

 

 

 

...-
...........
..........
.........
.......
...................
......
.......
.........

 

 

gauge bosons in the LP—T model.

143

 

HP-T
1 .0

0.8

 

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

___ (~P) (NP)
M H .777,
_ (SM) (SM)
M H 1m:
0.5
§
m an
8 o O.00<sin2(2[3)50.25
0.4 ..
D 0.25<sin2(20)s0.50
A 0.50<sin2(26)s0.75
02 0 0.75 < sin2(25)s1.00
0.0'1l1111l1141l11114
50 100 150 200 250 300 350
)7
HP-T
5

MW. (TeV)

Figure C.7: Bounds on the new physics parameters and the masses of the new heavy

Illllflllllllllllllllll

 

 

 

0 0.00 < sin2(2i§) s 0.25
:1 0.25 < sin2(2B') s 0.50
A 0.50 < sin2(26) s 0.75
0 0.75 < sin2(2i§) s 1.00

 
 

 

 

  
   
 

 

___ (NP) (NP)
M” ,111,

 

 

 

..............
..............

 

...........
................
........

 

gauge bosons in the HP-T model.

144

 

 

FP-T

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

1'0 ............................................ . ............................. . ...............................................................
02 ~—
‘ _ _ _ (NP) (NP)
5 M H I mt
)—
7311) (SM)
0.6 —- M H 1 mt
E r—
m — ~
8 — 0 0.00 < sin2(2[3) s 0.25 m
114 — ..
_ D 0.25 < sin2(2[5) s 0.50
‘ A 0.50 < sin2(2B) s 0.75 1‘
0.2 _ 0 0.75 < sin2(2)"1)s1.00 l A
,. ..............................................................................................................................................

 

0.0 l l 4 l l l l l l l l

 

 

FP-T
5

0 0.00 < sin2(2i§) s 0.25

4 1:1 0.25 < sin2(2§) s 0.50

A 0.50 < 9.1112(2)?) s 0.75

0 0.75 < sin2(2§) s 1.00

 

 

 

a

 

MW (TeV)

 

 

 

d

 

 

 

 

N
cTllllllllllIllllllllllll

 

Figure C.8: Bounds on the new physics parameters and the masses of the new heavy
gauge bosons in the FP-T model.

UU-D
1.0

0.8

sin(q5)

0.4

0.2

NU-D

1.0

0.8

Sth)

0.4

0.2

0.0

 

..................................................................................................................................................

 

_ ___ (NP) (NP)
MH 1 m1

_ _ ”if”: ”(Sm

 

 

 

_ allowed (95% CL)

.....................................................................................................................................................

 

 

 

 

..................................................................................................................................................

 
   

 

NP P)
—"M11 3777:”

(SM) (SM)
H ' 771',

_M

 

 

 

allowed (95% CL)

.....................................................................................................................................................

 

 

 

111l111l11111111111l111lm11J_111l11_1l111l111lL1nl
520 540 560 580 m 520 640
)7

Figure C.9: Bounds on the new physics parameters in the UU-D and the NU-D model.

146

 

 

 

C.2 3 Pull Distributions

For each model the pull distribution at a specific point in the excluded part of the
parameter space is shown. The goal in choosing the values of the parameters was to
find parameter configurations in which the respective features of the pull distributions
are clearly visible. The colors of the bars in Fig. C.10 to C. 14 indicate the signs of the

pulls: Blue bars stands for positive pulls; negative pulls correspond to orange bars.

147

I034) 02 (64) 02

  
  
  
  
  
 
 

  
    
  
   
    
  
  
 
 
 
 
 
 
 
 
 
 
  

-(63)C1 (63) Cl
(62) Qw(205 T1) (62) Qw(205Tl)
(61) ch33Cs) (61) ch—“Co

(53) @546)
7) 8,48
(56) 8v
(55) RV
(54) R1)
(53) RV
(52) Ry
(51) RV
omenz
)(gv )
(481111152
47) r,
(42) m1
(30) FW
(29) MW
(28) FW
(27) MW
(26) ALR(S)
0. ‘7 (25) Ame)
E . E1 (24) ALR(T)
(23) A1301) (23) A1110!)
(22) A112(6) (22) A111(6)
(21) A1.R(¢) (21) 4111(6)
(20) A1,R(C) (20) A111(0)
(19) A1120?) (19) A1.1103)
(18) A7110?) (18) A713(0)
(l7) A1:30)) (17) A130,)
(16) R(C) (16) R(C)
(15) R(b) (15) R(b)
(l4) 8(8) (14) NS)
(13) A1.R(S) (13) A111(8)
(12) Ql-‘B (12) Q13
(ll) (4111(6) (11) A111(6)
(10) A114(7) (10) A1110')
(9) Am(T) (9) (4158(7)
(3) AFB(#) (8) 1411101)
(7) (4113(6) (7) A713(6)
(6) R(1') (6) R(T)
(5) R(#) (5) R(fl)
(4) R(e) (4) R(e)
(3)9110. (3) (17:11:10
ammfiukuamg, 1() z 5.,m.-v..m...N-_,.. () 2

Figure C.10: Pull distributions in the LR models. Parameter values: 5: : 300,

~- 2 — - '~= ~_—_ 2~= -
c¢_0.90,325_0.00(LRD),x 70,9, 0.90,s2fi 0.00(LRT).

148

 

(63) Cl
(62) Qw(205Tl)
(61) Qw('—‘;Cs)

— g' )
(48) elf”?
47) 1',
(42) ”11
(30) FW
(29) MW
(28) FW
(27) M w
G (26) A112(3)
1 (25) A112(6)
E (24) A1.1{(T)
(23) A1110!)
(22) A111(6)
(21) AI.R(e)

  
   
  
 
 
    
  
 
  
  
   
   
  
  
  
  

   
   
  
  
   
 
  
    
  
  
  
   
  
 

(25) A1110:)
(24) (4111(7)
(23) ALRUI)
(22) ALR(6)
(21 ) [1112(6)
(20) ALR(C)

LP—T

(19) (4112(1)) (19) A130’)
(18) A11303) (18) A1713(0)
(17)/11430)) (17)AF13(b)
(16) 11(0) (16) R(C)
(15)R(b) (15) R03)
(14) 72(5) (14) R(S)
(13) ALR(S) (13) (4111(5)
(12) PB 2) QFB
(1 1) (4111(6) (11) A1120?)
(10) ALR(T) (10) ALR(T)
(9) AFB(T) (9) AFB(T)
(8) 471101) (8) 14171301)
(7) (4113(6) (7) (4113(6)
(6) R(T) (6) R(T)
(5) R(IJ) (5) R(ll)
(4) R(e) (4) R(e)
(3) 0'had. (3) (Thad.

g -. ... .(2) F2 2; - (2) F7.

0.3; "N“ . . . . ‘ (:1. '

Figure C.11: Pull distributions in the LP models. Parameter values: 5: = 300,

-_ 2 _ _ .~=-- -2 2-: -
c¢—0.71,s25~H0.OO(LPD),x ()0,c¢ 071,323 0.00(LPT).

149

 

W64) C:
EI<63) ,
l(62) Qw(‘05Tl)
11161) Qw(mCS)
_lgis) Qwe)

g1):
|(56)gv
E_ 11(55) R—
fl.(54) RV
fl(53) R7
M52) RV
27:. S 1(51)Rv
1.: .(50) K‘ ”N
..-... (49) (8“ ‘
5 (48) (.gf’v)2
Ell (47) Tr
‘1. 1(42)m1
K30) rw
=Fl(29) MW
. I(23) rW
E197) MW
1 (26) ALR(5)
I(25) A114(6)
f: (24) A1 R(T)
' (23) A1.1101)
"(22) ALR(e)
.(21) A111(6)
!(20) (4111(0)
f (19) ALR(b)
IV j(18) (4113(0)
l(16) R(C)
“(15) R(b)
I04) 8(8)
I13) Al R(S)
12 QFB
WJU l) (4111(6)
1.7 (10) ALR(T)
I_.I(9) (4113(7)
(“4(8) AFB(IJ)
,, (7) AFB(€)
[‘ i.-. (6) R(T)
1..&,_1(5) R01)
l(4) R(e)

HP-D
HP—T

l'

 

l

1

F

E 1
Pull

l"1(64) C2
El(63) 01
N62) Qw(2°5T1)
0(611ch3‘Cs)
_(58) Qw(e)

H197) MW
1; (26) ALR(S)
r !(25)ALR(€)
:6" (24) A1R(T)
l (23) ALRUI)
.(22) A111(6)
I(21) A111(6)
K20) ALR(C)
I“ 19) ALR(b)
r l(18) AFB(C)
1;..- . 1 .l(l7) AFB(b)
1(16) R(C)
“(15) R(b)
'(14) R(S)
1(13) A112(5)
E 1(12) QFB
...—J“ 1) (4112(6)
l 1(10) A111(7)
[3(9) (ll-8(7)
"1(8) AFBUI)
Ml? l(7) (4115(6)
E: 71(6) R(T)
1. 71(5) R(fl)
l l(4) R(e)
s. (3) 0'had

.__.._i(2)l"z
\5 In <1 6»; ‘N

 

Figure C.12: Pull distributions in the HP models. Parameter values: i' = 300,

04>

— ( 2 : — ' ~ : ~ : 2.. : -
_ 0.30, 323 0.00 (HP D), .r 70. 6,15 0.90, 623 0.00 (HP T).

 

 

FP—D

Full

Figure 0.13: Pull distributions in the FP models. Parameter values: ~
0.20, 333 = 0.00 (FP-D); 5c : 40, c

665:

(5. ..

 

  
   

_-

    

i164> 02

(64) C2
(63) C1 (63) Cl
(62) Qw(205T1) (62) Qw(205T1)
161) Qw1‘33Cs) 161) gw(l33CS)
(58) Qw(€) yW35)
(57) 85" 866
(56) gv" (56) gv‘
(55) R7 (55) R7
(54) R7 (54) R17
(53) RV (53) RV
(52) RV (52) RV
(5l)RV (51)RV
21112
( ) g" ) gV
148) 1111””)2 (48) 11:1”?
47) T1- 47) 1',
(42) m, (42) m,
(30) FW (30) FW
(29) MW (29) MW
(28) rw (28) FW
(27) MW (27) MW
(26) ALR(S) (26) 1413(5)
(25) 1413(6) [T (25) ALR(6)
(24) ALRU’) E: (24) ALR(T)
(23) ALRUI) (23) ALR(/~l)
(22) ALR(6) (22) ALR(6)
(21) ALR(C) (21) AIME)
(20) ALR(C) (20) ALR(C)
(19) A1-R(b) (19) ALR(b)
(13) 1413(0) (18) AFB(C)
(17) A1130?) (17) A1111b)
(16) R(c) (16) R(c)
(15) R(b) (15) R(b)
(14) KS) (14) NS)
(13) 1413(5) (13) ALR(S)
12) QFB QFB
(11)/4111(6) (11)/4111(6)
(10) ALR(T) (10) A1.R(T)
(9) AFB(T) (9) [41-13(7)
(8) 14171301) (8) A1301)
(7) AFB(6) (7) AFB(e)
(6) R(T) (6) R(T)
(5) R01) (5) R(fl)
(4) R(e) (4) R(e)
(3) 0’had (3) 0’had
(2) r2 = 1 . I I‘z
5 tr «1 (\l —
x = 1 0,

- _ 2 _ 1
¢ _ 0.20, 323 _ 0.00 (FP T).

151

 

EH(64) Cz

  
 
 
 

  
  
   
  
  
  
   
   
 
  
 
 
 
   
 

    
  
   
  
   
    
 
  
  
   
  
  
  
   

 

(64) 02
(63; 81 (20511) E23; 5‘ (205“)
w W
83 8”???” ‘E28 WELT“)
w e w
57) 8:3: (5 )83:
- 8v 8v
$63137
l(54) RT 7
-(53) RT
“52) RV
_188 RVN ‘50) ~
KV KV
m-Eigg W; E133 W3:
-(47) 5" MEL
"(42) mt
(30) I“w
(29) MW
8333;? (27) M
w W
D ' (26) ALR(S) Q (26) ALR(S)
n '05) ALR(e) I (25) ALR(e)
B (24) ALR(T) E (24) ALR(T)
(23) ALRUI) (23) ALRUI)
(22) 1413(6) (22) ALR(e)
(21) ALR(e) (21) ALR(e)
(20) ALR(C) (20) ALR(C)
(19) ALR(b) (19) ALR(b)
E§‘(13)AFB(C) (18) AFB(C)
(17)AFB(b) (17)AI-'B(b)_
(16) R(C) (16) R(C)
(15) am)
(14) 5)
£9183; 3111(5) E13; finds)
'3," FB FB
.(11) ALR(e) (11) ALR(e)
E(10)ALR(T) (10) ALR(T)
—(9) Ans(T) (9) AFB(T)
__(3) AFB(#) (8) AFBUI)
Iii-“7) AFB(€) (7) AFB(e)
E2358 8%?
~ [1 x1
:_ (4)R(e) =(4)R(e)
_ (3) 0' had. (3) 0' had.
5, .. .0 ,4” r2 ”3‘, M.“ .W‘. -, (2)1}
n. W. O. W. O. W. C. m: an In. 0 V! G "’2
m m N N — ... O (\l N -« -—4 o

   

Figure C.14: Pull distributions in the UU-D and NU-D models. Parameter values:
:7: = 400, 355 = 0.90 (UU-D); 5: = 500, Ci? 2 0.71 (NU—D).

 

Appendix D

Coupling Coefficients

D.1 Couplings in 022w,

D.1.1 Fermion Couplings to the Z Boson
Left-Handed Couplings fig (f)

The left—handed coupling fig (f) of a fermion f to the Z boson at the electroweak scale

can be written as the sum of the SM coupling 913%“ f) and various model—dependent

new physics corrections that are proportional to 1

Earl

4.

$3le

~Z _ Z l 12 . 3 12. r
9L(f)—9L,31\.1(f)+§: Nl+i823 N2+558q3 03+

Tabs. D.1 to D3 list the numerical values of the coefficients N1, N2, N3 and N4

in Eq. (D.1) for all fermions f and for all considered C(221) models.

Right-Handed Couplings gffi f)

The right-handed coupling 5112“ f) of a fermion f to the Z boson at the electroweak

scale can be written as the sum of the SM coupling gg SM( f) and various model-

153

 

 

 

f 938M 51:: %8§§ %8: .1158: Eqn

’11. 0.344 0.224 -0.224 41.015 0.391 (D2)
d -0.422 ~0.112 0.112 0.0574 0.0546 (13.3)
I/ 0.500 0 0 0.500 -0.500 (13.4)
6 41.200 41.330 0.336 1.17 -0.836 (13.5)

 

 

 

 

 

 

 

 

 

 

 

 

Table D1: Left-handed couplings 9% (f) of the fermions to the Z boson in the LR-
D, LP-D, HP-D and FP-D model. See text and Eq. (D.1) on the previous page for

details.

 

 

f 915,811 i? i 33 1% i323 Eqn

11. 0.344 0.0500 -0112 -0154 0.0977 (D6)
(1 -0.422 -0.0280 0.0500 0.0144 0.0137 (D.7)
z/ 0.500 0 0 0.125 -0.125 (D8)
6 -0.266 -0.0840 0.168 0.293 -0.209 (D.9)

 

 

 

 

 

 

 

 

 

 

 

 

 

Table D2: Left—handed couplings 57% (f) of the fermions to the Z boson in the LR-
T, LP-T, HP-T and FP-T model. See text and Eq. (D.1) on the previous page for

details.

 

 

 

f 915,310 gimme) %SE(NU—D) 3153:.) Eqn.
u 0.344 0.500 0.500 -0 276 (D.10)
d 0422 -0500 -0500 0388 (D.11)
y 0500 0 0500 0500 (D.12)
e 0266 0 -0500 0164 (D.13)

 

 

 

 

 

 

 

 

 

 

 

Table D.3: Left-handed couplings fig (f) of the fermions to the Z boson in the UU-
D and NU-D model. The SM values and the coffecients of the 313—5: term are the

same in both models. The results for the NU-D model apply to the first two fermion
generations. See text. and Eq. (D.1) on the previous page for details.

154

 

 

f 91231511 3i? i333 is: 3i?- :9 mm.

11 41.150 0.724 -0.224 —1.01 0.891 (D15)
(1 0.0779 -0.012 0.112 1.00 -0.445 (D.10)
u 0 0.500 0 0.500 0 (D17)
6 0.234 -0.830 0.330 2.17 -1.34 (D.18)

 

 

 

 

 

 

 

 

 

 

 

 

Table D4: Right-handed cmrplings fig”) of the fermions to the Z boson in the LR-D
model. See. text and Eq. (D.14) on the current page for details.

 

 

 

 

 

 

 

 

 

 

f 912?.sm % 1533 $5: 3155: 5‘1“"

u -(,).156 0.724 41.224 -1.01 0.2591 (D.19)
d 0.0770 -0012 0.112 1.00 044.5 (D20)
1/ 0 0 0 0 0 (D21)
6 0.234 -0.330 0.330 1.07 -1.34 (D22)

 

 

 

Table D5: Right-handed couplings QIZA f) of the fermions to the Z boson in the LP-D
model. See, text and Eq. (D.14) on the current. page for details.

dependent. new physics corrections that. are proportional to

hull—a

Tabs. D4 to D.12 list. the numerical values of the coefficients N 1, N2, N3 and N4

in Eq. (D.14) for all fermions f and for all considered C (221) models.

 

 

 

 

f 9121.83 I % %333 %32 315—5; Eqn

21 -0.156 0.224 -0.224 -1.11 0.891 (D23)
d 0.0779 -0.112 0.112 0.557 -0.445 (D24)
1/ 0 0.500 0 —0.500 0 (D25)
6 0.234 -0.836 0.336 2.17 -1.34 (D26)

 

 

 

 

 

 

 

 

 

 

 

Table D6: Right-handed couplings 612?“) of the fermions to the Z boson in the HP—D

model. See text and Eq. (D.14) on the previous page for details.

 

 

 

 

 

 

 

 

 

 

 

f 91%,SM Ei- %833 %s; is: Eqn.

u -0.156 0.224 -0.224 -1.11 0.891 (D27)
d 0.0779 -0.112 0.112 0.557 -0.445 (D28)
I/ 0 0 0 0 0 (D29)
6 0.234 -0.336 0.336 1.67 -1.34 (D.30)

 

 

 

Table D7: Right-handed couplings QIZ“ f ) of the fermions to the Z boson in the FP-D

model. See text and Eq. (D.14) on the previous page for details.

 

 

f 912?,SM :}: %833 313-3: %s:~; Eqn.

21. -0.156 0.181 -0.112 -0.404 0.223 (D31)
(1 0.0779 -0.153 0.0560 0.264 -0.111 (D32)
11 0 0.125 0 -0.125 0 (D33)
6 0.234 -0.209 0.168 0.543 -0.334 (D34)

 

 

 

 

 

 

 

 

 

 

 

Table D8: Right-handed couplings g}? f ) of the fermions to the Z boson in the LR—T

model. See text and Eq. (D.14) on the previous page for details.

 

 

f 95,8M % 3533 Eli-3:; 35:; 13‘1”
u. -O.156 0.181 -0.12 -0404 0.223 (n35)
d 0.0779 —0.153 0 0560 0.264 -0.111 (D36)
1/ 0 0 0 0 0 (p.37)
6 0.234 -0.0840 0.168 0.418 -0.334 (D38)

 

 

 

 

 

 

 

 

 

 

 

Table D9: Right-handed couplings QIZ“ f) of the fermions to the Z boson in the LP—T

model. See text and Eq. (D.14) on the previous page for details.

156

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f 9122,511 3 3933 3": 33; Equ-

21. -0.156 0.0560 —0.112 -0.279 0.223 (D39)
d 0.0779 -0.0280 0.0560 0.139 -0.111 (D40)
1/ 0 0.125 0 -0.125 0 (D41)
6 0.234 -0.209 0.168 0.543 -0.334 (D42)

 

 

Table D.10: Right-handed couplings 912% f ) of the fermions to the Z boson in the
HP-T model. See text and Eq. (D.14) on page 155 for details.

 

 

 

 

 

 

 

 

 

 

 

f 93,3111 3 3333 3 5?; 38:; E9“

11 -0.156 0.0560 -0.112 -0.279 0.223 (D43)
d 0.0779 -0.0280 0.0560 0.139 -0.111 (D44)
1/ 0 0 0 0 0 (p.45)
6 0.234 00840 0.168 0.418 -0334 (D46)

 

 

 

 

Table D.11: Right-handed couplings §g( f ) of the fermions to the Z boson in the
FP-T model. See text and Eq. (D.14) on page 155 for details.

 

 

f 9122,3111 35: Equ-
11 -0.156 0224 (D47)
d 0.0779 0112 (D48)
1/ 0 0 (p.49)
8 0.234 -0.336 (050)

 

 

 

 

 

 

 

 

Table D.12: Right-handed couplings §IZZ( f) of the fermions to the Z boson in the UU-
D and N U-D model. The results for the NU-D model apply to the first two fermion
generations. See text and Eq. (D.14) on page 155 for details.

157

D.1.2 Couplings of the New Physics Currents

Couplings of the Neutral Fermion Currents

The coupling coefficients 03E (f1,z’~ fg’j) of the neutral cru'rent four-fermion interac-
tions in the effective Lagrangian at the electroweak scale are defined such that the

Lagrangian $33119 2 —%1TT§,21{2K0*“' takes the following form:

NC_ C F _ . .
$8.0.— — f2: 263901.120(150023032); , m = we
f11f2 2 j
They can be written as the sum of various model—dependent new physics correc-
tions that are proportional to 3.

1 1 1 .
CNC (f12 fg’j)= 3 Nlegsg-Ng+§s -N3 (051)

Absolute terms that are independent of it do not appear in 0,;wa (f1,z‘= f2,j) since
the four-fermion interactions in few. represent a pure effect of the new physics in the
G (221) models. Tabs. D.13 to D22 list the numerical values of the coefficients N1, N2
and N3 in Eq. (D51) for all possible fermion pairs (fm', ng) and for all considered
C(221) models.

158

Couplings of the Charged Fermion Currents

The coupling coefficients Cg}; (fu. f3,j) of the charged current four-fermion inter-
actions in the effective Lagrangian at the electroweak scale are defined such that the

Lagrangian $62? = —fff‘;2,Kj KW“ takes the following form:

G — — _ . .
at? 2 ‘73 Z chS(fl,..f3,.-)(fl/33.03114); .m -—= we

H f1~f3 iaj

They can be written as the sum of various model-dependent. new physics correc-

tions that are proportional to 31.3.

055013.333): ~N1+ s .N2+ sg-Ng (0.412)

H’Il—I‘
Hill-d

all H
9453

1

Absolute terms that are independent of :7: do not appear in Cecil? (fm, f3,j) since
the four-fermion interactions in few, represent a pure effect of the new physics in
the G(221) models. Tabs. D23 to D26 list the numerical values of the coefficients
N1, N2 and N3 in Eq. (D412) for all possible fermion pairs (fLi, f3,J-) and for all
considered G (221) models. The couplings in the models of the first breaking pattern
have a very simple form which allows us to combine our results for the (BP-I,D) and

the (BF—LT) models in one table each.

159

 

 

 

 

 

 

 

 

 

 

(f1f1)L,R (f2f2)L,R 21; 38:3 5895 Eqn.
(flu) L (flu) L 0 0 0.0556 (D52)
(an) L (uu) R 0 -0.167 0.222 (D53)
(flu) L (it‘d) L 0 0 0.0556 (D.54)
(flu) L ( 11) R 0 0.167 -0111 (D55)
(flu) L (w) L 0 0 -0.167 (D56)
(an) L (w) R 0 -0.167 0 (D57)
(flu) L (66) L 0 0 -0.167 (D58)
(flu)L (ée)R 0 0.167 -0.333 (D59)
("u R (2117)]L2 0.500 —1.33 0.889 (D60)
(‘73 R (’d) L 0 -().167 0.222 (D61)
('u) R (_d) R 0500 1.00 -0.444 (D62)
('21) R (w) L 0 0.500 -0.667 (D63)
(m) R (w) R 0.500 -0.667 0 (D64)
(m) R (66) L 0 0.500 -0.667 (D65)
(’u) R (ée) R 0500 1.67 —1.33 (D66)
(‘3) L (6) L 0 0 0.0556 (D67)
(11) L (‘71) R 0 0.167 -0111 (D68)

dd) L (w) L 0 0 -0.167 (D69)

’d) L (w) R 0 -0.167 0 (D70)
(dd) L (6'6) L 0 0 -O.167 (D71)
( ’d) L (66) R 0 0.167 -0333 (D72)
(11) R (‘ ) R 0.500 -0.667 0.222 (D73)
(11) R (w) L 0 -0500 0.333 (D74)
('3) R (w) R -0500 0.333 0 (D75)
(11) R (ée) L 0 -0500 0.333 (D76)
( ’d) R (ée) R 0.500 -1.33 0.667 (D77)
(5) L (171/) L 0 0 0.500 (D78)
(‘11)L (DI/)R 0 0.500 0 (D79)
(—V)L (e'e)L 0 0 0.500 (D80)
(17!!) L (ée) R 0 0500 1.00 (D81)
(171/ R (271/) R 0.500 0 0 (D82)
(w R (ée) L 0 0.500 0 (D83)
(7») R (66) R 0500 1.00 0 (D84)
(56) L (ée) L 0 0 0.500 (D85)
(66) L (66) R 0 -0500 1.00 (D86)
(66) R (ée) R 0.500 -200 2.00 (D87)

 

 

 

 

Table D.13: Couplings 031w? (f1,i9 fgaj) of the neutral fermion currents at the elec—
troweak scale in the LR—D model. See text and Eq. (D51) on page 158 for details.

160

 

 

 

 

 

 

 

 

 

 

(f1f1)L,R (f2f2)L,R 315 33243 3:1:st Eqn.
(m) L (5.11) L 0 0 0.0556 (D88)
(”t—1,11,; L (511) R 0 -0.167 0.222 (D89)
(flu L (Jd) L 0 0 0.0556 (D90)
(au L ('d) R 0 0.167 0111 (D91)
(flu) L (w) L 0 0 -O.167 (D92)
(621 L (w) R 0 0 0 (D93)
(‘21) L (ée) L 0 0 -0.167 (D94)
(‘21)L (ée)R 0 0 -0.333 (D95)

‘11) R (flu R 0.500 -133 0.889 (D96)
(‘11) R (Juli L 0 -0.167 0.222 (D97)
(‘11) R ( ’d) R -0500 1.00 -0444 (D98)
(‘21) R (w) L 0 0.500 -0.667 (D99)
(11),? (w) R 0 0 0 (D100)
’11 R (66) L 0 0.500 -0.667 (D101)
(‘5) R (66) R 0 1.00 -133 (D102)
( _d) L (“(1) L 0 0 0.0556 (D103)
(it’d) L ('d) R 0 0.167 -0111 (D104)
(cfd) L (171/) L 0 0 -0.167 (D.105)
( 'd) L (171/) R 0 0 0 (D106)
(Jd) L (ée) L 0 0 -0.167 (D107)
(21) L (ée) R 0 0 -0333 (D108)
(Ed) R (11) R 0.500 -0.667 0.222 (D109)
(‘51) R (171/) L 0 -0500 0.333 (D110)
( 11) R (171/) R 0 0 0 (D.111)
(’ ) R (66) L 0 —0.500 0.333 (D112)
(‘3) R (ée) R 0 —1.00 0.667 (D113)
(w) L (171/) L 0 0 0.500 (D114)
(‘11 L (_V)R 0 0 0 (D115)
(7») L (66) L 0 0 0.500 (D116)
(w) L (ée) R 0 0 1.00 (D117)
Ev) R (171/)R 0 0 0 (D118)
w R (36) L 0 0 0 (D119)
(3») R (66) R 0 0 0 (D120)
(ée) L (66) L 0 0 0.500 (D121)
(66) L (66) R 0 0 1.00 (D122)
(ée) R (68) R 0 0 2.00 (D123)

 

 

 

Table D.14: Couplings 0ng (f1,i2 f2,j) of the neutral fermion currents at the elec-
troweak scale in the LP-D model. See text and Eq. (D51) on page 158 for details.

161

 

 

 

 

(f1f1)L R 2f2)L,R 3 38: 33:3 Eqn.

(an) L (m) L 0 0 0.0556 (D124)
(an) L (‘71) R 0 0 0.222 (D125)
(1:) L ("71) L 0 0 0.0556 (D126)
(‘21) L (‘71) R 0 0 6111 (D127)
(‘21) L (171/) L 0 0 -0.167 (D128)
(‘21) L (171/) R 0 -0.167 0 (D129)
(‘11) L (ée) L 0 0 -0.167 (D130)
(‘11) L (36) R 0 0.167 -0333 (D131)
‘21) R (m) R 0 0 0.889 (D132)
(m) R ('d) L 0 0 0.222 (D133)
(‘11) R ('d) R 0 0 -0444 (D134)
('11) R (w) L 0 0 -0.667 (D135)
(‘13; R (w) R 0 -0.667 0 (D136)
(m R (ée) L 0 0 -0.667 (D137)
(‘21) R (66) R 0 0.667 —1.33 “(D.138)
(dd) L (‘d) L 0 0 0.0556 (D139)
( ’d) L (’d) R 0 0 0111 (D140)
(it'd) L (w) L 0 0 -0.167 (D141)
_d) L (w) R 0 -0.167 0 (D142)
(Jd) L (56) L 0 0 -0.167 (D143)
(11) L (66) R 0 0.167 0333 (D144)
(11) R (’d) R 0 0 0.222 (D145)
('3) R (w) L 0 0 0.333 (D146)
('d) R (w) R 0 0.333 0 (D147)
(11) R (66) L 0 0 0.333 (D148)
( ’d) R (66) R 0 0333 0.667 (D149)
(w L (171/) L 0 0 0.500 (D150)
(w; L (171/) R 0 0.500 0 (D151)
(w) L (66) L 0 0 0.500 (D152)
(w) L (66) R 0 —0.500 1.00 (D153)
E—VgR (171/)R 0.500 0 O (D.154)
w R (66) L 0 0.500 0 (D155)
(w) R (56) R -0500 1.00 0 (D156)
((36) L (56) L 0 0 0.500 (D157)
(66) L (6’6) R 0 -0500 1.00 (D158)
(36) R (66) R 0.500 -200 2.00 (D159)

 

 

 

 

 

 

 

 

 

Table D.15: Couplings 031,? ”13‘: ng) of the neutral fermion currents at the elec-
troweak scale in the HP-D model. See text and Eq. (D51) on page 158 for details.

162

 

 

 

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.u _u .u .u _u _u _u .u .W _W _W _W _W _w _W_d.d _d _d _d .d 71% .uw _um _HMMW _V _V .V _V _W _W _W .206 _e _%

((((((((((((((((((((((((((((((((((((

 

 

 

Table D.16: Couplings 033E: (fu, f2,j) of the neutral fermion currents at the elec-

troweak scale in the FP—D model. See text and Eq. (D51) on page 158 for details.

163

 

 

 

 

(f1f1)0 (f2f2)C % 3153:; 1:382; 13911-
(021) L (flu) L 0 0 0.0139 (D196)
(an) L (m) R 0 -00417 0.0556 (D.197)
(flu) L (aid) L 0 0 0.0139 (D.198)
(m; L (21) R 0 0.0417 .0.0278 (D199)
(’0 L w) L 0 0 —0.0417 (D200)
(00) L (w) R 0 -00417 0 (D201)
(60) L (66) L 0 0 -0.0417 (D202)
(flu) L (66) R 0 0.0417 -0.0833 (D203)
(‘0) R (flu) R 0.125 0333 0.222 (D204)
('11 R (Jd)L 0 -00417 0.0556 (D205)
("0) R (11) R -0125 0.250 -0111 (D206)
(‘0) R (171/) L 0 0.125 -0.167 (D207)
(‘0; R (w) R 0125 -O.167 0 (D208)
71 R (66) L 0 0.125 -0.167 (D209)
(’11) R (66) R -0125 0.417 0333 (D210)
(Jd) L (Jd) L 0 0 0.0139 (D211)
11) L (21) R 0 0.0417 -0.0278 (D212)
Jd) L (171/) L 0 0 -0.0417 (D213)
‘71) L (171/) R 0 00417 0 (D214)
Edd) L (66) L 0 0 00417 (D215)
.d) L ('ée) R 0 0.0417 -0.0833 (D216)
(Jd R E’d) R 0.125 -0.167 0.0556 (D217)
(‘d R w) L 0 -0.125 0.0833 (D218)
(Ed) R (w) R -0125 0.0833 0 (D219)
(11; R (66) L 0 -0125 0.0833 (D220)
(“d R (66) R 0.125 -0.333 0.167 (D221)
W31: (w) L 0 0 0.125 (D222)
171/ L (‘12) R 0 0.125 0 (D223)
171/)L (ée) L 0 0 0.125 (D224)
(w) L (ée) R 0 -0125 0.250 (D225)
(w; R (171/) R 0.125 0 0 (D226)
‘1/ R (66) L 0 0.125 0 (D227)
(w) R (66) R 0125 0.250 0 (D228)
(66) L (ée) L 0 0 0.125 (D229)
(ée) L (66) R 0 -0.125 0.250 (D230)
(66) R (’e) R 0.125 -0500 0.500 (D231)

 

 

 

 

 

 

 

 

 

 

Table D.17: Couplings (3'wa (flR', f2,j) of the neutral fermion currents at the elec-
troweak scale in the LR—T model. See text and Eq. (D51) on page 158 for details.

164

 

 

 

 

 

 

 

 

 

 

 

 

 

(f1f1)C (f2f2)C % fig 5334; E911-
(flu)L (fiu)L 0 0 0.0139 (D232)
(flu) L (flu) R 0 00417 0.0556 (D233)
(flu) L (aid) L 0 0 0.0139 (D234)
('11) L ('d) R 0 0.0417 -0.0278 (D235)
(‘21) L (171/) L 0 0 -00417 (D236)
(‘21) L (w) R 0 0 0 (D237)
(‘21) L (ée) L 0 0 -00417 (D238)
(*0) L (66) R 0 0 -0.0833 (D239)
(‘11) R (flu) R 0.125 -0333 0.222 (D240)
‘11 R (cid)L 0 -00417 0.0556 (D241)
(‘11) R (’d) R 0125 0.250 0111 (D242)
(*0 R (w) L 0 0.125 -0.167 (D243)
(2.) R (‘11)R 0 0 0 (D244)
("11) R (ée) L 0 0.125 -0.167 (D245)
(‘11) R (66) R 0 0.250 0333 (D246)
(Jd) L (‘6) L 0 0 0.0139 (D247)
(11) L (’d) R 0 0.0417 .0.0278 (D248)
(6d) L (w) L 0 0 -00417 (D249)
(’d) L (w) R 0 0 0 (D250)
(Jd) L (ée) L 0 0 00417 (D251)
(11) L (68) R 0 0 -0.0833 (D252)
(‘8) R (' ) R 0.125 -0.167 0.0556 (D253)
:d R (w) L 0 —0.125 0.0833 (D254)
(_d) R (w) R 0 0 0 (D255)
( d) R (66) L 0 -0125 0.0833 (D256)
(11) R (66) R 0 -0250 0.167 (D257)
(’V)L (171/)L 0 0 0.125 (D258)
(171/ L (’1/ R 0 0 0 (D259)
E1701. (66) L 0 0 0.125 (D260)
171/)L (ée) R 0 0 0.250 (D261)
(71)}? (171/)R 0 0 0 (D262)
'12 R (ée) L 0 0 0 (D263)
(w) R (66) R 0 0 0 (D264)
(66) L (ée) L 0 0 0.125 (D265)
(ée) L (66) R 0 0 0.250 (D266)
(66) R (66) R 0 0 0.500 (D267)

 

Table D.18: Couplings Cg”? (fLi, f2”) of the neutral fermion currents at the elec-
troweak scale in the LP-T model. See text and Eq. (D51) on page 158 for details.

 

 

 

 

 

(f1f1)C, (f2f2)C % is: 3318:; 13911-
(71u)L (flu) L 0 0 0.0139 (D268)
(210.)L (210) R 0 0 0.0556 (D269)
(2171.) L (Jd) L 0 0 0.0139 (D270)
('21) L (dd) R 0 0 —0.0278 (D271)
(*0) L w) L 0 0 -00417 (D272)
(‘0) L (71/) R 0 -00417 0 (D273)
(’11) L (66) L 0 0 -00417 (D274)
("0) L (66) R 0 0.0417 -0.0833 (D275)
(‘11 R (flu) R 0 0 0.222 (D276)
(71 R (“(1) L 0 0 0.0556 (D277)
(’11 R (’d) R 0 0 0111 (D278)
(‘0) R (w) L 0 0 -0.167 (D279)
(713R (‘12) R 0 -0.167 0 (D280)
('0 R (66) L 0 0 -0.167 (D281)
(-u) R (ée) R 0 0.167 0333 (D282)
(aid) L (11) L 0 0 0.0139 (D283)
(11) L (11) R 0 0 -0.0278 (D284)
(”d) L (w) L 0 0 -00417 (D285)
(’d L (w) R 0 -00417 0 (D286)
( ‘d) L (66) L 0 0 -0.0417 (D287)
(Jd) L (66) R 0 0.0417 -0.0833 (D288)
(11) R (11) R 0 0 0.0556 (D289)
("d R 171/) L 0 0 0.0833 (D290)
(11) R (171/) R 0 0.0833 0 (D291)
( ’d) R (ée) L 0 0 0.0833 (D292)
( ’d) R (ée) R 0 -0.0833 0.167 (D293)
(‘VgL (171/)L 0 0 0.125 (D294)
71 L (171/) R 0 0.125 0 (D295)
(w) L (ée) L 0 0 0.125 (D296)
(7») L (ée) R 0 —0.125 0.250 (D297)
(w) R (171/) R 0.125 0 0 (D298)
(w R (66) L 0 0.125 0 (D299)
(w) R (ée) R -0125 0.250 0 (D300)
(66) L (66) L 0 0 0.125 (D301)
(66) L (ée) R 0 -0125 0.250 (D302)
(66) R (6'6) R 0.125 -0.500 0.500 (D303)

 

 

 

 

 

 

 

 

 

 

 

Table D.19: Couplings 0wa (flak fggj) of the neutral fermion currents at the elec-
troweak scale in the HP-T model. See text and Eq. (D.51) on page 158 for details.

166

 

(f1f1)C (mac .1. 51.33; 1s; Eqn
(flu) L (au) L 0 0 0.0139 (D304)
(au) L (210) R 0 0 0.0556 (D305)
(au) L (dd) L 0 0 0.0139 (D306)
(flu) L (dd) R 0 0 00278 (D307)
(flu) L (131/)L 0 0 00417 (D 308)
(au) L (171/) R 0 0 0 (D309)
(flu) L (ée) L 0 0 —0.0417 (D310)
(1711)]: (68) R 0 0 -().0833 (D311)
(flu) R (m) R 0 0 0.222 (D312)
(60) R (",d) L 0 0 0.0556 (D313)
(m) R (dd) R 0 0 0111 (D314)
(‘11) R (171/) L 0 0 -0.167 (D315)
(‘21.)3 (“I/)R 0 0 0 (D316)
(flu) R (ée) L 0 0 -0.167 (D317)
(flu) R (ée) R 0 0 0333 (D318)
(dd) L (dd) L 0 0 0.0139 (D319)
(dd) L ( “,d) R 0 0 0.0278 (D320)
( 'd) L (171/) L 0 0 —0.0417 (D321)
(dd) L (w) R 0 0 0 (D322)
(11) L (ée) L 0 0 —0.0417 (D323)
(dd) L (ée) R 0 0 00833 (D324)
( ‘d) R ( ‘d) R 0 0 0.0556 (D325)
(d) R (w) L 0 0 0.0833 (D326)
( d) R (71/) R 0 0 0 (D327)
(‘d) R (Ee) L 0 0 0.0833 (D328)
( ‘d) R (66) R 0 0 0.167 (D329)
(91/) L (171/) L 0 0 0.125 (D330)
(171/)L (171/)R 0 0 0 (D331)
(171/) L (ée) L 0 0 0.125 (D332)
(171/)L (66) R 0 0 0.250 (D333)
(7’)}? (171/)R 0 0 0 (D334)
(w) R (66) L 0 0 0 (D335)
('u) R ((56) R 0 0 0 (D336)
(ée) L (ée) L 0 0 0.125 (D337)
(ée) L (66) R 0 0 0.250 (D338)
(66) R (66) R 0 0 0.500 (D339)

 

 

 

 

 

 

 

 

Table D20: Couplings Cg”? (f1,iv f2,j) of the neutral fermion currents at the elec-
troweak scale in the FP-T model. See text and Eq. (D.51) on page 158 for details.

167

 

 

 

)))))))))))))))))))))))))))))
01234567890123456789012345678
44444444445555555555666666666
33333333333333333333333333333
DDDDDDDDDDDDDDDDDDDDDDDDDDDDD
(((((((((((((((((((((((((((((

 

 

 

1..
_

1.2
DIG—01.20000000

1.2
0120—000000000

0000000

 

1.2017000000000000120000000000000

0000000

 

 

 

LRLRLRLRRLRLRLRLRLRLRRLRLRLRLRRLRLRR

)) )) )\I/ \I./)))))))) )\u./ )\I/ ))))))) )) ))))) )))

u
.u

V V 6
.W .....w.....w .V .W _% .% WNW.“ .V .W .6

V
.w .uw .M .V .W

eeduueevue
.6 .6 ...a .V .V .6 .6 .V .V

.6 .6 .V

6V66666
6.6.6.6.6

(( (( (( ((((((((( (( (( ((((((( (( ((((( (((

 

 

LLLLLLLLRRRRRRRLLLLLLRRRRRLLLLRRRLLR

))))))))))))))))))))))))))))))))))))

U

.u

an

.. uuuuuuuuuuu
_U_u.u

U

.u .u .u .u .u .u .u .u .u .u .u .d...a ...a

ddddddd V V V

dd d.uw_d_d_d_d.d_d_d .V .V _V

VVUV666
.V .V .V .V .6 .6 .6

 

 

 

 

Table D21: Couplings 01“.“? ( f1 ’2', f2”) of the neutral fermion currents at the elec-

troweak scale in the UU—D model. See text. and Eq. (D.51) on page 158 for details.

168

Eqn.

47¢
S
1_.$

91¢
S

1.2

(f—1f1)L,R (f2f2)L,R

 

 

 

 

))))))))))))))))))))))))))))))))))))
6789012345678901234.1067009012345678901
7777888888888899999999990000000000111
33.3333333333333333333333444444444444
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
/|\((((/I\((((((((((((((((((((((((((((((
1.04 1,2 1.2 1.2

1.20 _01340 _00000000120 _012000000120 _00001200
1 1 1|. 1 1 1i

 

1.9“ 1.9. 1.9. 1?

 

 

 

\UMLMRLWMLRMMRLWMLRLRMRLRRLRLRLRLRRLRLRR
) )) ) )))) )\’.I\II\I.I\.I.I )1 l

uuddVV66uddVV66ddVV66dVV66VV6.6,V6.6,.6..6<6,
.u .u _d ...a _V .U .6 .6 _u .d _d .V .V .6 .6 ...a ...a _V .V .6 .6 .d _V .V .6 .6 .V .V .6 .6 .V .6 .6 .6 .6 .6

 

\}L\M\I.HD\}L\)L\M\)L\IMRRRRRRRLLLLLLRRRRRLLLLRRRLLR
. - - ))))))) )) )) ))))))) \ l\ I \ /\ /))) )))
uu UU uu UU dd V V VV

.u .u .u .u .u .u .u .u .W .W _W .W .W .W .W _M.M .....m_d .d.:1w.M.M_nw.M.nw .V .V .V _V .W .W .W .% .w .w

(( (( (( ((((((((( (( (( ((((((( (( ((((( (((

 

 

 

 

 

erations) at the electroweak scale in the NU-D model. See text and Eq. (D51) on

Table D22: Couplings Cg”? ( f1,z’» f2”) of the neutral fermion currents (first two gen-
page 158 for details.

169

 

 

(f1f2)C (f3f4)C % (LR-D) % (LP-D) % (HP-D) %(FP-D) Eqn.
(21d) L (du) L 0 0 0 0 (D413)
(ud) L (dd) R 0 0 0 0 (D414)
(ad) L (el/ L 0 0 0 0 (D415)
(27d) L (522; R 0 0 0 0 (D416)
(21d) R (du) R 1 1 0 0 (D417)
(ad) R (éu) L 0 0 0 0 (D418)
(ad) R (52) R 1 0 0 0 (D419)
(176 L (6V)L 0 0 0 0 (D420)
(176; L (651/) R 0 0 0 0 (D421)
(176)B (MR 1 0 1 0 (D422)

 

 

 

 

 

 

 

 

 

 

 

 

Table D23: Couplings GEE (fl’i, f3”) of the charged fermion currents at the elec-
troweak scale in the LR—D, LP-D, HP-D and FP—D model. See text and Eq. (D412)

011 page 159 for details.

D2 Couplings in .2“

D.2.1 Couplings of the Neutral Fermion Currents

The effective Lagrangian $4NC 4f that takes care of the neutral current four-fermion

interactions below the electroweak scale is given as:

 

2 .4
XNC = —- i JOJO’“— 26M ~Z———Z—’JOKO’“ + :LZZ’ JOJO 2“ 1 ————K0K0“
4” 2M; “ 1272, 143,117; “ 21712,

The coupling coefficients CEO (f1,ia f2”) are defined such that .204NC 4f takes the

following form:

KECLG ff; 2:041)?11.212901104402112); ;m'=L.R
1 2 1,]

we can write C EEC ( f1.z’2 fgd) as the sum of the SM coupling CLIFESM (f1,z'2 fgaj) and

170

 

 

(f1f2)C (f3f4 C 3 (LR-T) 4 (LP-T) 515 (HP T) 4 (FP-T) Eqn.
(ad) L (du) L 0 0 0 0 (D423)
(21d) L (du) R 0 0 0 0 (D424)
(fid)L (éu)L 0 0 0 0 (D425)
(‘d) L (522) R 0 0 0 0 (D426)
(27d R (”22) R 4 .3, 0 0 (D427)
(ad R (éu) L 0 0 0 0 (D428)
(ad) R (éu) R 5 0 0 0 (D429)
(176) L (éu) L 0 0 0 0 (D430)
(176)L (6V)R 0 0 0 0 (D431)
(272) R (512) R 4 0 4 0 (D432)

 

 

 

 

 

 

 

 

 

 

 

 

Table D24: Couplings Cg)? (f1,z‘» f3,j) of the charged fermion currents at the elec-
troweak scale in the LR—T, LP-T, HP-T and F P-T model. See text and Eq. (D412)

on page 159 for details.

 

 

(f1f2)c (f3f4)C :1; 3513: 43:3 Eqn.
(11d) L (du) L 1 -2 1 (D433)
ad) L (in) R 0 0 0 (D434)
(fld)L (El/)1] 0 -1 1 (D435)
(‘d) L (éu) R 0 0 0 (D436)
(71) R (9)14 0 0 0 (D437)
‘d R (éu) L 0 0 0 (D438)
(ad; R (éu) R 0 0 0 (D439)
(172) L (éu) L 0 0 1 (D440)
(272) L (6311);; 0 0 0 (D441)
(176)22 (Er/)2 0 0 0 (D442)

 

 

 

 

 

 

 

 

 

 

 

Table D25: Couplings 0ng ( f1,ia f3,j) of the charged fermion currents at the elec-
troweak scale in the UU-D model. See text and Eq. (D412) on page 159 for details.

171

 

 

(f1f2)C (f3f4)C El? Eli-’33; 3213335 Eqn-
(21d) L (du) L 1 —2 1 (D443)
(ad) L (d ) R 0 0 0 (D444)
(27d) L (6V) L 1 —2 1 (D445)
(27d) L (a) R 0 0 0 (D446)
('d) R ( 'u) R 0 0 0 (D447)
(‘d) R (611) L 0 0 0 (D448)
(‘d R (a) R 0 0 0 (D449)
(De L (éu)L 1 -2 1 (D450)
(27.2) L (a) R 0 0 0 (D451)
(176) R (a) R 0 0 0 (D452)

 

 

 

 

 

 

 

 

 

 

Table D26: Couplings CELL? (fLL', f3”) of the charged fermion currents (first two
generations) at the electroweak scale in the N U-D model. See text and Eq. (D412)
on page 159 for details.

various model-dependent new physics corrections that are proportional to 531:.

639 01.2.6.1) = CEESM (ft-2.122“)

1 4. 12 , 12 14
+ E-Nl+§S2B'N2+§S&'N3+§SL‘N4 (D453)

Tabs. D27 to DB6 list the numerical values of the coefficients N1, 1V2, N3 and N4
in Eq. (D453) for all possible fermion pairs ( f1,z'2 f2”) and for all considered G(221)

models.

172

 

 

 

D22 Couplings of the Charged Fermion Currents

The effective Lagrangiang’ if that takes (are of the charged (tune nt four— fer 1111011

interactions below the electroweak scale is given as:

~2 7’4

, 1 15 .11 , 6111. .,
ECG : _ ~‘ J+J—./1__1__i 11' .]+K'_”u + J—K+,/1 _11’_ W J+J— ,IJ
4f 1113., “’ .177, ( ”’ “ )+1172.,MZ. “
W 117
._ 1 ___.K‘l’K'—1H
1172. “‘
l1 ’

The coupling coefficients CELC (f11i1 f3’j) are defined such that $4CC 4f takes the

following form:

.Sfff'czf

f2; 20?} 01,2124)(1112),,,(1214);+.2.1=L.R
1 3 21.7

We can write CC fC"(f1 2'1 f3 3) as the sum of the SM coupling CLfC' SM (fL 1'1 1f-3j) and

various model-dependent new physics corrections that. are proportional to E

C4CfC (f11i1f30') = 03198111 (f1,i= hi)
1
i‘

1 1
1 N1+ N2 + —8~ 1V3+ —.S~ N4 (D814)

~523

Tabs. D37 to D45 list the 1111111erical values of the coefficients N1, N , N3 and N4

in Eq. (D814) for all possible fermion pairs (f1,i1 f3,j) and for all considered G (221)

models.

173

 

 

 

 

(flfl)c (f2f2)C CEESM ;%: 315833 $32. $34 Eqn.

 

 

 

 

 

 

 

 

 

 

(an) L (flu) L 0.237 0.546 -0.546 -1.32 0.831 (D454)
(flu) L (flu) R -0107 0.321 0.0228 -0.872 0.607 (D455)
(60) L (@d) L -0291 —0.557 0.557 1.14 -0527 (D456)
(W1L (d) R 0.0536 -0333 -00114 0.692 -0303 (D457)
(flu L (w) L 0.344 0.568 -0.568 -0959 0.224 (D458)
(an) L (w) R 0 0.344 0 -0511 0 (D459)
(flu) L (66) L -0.183 -0534 0.534 1.50 -113 (D460)
(flu) L (66) R 0.161 -0.310 -00342 1.05 0910 (D461)
(‘6) R (au) R 0.0485 0.0974 0.0911 -0424 0.382 (D462)
(*0 R (aid) L 0.132 -0.445 0.0228 0.916 0415 (D463)
(‘11) R (74) R -00243 -0221 -00455 0.468 0191 (D464)
(m) R (-u)L -0.156 0.568 -0.0683 -0959 0.224 (D465)
(‘11) R (w) R 0 0.344 0 -0.511 0 (D466)
(‘6) R (56) L 0.0830 -O.198 -0.0683 0.829 -O.798 (D467)
(‘6) R (66) R —0.0728 0.0261 -0137 0.381 -0574 (D468)
(‘ ) L (’d) L 0.356 0.546 -0.546 -0.810 0.320 (D469)
(dd) L (‘ ) R -0.0658 0.434 -0.0114 -0.586 0208 (D470)
(dd) L (w) L -0422 —0.534 0.534 0.480 -0112 (D471)
(301),; (w) R 0 -0422 0 0.255 0 (D472)
(44) L (66) L 0.225 0.568 -0.568 -147 0.735 (D473)
(‘d) L (ée) R -0197 0.456 -00342 -125 0.623 (D474)
€311) R E-d) R 0.0121 0.321 0.0228 -O.362 0.0956 (D475)
64) R w) L 0.0779 -0.534 0.0342 0.480 -0112 (D476)
(74) R (171/)R 0 —0.422 0 0.255 0 (D477)
(’ ) R (56) L -00415 0.232 0.0342 -0.798 0.399 (D478)
'6) R (ée) R 0.0364 0.120 0.0683 -0574 0.287 (D479)
51/) L (1711 L 0.500 0.500 -0.500 0 0 (D480)
1711 L (1711 R 0 0.500 0 0 0 (D481)
(w) L (ée) L -O.266 -0.602 0.602 1.44 -0.336 (D482)
(w) L (ée) R 0.234 -0.602 0.102 1.44 -0.336 (D483)
(7») R (Du) R 0 0.500 0 0 0 (D484)
(“u R (66) L 0 -O.266 0 0.766 0 (D485)
(w) R (66) R 0 -0.266 0 0.766 0 (D486)
(66) L (ée) L 0.142 0.500 -0500 -153 1.53 (D487)
(ée) L (5e) R -0124 0.164 0.102 -O.860 1.20 (D488)
(ée) R (5e)R 0.109 -0172 0.205 -0.188 0.860 (D489)

 

 

 

 

 

 

 

 

Table D27: Couplings (7ng (f1,i,f2,j) of the neutral fermion currents below the

electroweak scale in the LR-D model. See text and Eq. (D453) on page 172 for
details.

174

 

 

 

 

(f1f1)C (f2f2)c 04113178114 3 333); 333; 33:: Eqn-
(an) L (56) L 0.237 0.546 ~0.546 -132 0.831 (D490)
(1111) L (54) R -0107 0.321 0.0228 -0.872 0.607 (D491)
(flu) L (Ed) L -0291 -0557 0.557 1.14 0527 (D492)
("11) L (11) R 0.0536 —0.333 —0.0114 0.692 0303 (D493)
(‘4) L w) L 0.344 0.568 -0.568 -0959 0.224 (D494)
(‘14) L (w) R 0 0 0 0 0 (D495)
(-u) L (ée) L -0.183 -0534 0.534 1.50 -113 (D496)
("21) L (66) R 0.161 0.0342 —0.0342 0.542 0910 (D497)
E7014 (1737; R 0.0485 0.0974 0.0911 —0.424 0.382 (D498)
‘21) R ( d L 0.132 -0445 0.0228 0.916 0415 (D499)
('4) R (”d) R —0.0243 —0.221 00455 0.468 -0191 (D500)
("11)R (17V) L 0156 0.568 -0.0683 -0959 0.224 (D501)
Em; R (171;) R 0 0 0 0 0 (D502)
’11 R (66) L 0.0830 -0.198 -0.0683 0.829 -0.798 (D503)
(’21) R (é ) R -0.0728 0.370 -0137 -0130 0574 (D504)
( ’d) L ( 11) L 0.356 0.546 -0.546 -0.810 0.320 (D505)
( ’d) L ( ’d) R -0.0658 0.434 -00114 —0.586 0.208 (D506)
( 11) L (17V) L 0422 —0.534 0.534 0.480 0112 (D507)
(aid L (171/)R 0 0 0 0 0 (D508)
( 'd) L (ée) L 0.225 0.568 -0.568 -1.47 0.735 (D509)
(’d) L (ée) R 0197 0.0342 —0.0342 —0.990 0.623 (D510)
('3) R ('d) R 0.0121 0.321 0.0228 -0.362 0.0956 (D511)
(‘ R (-u)L 0.0779 —0.534 0.0342 0.480 0112 (D512)
( 21) R (w) R 0 0 0 0 0 (D513)
(_d) R (56) L -00415 0.232 0.0342 -0.798 0.399 (D514)
(‘ ) R (66) R 0.0364 -0302 0.0683 -0.318 0.287 (D515)
(w) L (171/)L 0.500 0.500 -0500 0 0 (D516)
(w) L (w) R 0 0 0 0 0 (D517)
(w) L (ée) L -0.266 -0.602 0.602 1.44 -0.336 (D518)
(171/)1; (6736) R 0.234 -0.102 0.102 1.44 -O.336 (D519)
(171/)R (171;) R 0 0 0 0 0 (D520)
(w) R (66) L 0 0 0 0 0 (D521)
(w) R (ée) R 0 0 0 0 0 (D522)
(56) L ((56) L 0.142 0.500 -0.500 -153 1.53 (D523)
(ée) L ée) R 0124 —0.102 0.102 -0.0941 1.20 (D524)
(ée) R ée) R 0.109 -0205 0.205 1.34 0.860 (D525)

 

 

 

 

 

 

 

 

 

 

 

Table D28: Couplings Cg? (1’13: ng) of the neutral fermion currents below the

electroweak scale in the LP-D model. See text and Eq. (D453) 011 page 172 for
details.

175

 

 

(f1f1)c ( 2f2)C 0%?SM % %839 $23 $333 Eqn.
(‘0) L (Tu) L 0.237 0.546 -().546 -132 0.831 (D526)
(m) L (721) R 0107 -0.0228 0.0228 -0.362 0.607 (D527)
(’u) L (10L -0.291 -0.557 0.557 1.14 0527 (D528)
(au) L (11) R 0.0536 0.0114 —0.0114 0.181 0303 (D529)
(‘11)L (w) L 0.344 0.568 -0.568 0959 0.224 (D530)
(‘11) L (w) R 0 0.344 0 -0 511 0 (D531)
(‘0) L (ée) L -0.183 —0.534 0.534 1.50 —1.13 (D532)
(m) L (ée) R 0.161 -0310 00342 1.05 0910 (D533)
{17) R (qu) R 0.0485 —0.0911 0.0911 0.598 0.382 (D534)
‘.u R (dd) L 0.132 -0.0228 0.0228 0.660 -0.415 (D535)
("7.7) R (aid) R —0.0243 0.0455 —0.0455 —0.299 0191 (D536)
(‘11) R (171/)L -0.156 0.0683 -0.0683 -0.959 0.224 (D537)
(‘13) R (91/) R 0 -U.156 0 —0.511 0 (D538)
(‘0) R (ée) L 0.0830 0.0683 -0.0683 0.0627 -0.798 (D539)
(-u) R (ée) R —0.0728 0.292 -0137 -0.385 0574 (D540)
(' ) L (11) L 0.356 0.546 -0.546 —0.810 0.320 (D541)
(@d) L ('d) R -0.0658 0.0114 .0.0114 -0330 0.208 (D542)
E6921: (w) L -0422 —0.534 0.534 0.480 -0112 (D543)
dd L (w) R 0 -0422 0 0255 0 (D544)
(Jd) L (66) L 0.225 0.568 ~0.568 -1.47 0.735 (D545)
(‘ ) L (656) R 0197 0.456 00342 -125 0.623 (D546)
(’6) R ER) R 0.0121 -0.0228 0.0228 0.149 0.0956 (D547)
(11) R w) L 0.0779 0.0342 0.0342 0.480 0112 (D548)
(‘71) R (w) R 0 0 0779 0 0.255 0 (D549)
(11) R (ée) L 0.0415 0.0342 0.0342 —0.0314 0.399 (D550)
(11 R (ée) R 0.0364 -0.146 0.0683 0.193 0.287 (D551)
(17V) L (1712) L 0.500 0.500 —0.500 0 0 (D552)
(1712 L (1712 R 0 0.500 0 0 0 (D553)
(171/)1: (66) L 0266 -0.602 0.602 1.44 -0.336 (D554)
(w) L (ée) R 0.234 —0.602 0.102 1.44 —0.336 (D555)
171/)3 (1712) R 0 0.500 0 0 0 (D556)
w) R (ée) L 0 —0.266 0 0.766 0 (D557)
1711 R (ée) R 0 -0.266 0 0.766 0 (D558)
(66) L (ée) L 0.142 0.500 —0.500 —1.53 1.53 (D559)
(ée) L (ée) R 0124 0.164 0.102 -0.860 1.20 (D560)
(ée) R (ée) R 0.109 -0172 0.205 -0.188 0.860 (D561)

 

 

 

 

 

 

 

 

 

 

Table D29: Couplings 0ng (f1,iv f”) of the neutral fermion currents below the
electroweak scale in the HP-D model. See text and Eq. (D453) on page 172 for
details.

176

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(f1f1)0 (f2f2)c 05585-1 3 3333 33:3 33: Eqn-
(flu) L (21.23 L 0.237 0.546 -0.546 -132 0.831 (D562)
(flu) L (flu) R -0107 -0.0228 0.0228 -0.362 0.607 (D563)
(an) L (dd) L 0291 —0.557 0.557 1.14 -0527 (D564)
(fiu)L (")R 0.0536 0.0114 -0.0114 0.181 -0303 (D565)
(*0) L (171/)L 0.344 0.568 -0.568 -0959 0.224 (D566)
('21) L (w) R 0 0 0 0 0 (D567)
(‘21) L (66) L ~0.183 —0.534 0.534 1.50 -113 (D568)
(“u)L (66) R 0.161 0.0342 -0.0342 0.542 0910 (D569)
(‘27) R (m) R 0.0485 -0.0911 0.0911 0.598 0.382 (D570)
(‘11 R (dd) L 0.132 —0.0228 0.0228 0.660 -0415 (D571)
(‘11) R (‘51) R -00243 0.0455 -0.0455 —0.299 0191 (D572)
(’u) R (171/)L -0.156 0.0683 -0.0683 -0959 0.224 (D573)
('21) R (171!) R 0 0 0 0 0 (D574)
(‘21 R (ée) L 0.0830 0.0683 -0.0683 0.0627 -0.798 (D575)
(’u) R (ée) R -0.0728 0.137 -0137 —O.896 -0574 (D576)
( ’d) L (dd) L 0.356 0.546 -0.546 -0.810 0.320 (D577)
(Jd)L ('d) R -0.0658 0.0114 -0.0114 -0330 0.208 (D578)
(dd) L (171;) L 0422 -0534 0.534 0.480 -0112 (D579)
(Jd) L (w) R 0 0 0 0 0 (D580)
(aid) L (ée) L 0.225 0.568 -0.568 -147 0.735 (D581)
(dd) L (58) R 0197 0.0342 .0.0342 -0990 0.623 (D582)
(11) R (11) R 0.0121 -0.0228 0.0228 0.149 0.0956 (D583)
('01 R (w) L 0.0779 0.0342 0.0342 0.480 -0112 (D584)
( ’d) R (w) R 0 0 0 0 0 (D585)
(“’d) R (66) L 00415 -00342 0.0342 .0.0314 0.399 (D586)
(_d) R (ée) R 0.0364 -0.0683 0.0683 0.448 0.287 (D587)
E1711) L (1712) L 0.500 0.500 -0.500 0 0 (D588)
171/)L (w) R 0 0 0 0 0 (D589)
(171/)L (66) L -0.266 -0.602 0.602 1.44 -0.336 (D590)
(171/)L (ée) R 0.234 —0.102 0.102 1.44 -0.336 (D591)
(w) R (171/)3 0 0 0 0 0 (D592)
(w) R (ée) L 0 0 0 0 0 (D593)
(w) R (ée) R 0 0 0 0 0 (D594)
(56) L (ée) L 0.142 0.500 —0.500 -153 1.53 (D595)
(68) L (ée) R 0124 -0102 0.102 —0.0941 1.20 (D596)
(éeLR (ée) R 0.109 -0205 0.205 1.34 0.860 (D597)

 

 

Table D.30: Couplings Cyfc (f1,i~ f2,j) of the neutral fermion currents below the

electroweak scale in the FP-D model. See text and Eq. (D453) on page 172 for
details.

177

 

 

( 1f1)c (f2f2)c 0.83m 3. 33:5 333-, 35:; Eqn-
(flu) L (flu) L 0.237 0.136 0.273 0.330 0.208 (D598)
(‘0) L (2111)};f 0.107 0.0804 0.0114 0.218 0.152 (D599)
(-u) L (dd) L 0.291 0.139 0.278 0.285 0.132 (D600)
(*0) L (aid) R 0 0536 0 0832 0.00569 0 173 0.0758 (D601)
(m) L (171/)1; 0.344 0.142 0.284 0.240 0.0560 (D602)
(76) L (w) R 0 0.0861 0 0.128 0 (D603)
(‘0) L (ée) L 0.183 0.134 0.267 0.375 0.283 (D604)
(‘0) L (ée) R 0.161 0.0775 0.0171 0.263 0.227 (D605)
(‘0) R (flu) R 0.0485 0.0243 0.0455 -0.106 0.0956 (D606)
(‘11) R (“ ) L 0.132 0.111 0.0114 0.229 0.104 (D607)
(‘0) R (‘71) R 0.0243 0.0552 0.0228 0.117 0.0478 (D608)
(‘0) R (17V) L 0.156 0.142 0.0342 0.240 0.0560 (D609)
(‘1; R (171/)R 0 0.0861 0 0.128 0 (D610)
(‘u R (56) L 0.0830 0.0495 0.0342 0.207 0.199 (D611)
(m) R ((36) R 0.0728 0.00652 0.0683 0.0952 0.143 (D612)
(0111) L (" ) L 0.356 0.136 0.273 0.202 0.0799 (D613)
( 11) L ('d) R 0 0658 0 108 0 00569 0.146 0.0519 (D 614)
( 11) L (w) L 0 422 0 134 0 267 0.120 0.0280 (D615)
(aid) L (w) R 0 0 106 0 0.0639 0 (D616)
(‘d) L (66) L 0 225 0 142 0 284 0 367 0.184 (D617)
( _d) L (ée) R 0 197 0 114 0.0171 0 311 0.156 (D618)
( 11) R ( _d) R 0 0121 0 0804 0.0114 0 0904 0.0239 (D619)
( “d R (w) L 0 0779 0134 0.0171 0120 0.0280 (D620)
(11) R (‘12) R 0 0 10b 0 0 0639 0 (D621)
(11) R (ée) L 0 0415 0 0580 0.0171 0 199 0.0997 (D622)
(‘d) R (56) R 0 0364 0 0300 0.0342 0 143 0.0717 (D623)
(w) L (m) L 0 500 0 125 0 250 0 0 (D624)
(w) L w) R 0 0125 0 0 0 (D625)
(72) L (ée) L 0.266 0.151 0.301 0.360 0.0840 (D626)
(1712) L (ée) R 0.234 0.151 0.0512 0.360 0.0840 (D627)
(171/)R (‘12) R 0 0.125 0 0 0 (D628)
(w) R (636) L 0 0.0666 0 0.192 0 (D629)
(w) R (ée) R 0 0.0666 0 0.192 0 (D630)
(66) L (66) L 0.142 0.125 0.250 0.383 0.383 (D631)
(ée) L (ée) R 0.124 0.0410 0.0512 0.215 0.299 (D632)
(66) R (ée) R 0.109 0.0431 0.102 0.0470 0.215 (D633)

 

 

 

 

 

 

 

 

 

 

Table D.31: Couplings 0ny (fu, f”) of the neutral fermion currents below the
electroweak scale in the LR—T model. See text and Eq. (D453) on page 172 for
details.

178

 

 

 

 

 

 

 

 

 

 

(mac (mac Cfifigm 31; 3ng 315333 31.33) Equ

(flu) L (flu) L 0.237 0.136 0.273 0.330 0 208 (D634)
(70) L (‘11) R 0.107 0.0804 0.0114 0.218 0 152 (D635)
('u) L (11) L 0.291 0.139 0.278 0.285 0.132 (D636)
( *u) L (‘d) R 0.0536 0.0832 0.00569 0.173 0.0758 (D637)
(“0) L (171/)L 0.344 0.142 0.284 0.240 0.0560 (D638)
(‘11) L (‘22) R 0 0 0 0 0 (D639)
(‘6) L (ée) L 0.183 0.134 0.267 0.375 0.283 (D640)
(‘21) L (68) R 0.161 0.00854 0.0171 0.136 0.227 (D641)
(‘0) R (‘11) R 0.0485 0.0243 0.0455 0.106 0.0956 (D642)
(‘11.) R ('d L 0.132 0.111 0.0114 0.229 0.104 (D643)
(‘0) R (11) R 0.0243 0.0552 0.0228 0.117 0 0478 (D644)
(‘0) R (17!!) L 0.156 0.142 0.0342 0.240 0.0560 (D645)
Em) R (w) R 0 0 0 0 0 (D646)

m) R (ée) L 0.0830 0.0495 0.0342 0.207 0.199 (D647)
(71)}? (ée) R 0.0728 0.0926 0.0683 0.0325 0.143 (D648)
(Jd) L ( ‘71) L 0.356 0.136 0.273 0.202 0.0799 (D649)
(21) L (11) R 0.0658 0.108 0.00569 0.146 0.0519 (D650)
(‘ ) L (Du) L 0.422 0.134 0.267 0.120 0.0280 (D651)
( ‘71) L (w) R 0 0 0 0 0 (D652)
(11) L (ée) L 0.225 0.142 0.284 0.367 0.184 (D653)

"d) L (ée) R 0.197 0.00854 0.0171 0.248 0.156 (D654)
(_d) R (’d) R 0.0121 0.0804 0.0114 0.0904 0.0239 (D655)
(aid) R (w) L 0.0779 0.134 0.0171 0.120 0.0280 (D656)
(Jd) R (w) R 0 0 0 0 0 (D657)
( 'd) R (ée) L 0.0415 0.0580 0.0171 0.199 0.0997 (D658)
( 7d) R (66) R 0.0364 0.0755 0.0342 0.0795 0.0717 (D659)
(w) L (171/)L 0.500 0.125 0.250 0 0 (D660)
(w) L (w) R 0 0 0 0 0 (D661)
(171;) L (56) L 0.266 0.151 0.301 0.360 0.0840 (D662)
(w) L (ée) R 0.234 0.0256 0.0512 0.360 0.0840 (D663)
(w) R (171/)R 0 0 0 0 0 (D664)
(w) R (ée) L 0 0 0 0 0 (D665)
(w) R (66) R 0 0 0 0 0 (D666)
(56) L (ée) L 0.142 0.125 0.250 0.383 0.383 (D667)
(ée) L (66) R 0.124 0.0256 0.0512 0.0235 0.299 (D668)
(ée) R (éehL 0.109 0.0512 0.102 0.336 0.215 (D669)

Table D.32: Couplings 03cc ( 131,2”, f“) of the neutral fermion currents below the

electroweak scale in the LP-T model. See text and Eq. (D453) on page 172 for

details.

179

 

 

 

(f1f1)0 ( 2f2)c 041138334 3 33:3 33:, 3335 Eqn~
(flu) L 0.237 0.136 0.273 0.330 0.208 (D670)
(flu) L 0.107 0.00569 0.0114 0.0904 0.152 (D671)
(60) L 0.291 0.139 0.278 0.285 0.132 (D672)
E1111) L 0.0536 0.00285 0.00569 0.0452 0.0758 (D673)
1123)]: 0.344 0.142 0.284 0.240 0.0560 (D674)
(‘14) L 0 0.0861 0 0.128 0 (D675)
(an) L 0.183 0.134 0.267 0.375 0.283 (D676)
(flu) L 0.161 0.0775 0.0171 0.263 0.227 (D677)
(‘10) R 0.0485 0.0228 0.0455 0.149 0.0956 (D678)
(~u) R L 0.132 0.00569 0.0114 0.165 0.104 (D679)
(m) R R 0.0243 0.0114 0.0228 0.0747 0.0478 (D680)
(‘0) R L 0.156 0.0171 0.0342 0.240 0.0560 (D681)
(’0) R R 0 0.0389 0 0.128 0 (D682)
(‘u R (e ) L 0.0830 0.0171 0.0342 0.0157 0.199 (D683)
(‘0) R (ée) R 0.0728 0.0731 0.0683 0.0964 0.143 (D684)
(Jd) L ( ’d) L 0.356 0.136 0.273 0.202 0.0799 (D685)
(Jd) L (’d) R 0.0658 0.00285 0.00569 0.0825 0.0519 (D686)
(33) L (w) L 0.422 0.134 0.267 0.120 0.0280 (D687)
(d‘d L (w) R 0 0.106 0 0.0639 0 (D688)
(11) L (6e) L 0.225 0.142 0.284 0.367 0.184 (D689)
(11) L (66) R 0.197 0.114 0.0171 0.311 0.156 (D690)
(Jd) R (11) R 0.0121 0.00569 0.0114 0.0373 0.0239 (D691)
(‘71) R 171/)L 0.0779 0.00854 0.0171 0.120 0.0280 (D692)
(_d) R (171/)R 0 0.0195 0 0.0639 0 (D693)
('d) R (ée) L 0.0415 0.00854 0.0171 0.00784 0.0997 (D694)
(’d) R (ée) R 0.0364 0.0366 0.0342 0.0482 0.0717 (D695)
(171/)L (17V) L 0.500 0.125 0.250 0 0 (D696)
(171/)L (171/)R 0 0.125 0 0 0 (D.697)
(171/)L (ée) L 0.266 0.151 0.301 0.360 0.0840 (D698)
(w) L (ée) R 0.234 0.151 0.0512 0.360 0.0840 (D699)
171/)R (171/)R 0 0.125 0 0 0 (D700)
171/)R (ée) L 0 0.0666 0 0.192 0 (D701)
(w) R (ée) R 0 0.0666 0 0.192 0 (D702)
(ée) L (ée) L 0.142 0.125 0.250 0.383 0.383 (D703)
(ée) L (ée) R 0.124 0.0410 0.0512 0.215 0.299 (D704)
(58) R (ée) R 0.109 0.0431 0.102 0.0470 0.215 (D705)

 

 

 

 

 

 

 

 

 

 

 

 

 

Table D.33: Couplings CE)? (f1,z" ng) of the neutral fermion currents below the

electroweak scale in the HP-T model. See text and Eq. (D453) on page 172 for
details.

180

 

 

(f1f1)C (f2f2)c CEESM 3 3533 35:3 33:3 Eqn-
(flu) L @101. 0.237 0.136 0.273 0.330 0.208 (D706)
(flu) L (66) R 0.107 0.00569 0.0114 0.0904 0.152 (D707)
(‘21) L (6111) L 0.291 0.139 0.278 0.285 0.132 (D708)
(‘11) L (Jd) R 0.0536 0.00285 0.00569 0.0452 0.0758 (D709)
(‘21) L (171/)L 0.344 0.142 0.284 0.240 0.0560 (D710)
(‘0) L (171/)R 0 0 0 0 0 (D711)
(‘11) L (ée) L 0.183 0.134 0.267 0.375 0.283 (D712)
(‘21.) L (66) R 0.161 0.00854 0.0171 0.136 0.227 (D713)
(‘0) R (m) R 0.0485 0.0228 0.0455 0.149 0.0956 (D714)
(‘14) R (Jd) L 0.132 0.00569 0.0114 0.165 0.104 (D715)
("21.) R (11) R 0.0243 0.0114 0.0228 0.0747 0.0478 (D716)
(’11) R (Du) L 0.156 0.0171 0.0342 0.240 0.0560 (D717)
(‘0) R (171/)R 0 0 0 0 0 (D718)
(‘0) R (ée) L 0.0830 0.0171 0.0342 0.0157 0.199 (D719)
(‘0) R (ée) R 0.0728 0.0342 0.0683 0.224 0.143 (D720)
( ‘d) L (dd) L 0.356 0.136 0.273 0.202 0.0799 (D721)
(dd) L ( 11) R 0.0658 0.00285 0.00569 0.0825 0.0519 (D722)
('d) L (w) L 0.422 0.134 0.267 0.120 0.0280 (D723)
(Jd) L (w) R 0 0 0 0 0 (D724)
( ‘d.) L (ée) L 0.225 0.142 0.284 0.367 0.184 (D725)
(d‘d) L (ée) R 0.197 0.00854 0.0171 0.248 0.156 (D726)
( 11) R E‘ ) R 0.0121 0.00569 0.0114 0.0373 0.0239 (D727)
(11) R (71/) L 0.0779 0.00854 0.0171 0.120 0.0280 (D728)
( 'd) R (171/)R 0 0 0 0 0 (D729)
('d) R (66) L 0.0415 0.00854 0.0171 0.00784 0.0997 (D730)
(11) R (ée) R 0.0364 0.0171 0.0342 0.112 0.0717 (D731)
(w) L (1712) L 0.500 0.125 0.250 0 0 (D732)
(w) L 171/)R 0 0 0 0 0 (D733)
(w) L (ée) L 0.266 0.151 0.301 0.360 0.0840 (D734)
(w) L (ée) R 0.234 0.0256 0.0512 0.360 0.0840 (D735)
(71/) R (171/)3 0 0 0 0 0 (D736)
(w R (56) L 0 0 0 0 0 (D737)
(w) R (66) R 0 0 0 0 0 (D738)
(ée) L (66) L 0.142 0.125 0.250 0.383 0.383 (D739)
(ée) L (ée) R 0.124 0.0256 0.0512 0.0235 0.299 (D740)
(ée) R (ée) R 0.109 0.0512 0.102 0.336 0.215 (D741)

 

 

 

 

 

 

 

 

 

 

 

 

 

Table D.34: Couplings Cyfc ( f1R-, f”) of the neutral fermion currents below the

electroweak scale in the FP-T model. See text and Eq. (D453) 011 page 172 for
details.

181

 

 

 

 

(fang (f2f2)c 033M 3 513-83; 38:; Eqn-
Rafi (an) L 0.237 0.500 0.312 0.357 (D742)
('11) L (‘21) R 0.107 0 0.156 0.133 (D743)
(“0) L (dd) L 0.291 0.500 0.234 0.291 (D744)
(‘21) L (dd) R 0.0536 0 0.0779 0.0665 (D745)
‘13) L (w) L 0.344 0 0 0.224 (D746)
("11.) L (w) R 0 0 0 0 (D747)
(-u) L (66) L 0.183 0 0.234 0.424 (D748)
(“21) L ((36) R 0.161 0 0.234 0.199 (D749)
Em) R (7111) R 0.0485 0 0 0.0911 (D750)
‘0) R (dd L 0.132 0 0.156 0.179 (D751)
(‘21) R (dd) R 0.0243 0 0 0.0455 (D752)
(‘13) R (171/)L 0.156 0 0 0.224 (D753)
27012 (171/)1? 0 0 0 0 (D754)
‘11) R (68) L 0.0830 0 0 0.0874 (D755)
("21) R (6‘36) R 0.0728 0 0 0.137 (D756)
(dd) L (' ) L 0.356 0.500 0.156 0.201 (D757)
(dd) L (’ ) R 0.0658 0 0.0779 0.0893 (D758)
Ed) L (w) L 0.42 0 0 0.112 (D759)
dd L (w) R 0 0 0 0 (D760)
(‘d) L (ée) L 0.225 0 0.234 0.380 (D761)
(dd) L (é_e) R 0.197 0 0.234 0.268 (D762)
( d) R E d) R 0.0121 0 0 0.0228 (D763)
(‘ R w) L 0.0779 0 0 0.112 (D764)
(’d) R (w) R 0 0 0 0 (D765)
("d R (66) L 0.0415 0 0 0.0437 (D766)
(‘d) R (56) R 0.0364 0 0 0.0683 (D767)
(w) L (w) L 0.500 0 0 0 (D768)
12 L (w) R 0 0 0 0 (D769)
(w) L (68) L 0.266 0 0 0.336 (D770)
(w) L (ée) R 0.234 0 0 0.336 (D771)
(w R (171/)R 0 0 0 0 (D772)
(w R (ée) L 0 0 0 0 (D773)
(w R (ée) R 0 0 0 0 (D774)
(ée) L (56) L 0.142 0 0 0.467 (D775)
(ée) L (ée) R 0.124 0 0 0.131 (D776)
(6e)R (56) R 0.109 0 0 0.205 (D777)

 

 

 

 

 

 

 

 

 

 

 

 

Table D.35: Couplings CFfC (fLi, f2,j) of the neutral fermion currents below the

electroweak scale in the UU-D model. See text and Eq. (D453) on page 172 for
details.

182

 

 

 

 

f1f1)c (f2f2)c 04121;?SM 3 "5897) 38:3 Eqn.
(flu) L (flu) L 0.237 0.500 0.312 0.357 (D778)
(‘21) L (1723)}? 0.107 0 0.156 0.133 (D779)
(*u) L ("d) L 0.291 0.500 0.234 0.291 (D780)
(“u L ('d) R 0.0536 0 0.0779 0.0665 (D781)
(‘11) L (w) L 0.344 0.500 0.156 0.224 (D782)
(‘u)L ("I/)R O 0 O 0 (D783)
(‘11.) L (6e) L 0.183 0.500 0.389 0.424 (D784)
("21) L (ée R 0.161 0 0.234 0.199 (D785)
(00) R (flu) R 0.0485 0 0 0.0911 (D786)
(‘21) R ('d) L 0.132 0 0.156 0.179 (D787)
(".u) R (’d) R 0.0243 0 0 0.0455 (D788)
E7012 (w) L 0.156 0 0.156 0.224 (D789)
'6) R (‘12) R 0 0 0 0 (D790)
(’21) R (66) L 0.0830 0 0.156 0.0874 (D791)
(‘0) R (ée) R 0.0728 0 0 0.137 (D792)
(’d) L (’d) L 0.356 0.500 0.156 0.201 (D793)
(’d L (‘d) R 0.0658 0 0.0779 0.0893 (D794)
(dd) L (w) L 0.422 0.500 0.0779 0.112 (D795)
(dd) L (w) R 0 0 0 0 (D796)
('d) L (56) L 0.225 0.500 0.312 0.380 (D797)
(dd) L (ée) R 0.197 0 0.234 0.268 (D798)
(‘d R ('d) R 0.0121 0 0 0.0228 (D799)
("d) R (171/)L 0.0779 0 0.0779 0.112 (D800)
(’ ) R (171;) R 0 0 0 0 (D801)
(’ ) R (86) L 0.0415 0 0.0779 0.0437 (D802)
(“ ) R (66) R 0.0364 0 0 0.0683 (D803)
(w) L (w) L 0.500 0.500 0 0 (D804)
(3) L (171/)R 0 0 0 0 (D805)
(7;) L (66) L 0.266 0.500 0.234 0.336 (D806)
(w) L (66) R 0.234 0 0.234 0.336 (D807)
(w) R (w) R 0 0 0 0 (D808)
(w) R (56) L 0 0 . 0 0 (D809)
(w) R ('66) R 0 0 0 0 (D810)
E86; L (6e) L 0.142 0.500 0.467 0.467 (D811)
ée L (66 R 0.124 0 0.234 0.131 (D812)
(86) R (6e)R 0.109 0 0 0.205 (D813)

 

 

 

 

 

 

 

 

 

 

 

 

Table D36: Couplings Cg? (fLi, f2,» of the neutral fermion currents (first two gen-

erations) below the electroweak scale in the NU-D model. See text and Eq. (D453)
on page 172 for details.

183

 

 

 

 

 

(f1f2)c (f3f4)c CffcfSM 3 3323 Eqn-
(ad) L (du) L 1 0 0 (D815)
(6d L (du)R 0 0 1 (D816)
(dd) L (612)}: 1 0 0 (D817)
(ad) L (éu) R 0 0 1 (D818)
('d) R (‘21) R 0 1 0 (D819)
(‘d) R (an) L 0 0 1 (D820)
(‘0! R (EV) R 0 1 0 (D821)
(De L (Eu) L 1 0 0 (D822)
De L (éu R 0 0 1 (D823)
(176)}? (El/)3 0 1 O (D824)

 

 

 

 

 

 

 

 

 

 

Table D37: Couplings 6'ng
electroweak scale in the LR-D model.
details.

(fm, f3,j) of the charged fermion currents below the
See text and Eq. (D814) on page 173 for

 

 

 

(f1f2)c(f3f4)c 088511 3 332); Eqn.
(dd) L (d d)L 1 0 0 (D825)
(u dL) (duR 0 0 1 (D826)
(u d)L (a) L 1 0 0 (D827)
( d) L (éu) R 0 0 0 (D828)
('d) R ( _u) R 0 1 0 (D829)
‘d R (EV) L 0 0 1 (D830)
(0) R (éu) R 0 0 0 (D831)
(176) L (611) L 1 0 0 (D832)
(De) L (611) R 0 0 0 (D833)
(De) R (611) R 0 0 0 (D834)

 

 

 

 

 

 

 

 

 

 

Table D.38: Couplings Cffc ( f1,z‘» f3,j) of the charged fermion currents below the
electroweak scale in the LP-D model. See text and Eq. (D814) on page 173 for

details.

184

 

 

 

(f1f2)C (f3f4)C GEESM 3 333325 Eqn.
(1101) L (du) L 1 0 0 (D835)
(ad) L (du) R 0 0 0 (D836)
(6d) L (éu) L 1 0 0 (D837)
(71d) L (éu) R 0 0 1 (D838)
(6d) R ( 'u) R 0 0 0 (D839)
(ad) R (éu) L 0 0 0 (D840)
(dd) R (EV) R 0 0 0 (D841)

176) L (éu) L 1 0 0 (D842)
176) L (631/) R 0 0 1 (D843)
(176) R (510R 0 1 0 (D844)

 

 

 

 

 

 

 

 

 

 

 

Table D.39: Couplings CE]? ( f1,is f3”) of the charged fermion currents below the

electroweak scale in the HP-D model. See text and Eq. (D814) on page 173 for
details.

 

 

(f1f2)C (f3f4)0 CEE'SM 51:; 35823 Eqn.
(dd) L (du) L 1 0 0 (D845)
(6d) L ( u) R 0 0 0 (D846)
ad L (611 L 1 0 0 (D847)
(ad) L (61!) R 0 0 0 (D848)
(‘d) R ( ’u) R 0 0 0 (D849)
(“d R (EV) L 0 0 0 (D850)
(‘d) R (EV) R 0 0 0 (D851)
(De) L (EV) L 1 0 0 (D852)
(178) L (éu) R 0 0 0 (D853)
(178) R (éu) R 0 0 0 (D854)

 

 

 

 

 

 

 

 

 

 

 

Table D40: Couplings Cffc (fm, f3,j) of the charged fermion currents below the

electroweak scale in the FP-D model. See text and Eq. (D814) on page 173 for
details.

 

 

C(f1f2)c C(f3f4)c CEfCESM 3 3823 Eqn.
(6d) L (du) L 1 0 0 (D855)
(21d) L (dd) R 0 0 L (D856)
(ad) L (181/) L 1 0 0 (D857)
(6d) L (év) R 0 0 L (D858)
(‘d) R ( u) R 0 L 0 (D859)
(17d) R (éu) L 0 0 4 (D860)
('d) R (EV) R 0 L 0 (D861)
(176) L (éu) L 1 0 0 (D862)
(176) L (EV) R 0 0 § (D863)
(De) R (El/)3 0 3 0 (D864)

 

 

 

 

 

 

 

 

 

 

Table D41: Couplings 040;: (fldv f3,j) of the charged fermion currents below the

electroweak scale in the LR—T model. See text and Eq. (D814) on page 173 for
details.

 

 

C(f1f2)C’ C(f3f4)c CEfCSM 3% 313-7823 Eqn.
(11d) L (du) L 1 0 0 (D865)
W) L ( 7U) R 0 0 L (D866)
6d) L (éll) L 1 0 0 (D867)
(W) L (€103 0 0 0 (D868)
(’60 R ( U) R 0 3 0 (D869)
(71);; (($11) L 0 0 L (D870)
(‘d) R (EV) R 0 0 0 (D871)
('76) L (év) L 1 0 0 (D872)
(176)L (éu) R 0 0 0 (D873)
(“)3 (El/)3 0 0 0 (D874)

 

 

 

 

 

 

 

 

 

 

 

Table D42: Couplings Cffc (f1,z‘» f3”) of the charged fermion currents below the

electroweak scale in the LP-T model. See text and Eq. (D814) on page 173 for
details.

186

 

 

C(f1f2)C C(f—3f4)C CcICESM 3 3825, Eqn.
(ad) L (du) L 1 0 0 (D875)
(ad) L (da) R 0 ' 0 0 (D876)
(ad) L (a) L 1 0 0 (D877)
(VOL (”DI/)1? O 0 3 ([1878)
(ad) R (du) R 0 0 0 (D879)

ad) R ('e‘u) L 0 0 0 (D880)
ad) R (éu) R 0 0 0 (D881)
(176)L (éu) L 1 0 0 (D882)
(178) L (6V) R 0 0 L (D883)
(ae) R (a) R 0 L 0 (D884)

 

 

 

 

 

 

 

 

 

 

 

Table D43: Couplings Cffc ( f1,iv f3,j) of the charged fermion currents below the

electroweak scale in the HP-T model. See text and Eq. (D814) on page 173 for
details.

 

 

C(f1f2)C C(f3f4)0 GEESM 33 33325 Eqn.
(ad) L (du) L 1 0 . 0 (D885)
(ad) L (‘21) R 0 0 0 (D886)
(ad) L (éu) L 1 0 0 (D887)
(ad) L (éu) R 0 0 0 (D888)
(“(1) R (da) R 0 0 0 (D889)
(’d) R (a) L 0 0 0 (D890)
("d) R (éu) R 0 0 0 (D891)
(De) L (6V) L 1 0 ' 0 (D892)
(ae L (éu R 0 0 0 (D893)
(De; R (a) R 0 0 0 (D894)

 

 

 

 

 

 

 

 

 

 

 

Table D44: Couplings Cgfc (fu, f3,j) of the charged fermion currents below the

electroweak scale in the F P—T model. See text and Eq. (D814) on page 173 for
details.

187

 

 

(Mac (Mac Offsa %(UU-D) 3(NU—D) Eqn.
(ad) L (da) L 1 1 1 (D895)
(ad) L (da) R 0 0 0 (D896)

‘d) L (éu) L 1 0 1 (D897)

’d) L (éu) R 0 0 0 (D898)
(ad) R (‘11) R 0 0 0 (D899)
(‘01) R (éu) L 0 0 0 (D900)
(ad) R (éu) R 0 0 0 (D901)
(ae) L (751/) L 1 0 1 (D902)
(De) L (812) R 0 0 0 (D903)
(ae) R (EV) R 0 0 0 (D904)

 

 

 

 

 

 

 

 

 

 

 

Table D45: Couplings 0ng ( f1,ia f3”) of the charged fermion currents below the

electroweak scale in the UU-D and N U-D model. The results for the NU-D model
apply to the first two fermion generations. See text and Eq. (D814) on page 17 3 for

details.

188

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