ABOVE AND BEYOND THE STANDARD MODEL:
ON PHENOMENOLOGY OF LEE–WICK THEORY AND MASSIVE VECTOR
COLOR–OCTET
By
Arsham Farzinnia

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

ABSTRACT
ABOVE AND BEYOND THE STANDARD MODEL:
ON PHENOMENOLOGY OF LEE–WICK THEORY AND MASSIVE
VECTOR COLOR–OCTET
By
Arsham Farzinnia

“A thesis has to be presentable... but don’t attach too much importance to it. If
you do succeed in the sciences, you will do later on better things, and then it will
be of little moment. If you don’t succeed in the sciences, it doesn’t matter at all! ”
—Paul Ehrenfest (1880 – 1933)

The present Thesis is dedicated to a formal and phenomenological investigation of extensions to two separate sectors of the Standard Model of particle physics (SM): the electroweak
sector and the strong sector. The Thesis is divided into two main parts: Part I focuses on the
Lee-Wick Standard Model (LW SM), which, by providing a solution to the Hierarchy problem, forms a natural extension of the electroweak sector, while Part II studies the coloron
theory, arising from extending the strong sector gauge group.
Providing a general introduction about the current state of the SM and the associated
challenges in Chapter , we proceed in Chapter to analyze the tension between naturalness
and isospin violation in the LW SM. Chapter discusses the global symmetries and the
renormalizability of LW scalar QED. A first complete calculation of QCD corrections to the
production of a massive color-octet vector boson (colorons) is reported in Chapter . Finally,
we conclude the Thesis in Chapter by summarizing the discussed results and presenting an
outlook for future research in the surveyed areas.

Copyright by
ARSHAM FARZINNIA
2012

‫ﭘ"ﺸﮑﺶ ﺑﻪ ﻣﺎدرﺗﺎی ﮔﺮاﻣ"ﻢ‬
‫ﻣﻬﺮی‬
‫‪To my dear aunt‬‬

‫‪Mehry‬‬

‫‪iv‬‬

ACKNOWLEDGMENTS

My personal interest in physics and astronomy developed a well-defined shape first when
I started reading the fascinating collection Library of the Universe by Isaac Asimov, as a
thirteen years old boy in my country of birth, Iran. I realized then where my true passion
resided and what I wanted to pursue in the coming years academically. Seventeen years later
and after traveling through several continents, I am finally on the verge of obtaining a Doctor
of Philosophy degree from Michigan State University in the United States of America.
On that account, several sincere gratitudes are in order. I would like to thank my dear
aunt, Mehry, for her unconditional love and encouragement throughout the difficult times in
my personal and professional life. Without her continuous support, I might not have been
where I am today. I would like to thank Dr. Wayne W. Repko for welcoming me into the
captivating world of physics research, introducing me to a first flavor of what lay ahead. I
owe my professional enthusiasm to his generous supervision. Finally I would like to thank
my advisors Dr. R. Sekhar Chivukula and Dr. Elizabeth H. Simmons for their patience with
me and their amazing professional guidance. I am truly honored to be their pupil.
Thinking about “Mother Nature” and trying to unlock her secrets is an incredibly exciting
experience, and I am humbled by having the opportunity to contribute, no matter how small,
to the world of human knowledge.

v

PREFACE

“The world needs to wake up from its long nightmare of religious belief; and
anything that we scientists can do to weaken the hold of religion should be done,
and may in fact be our greatest contribution to civilization! ”
—Steven Weinberg (1933 – )

The scientific method has been proven to be the only reliable method for unraveling the
mysteries of the natural world. Since the dawn of Homo sapiens’ domination of the earth,
curiosity has been the main thrust behind the slow but steady progress of this “Wise Man”.
The primitive man, himself a product of a long evolutionary process on a planet which from
the cosmological perspective is nothing but a speck of dust in the vast universe, looked with
awe at the heavens and their apparent order. Being inescapably confronted with the “big
questions” of the origin and the nature of life and the universe, this creature’s big brain saw
initially no way out but to speculate about the possible answers, desperately trying to find
temporary relief from curiosity. Indeed, curiosity was a curse from which there appeared to
be no escape.
In the bumpy course of human history, since reaching full behavioral modernity around
50,000 years ago, various cultures and societies have tried to dogmatize their speculative (and
often superstitious) resolutions in an effort to effectively terminate this curiosity “disease”.
To some it seemed appropriate to go so far as attempting to prosecute and eradicate freethought and intellectualism altogether. Despite ignorant opposition, a thirst for knowledge
has always compelled many individuals not to succumb to superficial speculations, but rather
vi

to try to inquire about the natural world through objective observations and evidence-based
empirical studies. Curiosity has played a central role in the development of this scientific
method of studying natural phenomena.
Scientists across different cultures, throughout centuries, have accumulated the acquired
knowledge of those before them and expanded upon it, “standing on the shoulders of giants”
as Isaac Newton put it. The scientific method stands in sharp contrast with the dogmatic
speculations preceding it, and as Steven Weinberg phrased, it has no prophet or authority
but it does have many heroes. In our modern perspective, the evidence-based quest for truth
about nature, inspired by the age-old curiosity and facilitated through falsifiable hypotheses and reproducible experiments, has culminated in what is called the Natural Sciences,
containing physics as a branch.
The magnificent success of science in unlocking the secrets of the nature has had a
tremendous impact on the development of the human intellect and its maturity. It has
enabled us, to a certain extent, to leave behind childlike wishful-thinking and elect the
rational deduction of facts as the correct approach towards understanding our own nature
and our relationship with the world we live in. Objective observation, lying at the heart
of science and the scientific method, provides us with the tools needed to tackle the global
problems facing us as a species. It implies the necessity of employing rationalism as an
alternative to tradition and the superstitious belief systems currently plaguing a substantial
percentage of the human population across the planet, including in the developed world.
My hope is that one day humanity as a whole will achieve a level of maturity to break
the millennia-old shackles of ignorance and embrace its own full intellectual potential which,
if applied correctly, can construct a far better world than the one we are currently living in!

vii

Physics vs. Physique
viii

TABLE OF CONTENTS

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

. . . . . . . . . . . . . . . . . . .

xi

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

xii

. . . . . . . . . . . . . . . . . . . . . . . . . . .

2

List of Figures
Chapter 1

INTRODUCTION

Chapter 2
2.1
2.2
2.3
2.4

2.5
2.6

CUSTODIAL ISOSPIN VIOLATION
STANDARD MODEL1 . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . .
The Lee-Wick Standard Model . . . . . . . . . . .
Effective Field Theory for ∆ρ and Zb¯ . . . . . .
b
Constraints from Post-LEP Parameters . . . . . .
2.4.1 Tree-Level Contributions . . . . . . . . . .
2.4.2 Fermionic One-Loop Contributions . . . .
2.4.3 Comparison with Data . . . . . . . . . . .
b
Constraints from the ZbL¯L Coupling . . . . . .
Conclusion & Discussion . . . . . . . . . . . . . .

IN
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .

THE LEE-WICK
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .

28
28
34
38
43
43
45
49
55
59

GLOBAL SYMMETRIES AND RENORMALIZABILITY OF
LEE-WICK THEORIES2 . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lee-Wick φ4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Symmetries of Lee-Wick Scalar QED . . . . . . . . . . . . . . . . . .
Gauge-Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Ordinary Gauge-Fixing . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 No-Mixing Gauge-Fixing . . . . . . . . . . . . . . . . . . . . . . . . .
One-Loop Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Vacuum Polarization Amplitudes . . . . . . . . . . . . . . . . . . . .
3.5.3 Self-Energy Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.4 φ4 Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61
61
65
69
72
73
75
77
79
82
83
88

Chapter 3
3.1
3.2
3.3
3.4

3.5

1 This chapter is based on the paper first published in [1].
2 This chapter is based on the paper first published in [2].
ix

3.6

3.5.5 Running of g and λ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4
4.1
4.2
4.3
4.4

4.5
4.6
4.7

PRODUCTION OF MASSIVE COLOR-OCTET VECTOR
BOSONS AT NEXT-TO-LEADING ORDER3 . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Minimal Theory for Spin-One Colorons . . . . . . . . . . . . . . . . . . .
LO Coloron Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NLO Coloron Production: Virtual Corrections . . . . . . . . . . . . . . . . .
4.4.1 Quark Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Abelian Vertex Corrections . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Non-Abelian Vertex Corrections a la Pinch-Technique: Unpinched Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.4 Form of the Vacuum Polarization Amplitudes . . . . . . . . . . . . .
4.4.5 Non-Abelian Vertex Corrections a la Pinch-Technique: Pinched Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.6 Full Propagator Correction . . . . . . . . . . . . . . . . . . . . . . . .
4.4.7 Cross Section at One-Loop . . . . . . . . . . . . . . . . . . . . . . . .
NLO Coloron Production: Real Corrections . . . . . . . . . . . . . . . . . .
NLO Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91
93

Chapter 5

97
97
102
106
108
111
113
114
118
122
123
125
127
131
137

CONCLUSION & DISCUSSION . . . . . . . . . . . . . . . . . . 145

APPENDICES . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . 153

Appendix A EVALUATION OF THE φ0 → bL¯L AMPLITUDE . . . . . . . 153
b
Appendix B FEYNMAN RULES OF THE COLORON THEORY . . . . . . 156
Appendix C NUMERICAL VALUES OF THE COLORON K-FACTOR . . 161
Bibliography . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . 166

3 This chapter is based on the paper first published in [3].
x

LIST OF TABLES

Table C.1

K-factors for colorons of various masses, and rL = rR = − tan θc . . 162

Table C.2

K-factors for colorons of various masses, and rL = rR . The classic
“axigluon” [5] corresponds to sin2 θc = 0.50. . . . . . . . . . . . . . 162

Table C.3

K-factors for colorons of various masses, and rL = rR = cot θc . . . . 163

Table C.4

K-factors for KK-gluons of various masses considered in [6]. This
calculation is based on the theoretical framework of [7], with the KKgluon coupling (specified in the column heading) varying between
−0.20gs and −0.40gs . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

xi

LIST OF FIGURES

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 2.1

Figure 2.2

An illustration of the Standard Model matter and force particle content and the Higgs boson. Quarks and leptons come in six different
flavors, and are each organized in three generations. The existence
of the Standard Model Higgs boson remains elusive as of yet. For
interpretation of the references to color in this and all other figures,
the reader is referred to the electronic version of this dissertation. . .

5

Relative strengths of the Standard Model interactions as a function
of energy. The electromagnetic coupling constant (α1 ) increases with
energy scale, while the weak force (α2 ) and the strong force (α3 )
coupling constants decrease, reflecting the asymptoticly free nature
of the latter interactions. . . . . . . . . . . . . . . . . . . . . . . . .

12

Standard Model Higgs potential illustrated. Upon spontaneous symmetry breaking, the extremum at zero vacuum expectation value
(VEV) becomes unstable and the system is forced to acquire a stable non-zero VEV at a randomly chosen point around the bottom of
the hat. The degree of freedom along the radial direction (along the
arrow) corresponds to the massive Higgs boson, while the degree of
freedom perpendicular to it (on the equipotential circle at the bottom
of the hat) is the massless Nambu-Goldstone boson. . . . . . . . . .

16

Experimentally measured values of various SM parameters. Columns
on right illustrate deviations from the SM theoretical prediction, ranging anywhere between 0 to 3 standard deviations. . . . . . . . . . .

18

ˆ
Diagrams contributing to the dimension-six four-φ operators in the
effective theory, with the LW fermions integrated out at tree-level. .

40

Diagrams contributing to the dimension-six operators with two external qL and two ϕ fields. The triangle diagrams lead to the second
ˆ
operator of Eq. (2.16), which contains non-universal corrections to
the ZbL¯L coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . .
b

41

xii

Figure 2.3

Figure 2.4

The XY VPA, defined as inverse of the full XY propagator, DXY
(see footnote 15 for details). . . . . . . . . . . . . . . . . . . . . . .

44

Dominant vacuum polarization amplitudes for the LW SM gauge
fields. These include the ordinary (t0 and b0 ) and the LW third
generation quarks (t1 , t2 , b1 , and b2 ) in the loop. These amplitudes
contribute to the two-point functions of Eq. (2.19). . . . . . . . . . .

46

Figure 2.5

Exclusion plot for the LW gauge-field masses M2 and M1 . These
bounds are due to the constraints on Y and W , as shown by Eq. (2.25)
and Eq. (2.26). For a light Higgs (mh = 115 GeV) the striped region
to the left of both curves is excluded. For a heavy Higgs (mh = 800
GeV) the additional yellow strip between the curves is excluded as well. 50

Figure 2.6

The 95% C.L. ellipses in the (Y ,W ) plane, and the LW prediction for
degenerate masses, M1 = M2 . The parametric plot is for 0.5 TeV <
M1 = M2 < 10 TeV, and the dots are equally spaced in mass every
0.95TeV (only two dots labeled). The lower bound on M1 = M2 is
approximately 2.4 TeV for a light Higgs. . . . . . . . . . . . . . . . .

51

ˆ
T as a function of Mq in the degenerate case, Mq = Mt . The experˆ
imental mean value for T is shown by the thick red line, along with
2
the ±2σ allowed region. Also shown are the all-order (in v 2 /Mq )
LW prediction (solid blue curve), the leading order LW prediction,
Eq. (2.30) (dashed blue curve), and the leading-log curve, Eq. (2.15)
(dotted blue curve), as functions of Mq , in the degenerate case. Note
that the leading order prediction is not valid below Mq ∼ 1 TeV. (See
text for details) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

The 95% C.L. exclusion plots for the LW fermion masses Mq and Mt .
These bounds come almost entirely from the experimental constraints
ˆ
on T . For a light Higgs the striped region to the left of the curve is
excluded, while a heavy Higgs is completely excluded. . . . . . . . .

53

ˆ ˆ
The 95% C.L. ellipses in the (S,T ) plane, and the LW prediction for
degenerate masses, Mq = Mt . The parametric plot is for 0.5 TeV <
Mq < 10 TeV and the dots are equally spaced in mass every 0.95TeV
(only two dots labeled). The lower bound on Mq is approximately
1.6 TeV for a light Higgs. . . . . . . . . . . . . . . . . . . . . . . . .

54

Figure 2.7

Figure 2.8

Figure 2.9

xiii

Figure 2.10

Figure 2.11

Figure 3.1

Figure 3.2

Figure 3.3

Diagram giving the largest contribution to the ZbL¯L coupling. The
b
latter is related through the Ward identity to the φ0 bL¯L coupling
b
0 is the Nambu-Goldstone boson eaten by the Z boson.
— where φ
The top quarks running in the loop are both ordinary and LW. . . .

57

Constraints from the ZbL¯L coupling. This graph features the experb
imental mean value (thick horizontal red line), the 2σ allowed region
below the mean value, the SM prediction (solid horizontal black line),
2
the all-order (in v 2 /Mq ) LW prediction (solid blue curve), the leading order LW prediction, Eq. (2.34) (dashed blue curve), and the
leading-log approximation (dotted blue curve), as functions of Mq , in
the degenerate case. . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

One-loop contribution to the vacuum polarization amplitudes. Each
external vector field is either a photon or a LW photon. . . . . . . .

82

One-loop contribution to the 1PI self-energy amplitude Σφφ in the
no-mixing ξ = 1 gauge. In the ordinary gauge there are also diagrams
involving internal mixed gauge propagators, P ˜. . . . . . . . . . .
AA
One-loop contribution to the 1PI amplitude with four external φ
fields. The number of diagrams is reduced by employing the hatfield propagators. (See text for details) . . . . . . . . . . . . . . . .

84

89

Figure 4.1

a
Tree-level contribution to coloron production. The coloron field, Cν ,
is represented by the zigzag line. . . . . . . . . . . . . . . . . . . . . 107

Figure 4.2

Structure of q q → C amplitude, to all orders in perturbation theory.
¯
Direct coloron production is illustrated on the left, while production
via mixing with the gluon is shown on the right. The gluon field is, as
usual, represented by the coiling line; the coloron field is represented
by the zigzag line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 4.3

Quark self-energy diagrams at one-loop. Particle notation as defined
in Fig. 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Figure 4.4

One-loop Abelian vertex correction to the q q → C amplitude. Parti¯
cle notation is as defined in Fig. 4.2. . . . . . . . . . . . . . . . . . . 113

Figure 4.5

One-loop non-Abelian vertex correction to the q q → C amplitude.
¯
Particle notation is as defined in Fig. 4.2. Each three-gauge boson
vertex is a full non-Abelian vertex Γµνρ in Eq. (4.37). . . . . . . . . 114
xiv

Figure 4.6

Non-Abelian unpinched vertex-correction diagrams for the q q → C
¯
amplitude at one-loop. Particle notation is as defined in Fig. 4.2.
The black disk indicates that each three-point gauge boson vertex in
µνρ
these diagrams has been replaced by the non-pinched portion ΓF ,
as described in Eqs. (4.38) and (4.39). . . . . . . . . . . . . . . . . 116

Figure 4.7

Non-Abelian pinched vertex-correction diagrams for the q q → C am¯
plitude at one-loop. Particle notation is as defined in Fig. 4.2. . . . . 117

Figure 4.8

Coloron-coloron vacuum polarization amplitude at one-loop. A gluon
field is, as usual, represented by a coiling line; a coloron field is represented by a zigzag line. The coloron ghost is represented by a sequence
of filled circles, and the eaten Goldstone bosons are represented by
dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 4.9

Gluon-coloron mixing amplitude at one-loop. Particle notation is as
defined in Fig. 4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 4.10

Fermion contributions (solid lines) to coloron-coloron vacuum polarization amplitude and gluon-coloron mixing amplitude. Particle notation is as defined in Fig. 4.8. . . . . . . . . . . . . . . . . . . . . . 119

Figure 4.11

The relevant contributions to the coloron Dyson series; as before, the
zigzag lines represent colorons. The first row is the sum of the coloron
VPA diagrams in the propagator, while the second row represents
the sum of the VPA diagrams on top of the one-loop contribution
from the pinched vertex correction (the double curly line illustrates
generically all the allowed gauge bosons in the original non-Abelian
vertices). The overall pinched amplitude factors out, and has no effect
on the coloron pole mass. . . . . . . . . . . . . . . . . . . . . . . . . 123

Figure 4.12

Diagrams contributing to the real emission processes. A gluon field is,
as usual, represented by a coiling line; a coloron field is represented by
a zigzag line. (a) Gluon emission. (b) Quark emission. (c) Antiquark
emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
xv

Figure 4.13

Figure 4.14

Behavior of the Q function defined in Eq. (4.84), for µ = MC : this
gives the contribution from the virtual corrections to the NLO cross
section for coloron production. The upper curve is for the rL = rR
scenario, whereas the almost identical lower curves are for rL = rR =
− tan θc , and rL = rR = cot θc . Note that Q, and, therefore, the
NLO corrections, become very large when sin2 θc is either too small
or too large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
√
Dependence of LO and NLO cross sections at the LHC ( s = 7
TeV), as a function of factorization scale µF for MC = 2.0 TeV,
sin2 θc |µ=2.0 TeV = 0.25, and the three possible flavor-universal
scenarios for the quark charge assignments. As expected, the NLO
cross section has a much weaker (formally, two-loop) residual scaledependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 4.15

NLO cross section times branching ratio to quarks for on-shell coloron
√
production at the LHC ( s = 7 TeV), corrected for acceptance as
described in the text. We consider the three possible flavor-universal
scenarios for the quark charge assignments, take the renormalization
scale µ to be equal to MC , and plot σ for sin2 θc |µ=M = 0.05
C
(dashed), 0.25 (dot-dashed), and 0.5 (dotted). We plot these cross
sections for µF ranging from MC /2 to 2 MC and, reflecting the weak
dependence of the NLO cross section on the factorization scale, the
resulting bands for each sin2 θc are very narrow. To give a sense of
current experimental reach, we plot the CMS [4] upper limit (solid
line) on the cross section times dijet branching ratio for a narrow
resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 4.16

“K-factor”, the ratio of the NLO to LO cross section for coloron
√
production at the LHC ( s = 7 TeV), plotted as a function of MC for
sin2 θc = 0.05 (dashed), 0.25 (dot-dashed) and 0.50 (dotted), µF =
MC , and the three different quark charge assignments. . . . . . . . 142

Figure 4.17

Fraction of colorons produced with a pT greater than pT min , as a
function of pT min . The curves are for MC = 1.2 (highest, thin
blue line), 2.0 (middle, medium purple line), and 3.0 TeV (lowest,
thick green line), for the vectorial case rL = rR = − tan θc and
sin2 θc = 0.05. Note that of order 30% of the colorons in this mass
range are produced with pT ≥ 200 GeV. As denoted by the red
shaded region, below a pT of 200 GeV the corrections become larger
2
than 30%, terms proportional to log(MC /p2 min ) become large, and
T
this fixed-order calculation becomes unreliable. . . . . . . . . . . . . 144
xvi

Figure B.1

Feynman rules for the trilinear vertices. In each diagram the momenta
are toward the vertex. A gluon field is, as usual, represented by
a coiling line; a coloron field is represented by a zigzag line. The
coloron ghost is represented by a sequence of filled circles, and the
eaten Goldstone bosons are represented by dashed lines. . . . . . . . 157

Figure B.2

Feynman rules for the trilinear vertices (continued). . . . . . . . . . 158

Figure B.3

Feynman rules for the quartic vertices. A gluon field is, as usual,
represented by a coiling line; a coloron field is represented by a zigzag
line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Figure B.4

Feynman rules for the quartic vertices (continued). . . . . . . . . . . 160

xvii

‫ﮔﻮ2ﻨﺪ ﮐﺴﺎن ﺑﻬﺸﺖ ﺑﺎ ﺣﻮر ﺧﻮش اﺳﺖ‬
‫ﻣﻦ ﻣﯽ ﮔﻮ2ﻢ ﮐﻪ آب اﻧﮕﻮر ﺧﻮش اﺳﺖ‬
‫ا2ﻦ ﻧﻘﺪ ﺑﮕ=ﺮ و دﺳﺖ از آن ﻧﺴ=ﻪ ﺑﺪار‬
‫ﮐﺂواز دﻫﻞ ﺷﻨ=ﺪن از دور ﺧﻮش اﺳﺖ‬
‫ﺧ=ّﺎم‬

“How sweet is mortal Sovranty!” -think some:
Others- “How blest the Paradise to come!”
Ah, take the Cash in hand and waive the Rest;
Oh, the brave Music of a distant Drum!
-Omar Khayyám (1048-1131)
Persian polymath, philosopher, mathematician, astronomer, and poet
Translation by Edward FitzGerald (1809-1883)

1

Chapter 1
INTRODUCTION

“The only thing that interferes with my learning is my education! ”
—Albert Einstein (1879 – 1955)

Modern particle physics revolves around the idea of all matter and energy being composed of elementary undividable discrete constituents. In the modern physics nomenclature,
this is referred to as “quantization”. This concept dates back to at least the 6th century BC,
proposed first by ancient Greek philosophers such as Leucippus, Democritus, and Epicurus,
ancient Indian philosophers such as Kanada, Dign¯ga, and Dharmakirti, and later studied
a
by medieval Persian scientists Alhazen and Avicenna, among others. Democritus was the
first to have coined the term ´tomos, meaning “indivisible”, to describe these elementary
a
constituents. The modern concept of atoms was introduced in the 19th century, through
the work of John Dalton in chemistry, who thought of atoms as the fundamental particles of
nature, and hence, adopted Democritus’ terminology. By the early 20th century, however,
2

it became evident through the experiments of Rutherford and others that atoms were not
indivisible, but instead were composed of even smaller components themselves.1
On the theoretical side, the special theory of relativity was discovered by Albert Einstein
in 1905. Furthermore, in his account for the photoelectric effect, Einstein proposed a radical
concept regarding the nature of light, describing it in terms of discrete packets of energy
(called photons). These prominent theoretical breakthroughs led to a profound change in
scientific understanding of the physical world in early 20th century. The notion of quantization of light, in particular, paved the way for the subsequent discovery and development
of quantum mechanics through the work of Niels Bohr, Wolfgang Pauli, Werner Heisenberg,
and Erwin Schr¨dinger, to name a few. Experimental discovery and identification of many
o
subatomic particles and their properties, such as the electron, proton, and neutron, together
with their quantum mechanical description, gave rise to ever more detailed models of atoms
in the first half of the 20th century.
The high energy scattering experiments throughout the 1950s and 1960s uncovered the
spectrum of a variety of new particles, initially all thought to be “fundamental” exhibiting
no further substructure, referred to as the “particle zoo”. Meanwhile, theoretical progress
concerning the incorporation of the special theory of relativity into quantum mechanics,
initiated by Paul Dirac in 1920s, gave birth to the development of a more comprehensive
quantum theory by the mid 20th century, called the quantum field theory. Quantum field
theory consistently describes all matter as being composed of point-like fundamental particles, and the interactions among them as being mediated by various quantized pockets
1 There are many books dedicated to reviewing the history of modern physics, with an
emphasis on particle physics and gravity. The enthusiastic reader is encouraged to consult
e.g. [8]-[11], among many other excellent reviews.
3

of energy, themselves also identified as fundamental particles. In light of this theory, the
particle zoo could be explained in terms of different bound-states of a limited number of
fundamental particles and their quantized interactions; this dramatically simplified the high
energy description of matter and interaction forces.2
Accordingly, many of the initially considered “fundamental” particles in the zoo turned
out to be described in terms of even more fundamental constituents, and hence, the notion
of being fundamental was now reserved to apply only to a limited number of particles from
which all others were composed. Initially, the existence of many of these truly fundamental
particles were merely postulated in the quantum field theoretical description of the particle
zoo; however, the experiments in the final decades of the 20th century unambiguously verified
the existence of all but one of these postulated particles, confirming their lack of substructure
up to the accessible energy scales.
According to the experiments, the currently accepted picture of the elementary constituents of matter, exhibiting no further substructure, contains the following: quarks existing in six different varieties commonly referred to as “flavors” (up, down, charm, strange,
top, and bottom), and six leptons consisting of electron, muon, and tau, each accompanied
by a neutrino of the corresponding flavor3 (Fig. 1.1). Quarks and leptons are organized in
three “generations” or “families”, and are all spin–

1
particles, which makes them fermions
2

(named after Enrico Fermi who made, among other things, important contributions towards
understanding the behavior of these particles). A direct prediction of quantum field theory
is that all of these particles have a partner with the same mass but opposite electric charge,
2 The gravitational force has not yet been described consistently by a quantized field
theory.
3 In light of the discovery of neutrino oscillations, the actual three neutrino particles have
tiny masses, and are linear combinations of the flavor neutrinos described above.
4

Figure 1.1: An illustration of the Standard Model matter and force particle content and the
Higgs boson. Quarks and leptons come in six different flavors, and are each organized in
three generations. The existence of the Standard Model Higgs boson remains elusive as of
yet. For interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this dissertation.
(Figure courtesy of Fermilab Visual Media Services. The small text inside each box denotes
the particle’s name; it is not intended to be necessarily readable and is for visual reference
only. http://www-visualmedia.fnal.gov/VMS_Site/gallery/stillphotos/2005/0400/
05-0440-01D.hr.jpg)

forming the fundamental constituents of antimatter.
Quarks tend to combine in special manners to form bound-states, called hadrons. Hadron
5

is a collective name for baryons (composed of three quarks) and mesons (composed of a quark
and an antiquark). The proton (two up quarks and one down quark) and neutron (one up
quark and two down quarks) are, therefore, baryons in this classification. Various numbers
of protons and neutrons bind together to form the nuclei of atoms of different elements in
nature, accompanied by an appropriate number of electrons in orbits around the nucleus to
make the atom electrically neutral. Hence, all known forms of matter, from simple elements
in the periodic table to complex molecules and structures made of them, can be traced back
to the bound-states of a handful of elementary particles.
Next, let us take a look at the quantized interactions in nature among the matter particles.
There are four known forces in nature: gravity (attractive between masses), electromagnetism
(attractive between opposite charges and repulsive between like charges), the strong nuclear
force (binding quarks inside the proton and neutron, and binding protons and neutrons inside
the nucleus), and the weak nuclear force (responsible for certain radioactive decays). In the
quantized description provided by quantum field theory, electromagnetism is mediated by
a massless, electrically neutral particle, called the photon, interacting only with electrically
charged particles. The strong interaction has eight massless, electrically neutral mediators,
called gluons, and is felt only by particles carrying the color quantum number, i.e. quarks
and gluons. Color can be perceived as the analogue “charge” appropriate for the strong
interaction; hence leptons, being colorless, are not affected by the strong force. The weak
interaction is mediated by three massive mediators: W + , W − , and Z vector bosons, where
superscripts represent the appropriate electric charges. The weak force is felt by all the
left-handed 4 quarks and leptons. All of these fundamental force-carriers are spin-1 particles,
4 In the relativistic limit, a particle is called right-handed if its direction of spin is parallel
to the direction of motion, and left-handed if the spin direction is anti-parallel to its motion.
6

which makes them bosons (named after Indian mathematician and physicist Satyendra Nath
Bose, for his work related to Bose-Einstein statistics and the theory of the Bose-Einstein
condensate).
A consistent quantum field theoretical description of gravity currently does not exist. A
proposed quantized mediator of gravity takes the form of a massless, electrically neutral,
spin-2 particle, called the graviton, with a tensor nature (as opposed to the spin-1 vector
mediators of the other three forces). This is motivated by the macroscopic properties of
gravitational fields as described by the general theory of relativity. However, since the rest
masses of the elementary particles are extremely small, and the gravitational interaction is
much weaker than the other forces, gravity can be safely neglected in the relevant energy
scales currently under study in particle physics.
The quantum field theoretical description of the electromagnetic, strong, and weak forces,
and their interactions with quarks and leptons is collectively called the Standard Model of
particle physics (SM). It represents the currently accepted theory explaining the particle zoo
and other observed high energy phenomena with high precision. A final elementary particle
postulated in the SM, but yet remaining undiscovered, is a massive, electrically neutral,
spin-0 particle, called the Higgs boson (named after Peter Higgs, one of the contributors
to the description of the so-called “Higgs mechanism” of spontaneous symmetry breaking).
This scalar particle has been postulated in order to explain the origin of the masses of the
quarks and leptons. Incorporation of the Higgs scalar into the SM leads, however, to the
so-called Hierarchy Problem. Solving the hierarchy problem is currently one of the main
topics of research in particle physics. A potential solution to the Hierarchy problem will be
discussed at length in Part I of this Thesis.
7

At this point, let us turn to a more comprehensive description of the SM. In the 1950s
and 1960s, it was realized by Chen Ning Yang, Robert Mills, Martinus Veltman, and many
others that quantized “gauge theories” provided a promising candidate for the quantum
description of the forces encountered in nature. A familiar classical example of a gauge
parameter would be the location where the gravitational potential field equals zero in the
Newtonian gravity. This represents a global free parameter in the theory, since its value
is a constant, independent of space-time. Depending on the problem at hand, the gauge
parameter may be freely chosen without affecting the outcome of the predictions for physical
observables, such as a particle’s kinetic energy in this potential field. The same observation
is true in quantum field theoretical description of interactions, although in a slightly more
complicated context. A proper isolation and treatment of the gauge parameter turns out to
be of extreme importance and leads to consistent theories which may describe a particular
interaction with properties obtained from experiment.

Let us examine more carefully how gauge theories emerge in context of quantum field
theory. Imagine a particular quantum field theory possessing a specific global continuous
mathematical symmetry; i.e. the theory remains invariant under a continuous symmetry
transforming operation. The branch of mathematics dealing with symmetries is called group
theory. As mentioned before, “global” means that the mathematical parameter in group
theory describing the symmetry is not a function of space-time; in other words, this symmetry
parameter is a constant across space-time of the theory. In that sense, the global continuous
symmetry of our quantum field theory is described by the particular mathematical group
associated with the symmetry under consideration, leaving the theory invariant. Given this,
if the ground state (vacuum) of the system does not exhibit the same global continuous
8

symmetry, the symmetry is said to be “spontaneously broken” in this quantum field theory.
In other words, non-invariance of the ground state serves as the condition for a spontaneous
symmetry breaking (SSB), even though the Hamiltonian or Lagrangian of theory is still
fully invariant under the symmetry transformation. Upon a spontaneous breaking of a
global continuous symmetry, according to Goldstone’s theorem [12] (named after Jeffrey
Goldstone who formulated and proved the theorem), a massless scalar particle, called the
Nambu-Goldstone boson, will be generated. This is a peculiarity of quantum field theories
with spontaneously broken global continuous symmetries.
The situation turns out to be more interesting if our continuous symmetry is “local”; i.e.
the symmetry parameter is a function of space-time of the theory. Such a quantum field
theory is then called a gauge theory. The reason for the terminology is that in order for the
theory to be invariant under the local continuous symmetry transformation, a new massless
vector degree of freedom, containing a free gauge parameter, must be introduced. Mathematically, this vector particle acts as if to “connect” different space-time points that have
different symmetry parameters.5 Hence, a quantum field theory possessing one or more local
continuous symmetries necessarily contains massless vector bosons (one for each symmetry
generator), which makes it quite tempting as a suitable candidate for describing quantized
interactions in nature, mediated by massless gauge bosons, such as electromagnetism and
the strong interaction.
What would happen if the local continuous symmetry of this gauge theory were spontaneously broken? According to Goldstone’s theorem, again a massless Nambu-Goldstone
boson is released. It turns out, however, that this Nambu-Goldstone boson combines with
5 Geometrically, the situation is analogous to “parallel transporting” a vector on a curved
surface, which leads necessarily to the introduction of the “Christoffel symbols”.
9

(or as is commonly referred to, is “eaten” by) the vector boson of the gauge theory, making the latter massive. The previously massless gauge boson with only two (transverse)
polarizations has now become massive and, consequently, has acquired a third (longitudinal) polarization component, which is exactly provided by the “eaten” Nambu-Goldstone
boson. Hence, spontaneously broken gauge theories contain massive gauge bosons (one for
each broken symmetry generator), making them again tempting as suitable candidates for
describing quantized interactions in nature, mediated by massive gauge bosons, such as the
weak interaction. Let us discuss how these ideas are adopted in practice.
The quantum field theoretical description of electromagnetism, involving photons, is
called Quantum Electrodynamics (QED), developed among others by Julian Schwinger [13,
14], Sin-Itiro Tomonaga [15], Freeman Dyson [16], and Richard Feynman [17]. Feynman
called it “the jewel of physics” for its extremely accurate predications of various physical
observables, such as the anomalous magnetic moment of the electron and the so-called Lamb
shift of the energy levels of hydrogen. It represents the first successful application of quantum field theory to the physics of elementary particles, describing the quantum nature of
the interaction between light and electrically charged matter. QED is a gauge theory with
a local continuous U (1) symmetry, representing an unobservable (local) phase. The U (1)
symmetry has only one generator and is unbroken; therefore, QED contains one massless,
electrically neutral gauge boson, identified with the photon. Since photons themselves do
not carry electric charge, they do not interact with one another. The Abelian 6 nature of the
U (1) group mirrors directly this physical property of the theory.
6 In group theory, a group is called Abelian if its generators commute with one another;
otherwise, it is referred to as non-Abelian. U (1) is a trivial example of an Abelian group,
since it has only one generator.
10

The QED vacuum is a dynamical “polarizable medium”; i.e. it contains virtual electronpositron pairs which are constantly created and annihilated for a short period of time, according to the Heisenberg uncertainty principle. In the presence of a “bare” charge (which is
formally infinite), these virtual pairs become polarized and induce a net screening of the bare
charge at low energies, corresponding to a finite observed charge at long distance scales.7
At high energies, shorter distance scales from the bare charge are probed, which are less
screened by the dynamical vacuum. This results in a larger observed charge and a stronger
perceived electromagnetic force. As a consequence, in QED the strength of the electromagnetic interaction increases (logarithmically) with energy (see α1 in Fig. 1.2); in other words,
the electromagnetic force becomes stronger at shorter distance scales.
Strong interaction field theory is based on an SU (3) unbroken local symmetry, and is
called Quantum Chromodynamics (QCD). It was developed by Chen Ning Yang and Robert
Mills [18], Murray Gell-Mann [19], and many others. The SU (3) group consists of the set of
Special Unitary 3 × 3 matrices, and is a non-Abelian theory. It involves the color quantum
number (analogous to the charge in QED), which is only present in quarks and gluons.
The symmetry group is, therefore, formally designated as SU (3)C (subscript C standing
for color). The QCD’s SU (3)C symmetry group contains eight generators, corresponding
to eight massless gluons as force-carriers. In contrast with the photon, which itself does not
carry electric charge, gluons do carry color quantum numbers and interact among themselves.
This fact is reflected in the non-Abelian nature of the theory. Moreover, contrary to the
Abelian QED, non-Abelian theories have as a peculiarity that the strength of the force
decreases logarithmically with increasing energy or equivalently at shorter distance scales (see

7 See the renormalization discussion later on in this chapter.
11

Figure 1.2: Relative strengths of the Standard Model interactions as a function of energy.
The electromagnetic coupling constant (α1 ) increases with energy scale, while the weak force
(α2 ) and the strong force (α3 ) coupling constants decrease, reflecting the asymptoticly free
nature of the latter interactions.
(Figure courtesy of c The Royal Swedish Academy of Sciences. “The 2004 Nobel Prize in
Physics - Popular Information”. Nobelprize.org. 6 Apr 2012. http://www.nobelprize.
org/nobel_prizes/physics/laureates/2004/popular.html)

α3 in Fig. 1.2); in other words, quarks and gluons are practically free (i.e. non-interacting)
inside hadrons at close distances. This phenomenon is referred to as Asymptotic Freedom
[20, 21], and makes a perturbative treatment of strong interaction possible at high energies.
Although not yet formally proven, it is generally assumed that QCD predicts the appearance
12

of only colorless bound-states of quarks in nature; i.e. all hadrons are colorless, in agreement
with observations. This is called color confinement.
As mentioned previously, the asymptoticly free nature lying at the heart of the strong
interaction makes it possible to invoke perturbation theory at high energies, since the strength
of the interaction decreases with an increasing energy. To be precise, in a perturbative
treatment the theory is expanded in terms of an asymptotic series as a function of its (small)
coupling constant (α3 for the strong force in Fig. 1.2). The leading order term (LO) in the
perturbative expansion corresponds to the classical level interaction, and is called the “treelevel” interaction. The subsequent terms in the expansion form the so-called “quantum
corrections” to the tree-level interaction with an increasing level of complexity. Each of
the higher order quantum corrections is, however, proportional to a higher power of the
perturbative coupling constant, and is, therefore, increasingly suppressed. A perturbative
analysis in quantum field theory is, hence, only meaningful so long as the coupling constant of
the theory under consideration remains small, increasingly suppressing the higher powers of
quantum corrections. Throughout this Thesis, we will extensively make use of perturbation
theory, confining the analyses to the energy regions where such a treatment remains valid.
A field theoretical description of the weak force with three massive gauge bosons, two of
which carry opposite electric charges, is quite subtle and hints at a gauge theory with spontaneously broken local continuous symmetries (to explain the masses), and involving some
degree of mixture with the electromagnetic U (1) symmetry (to explain the electric charges).
Sheldon Glashow [22], Steven Weinberg [23], and Abdus Salam [24] were independently able
to show that a gauge theory based on an SU (2)L × U (1)Y symmetry, spontaneously broken to U (1)EM , produces three massive gauge bosons (corresponding to the three broken
13

generators) with positive, negative, and neutral electric charges, along with another gauge
boson that is massless and electrically neutral (corresponding to the unbroken generator).
As previously noted, the weak interactions affect only left-handed fermions, or equivalently,
right-handed antifermions. The subscript L, thus, refers to the left-handed nature of the
weak force. The subscript Y represents “hyper-charge”, a U (1) quantum number for the
particles in the unbroken phase. Upon the spontaneous symmetry breaking, the surviving
U (1) symmetry is identified with the ordinary QED (hence, the subscript EM); the massless
gauge boson associated with it is identified with the photon, while the three massive gauge
bosons are the W + , W − , and Z vector bosons, the force-carriers of the weak interaction.
In that sense, SU (2)L × U (1)Y symmetry group provides a unified gauge description of
weak and electromagnetic forces, and is, therefore, called Electroweak theory. It should be
emphasized that electroweak theory represents only a unified gauge description, rather than
a true physical unification of weak and electromagnetic interactions, since the weak and
electromagnetic coupling strengths remain distinct (Fig. 1.2).
At this stage, let us define more precisely the Higgs mechanism [25]-[30] responsible for
electroweak symmetry breaking. In electroweak theory, the SSB is achieved by introducing
a potential energy function for a complex scalar SU (2) doublet (denoted as φ in Fig. 1.3),
effectively adding four extra degrees of freedom to the theory. As depicted in Fig. 1.3, the
potential develops a Mexican hat form, with the stable minima consisting of points along
a circle at the bottom of the hat, representing an infinite number of degenerate vacua with
non-zero values (note that the vacuum with the zero expectation value is unstable). One
may choose any point on this circle8 as the true vacuum of the theory, thereby, allowing the
8 The different points correspond to different choices of gauge and are, therefore, physically
equivalent.
14

development of a particular non-zero vacuum expectation value (VEV). This choice leads to
an explicit spontaneous breaking of electroweak symmetry as described above. Fluctuations
around this non-zero VEV redeploys three massless scalar degrees of freedom as the NambuGoldstone bosons (one for each broken symmetry generator), as well as one massive scalar
state, which is called the Higgs boson. In Fig. 1.3, the Nambu-Goldstone bosons correspond to
fluctuations along the equipotential circle at the bottom, which is reflected in their massless
nature. The Higgs boson, on the other hand, is generated by fluctuations in the radial
direction where the potential changes, inducing a mass term for this degree of freedom.
In electroweak theory, additionally, the coupling of the scalar doublet to fermions is
responsible for the fermion masses, with their masses proportional to the strength of the
associated couplings. This type of coupling is called the Yukawa coupling (named after
Hideki Yukawa who was the first to introduce a coupling between scalars and fermions in
context of a different theory). The massive weak gauge bosons acquire their masses by eating
the Nambu-Goldstone bosons, as discussed earlier. Therefore, in electroweak theory, gauge
bosons and fermions are massless in the unbroken electroweak symmetry phase, and become
massive in the broken phase. Even though the Higgs mechanism provides, among other
things, an economical solution to mass generation for both fermions and weak gauge bosons,
the Higgs boson itself remains elusive so far, and forms the last missing piece of the SM.9
To summarize, the SM has an SU (3)C × SU (2)L × U (1)Y gauge structure with a Higgs
mechanism for the SSB, and encompasses all known subatomic particles and their interac9 As of early 2012, the searches at the Large Hardon Collider (LHC) may indicate the
existence of a SM Higgs boson at around 2σ statistical significance. This implies that the
SM Higgs boson, if existed, is most likely to have a mass constrained to the range 116131 GeV by the ATLAS experiment [31]-[33], and 115-127 GeV by CMS [34]-[37]. These
findings are, however, statistically not strong enough to claim a discovery (the 5σ threshold)
and await the accumulation of more data.
15

Figure 1.3: Standard Model Higgs potential illustrated. Upon spontaneous symmetry breaking, the extremum at zero vacuum expectation value (VEV) becomes unstable and the system
is forced to acquire a stable non-zero VEV at a randomly chosen point around the bottom
of the hat. The degree of freedom along the radial direction (along the arrow) corresponds
to the massive Higgs boson, while the degree of freedom perpendicular to it (on the equipotential circle at the bottom of the hat) is the massless Nambu-Goldstone boson.
´
(Figure courtesy of Nature Publishing Group. “Eyes on a prize particle”, Luis AlvarezGaum´ & John Ellis, Nature Physics 7, 2-3 (2011) | doi:10.1038/nphys1874. http://www.
e
nature.com/nphys/journal/v7/n1/fig_tab/nphys1874_F1.html)
tions. Furthermore, using the experimentally determined values of a number of parameters
as input, the SM can, given its renormalizable nature, make specific predictions regarding
the outcome of many other experiments with, in principle, an arbitrary degree of accuracy.
In order to test the SM predictions in the electroweak sector, a number of high precision
experiments, the so-called electroweak precision tests, have been conducted. Data collected
through these experiments can be parametrized in various manners,10 in order to make com10 See Chapter for an elaborate discussion on these electroweak parametrizations.
16

parison with the theoretical predictions, and place tight constraints on the numerical values
of many of the SM input parameters.11 Fig. 1.4 summarizes the experimentally measured
values of a number of parameters predicted in the SM, and compares them with the theoretical prediction, with the highest deviation lying within a 3σ bound. As one may appreciate,
the SM is impressively successful in predicting the values of many observables, and passes
the most stringent experimental tests.
As impressive as the SM might sound, there are, nevertheless, a number of theoretical
and observational indications suggesting that it cannot represent a complete and final account of the quantum world, as the ultimate theory of nature. These indications demand
explicit extensions and improvement of the SM, requiring inclusion of some new physics. An
illustrative list of SM shortcomings contains the following observations:

• The SM does not contain a quantum field theoretical description of gravity;
• As indicated in Fig. 1.2, the strength of the electromagnetic, strong, and weak interactions do not meet at one single energy. If, based on theoretical considerations, one
is to take the idea of unifying the interactions seriously, the SM does not unify the
coupling constants of the three forces, neither does it provide a unified description of
all three as different manifestations of one fundamental interaction;

• There are over twenty free parameters in the SM, the origin of which remains unexplained within the framework of the theory, and their numerical values need to be
considered as input to the SM;
11 As we shall see, experimental deviations from the SM predicted value for these parameters can be attributed to new physics, allowing one to place (lower) bounds on the value of
various beyond the Standard Model (BSM) variables.
17

Measurement

Fit

meas

fit

91.1875 ± 0.0021
2.4952 ± 0.0023

0

41.540 ± 0.037

41.478

Rl

20.767 ± 0.025

1

2.4959

σhad [nb]

3

91.1874

ΓZ [GeV]

meas

0.02750 ± 0.00033 0.02759

mZ [GeV]

−O |/σ
1
2

0

(5)
Δαhad(mZ)

|O
0

20.742

0,l

Afb

Al(Pτ)
Rb

0.01714 ± 0.00095 0.01646
0.1465 ± 0.0032

0.1482

0.21629 ± 0.00066 0.21579

Rc

0.1721 ± 0.0030

0.1722

Afb

0,b

0.0992 ± 0.0016

0.1039

Afb

0,c

0.0707 ± 0.0035

0.0743

Ab

0.923 ± 0.020

0.935

Ac

0.670 ± 0.027

0.668

0.1513 ± 0.0021

0.1482

sin θeff (Qfb) 0.2324 ± 0.0012

0.2314

mW [GeV]

80.399 ± 0.023

80.378

ΓW [GeV]

2.085 ± 0.042

2.092

Al(SLD)
2 lept

mt [GeV]

173.20 ± 0.90

173.27

July 2011

2

3

Figure 1.4: Experimentally measured values of various SM parameters. Columns on right
illustrate deviations from the SM theoretical prediction, ranging anywhere between 0 to 3
standard deviations.
(Figure courtesy of LEP/TEV Electroweak Working Group (EW WG). “Preliminary constraints on the Standard Model” | preprint: arXiv:1012.2367 [hep-ex], updated for 2012 winter conferences. http://lepewwg.web.cern.ch/LEPEWWG/plots/summer2011/s11_show_
pull_18.pdf)

18

• As presented in Fig. 1.4, the SM predicted outcome for a number of experimental
parameters deviates up to 3σ from the experimentally determined value, entertaining
the possibility of some new physics substantially influencing those experiments;
• Dark matter, its existence inferred from various sources of astronomical and cosmological observations, is not explained within the context of the SM. In other words, the
particle content of the SM (Fig. 1.1) cannot account for the observed dark matter in
the universe;
• The SM prediction for the value of dark energy, presumably responsible for the observed
accelerating expansion of the universe, is far too large by many orders of magnitude
and is physically excluded;
• The SM cannot account for the observed dominance of matter over antimatter in our
universe;
• The SM provides no justification for the observed homogeneity and isotropy of the
universe at large distance scales;
• Quantum corrections to the mass of the predicted Higgs boson are quadratically divergent (see below), making its mass highly sensitive to the ultra high energy behavior
of the theory, around the Planck scale where gravity presumably becomes important.
These corrections, therefore, tend to generate a large mass for the Higgs boson. In
contrast, the actual mass of the Higgs boson must naturally lie near or below the electroweak symmetry breaking scale, many orders of magnitude smaller than the Planck
scale. This requires a severe fine-tuning of the Higgs mass (and other SM parameters
related to it), in order to remove the enormous contribution of the quadratically di19

vergent quantum corrections, and produce a natural weak scale “light” Higgs. This
spectacular cancellation is known as the Hierarchy problem. Another way to rephrase
the Hierarchy problem is to contemplate why the weak energy scale is so much smaller
than the Planck scale, or equivalently, why is gravity so much weaker than the other
forces.

Let us elaborate on this last bullet note. Within the context of quantum field theory, quantum corrections often introduce divergent contributions to observables. This is a consequence
of extrapolation of the quantum field theory to very high energies, where the current description presumably breaks down, and a new unknown and more fundamental theory takes
over. Nevertheless, in order to obtain physically meaningful quantities with our current
understanding, these unphysical infinities must be properly parametrized and isolated (the
so-called regularization procedure), and disposed of, by being reabsorbed into the definitions of physical quantities of the theory (the so-called renormalization procedure). In other
words, the unrenormalized theory contains bare parameters (such as bare mass and charge)
which are formally infinite due to the quantum corrections.
Upon renormalization, the properly regularized divergences of the theory are absorbed
into the definitions of these bare parameters, rendering finite results for the physical observables (such as renormalized mass and charge). As noted before, in case of the Higgs mass
the contributions from the quantum corrections are quadratically divergent. Reabsorbing
these gigantic contributions into the definition of the bare Higgs mass through the renormalization procedure requires a cancellation to a remarkable degree of accuracy, in order
to leave a finite, and in comparison tiny, renormalized weak scale mass for the Higgs. In
other words, the renormalized Higgs mass is severely fine-tuned, leading to the mentioned
20

Hierarchy problem.
Many theoretical extensions of the SM have been proposed in order to tackle and resolve
one or more of the aforementioned SM issues. The main effort in particle physics community
has been concentrated towards solving the Hierarchy problem, through somehow eliminating
the quadratic divergences in the Higgs mass by construction. This is usually achieved by
imposing some new symmetries, which result in a new set of (heavy) particles, inducing a
cancellation among the quantum contributions originating from the ordinary particles and
the new set of heavy particles. The first half of this Thesis is, accordingly, dedicated to
elaborate on by far the simplest, although as we will see rather peculiar, solution of the
Hierarchy problem; namely, the Lee-Wick Standard Model (LW SM), as discussed below.
There are various methods of regularization in use in particle physics, each possessing
certain advantages and shortcomings, depending on specific circumstances. A method suggested by Wolfgang Pauli and Felix Villars, published in 1949 [38], involves introducing a set
of auxiliary “fictitious” particles (see below) into the theory, the quantum contribution of
which is subtracted from that of ordinary particles. The mass parameter of these auxiliary
fields acts as the regulator, parameterizing the divergences of the original theory. Taking the
infinite limit of this mass parameter removes the fictitious particles from the theory, and the
original (divergent) theory is recovered.
In the original formulation of the Pauli-Villars (P-V) regularization method, the auxiliary particles were strictly introduced as a mathematical tool into the theory, in order to
regularize the infinities associated with the quantum corrections of ordinary particles. In
1969 Tsung-Dao Lee and Gian-Carlo Wick explored the possibility of these auxiliary fields’
being actual degrees of freedom, and the theoretical implications of this consideration [39].
21

They were motivated by the following observation: by construction, the divergences of the
theory are cancelled among the quantum contributions from ordinary particles and their
auxiliary “partners”. This feature made the resulting theory, from a formal perspective,
highly attractive. At that time, their considerations led to a finite theory of QED [40].12
P-V auxiliary fields inherently contain an overall negative sign as part of their description, originally inserted to cancel divergences of ordinary fields. Therefore, promoting these
auxiliary fields from fictitious to real particles turns them, by construction, into so-called
“ghosts”. Ghost fields, carrying this extra negative sign, contribute a negative probability to
the processes in particle physics, and hence, are considered unphysical. The P-V ghosts as
real degrees of freedom, therefore, violate unitarity, indicating that the sum of the probabilities of all processes in the theory would not equal one, and lead to an unphysical theory.
Coping with this problem, Lee and Wick were able to show that the issue could be averted
by relaxing causality at very high energies, having as a consequence that these auxiliary
partners decay before they are produced.13 As a result, the Lee-Wick (LW) theory inherently violates causality at small distance scales corresponding to very high energies, beyond
the reach of current collider probes.14
Lee and Wick also showed that, from a theoretical point-of-view, the addition of this
“auxiliary sector” to the original Lagrangian would be equivalent to invoking a “higherderivative term”. In the latter approach, the kinetic term in the Lagrangian, ordinarily
12 Modern finite theory of QED is based upon the renormalization formalism, in which
the infinities are reabsorbed in the definitions of the bare parameters, rather than being
cancelled by introduction of heavy partners.
13 For an elaborate discussion on peculiarities related to the Lee-Wick theory, see e.g. [41,
p. 282].
14 Odd as the theory might sound, one should bear in mind that causality in physics is an
axiom, and ultimately it is up to experiment to determine the energy bounds within which
this axiom is valid.
22

prescribed by a quadratic derivative term, is viewed as the first entity in an infinite expansion,
and is extended to contain the next entity in the expansion series,15 which will be of a quartic
form.16 Using the equations of motion, one can prove the equivalence of the “auxiliary-field
formulation” and “higher-derivative term formulation” of the theory. The Lagrangian with
a higher-derivative term naturally produces a propagator with higher powers of momentum
in the denominator for the ordinary fields, which, in calculations of quantum corrections,
leads to a softening or complete removal of any divergences. This is in accordance with the
auxiliary-field formulation, in which the auxiliary fields soften or cancel the infinities induced
by the ordinary fields.
In 2008 Benjamin Grinstein and Mark Wise examined the possibility of applying LW theory to the SM [42], since the resulting theory would be naturally free of any quadratic
divergences; hence, solving the Hierarchy problem in an economical way. As mentioned
above, the higher-derivative terms added to various sectors of the SM produce the higher
momentum propagators, softening or removing all SM divergences. Equivalently, in the
auxiliary-field formulation, all bosons of the SM are accompanied by a massive “Lee-Wick
partner”, while all left-handed and right-handed fermions have separately two corresponding
massive left-handed and right-handed LW partners, accomplishing the same result as the
higher-derivative formulation of theory.
In Part I of this Thesis, we take a deeper look into the LW SM, which, in the context of
resolving the Hierarchy problem, forms a natural extension to the SM electroweak sector. As
alluded to before, the electroweak precision data may be parametrized in particular manners,
15 From a formal perspective, the inclusion of more LW partners per SM field is equivalent
to adding more higher-derivative expansion terms.
16 Odd terms in the expansion are excluded by Lorentz invariance.
23

in order to make comparison with theory possible. One particular way of parametrizing these
data involves determining the quantum loop correction contributions to the self-energy of
massive gauge bosons (also called vacuum polarization amplitude (VPA)); i.e. W ± and
Z. The electroweak precision data place tight bounds on the value of the parametrizations
associated with these contributions. At low energies, the postulated existence of the heavy
LW partners introduces new (previously non-existing) corrections to these parameters; for
example, due to the LW partners manifesting themselves as “virtual particles” running in
quantum correction loops. As the contributions of LW particles are related to their masses,
one can utilize the tight constraints on the VPA electroweak parametrizations, deduced from
the electroweak precision tests, to place lower bounds on the LW masses. This analysis [1]
will be the subject of Chapter .
In 1971 Gerard ’t Hooft proved that the SM is a renormalizable theory [43]. This implies
that all divergences of the SM to all orders in perturbation theory are absorbed by renormalizing a limited number of the SM parameters, obtaining finite, physically meaningful
quantities for the observables. Generally, the renormalizability of a quantum field theory
may be qualitatively examined using a technique called power counting. In this technique,
one determines the powers of momentum in quantum correction loops, which in turn may
be used as an indication of the number of divergent quantum correction amplitudes (to all
orders) of the quantum field theory. If this number is finite, the theory under consideration is
renormalizable and all of its divergences may be absorbed in a redefinition of a finite number
of its bare parameters. If, however, by power counting the number of divergent amplitudes
is infinite, the theory is non-renormalizable.
The addition of higher-derivative terms to the SM Lagrangian, as prescribed by LW the24

ory, might naturally raise concerns regarding the renormalizability of the resulting LW SM,
since it potentially alters the powers of momentum in the quantum correction loops. Renormalizability has been previously explored in the higher-derivative formulation of the theory
by means of power counting arguments. It is, however, not clear a priori how renormalizability manifests itself in the auxiliary-field formulation. In Chapter , studying a LW scalar
QED theory as a toy model, we directly examine the symmetries and renormalizability of
this class of Abelian LW theories in the auxiliary-field formulation, identifying the relevant
symmetries which lead to a renormalizable theory.17
In Part II of the Thesis, we turn our attention to investigating the collider phenomenology of an extension to the strong sector of the SM described by QCD. As explained previously, QCD is based upon an SU (3)C local continuous symmetry. Since the late 1980s,
it has been of theoretical and phenomenological interest to extend the SM strong sector
to an SU (3)1C × SU (3)2C gauge theory, spontaneously broken to QCD’s SU (3)C [5][7]. This class of extensions represents an integral feature of theories in which the electroweak symmetry breaking is induced by the so-called strong dynamics, where a new type
of strongly-coupled gauge interaction forms a “composite Higgs” out of colored fermions.18
In accordance with Goldstone’s theorem, the spontaneous symmetry breaking releases eight
Nambu-Goldstone bosons, which are subsequently eaten by eight out of the sixteen originally
massless gauge bosons. Consequently, we obtain, in addition to eight massless colored gauge
bosons identified with ordinary QCD gluons, eight massive colored vector bosons which we
17 The analysis performed in the context of LW scalar QED is expected to generalize to
non-Abelian LW theories, and in particular to the LW SM, although a formal proof is yet to
be provided.
18 The situation is analogous to formation of “Cooper pairs” from electrons in a superconducting medium below the critical temperature.
25

generically refer to as the colorons.
From the phenomenological point-of-view, the main coloron production channel in a
hadron collider is the quark-antiquark annihilation process. To date, all theoretical and phenomenological analyses of colorons have been performed only at tree-level in perturbation
theory (LO). The next-to-leading order corrections (NLO) consist of the first order quantum corrections in the perturbative expansion, and additionally, the emissions of real soft
and collinear particles which are undetected due to the limited resolution of the detectors.
Chapter is devoted to the first complete and comprehensive study of coloron production at
NLO, taking into account the full corrections arising from the real emission of gluons and
light quarks, in addition to first order quantum corrections [3]. The NLO study dramatically
improves upon the previous LO calculations, and makes it possible to predict new coloron
kinematic variables, which emerge only after a complete NLO analysis is performed.
Finally, we conclude the Thesis in Chapter , by summarizing the main contributions
examined in the pervious chapters, and providing an outlook for future research in these
areas.

26

PART I

27

Chapter 2
CUSTODIAL ISOSPIN VIOLATION
IN THE LEE-WICK STANDARD
MODEL1

“If the facts don’t fit the theory, change the facts! ”
—Albert Einstein (1879 – 1955)

2.1

Introduction

The Lee-Wick Standard Model2 (LW SM) [42] forms a natural and simple extension of the ordinary Standard Model, which solves the Hierarchy problem in an economical
way. As discussed in Chapter , the Standard Model (SM) suffers from the quadratic diver1 This chapter is based on the paper first published in [1].
2 A Lee-Wick extension of the Higgs sector had been previously proposed in [53]-[55].
28

gences brought forth by quantum corrections to the mass of the SM Higgs boson. Absorbing
these contributions by means of renormalizing the bare Higgs mass requires a severe finetuning, and introduces the Hierarchy problem. The LW SM, as an extension to the SM,
introduces new fermions, among other particles, with exotic properties (see below) — the
Lee-Wick (LW) fermions. Within the context of the LW SM, the Hierarchy problem is
remedied by an induced cancellation among the ordinary fermion quantum contributions
and those generated by their LW counterparts. The bare Higgs mass will then contain no
quadratic divergences and can be renormalized in the usual way as explained in Chapter ,
remaining insensitive to the high energy behavior of the theory.
In the SM, the electroweak sector at one-loop may be fully parametrized using the following five observables: the mass of the Z vector boson mZ , the so-called Fermi constant
GF as a measure of the symmetry breaking vacuum expectation value (VEV), the so-called
Weinberg angle θW indicating the degree of mixture between the electromagnetic and weak
forces, the mass of the Higgs boson mh , and the mass of the top quark mt . The values of
these five electroweak observables are extremely well-measured.
In order to extract the values of the electroweak observables from the electroweak precision data, convenient parametrizations have been introduced in terms of one-loop fermionic
corrections to the electroweak gauge bosons’ vacuum polarization amplitudes (VPA) (see e.g.
Fig. 2.4). In four-fermion scattering processes, oblique corrections are defined as quantum
corrections to the gauge bosons’ VPA which do not depend on the identities of the initial
and final state fermions. If such a flavor dependence does exist, the corrections are referred
to as non-oblique corrections.
In general, the oblique corrections are parametrized using the Peskin-Takeuchi S and T
29

(or ∆ρ)3 parameters [56]-[58]. S measures, for example, the size of the electroweak symmetry
breaking sector, while T tracks isospin violation (defined below). If both non-oblique and
oblique contributions are present, one must employ, instead, the Barbieri et al. [59, 60] postˆ ˆ
LEP electroweak parameters S, T , W , and Y .4 The well-determined experimental values of
precision electroweak observables can subsequently be used to place tight constraints on any
contribution from new non-SM physics to the gauge bosons’ VPA.
Since none of the LW degrees of freedom have been observed in experiments, they must,
if they exist, be heavy with masses beyond the reach of previous collider searches.5 As
mentioned before, addition of these new heavy exotic particles to the spectrum in virtual
form risks running into conflict with precise experimental data on the electroweak observables: virtual LW fermions, running in quantum loops, contribute to the gauge bosons’ VPA;
thereby, modifying the values of the aforementioned (non-)oblique corrections. In conjunction with the electroweak experimental data, this information can be used to deduce lower
bounds for the LW particle masses. In what follows, we will discuss how the exotic properties
of the LW fermions solve the Hierarchy problem, and show, in detail, how corrections to the
post-LEP electroweak parameters can be utilized to constrain the LW masses.
The LW SM features higher-derivative kinetic terms for each SM field.6 As a consequence, the field propagators fall off to zero with momentum more rapidly than the ordinary
3 Within the context of the standard electroweak theory, the ρ parameter is defined as the
m2
W
zeroth-order ratio: ρ =
= 1. Small experimentally measured deviations from
m2 cos2 θW
Z
this value is attributed to the higher-order quantum corrections, and can place tight bounds
on the new physics contributions.
4 For the precise definition of these parameters, see Sec. 2.4.1.
5 As we will show in this chapter, a lower bound for the LW particle masses will be of the
order of a few TeV, in which case, they might lie within the reach of the LHC.
6 See Sec. 2.2 for the exact formal definition of the LW SM.
30

SM propagators, and the infinities associated with ultraviolet quantum fluctuations either
become less severe or are removed from the theory. In a scalar field theory all amplitudes
turn out to be finite by power counting. In a gauge theory the higher-derivative kinetic
terms generate new momentum-dependent interactions that prevent the theory from being
finite; however, a simple power counting argument shows that all possible divergences are
logarithmic. Thus, the LW SM offers a potential solution to the hierarchy problem. This
was the main motivation for studying the model [42] and analyzing its phenomenological
implications [61]-[64].
If a higher-derivative kinetic term is added to the Lagrangian, the propagator of a LW SM
field displays two poles,7 the lighter one corresponding to a SM-like particle, and the heavier
one corresponding to a new degree of freedom, the LW partner. An equivalent formulation
consists of separating the poles in such a way that to each field there corresponds only one
pole and one mass. The LW poles are then characterized by a negative residue, and, thus,
act as Pauli-Villar regulators. However, unlike mere regulators, the LW fields nontrivially
participate in gauge and Yukawa interactions.
In electroweak sector of the SM, the left-handed top and bottom quarks, tL and bL ,
(and their replicas in the other two quark generations, as well as the corresponding leptons)
form a doublet under the SU (2)L × U (1)Y group, while their right-handed partners, tR
and bR , are singlets. The doublet structure is referred to as isospin, in analogy with the
1
ordinary spin doublet for the spin– fermions. In contrast with the ordinary spin doublet,
2
however, members of these isospin doublets are different flavors and have different masses,
leading to a violation of the isospin symmetry. The mass difference among members of the
7 Note that in quantum field theory, the pole of a propagator (i.e. the zero-value of its
denominator) is identified with the mass of the particle the propagator is representing.
31

isospin doublet is most prominent in this third quark generation (top-bottom), compared to
the other two generations, with the top quark being about two orders of magnitude heavier
than the bottom quark. Hence, the isospin symmetry in the SM is broken, with the largest
violation in the top-bottom doublet.

Likewise, in the LW SM the largest one-loop contribution to the Higgs mass comes from
an isospin violating sector of the theory: the top Yukawa coupling. There are two heavy
partners of the top quark in the LW SM, one associated with the left-handed top-bottom
doublet, with mass Mq , and the other with the right-handed top, with mass Mt . The
contributions to the Higgs mass involving a single LW top are opposite in sign to those
from a single SM top, so they cancel the quadratic divergence associated with the Higgs
mass renormalization, δm2 . The net contribution is still logarithmically divergent, and for
h
degenerate LW top quarks, Mq = Mt , is of the form
2
2
2 = 3λt M 2 log Λ ,
δmh
q
2
8π 2
Mq

(2.1)

where Λ is the cutoff, i.e. the highest energy scale marking the limit of the validity of the
theory. In the limit Mq → ∞ the ordinary quadratic divergence reappears, with Mq acting
as a cutoff. Therefore, as already pointed out in Ref. [65], in order to avoid fine-tuning the
value of Mq cannot be too large.

Because the dominant correction to the Higgs mass is associated with an isospin violating
sector of the theory, it is important to check whether the LW tops cause a large contribution to the electroweak observables, which are usually protected by the so-called custodial
symmetry: ∆ρ, and, for theories with heavy replicas of the top quark, the ZbL¯L coupling
b
32

[66, 67].8 Large contributions to these quantities would lead to a stringent lower bound on
Mq , which would result in large corrections to m2 and thus the necessity of fine-tuning the
h
scalar sector of the theory.
In this chapter9 we analyze the potential conflict between naturalness and isospin violation, by computing the contribution of the top quark sector to the ρ parameter and to the
ZbL¯L coupling. Furthermore, we compute the Barbieri et al. [59, 60] post-LEP electroweak
b
ˆ ˆ
parameters (S, T , W , and Y ) to check for additional constraints. In terms of the post-LEP
parameters, we find a simple picture for the constraints on the LW SM. The dominant contriˆ
butions to T come from the fermion sector at one-loop, and limits on this parameter provide
the strongest constraints on the top quark sector.10 In contrast, the dominant contributions to Y and W arise from the gauge sector at tree-level, and limits on these parameters,
therefore, provide the strongest constraints on the gauge sector. These results imply that
ˆ
the bounds on the LW fermions, coming almost entirely from T , are essentially independent
of the LW gauge masses.
Our results differ from those in Refs. [65, 68] because their one-loop analysis of the
effects of LW top quarks on electroweak observables rests on the incorrect assumption that
the corrections are purely oblique [56]-[58]. As discussed in Ref. [69] important non-oblique
corrections arise at tree-level in the LW SM, in the form of non-zero values for W and
8 The electroweak gauge structure SU (2) × U (1) may also be realized from a more
L
Y
general approximate global symmetry structure SU (2)L × SU (2)R , in which only SU (2)L ,
and U (1)R subgroup of SU (2)R are gauged (i.e. are local symmetries) and are identified
with the electroweak theory. The latter SU (2)R approximate global symmetry is called the
custodial symmetry, and protects the electroweak observables, such as the W -Z mass ratio
and the isospin, from large quantum corrections.
9 Throughout this chapter, the timeline for the depicted Feynman diagrams is from left
to right.
10 The dominant contributions to S, likewise, come from the fermion sector at one-loop,
ˆ
but they are too small to provide strong constraints on the top quark sector.
33

Y . Therefore, one must use the Barbieri et al. parameters to compare the LW SM with
experiment.
In Sec. 2.2 we review the structure of the LW SM [42] and establish notation. In Sec. 2.3
we present an effective field theory analysis of the LW corrections to ∆ρ and to the ZbL¯L
b
coupling. In Sec. 2.4 we present our analysis of the post-LEP electroweak parameters and
the resulting constraints on the LW SM, while the constraints from the ZbL¯L coupling are
b
analyzed in Sec. 2.5. The leading logarithmic contributions to the electroweak observables
in the full theory and the effective theory have to match; thus the results of Sec. 2.3 provide
an important check for those of Sec. 2.4 and 2.5.
Global symmetries and renormalizability of LW theories will be discussed in detail in
Chapter , while questions concerning unitarity [70], causality [71], and Lorentz invariance
in LW theories, although potentially important, will not be considered in this analysis. A
complete analysis of the one-loop renormalization of the LW SM can be found in [72].

2.2

The Lee-Wick Standard Model

It is straightforward to write a higher-derivative extension of the SM electroweak Lagrangian
[42]. Adopting a non-canonical normalization for the gauge fields, the gauge Lagrangian reads
1 ˆ ˆ µν
1
ˆ
ˆ
Bµν B −
Tr Wµν W µν
2
2
4g1
2g2
1
1
ˆ
ˆ
ˆ ˆ
ˆ ˆ
+
∂ µ Bµν ∂λ B λν +
Tr Dµ Wµν Dλ W λν
2M 2
2M 2
2g1 1
g2 2

Lhd
gauge = −

34

(2.2)
,

where

ˆ
ˆ
ˆ
Bµν ≡ ∂µ Bν − ∂ν Bµ ,

ˆbˆc
ˆa
ˆ
ˆa
Wµν ≡ ∂µ Wν − ∂ν Wµ + f abc Wµ Wν τ a ,

a
a ≡ σ , and σ a the SU (2) generators (Pauli matrices). The “hat” notation indicates
with τ
2
that the field’s propagator contains not only the ordinary SM poles but also the LW poles.
ˆ
For example, in the limit of unbroken electroweak phase the Bµ propagator has a massless
pole, corresponding to the ordinary Bµ gauge field, and a mass-M1 pole, corresponding to
its LW counterpart. Notice also that additional dimension-six operators could, in principle,
be added to this Lagrangian. However, these would lead to scattering amplitudes for the
longitudinally polarized gauge bosons growing like E 2 , where E is the center-of-mass energy,
and thus to a rather early violation of unitarity [73]. We, therefore, do not include them
in this analysis. Notice also that we only include one higher-derivative term per SM field,
which introduces a single corresponding LW pole. This is certainly fine for our purposes, since
in this analysis we focus on the low momentum regime, where additional higher-derivative
terms are negligible. However, at large momenta additional poles in the propagator can have
important implications [74, 75].

The higher-derivative extension of the Higgs sector is
2
ˆ ˆ v
ˆ ˆ
Lhd
= |Dµ φ|2 − λ φ† φ −
Higgs
2

35

2
−

1
M2
h

ˆ ˆ
|D2 φ|2 ,

(2.3)

where as usual the Higgs doublet may be written in component form as



√ +
ˆ
1  i 2φ

ˆ
φ= √ 
 .
2
ˆ − iφ0
ˆ
v+h

(2.4)

Here and in Eq. (2.2) the covariant derivative written with a hat is built with the hatted
ˆ
gauge fields. We will find it convenient to have a compact way of denoting iσ 2 φ∗ as we
build operators that couple the Higgs to the right-handed top quark. Hence, we make the
definition



1 
ˆ
ϕ ≡ (iσ 2 φ∗ ) = √ 
ˆ
2

ˆ
ˆ
v + h + iφ0




.
√ − 
ˆ
i 2φ

(2.5)

ˆ
The field φ contains both the ordinary Higgs doublet and a massive doublet11 with mass ∼ Mh .
In the fermion sector we focus only on the third quark generation, since this is the
dominant source of isospin violation and gives the largest correction to the Higgs mass.12
The higher-derivative extension of the fermion Lagrangian is
¯ ˆˆ
¯ ˆˆ
ˆ /
¯ ˆq
ˆ /
Lhd
ˆ /
quark = qL iDˆL + tR iDtR + bR iDbR
1 ¯ ˆ3
1 ¯ ˆ 3ˆ
1 ¯ ˆ 3ˆ
ˆ iD b ,
ˆ /
+
qL i D q L +
ˆ / ˆ
tR iD tR +
b / R
2
2
2 R
Mq
Mt
M
b

(2.6)

ˆ b
where qL = (tL , ˆL ). Notice that the right handed fields have been primed because, for
ˆ
ˆ
ˆ
example, tL and tR are not left and right component of the same Dirac spinor. In the
11 If M is smaller than all other LW mass parameters, in a certain energy regime the
h
model behaves like a two-Higgs doublet model, although one doublet is of LW type. This
scenario was analyzed in [76].
12 Inclusion of the remaining flavors would introduce new mixing matrices, and, without the
assumption of minimal flavor violation, potential sources of flavor changing neutral currents
(FCNC). However, in Ref. [77] it was shown that for LW fermion masses in the TeV range
the FCNC are sufficiently small to satisfy experimental constraints.
36

ˆ ˆ
unbroken electroweak phase tL (tR ) contains the ordinary massless SM left-handed (righthanded) top as well as a massive Dirac fermion of mass Mq (Mt ).
Finally we consider the Yukawa Lagrangian, which in the SM has no derivatives. Therefore, we write
¯ ˆˆ
LYukawa = −yt qL ϕ tR + h.c. ,
ˆ
where the bottom Yukawa coupling has been ignored, since yb

(2.7)
yt .

As explained in the introduction, this “higher-derivative” formulation of the theory, in
which both the ordinary pole and the LW pole are contained in the same field, is equivalent
to an “ordinary formulation” in which, as follows: (i) the two poles belong to two different
fields, and (ii) the kinetic and mass terms for the LW fields have the wrong sign. This
alternative formulation is especially useful for calculating loop diagrams. In this chapter we
will compute loop diagrams with the top and bottom quarks in the loop. Thus we will find it
helpful to replace the higher-derivative fermion and Yukawa Lagrangians with the ordinary
formulation Lagrangians
¯ ˆ
/
/
/
Lquark = qL iDqL + tR iDtR + ¯R iDbR
¯ ˆ
b ˆ
¯
¯ ˆ
˜ ˜
/
− q iD − Mq q − t iD − Mt t − ¯
˜ /
˜ ˜ ˆ
b

(2.8)
ˆ
/
b
iD − Mb ˜ ,

and
¯
LYukawa = −yt qL − qL ϕ
¯
˜
ˆ

˜
tR − tR

+ h.c. ,

(2.9)

where
q L ≡ q L − qL ,
ˆ
˜

ˆ
˜
tR ≡ tR − tR ,

ˆ ≡ b −˜ ,
bR
R bR

(2.10)

and where the fields with (without) a tilde are LW (SM) fields. The equivalence between the
37

higher-derivative formulation, Eqs. (2.6) and (2.7), and the ordinary formulation, Eqs. (2.8)
and (2.9), can be easily proved; see, for example, [42]. Notice that the wrong sign in front
of the kinetic and mass terms makes the LW (tilde) fields act like Pauli-Villars regulators,
with the difference that they also participate nontrivially in gauge and Yukawa interactions.

2.3

Effective Field Theory for ∆ρ and Zb¯
b

The appearance of the LW fields in the Yukawa interactions, Eq. (2.9), suggest the presence of non-standard sources of custodial isospin violation at energies below the LW scale.
Dimension-six custodial violating operators can potentially arise from tree-level exchanges,
and from loop diagrams with one or more LW fermions in the loop. The leading contribution
to these operators, in inverse powers of the LW fermion masses, can be found by integrating
out the LW fermions at tree-level and computing loops in the resulting effective field theory.
For LW fermion masses much larger than both the Higgs vacuum expectation value (VEV)
and the external momenta, the effective Lagrangian can be computed in powers of ϕ/Mq,t
ˆ
ˆ
/
and D/Mq,t . Including the leading non-standard corrections, this leads to
¯ ˆ
¯ ˆ
/
/
/
Leff = qL iDqL + tR iDtR + ¯R iDbR − yt qL ϕ tR + tR ϕ† qL
¯ ˆ
b ˆ
¯ ˆ
y2
y2
¯ ˆ ˆ ˆ
/
/
− t qL iϕD ϕ† qL − t tR iϕ† D ϕ tR .
¯ ˆˆ ˆ
2
2
Mt
Mq

(2.11)

Notice that the primes have been removed from the right-handed fermion fields, because
now left-handed and right-handed components are Dirac partners. Notice also that this Lagrangian assumes Mq and Mt to be of the same order, with no hierarchy between them. The
2
leading logarithmic correction to observables will, therefore, be proportional to log Mq /v 2 ∼
38

2
log Mt /v 2 . In what follows, we compute these leading-log corrections by constructing the
operators which arise in the effective theory appropriate for energy scales below Mt

Mq ,

in which the LW partners have been “integrated out” but the top quark remains in the
spectrum.
After electroweak symmetry breaking, the higher-derivative operators lead to a renormalization of the fermion kinetic terms. An alternative approach consists of redefining qL and
tR to make their kinetic terms canonically normalized in both the broken and the unbroken
electroweak phase. This is achieved by the replacements
2
yt

3
qL → 1 +
ϕϕ† + O(1/Mt ) qL ,
ˆˆ
2
2Mt

2
yt †
3
ϕ ϕ + O(1/Mq ) tR , (2.12)
ˆ ˆ
tR → 1 +
2
2Mq

which leads to a new Lagrangian, equivalent to Leff
y2
ˆ
ˆ
ˆ
¯ iDt + ¯ iDb − yt q ϕ 1 + t
Leff = qL iDqL + tR / R bR / R
¯ /
¯L ˆ
2
2
yt
2
2Mt
2
yt
+
2
2Mq
+

1
2
Mq

+

1
2
Mt

ϕ† ϕ tR + h.c.
ˆ ˆ

ˆ ˆ ˆ
qL i (Dµ ϕ)ϕ† − ϕ(Dµ ϕ)† γ µ qL
¯
ˆ ˆ ˆ

(2.13)

ˆ ˆ ˆ ˆ ˆ ˆ
¯
tR γ µ tR i (Dµ ϕ)† ϕ − ϕ† (Dµ ϕ) .

As expected, there are custodial symmetry violating dimension-six operators. However, at
tree-level there is no non-standard contribution to ∆ρ or the ZbL¯L coupling.
b
Leff features terms coupling one, two, or three ϕ fields to a pair of fermions. Therefore,
ˆ
dimension-six four-ϕ operators arise both from vacuum polarization amplitudes and triangle
ˆ
diagrams, as shown in Fig. 2.1. The log-divergent parts of these diagrams (which yield
2
the log(Mt,q /m2 ) contributions) can be computed in the unbroken electroweak phase, with
t
39

'†
ˆ

'
ˆ

+

tR

qL

'
ˆ

'†
ˆ

tR

qL

+

qL

tR
tR

tR

qL

qL

qL
tR

+

qL
tR

+

tR

qL

+

+

tR

qL

tR

tR

+

qL

tR
tR

tR

qL

+

tR

tR

qL

qL

+

qL
qL

qL

ˆ
Figure 2.1: Diagrams contributing to the dimension-six four-φ operators in the effective
theory, with the LW fermions integrated out at tree-level.

fermions in the loop. The logarithmically divergent part of the amplitude is reproduced by
the operators13
4
4
3yt
3yt
2
1
1
1
1
2
ˆ ˆˆ
ˆˆ ˆ
|Dφ|2 |φ|2 · +
|φ† Dφ|2 · ,
+
+
2 M2
2 M2
16π 2 Mt
16π 2 Mt
q
q

(2.14)

where as usual = 2−d/2 in dimensional regularization. The first operator respects custodial
symmetry, but the second operator does not, since it contributes only to the Z boson mass.
The second operator gives the dominant contribution to ∆ρ, which is, therefore, of the order
3 2m4
t
(∆ρ)LW ∼ −
2 v2
16π

1
2
Mt

+

2
2
Mq

log

2
Mq
,
m2
t

(2.15)

where the 1/ is replaced by the large log which arises in the effective theory scaling from
13 There are also quadratic divergences which are completely absorbed by a counterterm
of the form |ϕ|4 , with no derivatives.
ˆ
40

'†
ˆ

qL

'
ˆ
tR

+

(1 + Z + ZqL )

+

tR

'
ˆ

qL

'
ˆ

tR
tR

'
ˆ

+

+

qL

'
ˆ

+

qL

'
ˆ

tR
tR
Figure 2.2: Diagrams contributing to the dimension-six operators with two external qL and
two ϕ fields. The triangle diagrams lead to the second operator of Eq. (2.16), which contains
ˆ
b
non-universal corrections to the ZbL¯L coupling.

the scale Mq ∼ Mt to the weak scale mt ∼ v. For LW fermions lighter than 1 TeV this is
a large negative isospin violating effect. For example, taking Mq = Mt = 500 GeV gives
∆ρ ∼ −1.4%. Furthermore, since ∆ρ is always negative, a heavy Higgs is strongly disfavored
in the LW SM.
The diagrams contributing to the left-handed fermionic gauge couplings up to one-loop
order are shown in Fig. 2.2. The tree-level diagram (corrected by the field strength renor2
2
malizations) corresponds to the custodial violating operator proportional to yt /2Mt , in
¯
Eq. (2.13). This operator contributes to the ZtL tL coupling, not to ZbL¯L .14 The reb
maining diagrams contain non-standard logarithmic divergences which are reproduced by

14 Including the bottom Yukawa coupling would lead to a tree-level operator contributing
2
to ZbL¯L . However, the top loop contribution is dominant, since 16π 2 yb ∼ 0.1.
b
41

the operators
←
−
1
†
ˆ
i qL γ µ D µ qL − q L D µ γ µ qL ϕ † ϕ · +
¯
¯ ˆ
ˆ ˆ
2
16π 2 4Mq
4
yt
1
1
1
ˆ ˆ ˆ ˆ ˆ ˆ
q γ µ qL i (Dµ ϕ)† ϕ − ϕ† (Dµ ϕ) · .
¯
+
2 M 2 4M 2 L
16π
q
t
4
yt

1

(2.16)

In this expression the first (custodially symmetric) operator amounts to a renormalization
of the standard gauge interactions, and does not contribute to non-standard fermionic gauge
couplings. The second operator violates custodial symmetry, and is only due to the triangle
¯
b
diagrams in Fig. 2.2. This contributes both to the ZtL tL coupling and the ZbL¯L coupling.
Expressing the latter in the form
e
b¯
b
b
gL Zµ ¯L γ µ bL
cw s w
e
1 1
b¯
b
b¯
b
≡
− + sin2 θW + (δgL )SM + (δgL )LW Zµ ¯L γ µ bL ,
b
cw s w
2 3

(2.17)

b¯
b
where (δgL )SM includes all higher order SM corrections, and replacing the 1/ poles with
the large log arising from scaling in the theory, we find that the second operator of Eq. (2.16)
b¯
b
gives the dominant non-universal LW contribution to gL
2
Mq
m4
4
1
b¯)
b
t
(δgL LW ∼ −
+
log
.
2 M2
32π 2 v 2 Mt
m2
q
t

(2.18)

The SM prediction is already 1.96σ below the observed central value. Hence, the additional
negative correction in the LW theory goes in the direction opposite to what is favored by
experiment.
2
2
In the next two sections we compute perturbatively (in v 2 /Mq and v 2 /Mt ) and numer42

ically the values of ∆ρ and the ZbL¯L coupling in the full LW theory. Our effective field
b
theory results, Eq. (2.15) and Eq. (2.18), provide a check of these full calculations, since
the leading logarithmic contributions have to match. More generally, below we compute the
top sector one-loop contribution to all Barbieri et al. [59, 60] electroweak parameters, and
provide lower bounds on Mq and Mt from comparison with experiment.

2.4

Constraints from Post-LEP Parameters

ˆ ˆ
In the language of Barbieri et al. [59, 60], the observables S, T , Y , and W parametrize the
flavor-universal deviations from the SM at low energies. We now analyze the tree-level and
the fermionic one-loop contributions to these parameters and use them to obtain constraints
on the masses of the LW states.

2.4.1

Tree-Level Contributions

At tree-level, it is straightforward to read from Eq. (2.2) the vacuum polarization amplitudes15
q2
(q 2 )2
v2
Π ˆ + ˆ − (q 2 ) =
−
−
,
2
2 2
W W
4
g2 g2 M2
v2
Π ˆ 3 ˆ (q 2 ) =
,
W B
4

q2
(q 2 )2
v2
Π ˆ 3 ˆ 3 (q 2 ) =
−
−
,
2
2 2
W W
4
g2 g2 M2
q2
(q 2 )2
v2
−
−
Π ˆ ˆ (q 2 ) =
.
2
2 2
BB
4
g1 g1 M1

(2.19)

15 The full Lorentz structure of a gauge boson’s VPA contains a term proportional to g µν
and another term proportional to the external momentum q µ q ν . For a massless gauge boson,
these two coefficients are the same, and lead to a transverse VPA Lorentz structure. This
is, however, generally not the case for a massive gauge boson, such as W ± and Z, where
the two coefficients are different. In (2.19), ΠXY (q 2 ) represents the coefficient of g µν in
the XY VPA, since only this piece is relevant for our calculation. B and W 3 are U (1)Y
and the third component of SU (2)L group generators, respectively; within the context of
the electroweak theory, they mix to produce massive Z and massless photon.
43

X

1
= DXY

Y

Figure 2.3: The XY VPA, defined as inverse of the full XY propagator, DXY (see footnote 15 for details).

Following [59], we see that there is no tree-level correction to the Fermi constant

√

1
= −4Π ˆ + ˆ − (0) = v 2 .
W W
2GF

(2.20)

Barbieri et al. define the approximate electroweak gauge couplings

1
≡ Π ˆ + ˆ − (0) ,
W W
g2
1
≡ Π ˆ ˆ (0) ,
BB
g2

(2.21)
(2.22)

which in the LW SM gives g = g1 and g = g2 . We then compute the tree-level electroweak
parameters [69],

ˆ
S ≡ g 2 Π ˆ 3 ˆ (0) = 0 ,
W B
ˆ
T ≡ g 2 Π ˆ 3 ˆ 3 (0) − Π ˆ + ˆ − (0) = 0 ,
W W
W W
m2
1 2 2
Y ≡ g mW Π ˆ ˆ (0) = − W ,
2
2
BB
M1
m2
1
W ≡ g 2 m2 Π ˆ 3 ˆ 3 (0) = − W ,
W W W
2
2
M2

(2.23)
(2.24)
(2.25)

(2.26)

where in each equation the first equality is the definition of the corresponding post-LEP
parameter [59]. The VPAs (ΠXY ) are defined as inverse of the full XY propagator (Fig. 2.3).
44

2.4.2

Fermionic One-Loop Contributions

The gauge current correlators receive important loop corrections from the top-bottom sector,
through the diagrams shown in Fig. 2.4. These vacuum polarization amplitudes contain two
infinities, which are absorbed in the definitions of g and g given in Eqs. (2.21) and (2.22),
respectively. As a consequence the non-canonical normalization adopted in Eq. (2.2) forces
us to define renormalized LW gauge masses. A convenient scheme consists of defining M
and M by
−

2
≡ Π ˆ + ˆ − (0) ,
2M 2
W W
g

−

2
≡ Π ˆ ˆ (0) ,
2M 2
BB
g

(2.27)

which simplify the one-loop calculations below. At tree-level, from Eq. (2.19), we see that
M = M2 and M = M1 , and both are related to the masses of the LW partners of the gauge
bosons. Because of the power counting properties of LW theories, after the usual16 couplingconstant and mass renormalizations, all physical quantities remain finite [42]. Hence, M
and M remain finite at one-loop (and beyond). However, since they are defined by the
zero-momentum properties of the gauge boson two-point functions, their values only approximately equal the masses of the LW partners of the gauge bosons. This suffices for our
purposes, since we are interested in low-energy observables; if we were studying quantities
measured at higher energies, we would want to define M and M based on propagators
renormalized at high q 2 instead.
The propagators in the loops of Fig. 2.4 correspond to mass eigenstates, where the masses
are obtained by diagonalizing the mass matrices by means of symplectic rotations: in this way
16 Notice that the vacuum polarization diagrams involving only one LW fermion carry an
overall negative sign. In fact this happens to make all zero-derivative functions, at q 2 = 0,
finite. For this reason there is actually one less infinity compared to the ordinary SM, and
the bare v is finite [42].
45

⇧fˆ + W
W ˆ

(q 2 )

=

X

ti

q

ˆ
W

ˆ
W+

ij

⇧fˆ 3 W 3 (q 2 )
W ˆ

=

X h

bj

q
ˆ
W3

=

X h

q
ˆ
W

=

X h

q

ˆ
B

+

ˆ
W

ij

q

ˆ
B
tj

ˆ
B

bj

ti

ˆ
B

i

bi

3

tj

q

i

bj

ti

3

i

ˆ
B

ˆ
W3

tj

ij

⇧f B (q 2 )
ˆ
Bˆ

+

bi

ˆ
W3

q
ˆ
W3

ij

⇧fˆ 3 B (q 2 )
W ˆ

ti

+

bi

ˆ
B
bj

Figure 2.4: Dominant vacuum polarization amplitudes for the LW SM gauge fields. These
include the ordinary (t0 and b0 ) and the LW third generation quarks (t1 , t2 , b1 , and b2 ) in
the loop. These amplitudes contribute to the two-point functions of Eq. (2.19).

the LW fields maintain the “wrong-sign” kinetic and mass terms. A perturbative diagonaliza2
2
tion in v 2 /Mq and v 2 /Mt [65] requires considering two different scenarios: non-degenerate
2
2
2
2
2
LW masses, |Mq − Mt | ∼ Mq , and (near) degenerate LW masses, |Mq − Mt |

2
Mq . For

non-degenerate LW top quarks the contributions to the electroweak parameters are quite
lengthy. To leading order we obtain

46

2 1 − 3r2 + 6r4 − r6 + 3r8
Mq
1
t
t
t
t log r2
log
+
t
2
2
2 (1 − r2 )5
rt
mt
rt
t
10
8
6
4
2
5 − 17rt + 4rt + 12rt − 23rt + 7rt
,
−
2
2
2rt (1 − rt )4
2 1 − 3r2 + 6r6
Mq
3g 2 m4
1
t
t log r2
t
ˆ= −
2+
log
+
T
t
2
2
2 (1 − r2 )5
2 m2 M 2
32π W q
rt
mt
rt
t
2
4
6
8
10
9 − 12rt − 21rt + 46rt − 68rt + 22rt
−
,
2
2
6rt (1 − rt )4
2 2
m2
m2
9
3
W ,
W + g mW
Y = −
−
,
W = −
−7+
2
2
M2
M2
640π 2 Mq
r2 rt
b
ˆ
S= −

g 2 m2
t
2M 2
48π
q

2+

(2.28)

where
rt ≡ Mt /Mq ,

rb ≡ Mb /Mq .

(2.29)

The electroweak parameters in the (near) degenerate case cannot simply be obtained by
taking the rt , rb → 1 limit in Eqs. (2.28), since the corresponding expressions diverge.
2
2
Instead, we must diagonalize the mass matrices perturbatively in 1/Mq (or 1/Mt ) and
2
2
2
|Mq − Mt |/Mq , and then compute the electroweak parameters. For exact degeneracy,
Mq = Mt , this gives
2
Mq
g 2 m2
12
t
log
−
,
2
5
16π 2 Mq
m2
t
2
2 m4
Mq
3g t
141
ˆ= −
T
3 log
−
2
2 m2 M 2
20
32π W q
mt
m2
Y = − W ,
M2
m2
7g 2 m2
W .
W = − W −
2
2M 2
M
640π
q
ˆ
S= −

,
(2.30)

Note that the absence of fermionic one-loop corrections to the tree-level value of Y is a
47

direct consequence of the second definition in Eq. (2.27): a different scheme choice would
lead to an additional contribution. In the same way, changing the definition of M would
lead to a different fermionic one-loop expression for W ; in any case, the second term17 in
W is numerically very small and can be neglected. We, therefore, conclude that the leading
contributions to Y and W are those arising from the LW gauge-sector at tree-level, Eqs. (2.25,
2.26).
ˆ
ˆ
ˆ
Since the tree-level values of S and T vanish, the leading LW contributions to both S
ˆ
ˆ
and T arise from the top quark sector at one-loop. In the case of T this is not surprising
since the dominant locus of isospin violation in the model is the splitting between the top
ˆ
and bottom quark masses. Because T is the same as ∆ρ [60], we may compare the leading
logarithmic correction in Eq. (2.28) with the result obtained in the effective theory, Eq. (2.15);
ˆ
we see that they agree. In the case of S, the situation is more subtle. The LW gaugeeigenstate fermion partners, being massive, are not chiral and therefore, in the absence of
electroweak contributions to the masses that mix them with the light chiral gauge-eigenstates,
ˆ
ˆ
their contribution to S vanishes. Hence, the dominant LW contributions to S also arise
predominantly from the top sector of the theory.
Therefore, at tree-level plus one fermion loop we obtain a very simple conclusion: the
ˆ
ˆ
fermion sector contributes to S and T only, while the gauge sector contributes to Y and W
only. It is true that when gauge loops are included, there will be additional contributions.
However, the gauge loop contributions are generally sub-dominant compared to the quantities
ˆ
we have already calculated. The only potential exception is S, for which the fermionic oneloop contribution is small. However, in Ref. [74] a numerical computation shows that the
17 Note that the first definition in Eq. (2.27) pertains to Π
ˆ ˆ , whereas W is defined
W +W −
in terms of Π ˆ 3 ˆ 3 .
W W
48

gauge loop contribution to S is suppressed (see also [65]), since |Stree − Sloop |

0.01.

ˆ
Using the results of Ref. [60], this allows us to estimate the gauge loop contribution to S
ˆ
to be 103 S

0.1, which is negligibly small. Thus, our existing results suffice for extracting

constraints on the LW fermions from the experimental data.

2.4.3

Comparison with Data

We begin with constraints on the masses of the LW partners of the gauge bosons. The
previous subsection found that the only post-LEP parameters affected by the LW gauge
boson masses are W and Y , and also that the tree-level expressions for W and Y , Eqs. (2.25,
2.26), suffice for comparison with data. The experimental constraints on Y and W are rather
tight and almost independent of the value of the Higgs mass [59]. These translate into the
95% C.L. lower bounds on M2 and M1 shown18 in Figs. 2.5 and 2.6. Fig. 2.5 shows the
bounds for arbitrary values of M1 and M2 : for mh = 115 GeV the striped region is excluded,
while for mh = 800 GeV the additional narrow yellow region is excluded as well. Fig. 2.6
shows the 95% C.L. ellipses in the (Y, W ) plane from the global fit to data [59], for mh = 115
and mh = 800, as well as the LW prediction for degenerate LW masses, M1 = M2 . All this
is in agreement with the results of Ref. [69] and gives the constraints M1 , M2

2.4 TeV.

Next, we seek constraints on the masses of the LW partners of the top quark. The
previous subsection found that the post-LEP parameters sensitive to the LW fermion masses
ˆ
ˆ
are S and T , which do not depend on the LW gauge masses at the one-loop level. We should
ˆ
also note that, for a light Higgs, the LW prediction of S is very close to its central value,
ˆ
S

0. Furthermore, from the global fit to the experimental data in Ref. [59], we conclude
18 These bounds are derived using the errors and correlation matrix given in Ref. [59].
49

10

95

CL 2 d.o.f.

8

M2 TeV

6

4

2

0
0

2

4

6

8

10

M1 TeV
Figure 2.5: Exclusion plot for the LW gauge-field masses M2 and M1 . These bounds are
due to the constraints on Y and W , as shown by Eq. (2.25) and Eq. (2.26). For a light Higgs
(mh = 115 GeV) the striped region to the left of both curves is excluded. For a heavy Higgs
(mh = 800 GeV) the additional yellow strip between the curves is excluded as well.

ˆ
that T is only mildly correlated to Y and W , the parameters that are most sensitive to the
LW gauge boson masses in the LW SM. This confirms that the bounds on the LW fermions
should be essentially independent of the LW gauge masses, and should come almost entirely
ˆ
from T .
50

4

1000 W

2

800 GeV

mh

0
115 GeV

mh

2.4 TeV

2

1.45 TeV

4

4

2

0

2

4

1000 Y
Figure 2.6: The 95% C.L. ellipses in the (Y ,W ) plane, and the LW prediction for degenerate
masses, M1 = M2 . The parametric plot is for 0.5 TeV < M1 = M2 < 10 TeV, and the
dots are equally spaced in mass every 0.95TeV (only two dots labeled). The lower bound on
M1 = M2 is approximately 2.4 TeV for a light Higgs.

ˆ
In Fig. 2.7 we show the experimental mean value for T (thick red line), the ±2σ al2
lowed region, the all-order (in v 2 /Mq ) LW prediction (solid blue curve), the leading order
LW prediction from Eq. (2.30) (dashed blue curve), and the leading-log approximation (dotted blue curve), as functions of Mq , in the degenerate case. This figure reveals the bound
51

3
2

103 T

1
0
1
2
3
1

2

3

4

5

Mq TeV
ˆ
Figure 2.7: T as a function of Mq in the degenerate case, Mq = Mt . The experimental mean
ˆ
value for T is shown by the thick red line, along with the ±2σ allowed region. Also shown are
2
the all-order (in v 2 /Mq ) LW prediction (solid blue curve), the leading order LW prediction,
Eq. (2.30) (dashed blue curve), and the leading-log curve, Eq. (2.15) (dotted blue curve), as
functions of Mq , in the degenerate case. Note that the leading order prediction is not valid
below Mq ∼ 1 TeV. (See text for details)

Mq = Mt

1.6 TeV on the LW fermion masses in the degenerate case. Note that although

ˆ
Eq. (2.30) appears to predict a positive T for small Mq (dashed blue curve), the complete
ˆ
numerical evaluation (solid blue curve) shows that T is always negative, as Fig. 2.7 shows
explicitly; below Mq = 1 TeV the perturbative diagonalization of the mass matrix is no
longer valid, rendering the leading order LW prediction unreliable in that mass regime.
If we relax the requirement of degenerate LW fermion masses, we obtain the 95% C.L.
bounds on Mq and Mt shown in Fig. 2.8. For a light Higgs the striped region in Fig. 2.8 is
52

10

95

CL 2 d.o.f.

Mt TeV

8

6

4

2

2

4

6

8

10

Mq TeV
Figure 2.8: The 95% C.L. exclusion plots for the LW fermion masses Mq and Mt . These
ˆ
bounds come almost entirely from the experimental constraints on T . For a light Higgs the
striped region to the left of the curve is excluded, while a heavy Higgs is completely excluded.

excluded, while for a heavy Higgs the whole (yellow) region is excluded. Note from Figs. 2.5
and 2.8 that the mildest constraints on the LW masses are obtained in the fully degenerate
case, M = M and Mq = Mt .
Returning to the case of degenerate LW fermion masses, we show in Fig. 2.9 the values
ˆ
ˆ
of S and T as a function of Mq = Mt for 0.5 TeV < Mq < 10 TeV; the dots representing
53

4
800 GeV

mh

2

1000 T

mh

115 GeV

0
2.4 TeV
1.45 TeV

2

4

4

2

0

2

4

1000 S
ˆ ˆ
Figure 2.9: The 95% C.L. ellipses in the (S,T ) plane, and the LW prediction for degenerate
masses, Mq = Mt . The parametric plot is for 0.5 TeV < Mq < 10 TeV and the dots are
equally spaced in mass every 0.95TeV (only two dots labeled). The lower bound on Mq is
approximately 1.6 TeV for a light Higgs.

different values of Mq are placed at regular intervals. The 95% C.L. ellipses from the global
fit to the data [59] confirm the constraint Mq

1.6 TeV for a light Higgs, while a heavy

ˆ
Higgs scenario is disfavored for any LW fermion mass. In fact, for a heavy Higgs the T
ˆ
parameter is expected to be positive, while the LW SM predicts a negative T . This is a
54

direct consequence of the negative sign in the LW fermion propagators, which results in an
overall negative sign from the (dominant) diagrams involving a single LW fermion in the
loop.
Our results disagree with those of [65, 68] in two ways: their bounds on the LW fermion
masses appear more stringent for a light Higgs and their limits appear to depend on the
masses of the LW gauge boson partners. The disagreement arises because their study of
one-loop electroweak corrections in the LW SM assumes the corrections to be purely oblique
and derives constraints by comparing the Peskin-Takeuchi S and T [56] parameters to data.
However, as clearly discussed in Ref. [69], and confirmed above in Eqs. (2.25) and (2.26),
the LW SM features large non-oblique corrections, in the form of non-zero values for Y and
W at tree-level. Hence, one must use the Barbieri et al. parameters to compare the LW SM
with experiment, as we have done.

2.5

Constraints from the ZbL¯L Coupling
b

b
The leading contribution to the ZbL¯L coupling (in the gauge coupling expansion) can be
obtained in the gaugeless limit from the φ0 bL¯L coupling [78]-[81], where φ0 is the Nambub
Goldstone boson eaten by the Z. The loop diagram giving the largest correction involves
the SM and LW top quarks19 and is shown in Fig. 2.10. A detailed computation of the
loop integral, valid for arbitrary models with heavy replicas of the top quark, is given in
Appendix . At zero external momentum the amplitude corresponding to the diagram has
19 In the gaugeless limit of the LW SM, as in the SM itself, all external b-quark wavefunction
2
renormalization corrections are proportional to yb and are, therefore, negligible. This should
be contrasted with the situation in the three-site Higgsless model [81].
55

the form
iM = −A pPL ,
/

(2.31)

where PL ≡ (1−γ 5 )/2 is the left-handed projector, p is the incoming φ0 momentum, and the
external fermion wavefunctions have been omitted. Then to leading order in g the correction
to the ZbL¯L coupling is [78]-[81]
b
b¯ v
b
δgL = A .
2

(2.32)

2
Expanding the amplitude in powers of m2 /Mq , we obtain
t
b¯
b
(δgL )LW = −

2
Mq
m4
4
t
+ 1 log
2
2
32π 2 v 2 Mq
rt
m2
t
2 + 9r4
2
4
4 − 11rt
t log r2 − 6 − 10rt + 2rt
+
t
2
2
2
2
rt (1 − rt )3
rt (1 − rt )2

(2.33)
,

for non-degenerate LW fermion masses, and

b¯
b
(δgL )LW = −

2
Mq
m4
49
t
5 log
−
2
6
32π 2 v 2 Mq
m2
t

,

(2.34)

for degenerate LW masses. Both of these expressions agree with the dominant contribution
found in the effective theory, Eq. (2.18).

b¯
b
The experimental value of gL is derived from measurements of Rb , the ratio of the Z → b¯
b
width to the width of the hadronic decays, and Ab , the forward-backward asymmetry for Z
decays into b¯ [82]
b
b¯
b
(gL )exp = −0.4182 ± 0.0015 .

56

(2.35)

bL

p

ti

X

p
+

0

ij

¯
tj

¯L
b
b
Figure 2.10: Diagram giving the largest contribution to the ZbL¯L coupling. The latter
0 b ¯ coupling — where φ0 is the Nambuis related through the Ward identity to the φ L bL
Goldstone boson eaten by the Z boson. The top quarks running in the loop are both ordinary
and LW.

The SM value was computed using ZFITTER [83, 84] in Ref. [67], leading to

1 1
b¯
b
b¯
b
(gL )SM = − + sin2 θW + (δgL )SM = −0.42114 ,
2 3

(2.36)

while the LW prediction is given by Eq. (2.33) and Eq. (2.34). In Fig. 2.11 we show the
experimental mean value (thick horizontal red line), the 2σ allowed region below the mean
2
value, the SM prediction (solid horizontal black line), the all-order (in v 2 /Mq ) LW prediction
(solid blue curve), the leading order LW prediction, Eq. (2.34) (dashed blue curve), and the
leading-log approximation (dotted blue curve), as functions of Mq , in the degenerate case.
Note that the dashed curve and Eq. (2.34) are not reliable for Mq
57

1 TeV, because the

0.418
0.419

gLb

0.420
0.421
0.422
0.423
1

2

3

4

5

Mq TeV
Figure 2.11: Constraints from the ZbL¯L coupling. This graph features the experimental
b
mean value (thick horizontal red line), the 2σ allowed region below the mean value, the SM
2
prediction (solid horizontal black line), the all-order (in v 2 /Mq ) LW prediction (solid blue
curve), the leading order LW prediction, Eq. (2.34) (dashed blue curve), and the leading-log
approximation (dotted blue curve), as functions of Mq , in the degenerate case.

perturbative diagonalization of the mass matrix is not valid in that mass regime.
As anticipated by the effective field theory calculation, the LW correction is always negative: this is essentially due to the negative sign in front of the dominant non-standard
triangle diagrams with one LW top and one SM top. It is large (for small values of Mq )
because of the explicit breaking of custodial isospin symmetry. Since the SM value is already
1.96σ below the experimental mean value, this correction goes in the direction opposite to
what is needed. Agreement at the 2σ level requires Mq
to Mq

700 GeV.
58

4 TeV; at 2.5σ this bound relaxes

2.6

Conclusion & Discussion

There is significant tension between naturalness and isospin violation in the LW SM. While
corrections to the Higgs mass are smallest when the LW partners of the gauge bosons and
fermions are light, isospin violation that must be present in the top sector to account for the
large splitting between mt and mb tends to constrain the LW partners to have masses over
ˆ
a TeV. We have performed an effective field theory analysis of the corrections to T and the
ZbL¯L coupling in the LW SM, and used it to confirm our full calculation of the LW effects
b
b¯
b
ˆ ˆ
on S, T , W , Y , and gL , including tree-level and fermionic one-loop contributions. The
post-LEP parameters yield a simple picture in the LW SM: the gauge sector contributes
to Y and W only, with leading contributions arising at tree-level, while the fermion sector
ˆ
ˆ
contributes to S and T only, with leading corrections arising at one-loop.
In agreement with [69], we find that experimental limits on W and Y jointly constrain
the masses of the LW gauge bosons to satisfy M1 , M2

2.4 TeV at 95% C.L., with relatively

little sensitivity to the Higgs mass.
ˆ
We also conclude that the experimental limits on T require the masses of the LW fermions
to satisfy Mq , Mt

1.6 TeV at 95% C.L. if the Higgs mass is light and tend to exclude the

LW SM for any LW fermion masses if the Higgs mass is heavy. This is because a model
containing a heavy Higgs can be rendered consistent with the data only if some other sector
ˆ
of the model provides a large positive correction to T . However, in the LW SM, the fermionic
ˆ
ˆ
loops that provide the dominant contribution to T always make T more negative, due to the
negative sign in the LW fermion propagators. The LW fermions simply cannot compensate
for the presence of a heavy Higgs. Our results differ from those in Refs. [65, 68] because their
analysis incorrectly assumes that the electroweak corrections due to LW states are purely
59

oblique. As explained in Ref. [69] the LW states actually induce important non-oblique
corrections, and one must, therefore, use the Barbieri et al. [59, 60] post-LEP parameters to
compare the LW SM with experiment, as we have done.
Weak isospin violation in the top quark sector also manifests itself through corrections
b¯
b
to the ZbL¯L coupling. The SM prediction for gL lies at the lower end of the range allowed
b
by experiment at 95% C.L., so that new physics making negative contributions to the value
b¯
b
ˆ
of gL would decrease the agreement with the data. As in the case of T , however, we find
that the negative sign in the LW fermion propagators translates into a negative contribution
b¯
b
to gL ; the lighter the LW fermions, the greater the disagreement between prediction and
b
data. We find that the ZbL¯L coupling places a lower bound of 4 TeV on the LW fermion
masses at 95% C.L.
On this note, we conclude our phenomenological analysis of the LW SM, reflecting various
constraints from the current experimental data. Having completed our phenomenological
study of the LW SM, we are going to focus, in the next chapter, on some of the theoretical
aspects concerning the global symmetries and renormalizability of LW theories, since they
are in general poorly understood. Understanding renormalizability of this class of theories,
in particular, is of great importance in order for them to be considered as reliable beyond the
Standard Model (BSM) candidates with accurate all-order predictions for the experiments.
We will examine these issues in both the higher-derivative and auxiliary-field formulations
of the theory by means of a relatively simple Abelian example.

60

Chapter 3
GLOBAL SYMMETRIES AND
RENORMALIZABILITY OF
LEE-WICK THEORIES1

“Science is a way of trying not to fool yourself. The first principle is that you
must not fool yourself — and you are the easiest person to fool! ”
—Richard Feynman (1918 – 1988)

3.1

Introduction

Renormalizabilty of Lee-Wick theories (LW) is an important theoretical aspect
which needs to be well-stablished, in order for them to be considered as reliable beyond the
Standard Model (BSM) candidates. In Chapters and , we explained how the LW theory
1 This chapter is based on the paper first published in [2].
61

may be formulated in terms of two separate but equivalent formal descriptions. Depending
on the problem at hand, one may choose to utilize either of the two formulations, in order
to facilitate the calculations and make certain formal aspects of the theory more transparent. Renormalizability has been previously explored in the higher-derivative formulation of
the theory (see below), but its precise theoretical description in the auxiliary-field formulation remained unknown. In this chapter, we address this nontrivial issue along with global
symmetries possessed by LW theories in the auxiliary-field formulation.
To summarize our discussion so far, we saw that in the higher-derivative formulation,
the higher-derivative kinetic term gave rise to propagators that fall off with momentum
more rapidly than the ordinary Standard Model (SM) field propagators, thereby reducing
the degree of divergence of loop diagrams. On the other hand, higher covariant derivatives
also introduce new momentum-dependent interactions, which raise the degree of divergence
of quantum fluctuations. Power counting arguments [42] show that these two competing
effects conspire to make all loop diagrams at most logarithmically divergent. If the scale
associated with the higher-derivative terms is of the order of the electroweak scale, then the
latter becomes stable against radiative corrections: no quadratic divergences are present at
any order in perturbation theory, and no unnatural fine-tuning of parameters is required. In
addition, power counting arguments [42] show that this higher-derivative formulation of the
theory is renormalizable.
The higher-derivative kinetic terms in the Lee-Wick Standard Model (LW SM) result
in propagators with more than one pole. In minimal (so-called N = 2) LW theories2 [75],
2 We will focus on N = 2 theories throughout this Thesis, though our results can be
potentially generalized to arbitrary-N LW theories, with N − 1 higher-derivative kinetic
terms associated with each field.
62

there is only one higher-derivative kinetic term for each field, corresponding to two-pole
propagators. In the N = 2 LW SM, the lighter pole can be identified as a SM-like state
already seen in experiment, while the heavier pole corresponds to a new LW ghost state with
the same quantum numbers and negative norm. As explained in Chapter , this is dangerous,
since it would lead to a violation of unitarity. In order to avoid this scenario, the LW ghosts
must appear only as virtual states [39]. Furthermore, the integration contour in momentum
integrals involving ghost propagators must be modified so as to preserve unitarity [70]. The
price to be paid for these modifications is the presence of unobservable acausal effects in
scattering processes [39, 71].
Alternatively, in the auxiliary-field form of the theory, in addition to the SM fields there
are “LW fields” with kinetic energy terms with the opposite sign from their SM counterparts.
The opposite sign for the kinetic energy terms enforces the cancellations that soften the
divergences in the theory. The main advantage of the auxiliary-field approach lies in the
computation of loop diagrams, since aside from the overall sign the propagators are just
ordinary propagators. There are, however, a number of open field-theoretic issues with this
formulation, which require clarification.
In this chapter3 we clarify two issues in the auxiliary-field description of the theory in the
context of a simple but nontrivial theory — LW scalar quantum electrodynamics (QED).4
First, the interaction terms involving the LW fields have a very particular form, which is
not the most general one allowed by gauge invariance. For example, the couplings of the
LW vector fields are identical to the gauge couplings of the corresponding SM gauge fields.
3 Throughout this chapter, the timeline for the depicted Feynman diagrams is from left
to right.
4 Our analysis extends immediately to LW QED with an arbitrary number of matter fields,
either scalars or fermions.
63

This equality and others are essential if the cancellations softening or removing the infinities
are to hold. On the one hand, it is not clear why, a priori, this special form of the interactions
should be preserved to all orders in perturbation theory. On the other hand, we know that it
must be preserved since power counting shows that the equivalent higher-derivative theory
is free of quadratic divergences.5 Here we identify approximate SO(1, 1) global symmetries
of the auxiliary-field description of the theory that allow us to understand its structure.
Second, we clarify the renormalizability of LW scalar QED in the presence of the massive
2
ghost LW vector field. Because of the q µ q ν /MA term in a heavy vector boson propagator
(where MA is the heavy vector mass), power counting in the auxiliary-field formalism is
difficult. We will identify two SO(1, 1) symmetric gauge-fixing conditions that simplify the
2
auxiliary-field LW analysis. In one case (ordinary), the gauge-fixing forces the q µ q ν /MA
terms to appear with canceling signs in the gauge-LW propagator matrix. In the other case
2
(“no-mixing”), the gauge-fixing eliminates the q µ q ν /MA term in the vector field propagators.
Working in the no-mixing gauge allows us to show that the number of superficially divergent
amplitudes in an Abelian gauge theory is finite, and the theory is, therefore, renormalizable.
Finally, to illustrate the renormalizability of LW scalar QED, we explicitly carry out
the one-loop renormalization program and demonstrate how the counterterms required are
constrained by the joint conditions of gauge and SO(1, 1) invariance. As a by-product of
these discussions, we compute the one-loop beta functions in LW scalar QED and contrast
them with those of ordinary scalar QED.
In Sec. 3.2 we introduce and illustrate the SO(1, 1) symmetries of a LW theory in the
context of φ4 theory. In Sec. 3.3 we consider LW scalar QED and derive the equivalent
5 In the auxiliary-field formulation power counting is more difficult because of the cancellations involved between different diagrams.
64

auxiliary-field description. We then analyze the global symmetries of the theory and explain
how these protect the form of the Lagrangian against radiative corrections. In Sec. 3.4 we
show how gauge-fixing can be implemented in an SO(1, 1) invariant fashion and derive the
corresponding propagators. In Sec. 3.5 we show that the number of superficially divergent
amplitudes is finite, and the theory is, therefore, renormalizable. Then we illustrate these
results at one-loop by carrying out the renormalization program and computing the beta
functions. Finally, in Sec. 3.6 we offer our conclusions and we sketch why a modified approach
is needed for the case of non-Abelian gauge theories.

3.2

Lee-Wick φ4 Theory

We first consider LW φ4 theory for a complex scalar field in order to introduce the auxiliaryfield formalism and the SO(1, 1) global symmetry of the model, as well as to set our notational
conventions.6 Lee-Wick φ4 theory is defined by the higher-derivative (hd) Lagrangian

1 2ˆ 2
ˆ
|∂ φ| − m2 |φ|2 −
ˆ ˆ
Lhd = |∂µ φ|2 −
ˆ
M2

ˆ
λ ˆ4
|φ| ,
4

(3.1)

ˆ
ˆ
where φ is a complex scalar field, and the Lee-Wick scale M parameterizes the energy at
ˆ
which the model deviates substantially from the standard φ4 model. As we will see, M
also characterizes the mass scale of the LW ghosts, so long as m
ˆ

ˆ
M . This Lagrangian is

˜
equivalent to one in which we introduce an “auxiliary” complex scalar field φ (the reason

6 In this chapter we follow closely the conventions of Ref. [42].
65

for the “prime” will become clear in what follows)
ˆ
λ ˆ
˜
ˆ
ˆ
ˆ ˜
ˆ ˜
ˆ ˆ
L = |∂µ φ|2 + M 2 |φ |2 + ∂µ φ ∂ µ φ ∗ + ∂µ φ∗ ∂ µ φ − m2 |φ|2 − |φ|4 .
4

(3.2)

Making the change of variable
ˆ
˜
φ=φ −φ ,

(3.3)

ˆ
λ
˜
˜
˜
ˆ ˜
L = |∂µ φ |2 − |∂µ φ |2 + M 2 |φ |2 − m2 |φ − φ |2 − |φ − φ |4 .
ˆ
4

(3.4)

we find

The symplectic rotation7
 
  
φ  cosh θ sinh θ  φ
  ,
 =
˜
˜
φ
sinh θ cosh θ
φ

(3.5)

where
tanh 2θ =

ˆ
−2m2 /M 2
ˆ
,
ˆ
1 − 2m2 /M 2
ˆ

(3.6)

diagonalizes the scalar field mass terms while preserving the symplectic structure of the
kinetic terms [42]. Hence, we arrive at the auxiliary-field description of the LW φ4 theory
λ
˜
˜
˜
L 4 = |∂µ φ|2 − |∂µ φ|2 + M 2 |φ|2 − m2 |φ|2 − |φ − φ|4
φ
4
λ
λ
˜
˜
˜
˜
= |∂µ φ|2 − |∂µ φ|2 + M 2 |φ|2 − m2 |φ|2 − |φ|4 + |φ|2 φφ∗ + φ∗ φ
4
2
λ 2 ˜∗2
λ ˜
λ ˜
˜
˜
˜
˜
− λ|φ|2 |φ|2 −
φ φ + φ∗2 φ2 + |φ|2 φφ∗ + φ∗ φ − |φ|4 ,
4
2
4

7 Note the contrast with an orthogonal rotation,
66

cos θ − sin θ
.
sin θ cos θ

(3.7)

where
ˆ
M 2 = cosh2 θ M 2 − e−2θ m2 ,
ˆ
ˆ
m2 = e−2θ m2 − sinh2 θ M 2 ,
ˆ

(3.8)

ˆ
λ = e−4θ λ .
˜
Note that the kinetic term of the φ field has the opposite sign to the usual one, and, hence,
the corresponding particle has negative norm and is the LW ghost field. Furthermore, the
mass of the LW ghost M is, in the limit m
ˆ

ˆ
ˆ
M , approximately the LW scale M introduced

in Eq. (3.1).
This theory has an exact global U (1) symmetry, but is not the most general U (1) symmetric renormalizable Lagrangian that can be built out of the ordinary field φ and the ghost
˜
field φ charged under the U (1) symmetry. In particular, the six interaction terms in the second line can in principle have six independent couplings. However, the dimension-four terms
in Eq. (3.7) do have an additional SO(1, 1) symmetry, under which the fields transform as

 
 
cosh β sinh β  φ
φ
  ,
 →
˜
˜
φ
sinh β cosh β
φ

(3.9)

so long as we also promote λ to a spurion field8 that transforms as

λ → e4β λ .

(3.10)

8 A spurion is a (fictitious) field that parametrizes the symmetry breaking. Its initial
symmetry invariant transformation can be used to construct the invariant operators of the
theory. Setting this field equal to its actual constant value will, subsequently, capture all of
the symmetry-breaking operators.
67

Furthermore, the Lagrangian of Eq. (3.7) is the most general renormalizable and U (1)symmetric Lagrangian with SO(1, 1)-symmetric dimension-four terms. The different mass
˜
terms for φ and φ break the SO(1, 1) symmetry, but do so only softly. They are also the
only U (1)-preserving soft terms that break SO(1, 1). Thus in the LW φ4 theory, the global
SO(1, 1) symmetry of the dimension-four terms implies that loop corrections can only modify
˜
the structure of the mass terms, introducing a mixing term between φ and φ with infinite
coefficient. This can always be diagonalized via a symplectic rotation (of the form given in
Eq. (3.5)), which leaves the rest of the Lagrangian unchanged, except for a redefinition of
the coupling. Hence, Lee-Wick φ4 theory is renormalizable by power-counting.

LW φ4 theory is rather simple, because aside from mass renormalization the theory is finite. The LW scenario is, however, much less trivial in LW gauge theories, because of the new
momentum-dependent interactions in the higher-derivative formulation. In this case global
symmetries are important to understand the full structure of the theory. In the following
we will show that Abelian N = 2 LW theories have a softly broken SO(1, 1)m+1 symmetry,
where m is the number of matter fields, and the remaining SO(1, 1) transformation acts on
the vector fields. Since the SO(1, 1)m+1 breaking is soft, the special relation between the
LW couplings and the ordinary couplings is protected against radiative corrections.

68

3.3

Global Symmetries of Lee-Wick Scalar QED

Let us now study an N = 2 LW theory of scalar electrodynamics. In the higher-derivative
formulation, the Lagrangian is9
ˆ
2
λ ˆ
1 ˆ2
1
1 ˆ2 ˆ 2
ˆ
ˆ ˆ
Lhd = − Fµν +
∂ µ Fµν + |Dµ φ|2 −
ˆ ˆ
|D φ| − m2 |φ|2 − |φ|4 ,
2
ˆ
4
4
M2
2MA

(3.11)

where
ˆ
ˆ
Dµ ≡ ∂µ − ig Aµ .

(3.12)

The scalar sector is simply that of φ4 theory as shown in Eq. (3.1), and, hence, our analysis
of this Lagrangian will parallel the discussion of Sec. 3.2. Introducing auxiliary fields, now
for both the vector and the scalar and using the notation described above, we see that the
Lagrangian of Eq. (3.11) is equivalent to
2
1 ˆ2
µ Aν F − MA A2 + |D φ|2 + M 2 |φ |2
˜ ˆµν
˜
ˆµ ˆ
ˆ ˜
L = − Fµν − ∂
4
2 µ
ˆ
λ ˆ
ˆ ˆˆ ˜
ˆ ˆ ˆ ˜
+ Dµ φDµ φ ∗ + Dµ φ∗ Dµ φ − m2 |φ|2 − |φ|4 ,
ˆ ˆ
4

(3.13)

ˆ ˜ ˆ ˜
˜
to all orders in perturbation theory. Changing variables from Aµ , Aµ , φ, φ to Aµ , Aµ , φ ,
˜
φ , where

ˆ
˜
Aµ = Aµ − Aµ ,

(3.14)

ˆ
˜
φ= φ −φ ,

(3.15)

9 In non-Abelian theories there can be additional dimension-six higher-derivative operators, which lead to heavy vector scattering amplitudes growing like E 2 , where E is the
center-of-mass energy [73]. For N ≥ 2 LW theories see, for example, Ref. [74].
69

and substituting in Eqs. (3.12) and (3.13), gives
2
MA 2
1 2
1 ˜2
˜
˜
˜
ˆ ˜
L = − Fµν + Fµν −
Aµ + |Dµ φ |2 − |Dµ φ |2 + M 2 |φ |2 − m2 |φ − φ |2
ˆ
4
4
2
ˆ
λ
˜
˜
˜
˜
˜
˜
˜
− |φ − φ |4 − ig Aµ φ Dµ φ ∗ − φ ∗ Dµ φ + ig Aµ φ Dµ φ ∗ − φ ∗ Dµ φ
4
˜
˜µ
+ g 2 A2 |φ |2 − |φ |2 ,

(3.16)

where now the covariant derivative is in terms of Aµ ,

Dµ = ∂µ − ig Aµ .

(3.17)

The symplectic rotation of Eq. (3.5) again diagonalizes the scalar field mass terms while
preserving the symplectic structure of the kinetic terms [42]. Since the gauge interactions
stem from kinetic terms, and the φ4 -interaction has a symplectic structure as well, it follows
that Eq. (3.5) only diagonalizes the mass terms leaving the rest of the Lagrangian invariant
˜
in form. In terms of φ and φ the Lagrangian now reads
2
MA 2
1 2
1 ˜2
˜
˜
˜
L = − Fµν + Fµν −
A + |Dµ φ|2 − |Dµ φ|2 + M 2 |φ|2 − m2 |φ|2
4
4
2 µ
λ
˜
˜
˜
˜
˜
˜
˜
− |φ − φ|4 + ig Aµ φ Dµ φ∗ − φ∗ Dµ φ − ig Aµ φ Dµ φ∗ − φ∗ Dµ φ
4
˜
˜
+ g 2 A2 |φ|2 − |φ|2 ,
µ

(3.18)

where we redefine parameters as in Eq. (3.8).
The Lagrangian of Eq. (3.18) has an exact U (1) gauge symmetry. In the limit λ → 0
˜
the global symmetry is promoted to U (1) × U (1), because the φ and φ fields can now
rotate independently, and only the diagonal U (1) subgroup is gauged. Thus, we expect
loop corrections to generate U (1)-symmetric terms — some with infinite coefficients — that
70

will be U (1) × U (1)-symmetric in the λ → 0 limit. Eq. (3.18) is not the most general
renormalizable Lagrangian with this symmetry structure; for example, the coefficients of the
˜
interactions involving Aµ could be arbitrary. Notice, however, that this Lagrangian can be
re-arranged in the form
2
MA 2
1 2
1 ˜2
λ
˜
˜
˜
˜
L = − Fµν + Fµν −
Aµ + |∂µ φ|2 − |∂µ φ|2 + M 2 |φ|2 − m2 |φ|2 − |φ − φ|4
4
4
2
4
˜
˜ ˜
˜
˜
− ig(Aµ − Aµ ) φ ∂ µ φ∗ − φ ∂ µ φ∗ − h.c. − g 2 (Aµ − Aµ )2 |φ|2 − |φ|2 . (3.19)
In the limit MA → 0, and treating the gauge coupling as a spurion field, the Lagrangian
respects a global SO(1, 1) symmetry under which










Aµ 
cosh α sinh α  Aµ 
 →
 ,
˜µ
˜
A
sinh α cosh α
Aµ

g → eα g .

(3.20)

As mentioned in Sec. 3.2, an additional SO(1, 1) global symmetry for the scalar field arises
in the limit M → m, when λ is treated as a spurion field
 

 
φ
cosh β sinh β  φ
 →
 ,
˜
˜
φ
sinh β cosh β
φ

λ → e4β λ .

(3.21)

In Sec. 3.5 we will argue that this theory is renormalizable, because the number of superficially divergent amplitudes is finite, and no operators of dimension greater than four are
present in the auxiliary-field formulation. We may wonder whether radiative corrections
require introducing dimension-four SO(1, 1) × SO(1, 1)-breaking counterterms. However, as
in the LW φ4 theory described above, the answer is no: since SO(1, 1) × SO(1, 1) is only

71

softly broken by mass terms,10 the SO(1, 1)×SO(1, 1)-breaking corrections to the renormalizable terms are finite. Furthermore, Eq. (3.19) is the most general U (1) gauge Lagrangian
with dimension-four SO(1, 1) × SO(1, 1)-symmetric terms. Since renormalizability prevents
higher dimensional operators from being generated, we conclude that the form of the Lagrangian is protected to all orders against radiative corrections, with the exception of the
scalar field mass terms. However, as we have already seen, these can be diagonalized with a
symplectic rotation, without affecting the rest of the Lagrangian. In the simple example we
have shown there is only one matter field: for an arbitrary number m of matter fields the
global symmetry is promoted to SO(1, 1) × SO(1, 1)m , since each field is acted upon with
a different SO(1, 1) symmetry transformation, and the conclusions about renormalizability
persist, mutatis mutandis.

3.4

Gauge-Fixing

In order to quantize the electromagnetic field, one must introduce a gauge-fixing term. To
facilitate our subsequent analyses of divergences and renormalizability, we will find it most
convenient to employ gauge-fixing functions that respect the SO(1, 1) symmetry; otherwise it
can be unnecessarily difficult to recognize when significant cancellations occur. For example,
Ref. [42] employs an SO(1, 1) violating gauge-fixing term −

1
∂ µ Aµ 2 which leads to
2ξ

diagonal propagators of the form
qµqν
−i µν
µν
PAA =
g − (1 − ξ)
q2
q2

i
qµqν
µν
P˜˜=
g µν −
2
AA q 2 − M 2
MA
A

,

.

10 The renormalizability of massive Abelian gauge theory [85] — arising from the coupling
of the Abelian gauge boson to a conserved current — insures that the gauge boson mass
term is “soft”.
72

2
Because the q µ q ν /MA term is only present in the LW photon propagator, there is no simple
cancellation of the badly-behaved terms and the theory appears to suffer from quadratic
divergences and non-renormalizability at one-loop. The reason that Ref. [42] found no
quadratic divergences when computing the self-energy amplitudes for a massless scalar field
at zero momentum is that the quadratic divergence vanishes in the limit m → 0 and q → 0,
2
2
since it is necessarily of the form Λ2 m2 /MA or Λ2 q 2 /MA . These quadratic divergences are
“gauge artifacts” [72] in the sense that they contribute to both scalar wavefunction and mass
renormalization in such a way that the pole mass of the scalar is not quadratically sensitive
to the cutoff.
To ensure that the symmetry will be preserved, it is sufficient to write the gauge-fixing
√
ˆ
term in terms of Aµ / ξ, where ξ is the gauge-fixing parameter. If one treats ξ as a spurion
field, the SO(1, 1) transformation

ˆ
ˆ
Aµ → e−α Aµ ,

ξ → e−2α ξ ,

(3.22)

√
ˆ
clearly leaves Aµ / ξ invariant. We will consider two different SO(1, 1) symmetric gaugefixing scenarios that are each convenient in different circumstances, and will denote them as
ordinary and “no-mixing” gauge-fixing.

3.4.1

Ordinary Gauge-Fixing

ˆ
ˆ
First, let us consider a gauge-fixing function of the typical form G(A) = ∂ µ Aµ . In Rξ gauge,
this amounts to adding to the Lagrangian the SO(1, 1) symmetric gauge-fixing term

L

2
1
ordinary
ˆ
∂ µ Aµ
=−
.
fixing
2ξ
73

(3.23)

˜
Using Eq. (3.14) to rewrite this in terms of Aµ and Aµ , one obtains

L

2 1
1
1
ordinary
˜
˜
=−
∂ µ Aµ 2 −
∂ µ Aµ + ∂ µ Aµ ∂ ν Aν .
fixing
2ξ
2ξ
ξ

(3.24)

With the gauge-fixing included, and after integrating by parts, the gauge field Lagrangian
reads
1
ordinary
Lgauge = Aµ g µν ∂ 2 − (1 − 1/ξ)∂ µ ∂ ν Aν
2
1
1˜
2
˜
˜
− Aµ g µν (∂ 2 − MA ) − (1 + 1/ξ)∂ µ ∂ ν Aν − Aµ ∂ µ ∂ ν Aν .
2
ξ

(3.25)

We can invert the diagonal terms, in momentum space, to find the partial propagators
qµqν
qµqν
−i µν
i
µν
µν
DAA =
g µν − (1 + ξ)
g − (1 − ξ)
, D˜˜=
2
AA q 2 − M 2
q2
q2
q 2 + ξMA
A

. (3.26)

Then the full tree-level photon and LW photon propagators, as well as the mixed-propagators,
can be computed by resumming the Dyson series to obtain
−i µν
qµqν
qµqν
µν
PAA =
,
g − (1 − ξ)
+
2
q2
q2
MA
i qµqν
µν
µν
P˜ =P ˜=−
.
2
AA
AA
q 2 MA

µ ν
i
µν
µν − q q ,
P˜˜=
g
2
AA q 2 − M 2
MA
A

(3.27)

Notice that only the photon propagator depends on the gauge-fixing parameter ξ, since the
photon is the only true gauge field in this theory.
˜
Up to an overall sign, the Aµ propagator is identical to the unitary-gauge propagator of
a massive gauge boson, in a spontaneously broken gauge theory. In particular, it contains
2
the q µ q ν /MA term which would apparently render the theory non-renormalizable and rein74

troduce quadratic divergences. However, the photon propagator and the mixed-propagators
contain the same term; when the PAA , P ˜ ˜, and P ˜ propagators are all included in loop
AA
AA
integrals, the badly behaved terms cancel, and quadratic divergences are avoided. This can
be seen even more clearly from the form of the P

µν
propagator. Recalling that all gauge
ˆˆ
AA

˜
ˆ
interactions depend on Aµ − Aµ = Aµ and working in Feynman gauge (ξ = 1), we obtain
P

qµqν
−i
µν
µν
µν
µν
= PAA + P ˜ ˜ − 2P ˜ =
g µν −
ˆˆ
2
AA
AA q 2 − q 4 /M 2
AA
MA
A

,

(3.28)

which decays like 1/q 2 for large values of the momentum.

3.4.2

No-Mixing Gauge-Fixing

2 1/2 ∂ µ A .
ˆµ
ˆ
Next, we consider the alternative11 gauge-fixing function G(A) = 1 + ∂ 2 /MA
The resulting SO(1, 1)-symmetric gauge-fixing Lagrangian is

L

2
2
1
1
no-mixing
ˆ
ˆ
=−
∂ µ Aµ +
∂ µ ∂ ν Aν
.
fixing
2
2ξ
2 ξMA

(3.29)

Adding this to the original higher-derivative gauge Lagrangian gives, after integration by
parts,
no-mixing 1 ˆ
= Aµ g µν ∂ 2 − (1 − 1/ξ)∂ µ ∂ ν
Lgauge
2

2 ˆ
1 + ∂ 2 /MA Aν .

(3.30)

11 In non-Abelian theories this introduces a q 2 /M 2 expansion in the gauge-ghost interacA
tion, which renders it less interesting. In Abelian gauge theories, however, the ghosts are
decoupled.
75

This is equivalent to
1ˆ
no-mixing
ˆ
Lgauge
= Aµ g µν ∂ 2 − (1 − 1/ξ)∂ µ ∂ ν Aν
2

M2
ˆ
˜
˜
+ Aµ g µν ∂ 2 − (1 − 1/ ξ)∂ µ ∂ ν Aν − A A2 .
2 µ

(3.31)

˜
in the sense that solving the equations of motion for Aµ and inserting the solution in (3.31)
recovers the form of (3.30).
ˆ
At this point we can eliminate the Aµ field from Eq. (3.31) via Eq. (3.14); the Lagrangian
˜
will include both diagonal and mixing terms in Aµ and Aµ , and the full tree-level propagators
can again be computed by summing the Dyson series. However, for ξ = 1 the mixing term
vanishes,12 and we obtain the simpler diagonal Lagrangian

1
1˜
no-mixing
2 ˜
L
= Aµ ∂ 2 Aµ − Aµ (∂ 2 − MA )Aµ .
gauge,ξ=1 2
2

(3.32)

The corresponding propagators
−i g µν
µν
PAA =
,
q2

i g µν
µν
P˜˜=
,
AA q 2 − M 2
A

(3.33)

have no q µ q ν terms so they are well-behaved at high energies. This is equally clear if we

12 As an alternative to the ξ = 1 gauge, one could replace Eq. (3.14) with
ν
ˆ
Aµ = Aµ − δµ − (1 −

qµ q ν ˜
ξ)
Aν ,
q2

in momentum space. This cancels the off-diagonal terms for any value of ξ, at the price of
introducing non-local interactions in coordinate space for ξ = 1.
76

ˆ
construct the A propagator,

P

−ig µν
µν
µν
µν
µν
= PAA + P ˜ ˜ − 2P ˜ =
,
ˆˆ
AA
AA q 2 − q 4 /M 2
AA
A

(3.34)

which falls off like q −4 in the ultraviolet.
We will now use the two convenient gauges introduced in this section to explore LW scalar
QED at one-loop.

3.5

One-Loop Renormalization

We will start by establishing an upper bound on the superficial degree of divergence of
Feynman diagrams in N = 2 LW scalar QED. For specificity, we work in the auxiliaryfield formulation and employ the no-mixing ξ = 1 gauge. Recalling that each loop integral
introduces four powers of momentum in the numerator, each trilinear gauge-scalar-scalar
vertex introduces one power of momentum in the numerator, and the propagator has two
powers of momentum in the denominator, we arrive at

D ≤ 4L − 2PA − 2P ˜ − 2Pφ − 2P ˜ + Vgss ,
A
φ

(3.35)

where L is the number of loops, Pf is the number of propagators of the f field, and Vgss is
the number of trilinear gauge-scalar-scalar vertices. The number of loop integrals is, in turn,
given by the total number of propagators (each carrying its own momentum-space integral)
minus the total number of vertices (each carrying a momentum-space delta function) plus
one, since an overall delta function ensures momentum conservation for the external fields.
77

Therefore, denoting the number of quartic gauge-gauge-scalar-scalar vertices by Vggss and
the number of four-point scalar vertices by Vssss , we have

L = PA + P ˜ + Pφ + P ˜ − Vgss − Vggss − Vssss + 1 .
A
φ

(3.36)

Finally we can relate the number of lines attached to a vertex to the number of propagators
(each connecting two vertices) and the number of external lines Nf

Vgss + 2Vggss = 2PA + 2P ˜ + NA + N ˜ ,
A
A

(3.37)

2Vgss + 2Vggss + 4Vssss = 2Pφ + 2P ˜ + Nφ + N ˜ ,
φ
φ
where the first relation deals with gauge lines and the second with scalar lines. Inserting
Eqs. (3.36) and (3.37) in Eq. (3.35) yields

D ≤ 4 − NA − N ˜ − Nφ − N ˜ .
A
φ

(3.38)

This equation tells us that the number of superficially divergent amplitudes is finite; since
no operators of dimension greater than four are present in the auxiliary-field Lagrangian, we
conclude that the theory is renormalizable.
In order to confirm renormalizability explicitly and to verify how the SO(1, 1) × SO(1, 1)
structure of the theory is protected against radiative corrections, we will now compute the
infinite13 part of the divergent one-particle irreducible (1PI) diagrams at one-loop. As a
way of checking our results and exploring the detailed symmetry structure, we will compute
13 If we were to compute the finite part as well, we would need to employ the CutkoskyLandshoff-Olive-Polkinghorne prescription in order to avoid unitarity violation [70]. However, the infinite part is not affected by this subtlety [72].
78

the diagrams in both the ordinary and no-mixing gauges, with ξ = 1. As we shall see, in the
no-mixing gauge only the vacuum polarization and self-energy amplitudes are infinite, while
in the ordinary gauge infinities also arise in the vertex corrections. Therefore, the way the
counterterms preserve the symplectic structure of the theory is different in the two gauges.

3.5.1

Counterterms

Radiative corrections renormalize the fields and mix the ordinary fields with the LW partners.
This not only preserves the U (1) gauge symmetry, but also does not generate any hard
breaking of the global SO(1, 1) × SO(1, 1) symmetry. Let us derive the most general relation
between bare and renormalized fields satisfying these requirements. We will employ the
standard QED nomenclature for the counterterms by using the subscript “3” for the photon
wavefunction renormalization, “2” for the matter field wavefunction renormalization, and
“1” for gauge vertex renormalization.
For the vector fields we have in general




Aµ
˜
Aµ





 
=

Z3
Z3


µ
Z3  Ar 
  .
˜µ
Z3
Ar


Gauge invariance requires Z3 = 0, lest a photon mass term be generated. Preserving the
˜
form of the symplectic combination Aµ − Aµ demands
Z3 =

Z3 −

Z3 .

Finally, substituting in the kinetic term Lagrangian and imposing the SO(1, 1) symmetry
79

on the counterterms gives
1
Z3 =
.
Z3
Therefore, the relation between bare and renormalized vector fields consistent with the symmetries of the theory is
 
µ
A 
 =
˜
Aµ



1
Z3 
0

 
µ
−1 − 1
Z3
 Ar 
 ,
−1
˜µ
Ar
Z3

Z3 ≡ 1 + δ3 .

(3.39)

Similarly, in order to preserve the SO(1, 1) symmetry on the scalar fields, the relation between
bare and renormalized scalar fields must be a symplectic rotation times a wavefunction
renormalization
 
φ
 =
˜
φ







cosh η sinh η  φr 
Z2 
 ,
˜
sinh η cosh η
φr

Z2 ≡ 1 + δ2 .

(3.40)

Substituting Eqs. (3.39) and (3.40) in the Lagrangian, Eq. (3.19), and denoting the gaugescalar-scalar vertex and gauge-gauge-scalar-scalar vertex renormalizations, respectively, by
Z1 ≡ 1 + δ1 and Z1 = 1 + δ1 leads to
2
MAr 2
1 ˜2
1 2
2 ˜
˜
˜
Arµ + |∂µ φr |2 − |∂µ φr |2 + Mr |φr |2 − m2 |φr |2
L = − Frµν + Frµν −
r
4
4
2
λr
˜
˜
˜
˜
|φr − φr |4 − igr (Arµ − Arµ ) φr ∂ µ φ∗ − φr ∂ µ φ∗ − h.c.
−
r
r
4
2
˜
˜
− gr (Arµ − Arµ )2 |φr |2 − |φr |2 + Lct ,

80

(3.41)

where the counterterm (ct) Lagrangian is
δ 2
MA 2
δ3 2
δ3
µν δ3 ˜ 2
˜
˜
Lct = − Frµν + Frµν Fr − Frµν −
Arµ
4
2
4
2
˜
˜
˜
˜
+ δ2 |∂µ φr |2 − δ2 |∂µ φr |2 + δ 2 |φr |2 − δ 2 |φr |2 − δmM φ∗ φr + φr φ∗
r
r
M
m

(3.42)

δ
˜
˜
˜
˜
− λ |φr − φr |4 − iδ1 gr (Arµ − Arµ ) φr ∂ µ φ∗ − φr ∂ µ φ∗ − h.c.
r
r
4
2
˜
˜
− δ1 gr (Arµ − Arµ )2 |φr |2 − |φr |2 .

The renormalized trilinear and quartic gauge-scalar couplings are related to the bare
couplings by
g

2
g 2 Z3 Z2 = gr Z1 ,

Z3 Z2 = gr Z1 ,

(3.43)

where gauge invariance guarantees

Z1 = Z1 = Z2 ,

(3.44)

to all orders in perturbation theory. The renormalized mass parameters are related to the
bare masses by
Z2 (cosh η)2 m2 − (sinh η)2 M 2 = m2 + δ 2 ,
r
m
2
Z2 (cosh η)2 M 2 − (sinh η)2 m2 = Mr + δ 2 ,
M

(3.45)

Z2 (M 2 − m2 ) cosh η sinh η = − δmM ,
whereas the renormalized and bare four-scalar couplings are related by

2
λ Z2 e−4η = λr + δλ .
81

(3.46)

=

+

˜

˜
+

+

˜
Figure 3.1: One-loop contribution to the vacuum polarization amplitudes. Each external
vector field is either a photon or a LW photon.
The vector field kinetic terms in the counterterm Lagrangian are now mixed. However, it
can be easily shown that these are still invariant under an SO(1, 1) transformation, provided
that δ3 is promoted to a spurion field.
We shall now prove that this set of counterterms is sufficient to absorb all infinities at
one-loop. In the process, the SO(1, 1) × SO(1, 1) global symmetry leads to cancellation of
the quadratic divergences in the scalar field self-energy amplitudes. In order to simplify our
notation we will drop the subscript r everywhere, but it should be kept in mind that all
fields and parameters involved in the calculations below are the renormalized ones.

3.5.2

Vacuum Polarization Amplitudes

We begin our examination of the infinite part of the divergent 1PI diagrams of LW scalar
QED by computing the one-loop contributions to the vacuum polarization amplitudes for
the vector fields. The relevant diagrams are illustrated in Fig. 3.1, where each external field
82

is either a photon or a LW photon. Since no gauge field propagators are involved in the
one-loop diagrams, the results are manifestly gauge independent. We find

µν
µν
µν
iΠAA = iΠ ˜ ˜ = −iΠ ˜ = iΠ(q 2 ) (q 2 g µν − q µ q ν ) ,
AA
AA

(3.47)

where, in dimensional regularization,

Π(q 2 ) = −2 ×

e2 1
+ finite terms ,
48π 2

(3.48)

with ≡ 2−d/2 as usual. The explicit factor of two arises from the presence of the LW scalar
µν
loops, and the remaining factor is the ordinary scalar QED contribution. Since Π ˜ ˜ contains
AA
no mass term, we have
δ

2
MA

=0.

(3.49)

The relevant counterterm contributions from the field-strength terms in Eq. (3.42) are

µν
µν
µν
iδΠAA = iδΠ ˜ ˜ = −iδΠ ˜ = −iδ3 (q 2 g µν − q µ q ν ) ,
AA
AA

(3.50)

which are precisely of the form required to cancel the infinities in Eq. (3.47). In the minimal
subtraction scheme we obtain
δ3 = −

3.5.3

e2 1
.
24π 2

(3.51)

Self-Energy Amplitudes

We will calculate the one-loop contribution to the self-energy amplitude Σ for a scalar field
in the no-mixing ξ = 1 gauge and then will repeat the calculation in the ordinary ξ = 1
83

A

A
+

=

˜
A

˜
A
+

+
˜

+

+

Figure 3.2: One-loop contribution to the 1PI self-energy amplitude Σφφ in the no-mixing
ξ = 1 gauge. In the ordinary gauge there are also diagrams involving internal mixed gauge
propagators, P ˜.
AA

gauge as a check. The relevant diagrams for the φ field, in the no-mixing gauge, are shown
˜
˜
in Fig. 3.2. Those for the φ field are obtained by replacing φ with φ; given the form of
the Lagrangian (3.19), we expect that the contributions of the diagrams involving internal
gauge bosons will change sign. The mixed self-energy amplitude Σ ˜ explicitly breaks the
φφ
U (1) × U (1) symmetry to diagonal U (1), and must vanish in the limit λ → 0; therefore, only
the diagrams with scalar loops will contribute to Σ ˜.
φφ
We begin our calculation of Σφφ in the no-mixing ξ = 1 gauge by considering potential
quadratic divergences. First, we examine the gauge-scalar diagrams on the top and middle
lines of Fig. 3.2. The first two diagrams correspond to the gauge-sector contribution in
ordinary scalar QED, which is quadratically divergent. That quadratic divergence is canceled
˜
by the A diagrams, as we now demonstrate. The SO(1, 1) symmetry acting on the vector
84

fields guarantees that: (i) the gauge-gauge-scalar and gauge-gauge-scalar-scalar couplings
involving the photon and LW photon are identical (up to an unphysical minus sign in the
gauge-gauge-scalar coupling14 ), and (ii) the LW photon propagator has a minus sign, relative
to the photon propagator. As a result, each diagram with an internal LW photon is opposite
in sign to its counterpart with an ordinary photon, and in the UV (where the LW photon mass
becomes irrelevant), there is an exact cancellation of the quadratic divergences. Likewise,
moving to the diagrams in the bottom row of Fig. 3.2, we recognize that the first diagram is
familiar from the ordinary φ4 theory, and is of course quadratically divergent. The second
diagram exactly cancels the quadratic divergence, as the SO(1, 1) symmetry acting on the
˜
scalar fields guarantees the equality of the |φ|4 and |φ|2 |φ|2 couplings, as well as the negative
˜
sign in the φ propagator.
Having established that Σ is free from quadratic divergences, we may proceed to complete
the one-loop calculation in no-mixing ξ = 1 gauge. In dimensional regularization, near d = 4,
the result is
2
λ(M 2 − m2 ) + 3g 2 MA 1
+ finite terms ,
16π 2
2
λ(M 2 − m2 ) − 3g 2 MA 1
−iΣ ˜ ˜ = − i
+ finite terms ,
φφ
16π 2
λ(M 2 − m2 ) 1
−iΣ ˜ = i
+ finite terms .
φφ
16π 2
−iΣφφ = − i

(3.52)

Notice that there is mass renormalization but not wavefunction renormalization, as the 1/
coefficients are q 2 independent in this gauge. Moreover, in the limit of exact SO(1, 1) ×
SO(1, 1) symmetry (where M → m and MA → 0) the self-energy amplitudes vanish exactly
14 This minus sign comes from the A − A dependence. However, we can always redefine
µ ˜µ
˜
˜
˜
˜
Aµ to −Aµ , which turns Aµ − Aµ into Aµ + Aµ . This, for example, is the convention
adopted in Ref. [42].
85

in the no-mixing ξ = 1 gauge, because the LW and ordinary propagators are then of equal
magnitude and opposite sign. The theory is still not finite, because, for example, the vacuum
polarization amplitudes do not vanish.
From Eq. (3.42) we find that the relevant counterterm contributions are of the form
−iδΣφφ = iδ2 q 2 − iδ 2 ,
m
−iδΣ ˜ ˜ = − iδ2 q 2 + iδ 2 ,
φφ
M

(3.53)

−iδΣ ˜ = − iδmM ,
φφ
and in the minimal subtraction scheme, we conclude

δ2 = 0 ,

(3.54)

2
λ(M 2 − m2 ) + 3g 2 MA 1
,
δ 2= −
m
16π 2
2
λ(M 2 − m2 ) − 3g 2 MA 1
δ 2=
,
M
16π 2
λ(M 2 − m2 ) 1
δmM =
.
16π 2

(3.55)

and

Inserting these results into Eq. (3.45) yields the following expression for the mixing angle

η=−

λ 1
.
16π 2

(3.56)

Note that this vanishes in the λ → 0 limit, as expected: for λ → 0 the theory acquires a
˜
global U (1) × U (1) symmetry (with the diagonal U (1) gauged) under which φ and φ rotate
86

˜
independently. This prevents φ-φ mixing terms from being radiatively generated.

The calculation in the ordinary ξ = 1 gauge proceeds somewhat differently because
mixed gauge propagators (P ˜) are present. Once their effects are included, the quadratic
AA
divergences still cancel among the diagrams involving internal gauge propagators. A direct
computation of the self-energy functions then yields
2
λ(M 2 − m2 ) + g 2 (3MA + m2 − q 2 ) 1
−iΣφφ = − i
+ finite terms ,
16π 2
2
λ(M 2 − m2 ) − g 2 (3MA + M 2 − q 2 ) 1
−iΣ ˜ ˜ = − i
+ finite terms ,
φφ
16π 2
λ(M 2 − m2 ) 1
−iΣ ˜ = i
+ finite terms .
φφ
16π 2

(3.57)

In this gauge, both mass renormalization and wavefunction renormalization are present. The
counterterm contributions are, of course, still given by Eq. (3.53), which have the right form
to cancel both the q 2 -dependent and q 2 -independent infinities in Eq. (3.57). In minimal
subtraction scheme, one obtains the relationships
2
λ(M 2 − m2 ) + g 2 (3MA + m2 ) 1
,
16π 2
2
λ(M 2 − m2 ) − g 2 (3MA + M 2 ) 1
δ 2=
,
M
16π 2
λ(M 2 − m2 ) 1
δmM =
,
16π 2
δ 2= −
m

(3.58)

which lead to the same expression for the mixing angle as in the no-mixing ξ = 1 gauge,
Eq. (3.56). However, this time the wavefunction renormalization counterterm is

δ2 = −

g2 1
,
16π 2

87

(3.59)

which differs from the result in the other gauge.
Since universality of the U (1) gauge coupling insures that the scalar field wavefunction
renormalizations are always exactly compensated by the vertex corrections, as Eq. (3.44)
shows explicitly, all that remains to show at one-loop, is the cancellation of all SO(1, 1)
breaking amplitudes in the φ4 sector.

3.5.4

φ4 Vertex

There are many diagrams contributing to the 1PI amplitudes with four external scalar fields.
As with the self-energy amplitudes, there are significant SO(1, 1) × SO(1, 1) cancellations
involved. However, now that these are well understood, we can reduce the number of diaˆ
grams by using the hat-field propagators on all internal lines: the A propagator of Eq. (3.28)
ˆ
when working in the ordinary ξ = 1 gauge, the A propagator of Eq. (3.34) when employing
ˆ
the no-mixing ξ = 1 gauge, and the gauge-independent φ propagator which is constructed
˜
by summing the simple propagators of the φ and φ fields
i (M 2 − m2 )
P ˆ ˆ = Pφφ + P ˜ ˜ =
.
φφ (q 2 − m2 )(M 2 − q 2 )
φφ

(3.60)

When we use the hat-field propagators, the expected cancellations occur within single diagrams, and are due to denominators with higher powers of loop momenta.
Let us denote by Γf f f f the one-loop contribution to the 1PI amplitude with external
1 2 3 4
scalar fields f1 , f2 , f3 , and f4 and start by working within the no-mixing ξ = 1 gauge. As a
concrete example, we will consider Γφφφ∗ φ∗ , for which the relevant diagrams are those shown
in Fig. 3.3. The first two diagrams are entirely due to the φ4 interaction, and involve only
ˆ
ˆ
internal φ fields. These diagrams are, thus, finite by power counting, since the φ propagator
88

φ∗

φ

ˆ
φ
ˆ
+ φ

=

1PI

ˆ
A

ˆ
φ +

φ

ˆ
φ

ˆ
A

φ

+ φ
ˆ
A

+ ˆ
A

ˆ
+ A

φ
φ +

φ

ˆ
A

φ

φ

φ
ˆ
A

+

φ

ˆ
A
+ φ

φ
ˆ
A

φ
+

φ
φ
ˆ
A

ˆ
A
φ

ˆ
A

φ
ˆ
+A

ˆ
A

ˆ
A

φ
ˆ
A
ˆ
A + φ

φ
ˆ
φ + A

ˆ +
A

φ

+

ˆ
A

ˆ
A

ˆ
A

∗

ˆ
+ A

+

ˆ
A
ˆ
A+ φ

φ

φ

ˆ
A

φ

Figure 3.3: One-loop contribution to the 1PI amplitude with four external φ fields. The
number of diagrams is reduced by employing the hat-field propagators. (See text for details)
decays like q −4 , at large momenta, and the vertices are momentum independent. The
remaining two diagrams of the first row, as well as the diagrams of the second and third row,
˜
ˆ
are entirely due to the gauge-scalar interactions, which depend on Aµ − Aµ ≡ Aµ . Although
the trilinear gauge-scalar-scalar vertices are momentum-dependent, in the no-mixing gauge
all these diagrams are finite by power counting. Finally, there are the diagrams in the last
two rows, which involve both φ4 and gauge vertices. Once again these are finite by power
89

counting in the no-mixing gauge. The same reasoning holds for all of the U (1) × U (1)
symmetric amplitudes; for the U (1) × U (1) violating amplitudes Γ

˜
φφφ∗ φ∗

, Γ ˜ ˜∗ ˜∗ , and
φφφ φ

Γ ˜∗ ˜∗ , the only difference is that the diagrams involving only gauge-scalar vertices do
φφφ φ
not contribute. We conclude that the amplitudes are purely finite in the no-mixing gauge
and
δλ = 0 .

(3.61)

The situation is different in the ordinary ξ = 1 gauge. If we, again, start by considering
Γφφφ∗ φ∗ , we still conclude that the first two diagrams, which involve only the φ4 interaction,
ˆ
and involve only internal φ fields are finite. The ten diagrams involving only gauge-scalar
interactions are another story. Power counting now predicts a logarithmic divergence for
2
ˆ
each of these diagrams, because of the q µ q ν /MA term in the Aµ propagator. However, we
expect that these divergences must cancel against one another for symmetry reasons. Recall
that the U (1) × U (1) violating amplitudes Γ

˜ , Γ ˜ ˜ ˜ , and Γφφφ∗ φ∗ receive no
˜ ˜
φφφ∗ φ∗ φφφ∗ φ∗

contribution at all from the diagrams with only gauge vertices. Then those diagrams cannot
make an infinite contribution to the U (1) × U (1)-symmetric amplitudes like Γφφφ∗ φ∗ either,
since this would correspond to a hard breaking of the SO(1, 1) symmetry acting on the scalar
fields. Explicit calculation confirms that the infinities arising from the diagrams with an even
number of gauge vertices (last two diagrams of the first row, and diagrams of the third row)
are precisely cancelled by the infinities from the diagrams with an odd number of gauge
vertices (diagrams of the second row). Therefore, the only possible infinite contribution to
Γφφφ∗ φ∗ and the other 1PI amplitudes in this gauge must arise from the diagrams in the
last two rows, which involve both φ4 and gauge-scalar vertices. Note that this implies that
in the λ → 0 limit all amplitudes with four external scalar fields are finite.
90

Computing the φ4 vertex correction diagrams from the last two rows of Fig. 3.3 in
ordinary ξ = 1 gauge yields the following infinite contributions

iΓφφφ∗ φ∗ = iX + finite, iΓ ˜ ∗ ˜∗ = i 4X + finite, iΓ ˜ ˜ ˜∗ ˜∗ = iX + finite,
φφφ φ
φφφ φ

(3.62)

iΓ
˜ = −i 2X + finite, iΓφφφ∗ φ∗ = −i 2X + finite, iΓφφφ∗ φ∗ = iX + finite,
˜˜ ˜
˜ ˜
φφφ∗ φ∗
where
X≡−

λ e2 1
.
8π 2

(3.63)

At the same time, Eq. (3.42) provides the counterterm contributions

iδΓφφφ∗ φ∗ = −iδλ ,

iδΓ ˜ ∗ ˜∗ = −i 4δλ ,
φφφ φ

iδΓ
˜ = i 2δλ ,
φφφ∗ φ∗

iδΓ ˜ ˜∗ ˜∗ = i 2δλ ,
φφφ φ

iδΓ ˜ ˜ ˜∗ ˜∗ = −iδλ ,
φφφ φ

(3.64)

iδΓ ˜∗ ˜∗ = −iδλ ,
φφφ φ

which are precisely of the form required to cancel the infinities in Eq. (3.62). In the minimal
subtraction scheme we obtain
δλ = −

3.5.5

λ e2 1
.
8π 2

(3.65)

Running of g and λ

We will now determine the β functions of the LW theory and compare them with the results
for ordinary scalar QED. To lowest order, the β functions for the electromagnetic and φ4
couplings in the LW theory are given by
g2 ∂
µ δ ,
βg =
2 ∂µ 3

βλ = µ
91

∂
−δλ + 2λ δ2 ,
∂µ

(3.66)

where µ is the scale we must introduce in dimensional regularization to make the log arguments dimensionless, and 1/ in the counterterms is interpreted as

1

→ log

Λ2
.
µ2

(3.67)

Since we are employing a mass-independent renormalization scheme, below the LW mass
scale (MLW ) we must impose the decoupling theorem and integrate out the LW fields. Since
what remains is identical to ordinary scalar QED, the counterterms are

δ3 = −

g2 1
,
48π 2

δ2 =

g2 1
,
8π 2

λ g2 1
λ2
δλ =
.
−
8π 2
8π 2

(3.68)

These, in turn, yield the low-energy leading order β functions

βg (µ < MLW ) =

g3
,
48π 2

λ2
3λ g 2
βλ (µ < MLW ) =
−
,
4π 2
4π 2

(3.69)

that are characteristic of ordinary scalar QED.
Above the LW mass scale, the appropriate counterterms are those we have derived for
the full LW theory. For the vector coupling, the counterterm value in Eq. (3.51) leads to

βg (µ > MLW ) =

g3
,
24π 2

(3.70)

which is twice the ordinary scalar QED βg function. In other words, the contribution from
loops of the LW scalar is identical to that from loops of the ordinary scalar; since there are
no internal gauge fields, the calculation is manifestly gauge invariant.
For the βλ function above the LW scale we consider the no-mixing and ordinary gauges
92

separately. If we employ the no-mixing ξ = 1 gauge, then Eqs. (3.54) and (3.61) tell us
that δ2 and δλ are each separately zero. In this gauge the LW scalar and vector fields make
contributions to the counterterms that are equal and opposite to those of the ordinary scalar
and vector fields. As a result, we find

βλ (µ > MLW ) = 0 .

(3.71)

In the ordinary ξ = 1 gauge, the values of δ2 are δλ are non-zero, as shown, respectively, by
Eqs. (3.59) and (3.65); however, the final result for βλ is the same, which provides a useful
check of our calculations.

3.6

Conclusion & Discussion

In this chapter we have discussed the global symmetries and the renormalizability of LeeWick scalar QED. The combination of SO(1, 1) global symmetry, U (1) gauge invariance,
and an SO(1, 1) invariant gauge-fixing condition allows us to show directly in the auxiliaryfield formalism that the number of superficially divergent amplitudes in a LW Abelian gauge
theory is finite. To illustrate the renormalizability of the theory, we have explicitly carried
out the one-loop renormalization program in LW scalar QED and demonstrated how the
counterterms are constrained by the joint conditions of gauge and SO(1, 1) invariance. We
have also computed the one-loop beta functions in LW scalar QED.
It would be interesting to generalize the discussion presented here to the case of nonAbelian gauge theories. However, this is not immediately possible. Notice that the SO(1, 1)
˜
transformation of Eq. (3.20) mixes a gauge field, Aµ , with a non-gauge vector field, Aµ .
93

˜
In an Abelian theory, we have the freedom to promote Aµ ≡ cosh αAµ + sinh αAµ to a
˜
gauge field for two reasons. First, the requirement that Aµ − Aµ transform like a gauge
field gives us the freedom to choose which field should bear the transformation. Second,
˜
all gauge interactions depend solely on e(Aµ − Aµ ), which is an SO(1, 1) invariant. That
these conditions are satisfied in Abelian gauge theory is perhaps not surprising, given that
a massive Abelian gauge theory is renormalizable [85].

In a non-Abelian gauge theory, however, interactions do not depend solely on g(Aa − Aa ),
µ ˜µ
and the SO(1, 1) symmetry is violated by the gauge interactions themselves. To see this
consider the generalization of the gauge kinetic energy terms of Eq. (3.13) to non-Abelian
interactions,
1
ˆ2
˜ ˆ
Lgauge = − Tr Fµν − 2 Tr Dµ Aν Fµν ,
2

(3.72)

˜
˜
ˆ ˜
Dµ Aν = ∂ µ Aν − ig[Aµ , Aν ] .

(3.73)

where

An SO(1, 1) transformation on the hat and tilde fields, and the gauge coupling, reads

ˆ
ˆ
Aµ → e−α Aµ ,

˜
ˆ
˜
Aµ → sinh α Aµ + eα Aµ ,

g → eα g .

(3.74)

Applying this to Lgauge gives
ˆ
ˆ ˆ
Lgauge → Lgauge + ig sinh α e−α Tr F µν [Aµ , Aν ] .

(3.75)

Thus the SO(1, 1) symmetry associated to the vector fields is explicitly broken by dimensionfour gauge boson self-interactions.
94

In principle we would, therefore, expect the SO(1, 1) breaking to propagate to other
sectors of the theory, and spoil the special relations between couplings that guarantee the
cancellation of quadratic divergences. However, both power counting in the higher-derivative
formulation [42] and the high energy behavior of massive vector meson scattering in LeeWick gauge theory [73] suggest that the number of superficially divergent diagrams remains
finite and that non-Abelian LW gauge theory may be renormalizable. A more thorough
understanding of non-Abelian LW gauge theories is, therefore, necessary in order to extend
the results demonstrated here for Abelian theories.
As we have seen, Part I of this Thesis has focused on an extension of the electroweak sector of the SM in which new (heavy) fermions, among other degrees of freedom, are introduced
in order to tackle and resolve the Hierarchy Problem, canceling the quadratic divergences
associated with the SM Higgs boson’s mass. We investigated in detail some of the interesting
and important theoretical aspects of this BSM theory, and analyzed how current experimental data constrains the new heavy degrees of freedom. In Part II of this Thesis, we proceed
to discuss a different type of BSM theory and some of its thought-provoking field theoretical
issues. This type of BSM theory would form an extension to the strong sector of the SM,
one in which new (heavy) colored gauge bosons (rather than new fermions) appear. Once
more, we will carefully explore both the formal and phenomenological aspects of this BSM
theory in detail.

95

PART II

96

Chapter 4
PRODUCTION OF MASSIVE
COLOR-OCTET VECTOR BOSONS
AT NEXT-TO-LEADING ORDER1

“Everything should be made as simple as possible, but no simpler! ”
—Albert Einstein (1879 – 1955)

4.1

Introduction

Massive color-octet vector bosons are present in theories which constitute an extension of the QCD sector of the Standard Model (SM). As explained in Chapter , since
late 1980s it has been of theoretical and phenomenological interest to extend the SM strong
sector in order to accommodate, among others, theories in which the electroweak symmetry
1 This chapter is based on the paper first published in [3].
97

breaking is induced by strong dynamics, where a new type of strongly-coupled gauge interaction forms composite Higgs out of colored fermions. Just as in the case of ordinary QCD,
one may apply perturbation theory to study this class of theories at high energies. This
may be again achieved through an asymptotic expansion of the theory in terms of the strong
coupling, αs ,2 since the coupling decreases in strength as a function of increasing energy,
reflecting the asymptotically free nature of the theory.
The analyses of the production of the massive colored gauge bosons to date, however,
capture only the leading order (LO) in perturbation theory. In this chapter, we extend the
production analysis to the next-to-leading order (NLO) in perturbation theory, thereby, improving dramatically upon the previous LO studies, in addition to predicting new kinematic
variables important for comparison with experiments.
Massive color-octet vector bosons are predicted in a variety of models, including axigluon
models [5, 44], topcolor models [45]-[48], technicolor models with colored technifermions [86,
p. 352-382], flavor-universal [49, 50] and chiral [51] coloron models, and extra-dimensional
models with KK gluons [7, 52]. These states have also recently been considered as a potential source [87, 88] of the top quark forward-backward asymmetry observed by the CDF
collaboration [89, 90].3 Recent searches for resonances in the dijet mass spectrum at the
LHC imply that the lower bound on such a boson is now 2-3 TeV [4],[94]-[96].4 If there
2 In previous chapters, we defined α to indicate the strong coupling. In this chapter,
3
however, we employ the more conventional term αs . Both conventions are interchangeably
used in the literature.
3 Note, however, that the observation of a top quark forward-backward asymmetry is not
confirmed by results of the D0 collaboration [91, 92]. Furthermore, if the observed top quark
forward-backward asymmetry is confirmed, explaining this using color-octet vector bosons
is problematic given the tight constraints on flavor-changing neutral-currents [93].
4 At least for the fermion charge assignments considered, and in the case where the resonance is narrow compared to the djiet mass resolution of the detector.
98

are color-octet vector bosons associated with the electroweak symmetry breaking sector, as
suggested by several of the models discussed above, their presence should be uncovered by
the LHC in the future.
In this chapter5 we report the first complete calculation6 of QCD corrections to the
production of a massive color-octet vector boson. We will refer to these massive coloroctet vector states generically as “colorons”. We treat the coloron as an asymptotic state
in our calculations, employing the narrow width approximation. Our next-to-leading order
(NLO) calculation includes both virtual corrections, as well as corrections arising from the
emission of gluons and light quarks, and we demonstrate the reduction in factorization-scale
dependence relative to the leading order (LO) approximation used in previous hadron collider
studies.
The QCD NLO calculation of coloron production reported here differs substantially from
the classic computation of the QCD NLO corrections to Drell-Yan production [99], because
the final state is colored. In particular, Drell-Yan production involves the coupling of the
light quarks to a conserved (or, in the case of W - or Z-mediated processes, conserved up to
quark masses) current. Hence, in computing the NLO corrections to Drell-Yan processes,
the current conservation Ward identity insures a cancellation between the UV divergences
arising from virtual quark wavefunction and vertex corrections. These cancellations do not
5 Throughout this chapter, the timeline for the depicted Feynman diagrams is from bottom to top, with exception of the fermion self-energy (Fig. 4.3) and the gauge boson VPA
(Figs. 4.8-4.10) diagrams where it is from left to right.
6 As this work was being completed, a computation of the NLO virtual corrections of top
quark pair production via a heavy color-octet vector boson has been reported in [97]. That
work is complementary to ours in that it does not employ the narrow width approximation
for the color-octet boson, but neither does it include real gluon or quark emission. After this
work was submitted for publication, real emission has also been considered by those authors
[98].
99

occur in the calculation of the NLO corrections to coloron production, because of vertex
corrections involving the three-point non-Abelian colored-boson vertices. As we describe
in Section 4.4, we use the “pinch technique” [100] to divide the problematic non-Abelian
vertex corrections into two pieces — a “pinched” piece whose UV divergence contributes
to the renormalization of the coloron wavefunction (and, ultimately, a renormalization of
the coloron coupling) and an “unpinched” part whose UV divergence (when combined with
an Abelian vertex correction) cancels against the UV divergences in quark wavefunction
renormalization. As we show, once the UV divergences are properly accounted for, the IR
divergences cancel in the usual way: the IR divergences arising from real quark or gluon
emission cancel against the IR divergences in the virtual corrections, and the IR divergences
arising from collinear quarks or gluons in the initial state are absorbed in the properly defined
parton distribution functions (PDFs).

We compute the gauge-, quark-, and self-couplings of the coloron from a theory with an
extended SU (3)1C × SU (3)2C → SU (3)C gauge structure, where SU (3)C is identified with
QCD. The calculation yields the minimal coupling of gluons to colorons, and allows for the
most general couplings of quarks to colorons. The cancellation of UV divergences described
above, however, occurs only when the three-coloron coupling has the strength that arises from
the dimension-four gauge-kinetic energy terms of the extended SU (3)1C × SU (3)2C gauge
structure. Our computation applies directly to any theory with this structure, i.e. to massive
color-octet vector bosons in axigluon, topcolor, and coloron models. In general, the triple
coupling of KK gluons in extra-dimensional models, or of colored technivector mesons in
technicolor models, will not follow this pattern. However, our results apply approximately
to these cases as well, to the extent that the SU (3)1C × SU (3)2C model is a good low100

energy effective theory for the extra-dimensional model (a “two-site” approximation in the
language of deconstruction [101, 102]) or for the technicolor theory (a hidden local symmetry
approximation for the effective technivector meson sector [103, 104]).7
This chapter is structured as follows: in Sec. 4.2 we introduce the formalism of a minimal
vector coloron theory, deriving all the Feynman rules, and setting the stage for the subsequent
calculations. In Sec. 4.3 we review the leading order computations of the amplitude and cross
section for coloron production due to q q pair annihilation. Sec. 4.4 describes in detail the one¯
loop virtual corrections to the q q pair annihilation process, elaborating on the contributions
¯
from the quark self-energy, coloron-coloron and gluon-coloron mixed vacuum polarization
amplitudes, and the vertex corrections. We employ the pinch technique [100], described
above, in order to consistently treat the UV divergences, and obtain a gauge-invariant,
mutually independent set of counterterms. The one-loop cross section is constructed, and
the IR singularities of the virtual correction properly extracted. In Sec. 4.5 we consider
the real emission processes, consisting of real (soft and collinear) gluon and (collinear) quark
emission. In Sec. 4.6 we put all the pieces together, exhibiting the explicit cancellation of the
IR divergences among the real and virtual corrections, and demonstrate the renormalization
of the quark and gluon PDFs. We give a finite expression for the NLO-corrected production
cross section. Finally, in Sec. 4.7 we plot the cross section, demonstrate that the QCD NLO
corrections are as large as 30%, and show that the residual factorization-scale dependence is
at the 2% level. We also calculate the K-factor and the pT spectrum for coloron production,
7 Arbitary three- and four-point coloron self-couplings can be incorporated in the
SU (3)1C × SU (3)2C by adding O(p4 ) terms in the effective chiral Lagrangian of Eq. (4.1),
2
and deviations in these couplings are, therefore, of O(MC /Λ2 ), where Λ is the cutoff of the
effective coloron theory. The three- and four-point self-couplings, however, are neither relevant to the leading order q q , nor to the IR divergent NLO coloron production contributions,
¯
and, therefore, are numerically insignificant.
101

since these are valuable for comparison with experiment. Appendix contains all the Feynman
rules of the coloron theory.

4.2

A Minimal Theory for Spin-One Colorons

In this section, we introduce colorons8 as the massive color-octet bosons arising when an
extended SU (3)1C × SU (3)2C gauge symmetry is spontaneously broken by a non-linear
sigma model field to its diagonal subgroup, SU (3)C , which we identify with QCD. The
symmetry breaking results in a low-energy spectrum that includes both a massless spin-one
color-octet of gauge bosons, the gluons, and a massive spin-one color-octet of gauge bosons,
the colorons.
In detail, we replace the QCD Lagrangian with
2
1
µν 1
µν f
Lcolor = − G1µν G1 − G2µν G2 +
Tr Dµ Σ Dµ Σ†
4
4
4

(4.1)

+ Lgauge-fixing + Lghost + Lquark .
Here, Σ is the non-linear sigma field breaking SU (3)1C × SU (3)2C to SU (3)C ,
Σ = exp

2iπ a ta
f

,

a = 1, . . . , 8 ,

(4.2)

where π a are the Nambu-Goldstone bosons “eaten” by the coloron, f is corresponding
8 Colorons can, in principle, be introduced as matter fields in the adjoint of SU (3) . This
C
approach, however, would lead to an early violation of tree-level unitarity, as the scattering
amplitude of longitudinally polarized massive spin-one bosons can grow, by power counting,
like E 4 , where E is the center-of-mass (CM) energy. The only way to avoid this is to
“promote” the coloron to the status of gauge field of a spontaneously broken gauge theory:
then the special relation between trilinear and quartic gauge couplings will lead to an exact
cancellation of the terms growing like E 4 , as happens in the standard electroweak theory.
102

“decay-constant”, and ta are the Gell-Mann matrices, normalized as Tr ta tb = δ ab /2. The
Σ field transforms as the bi-fundamental of SU (3)1C × SU (3)2C ,
†
Σ → u1 Σ u2 ,

a
ui = exp iαi ta ,

(4.3)

a
where the αi are the parameters of the SU (3)iC transformations. This leads to the covariant
derivative
Dµ Σ = ∂µ Σ − igs1 Ga ta Σ + igs2 Σ Ga ta ,
2µ
1µ

(4.4)

where gsi is the gauge coupling of the SU (3)iC gauge group. Up to a total divergence, the
quadratic terms in the Lagrangian are
2
2 1
1 a
(2)
a 2
µν ∂ 2 − ∂ µ ∂ ν Ga + f
L
= Giµ g
gs1 Ga − gs2 Ga
1µ
2µ + 2 ∂µ π
iν
color
2
8
f
(2)
(2)
(2)
−
gs1 Ga − gs2 Ga ∂ µ π a + L
+L
+L
,
1µ
2µ
gauge-fixing
ghost
quark
2

(4.5)

where a sum over i = 1, 2 in the gauge kinetic terms is implied.
The gauge-Goldstone mixing term can be removed, up to a total divergence, by choosing
the gauge-fixing Lagrangian to be

1
a
Lgauge-fixing = − Fi 2 ,
2

(4.6)

where the gauge-fixing functions are

1
a
F1 ≡ √
ξ

gs f
∂ µ Ga + ξ 1 π a
1µ
2

,

1
a
F2 ≡ √
ξ

gs f
∂ µ Ga − ξ 2 π a
2µ
2

.

(4.7)

The Faddeev-Popov ghost Lagrangian is obtained by taking the functional determinant of
103

a
b
δFi /δαj . This leads to
Lghost = ca − ∂ µ δij δ ab ∂µ − gsi f abc δij Gc
¯i
iµ
2
gs f 2
−ξ i
δi1 − δi2 δ1j − δ2j δ ab + O(π) cb ,
j
4

(4.8)

where f abc are the SU (3) structure constants, and a sum over i, j = 1, 2 is implied. Notice
that we have included only the inhomogeneous terms in the transformation of the eaten
Goldstone boson, whence, the unspecified O(π) term in the ghost Lagrangian which are
unnecessary for our computation. Up to a total divergence, the quadratic Lagrangian now
reads
g2 f 2
µ ∂ ν + si
∂
δi1 − δi2
4

1
1
(2)
L
= Ga δij g µν ∂ 2 − δij 1 −
color
2 iµ
ξ

δ1j − δ2j

1
ξ 2
2
g + gs f 2 π a
− πa ∂ 2 +
2
2
4 s1
g2 f 2
a δ ∂ 2 + ξ si
− ci ij
¯
δi1 − δi2
4

Ga
jν
(4.9)

δ1j − δ2j

ca + L
j

(2)
.
quark

Aside from a factor of the gauge-fixing parameter ξ, the gauge and ghost fields share the
same mass matrix, as expected. This is diagonalized by


 
a
a
G1µ 
Gµ 

 = R  ,
a
a
G2µ
Cµ
where





cos θc − sin θc 
R≡
 ,
sin θc cos θc
104

 
 
a
a
 c1 
 cG 
  = R  ,
ca
ca
2
C

sin θc ≡

gs1
.
2 + g2
gs
s2
1

(4.10)

(4.11)

a
In Eq. (4.10) Ga is the gluon field and Cµ is the coloron field, whereas ca and ca are the
µ
C
G
corresponding ghost fields. Inserting these expressions in Eq. (4.9) gives, for the coloron
mass,
MC =

2
2
gs + gs f
1
2 ≡ gs f ,
2
sin 2θc

(4.12)

where gs is the SU (3)C coupling,
1
1
1
=
+
.
2
2
2
gs
gs
gs
1
2

(4.13)

The gluon ghost is massless, whereas both the coloron ghost and the eaten Goldstone boson
have mass

√

ξMC . The interaction vertices and the corresponding Feynman rules can be

found in Appendix .
We will leave the quark charge assignments under SU (3)1C × SU (3)2C arbitrary, for
greater generality. In the mass eigenstate basis we write

/
/
Lquark = q i i ∂ − igs Ga ta − iC a ta gL PL + gR PR
¯ /

qi ,

(4.14)

where PL and PR are the helicity projection operators,

PL ≡

1 − γ5
,
2

PR ≡

1 + γ5
,
2

(4.15)

and i is a flavor index.9 The coupling to the gluon is dictated by charge universality,
whereas the gL and gR couplings to the coloron depend on the original charge assignments
of the quarks. For example, if both left-handed and right-handed quarks are only charged
9 Here we work in the broken electroweak phase, and only employ fermion mass eigenstates.
105

under SU (3)1C , then gL = gR = −gs tan θc , while the axigluon [5, 44] corresponds to
gL = −gR = gs (i.e. θc = π/4). In general, gL and gR can each take on the values
−gs tan θc or gs cot θc in any specific model,10
gL , gR ∈ {−gs tan θc , gs cot θc } .

4.3

(4.16)

LO Coloron Production

The dominant channel for coloron production at a hadron collider is given by the tree-level
diagram of Fig. 4.1, in which a q q pair annihilates into a coloron. The tree-level diagram with
¯
gluon-gluon fusion into a coloron does not exist in the Lagrangian of Eq. (4.1): in general
there are no dimension-four terms with two gauge bosons of an unbroken symmetry and a
spin-one field charged under the same symmetry. We use the narrow width approximation for
the coloron, take the quarks to be on-shell, and set their masses to zero:11 this is certainly a
good approximation, as current experimental bounds [4],[94]-[96] constrain the coloron mass
to be in the TeV range.
The leading order (LO) amplitude corresponding to the diagram of Fig. 4.1 is

(0)
iMq q →C = gs v r (¯) iγ µ rL PL + rR PR ta us (p) εaλ∗ (r) ,
¯ p
µ
¯

(4.17)

where the superscripts r and s denote quark spin projections, λ is the coloron polarization,
10 It is possible to generalize this setup to non-universal charge assignments: in this case
flavor-diagonal chiral couplings to the coloron would depend on a generation index. Flavorchanging couplings are strongly constrained [93].
11 Note that the Yukawa couplings of quarks to the eaten Goldstone bosons are proportional
to the quark masses, and hence vanish in the zero-mass limit.
106

→

a, ⌫

r

p

→

p
¯

a
Figure 4.1: Tree-level contribution to coloron production. The coloron field, Cν , is represented by the zigzag line.

and
g
rR ≡ R ,
gs

g
rL ≡ L ,
gs

rL , rR ∈ {− tan θc , cot θc } .

(4.18)

In d = 2(2 − ) dimensions, the squared amplitude averaged over initial spins and colors,
and summed over final polarization states, is

(0)
|Mq q →C |2 ≡
¯
=

2
1
dim(r)

1 2
2

spin & color

(0)
|Mq q →C |2
¯

C2 (r)(1 − ) 2 2
2 ˆ
gs rL + rR s ,
2 dim(r)

(4.19)

where dim(r) = 3 and C2 (r) = 4/3 are respectively the dimension and Casimir of the
fundamental representation of SU (3), and s ≡ (p + p)2 = 2 p · p is the partonic center-ofˆ
¯
¯
mass (CM) energy. This gives the LO cross section [44] for q q → C,
¯
2
2
α A(rL + rR )
π
(0)
(0)
2 δ(1 − χ) = s
σq q →C =
ˆ ¯
|Mq q →C |
δ(1 − χ) ,
¯
s
ˆ
s2
ˆ
107

(4.20)

2
where αs ≡ gs /4π,
A≡

2π 2 C2 (r)(1 − )
,
dim(r)

(4.21)

and
2
MC
χ≡
.
s
ˆ

(4.22)

The full LO cross section for pp → C is given by the convolution of the LO partonic

(0)
cross section σq q →C with the parton distribution functions (PDFs) for the quarks within
ˆ ¯
the protons, and a sum over all quark flavors,

σ LO =

dx1

dx2

q

(0)
fq (x1 )fq (x2 ) + fq (x1 )fq (x2 ) σq q →C ,
ˆ ¯
¯
¯

(4.23)

where fq (x) is the PDF of parton q, and x the momentum fraction of the corresponding
parton. Taking the collision axis to be the 3-axis, the four-momenta of the partons are
√

s
(x1 , 0, 0, x1 ) ,
p=
2

p=
¯

√

s
(x2 , 0, 0, −x2 ) ,
2

(4.24)

where s is the CM energy of the colliding hadrons. This gives

s = x1 x2 s ,
ˆ

4.4

2
MC
χ=
.
x1 x2 s

(4.25)

NLO Coloron Production: Virtual Corrections

In this section we compute the next-to-leading order (NLO) virtual QCD corrections to the
q q → C amplitude. These include one-loop wavefunction and vertex corrections, which we
¯
choose to compute in ’t Hooft-Feynman gauge, ξ = 1. The non-Abelian vertex corrections
108

are computed by employing the pinch technique: this allows us to obtain QED-like Ward
identities, and absorb all UV infinities in the renormalization of the gauge field propagators.
After inclusion of the counterterms, the virtual corrections are UV-finite, yet IR infinite.
In Sec. 4.6 we show that the IR divergences cancel once the real corrections, corresponding
to the emission of soft and collinear gluons and quarks, are included in the calculation
of the inclusive production cross section. Our loop integrals are computed in dimensional
regularization, with d = 2(2 − ) dimensions. We first regulate the IR divergences by giving
the gluon a small mass (mg → 0+ ): in this way all infinities are in the UV, and regularization
requires > 0. After all of the UV infinities are removed, by cancellation and inclusion of the
counterterms, we let the gluon mass approach zero. This will make the virtual corrections
IR divergent, with the infinities being regulated by taking < 0.
Since the quark couplings to the coloron are chiral, in general, we need a prescription
for treating γ5 in d = 4. Here we take γ5 to always anticommute with γ µ . Choosing an
alternative prescription, such as ’t Hooft-Veltman in which γ5 anticommutes with γ µ for
µ = 0, 1, 2, 3 and commutes for other values of µ, would lead to a cross section for q q → C
¯
which differs from ours by only a finite renormalization of the coupling(s).
The general structure of the q q → C amplitude, illustrated in Fig. 4.2, is
¯
(ˆ)
s
1/2 aµ
aµ Π
iMq q →C = gs v r (¯) i ZC ΓqqC + ΓqqG GC
¯ p
Zq us (p) εaλ∗ (r) ,
µ
¯
s
ˆ

(4.26)

aµ
aµ
where ΓqqC (ΓqqG ) is the one-particle irreducible (1PI) quark-quark-coloron (quark-quarkgluon) vertex and ΠGC is the coefficient of g µν in the gluon-coloron vacuum polarization
mixing amplitude (VPA). The factors Zq and ZC are, respectively, the residues of the full
quark and coloron propagators at the mass pole; they are obtained from the quark self-energy
109

1PI

1PI

amputated

amputated

1PI

1PI

1PI

(a)

1PI

(b)

Figure 4.2: Structure of q q → C amplitude, to all orders in perturbation theory. Direct
¯
coloron production is illustrated on the left, while production via mixing with the gluon is
shown on the right. The gluon field is, as usual, represented by the coiling line; the coloron
field is represented by the zigzag line.

amplitude, Σ(p), and the coefficient of g µν in the coloron-coloron VPA, ΠCC (q 2 ), as follows
/

Zq =

1
,
1 − Σ (0)

ZC =

1
1 − ΠCC (M 2
)
C phys

,

(4.27)

where the prime denotes a derivative with respect to the argument, and MC phys is the
aµ
coloron’s physical mass. To lowest order, Zq = 1, ZC = 1, ΠGC = 0, and iΓqqC =
γ µ rL PL + rR PR ta ; inserting these expressions in Eq. (4.26) recovers the tree-level amplitude of Eq. (4.17).

110

p

p

p

p

Figure 4.3: Quark self-energy diagrams at one-loop. Particle notation as defined in Fig. 4.2.

4.4.1

Quark Self-Energy

The NLO quark self-energy correction to the q q → C amplitude is found, from Eqs. (4.26)
¯
and (4.27), to be

iQ = v r (¯) iγ ν gL PL + gR PR ta δZq us (p) εaλ∗ (r) ,
¯ p
ν

(4.28)

δZq = Σ (0) .

(4.29)

where

At one-loop, the Σ(p) amplitude is given by the diagrams of Fig. 4.3. These lead to the
/
expression
g 2 C (r) 2(1 − )Γ( )
Σ(p) = − p s 2
/
/
16π 2
1
4πµ2
×
dx (1 − x)
∆Gq
0

+

4πµ2
∆Cq

(4.30)
2
2
(rL PL + rR PR )

,

where Γ( ) is the Euler Gamma-function evaluated at infinitesimal , and

∆Gq = (1 − x)m2 − x(1 − x)p2 − iη ,
g

2
∆Cq = (1 − x)MC − x(1 − x)p2 − iη .

(4.31)

The parameter µ is the mass scale introduced by the loop integral in d dimensions, and η
111

is the positive infinitesimal parameter giving the appropriate prescription for computing the
integral in momentum space. As previously anticipated, we have introduced a small gluon
mass, mg , in order to regulate the IR divergences and isolate the UV infinities: with mg = 0,
Σ(p) and Σ (p) contain only UV divergences. Inserting Eq. (4.30) in Eq. (4.29) gives
/
/
g 2 C (r) 2(1 − )Γ( ) 1
δZq = − s 2
dx (1 − x)
16π 2
0
4πµ2
4πµ2
×
+
2
(1 − x)m2 − iη
(1 − x)MC − iη
g

(4.32)
2
2
(rL PL + rR PR )

.

The amplitude of Eq. (4.28) becomes

iQ = −

1
1−x
αs
2C2 (r)(1 − )Γ( )
dx
dy
4π
0
0
4πµ2
4πµ2
(0)
iMq q →C +
¯
2
(1 − x)m2 − iη
(1 − x)MC − iη
g

(0)
iMq q →C
¯

(4.33)
,

(0)
where Mq q →C is given by Eq. (4.17), and
¯
(0)
3
3
iMq q →C = gs v r (¯) iγ ν rL PL + rR PR ta us (p) εaλ∗ (r) .
¯ p
ν
¯

(4.34)

For later convenience we have traded the 1 − x factor, in Eq. (4.32), for an integral over
dy: this will allow us to directly add the self-energy correction to the vertex correction and
explicitly show the cancellation of the UV divergences.

112

Figure 4.4: One-loop Abelian vertex correction to the q q → C amplitude. Particle notation
¯
is as defined in Fig. 4.2.

4.4.2

Abelian Vertex Corrections

The one-loop Abelian vertex correction to the q q → C amplitude is given by the diagrams
¯
of Fig. 4.4. These lead to the amplitude

iVAbelian =

1
1−x
αs
[2C2 (r) − C2 (G)] Γ(1 + )
dx
dy
4π
0
0
(1 − )2
s
ˆ
4πµ2
(0)
iMq q →C (4.35)
− (xy − (1 − x)(1 − y))
¯
∆Gqq
∆Gqq
s
ˆ
4πµ2
(1 − )2
(0)
− (xy − (1 − x)(1 − y))
iMq q →C ,
+
¯
∆Cqq
∆Cqq

where C2 (G) = 3 is the Casimir of the adjoint representation, and
∆Gqq = (1 − x − y)m2 − xyˆ − iη ,
s
g

(4.36)

2
∆Cqq = (1 − x − y)MC − xyˆ − iη .
s
Once again, we have included a small gluon mass mg in order to regulate the IR divergences.

113

→

r =p+p
¯

µ

⇢

p

k

p
¯

→

⌫

Figure 4.5: One-loop non-Abelian vertex correction to the q q → C amplitude. Particle
¯
notation is as defined in Fig. 4.2. Each three-gauge boson vertex is a full non-Abelian vertex
Γµνρ in Eq. (4.37).

4.4.3

Non-Abelian Vertex Corrections a la Pinch-Technique: Unpinched Diagrams

The non-Abelian vertex corrections are given by the diagrams of Fig. 4.5. When added to the
overall Abelian vertex correction, Eq. (4.35), these give the one-loop total vertex correction
to q q → C. Unlike in QED, the UV divergences in the vertex correction do not cancel the
¯
UV divergences arising from the self-energy amplitudes. The reason for this is that the QED
a
Ward identity ∂ µ jµ = 0 is now replaced by its non-Abelian counterpart Dµ jµ = 0, which
does not imply the equality of vertex and quark-wavefunction renormalization constants. It
a
is possible, though, to recover QED-like Ward identities for the currents jµ by employing
the pinch technique. This consists of breaking up the gauge boson internal momenta of
a Feynman diagram into “pinching” and “non-pinching” pieces. The pinching momenta
are those which cancel some internal propagators, leading to a simpler diagram with the
external-momentum structure of a propagator. The non-pinching momenta will instead give
overall amplitudes satisfying QED-like Ward identities. A formal proof of these statements,
for an arbitrary non-Abelian gauge theory, can be found in the review of Ref. [100] (and
114

references therein).
In our vertex computation the pinch technique works as follows. The non-Abelian vertex
structure in each of the diagrams in Fig. 4.5 is

Γµνρ (k, p, p) = g µν (−2p − p + k)ρ + g νρ (p − p − 2k)µ + g ρµ (k + p + 2¯)ν .
¯
¯
¯
p

(4.37)

We can break this into two parts,

µνρ
µνρ
Γµνρ (k, p, p) = ΓF (k, p, p) + ΓP (k, p, p) ,
¯
¯
¯

(4.38)

where

µνρ
ΓF (k, p, p) = − 2g µν (p + p)ρ + 2g ρµ (p + p)ν + g νρ (p − p − 2k)µ ,
¯
¯
¯
¯

(4.39)

µνρ
ΓP (k, p, p) = g µν (¯ + k)ρ + g ρµ (k − p)ν .
¯
p

(4.40)

µνρ
Unlike Γµνρ (k, p, p), the ΓF (k, p, p) vertex satisfies a QED-like Ward identity for the
¯
¯
gC → C and CC → C amplitudes,
µνρ
(p + p)µ ΓF (k, p, p) = g νρ (p − k)2 − (¯ + k)2 .
¯
¯
p

(4.41)

µνρ
As shown below, when ΓF (k, p, p) is used to compute the integral in momentum space
¯
(instead of Γµνρ (k, p, p)), its UV divergences, added to the UV divergences of the Abelian
¯
vertex corrections, exactly cancel the UV divergences of the quark self-energy amplitudes.
As mentioned above, this occurs because a QED-like Ward identity for q q → C holds, as one
¯
can prove by using the QED-like Ward identity for the gC → C and CC → C amplitudes
115

Figure 4.6: Non-Abelian unpinched vertex-correction diagrams for the q q → C amplitude
¯
at one-loop. Particle notation is as defined in Fig. 4.2. The black disk indicates that each
three-point gauge boson vertex in these diagrams has been replaced by the non-pinched
µνρ
portion ΓF , as described in Eqs. (4.38) and (4.39).

µνρ
given in Eq. (4.41). The three diagrams which correspond to using ΓF (k, p, p) instead of
¯
Γµνρ (k, p, p) are symbolically denoted with a black disk over the non-Abelian vertex, and
¯
are shown in Fig. 4.6. These lead to the following contribution to the q q → C amplitude
¯
1−x
1
αs
dy
dx
C2 (G)Γ(1 + )
4π
0
0
1−
s
ˆ
4πµ2
1−
s
ˆ
− (x + y)
+
− (x + y)
∆GCq
∆GCq
∆CGq

iVnon-Abelian =

−
+

1−
1−

− (x + y)

− (x + y)

s
ˆ
∆CCq
s
ˆ

∆CCq

4πµ2
∆CCq
4πµ2
∆CCq

(0)
iMq q →C
¯
(0)
iMq q →C
¯

4πµ2
∆CGq
(4.42)

,

where
2
∆GCq = xm2 + yMC − xyˆ − iη ,
s
g
2
∆CGq = xMC + ym2 − xyˆ − iη ,
s
g
2
∆CCq = (x + y)MC − xyˆ − iη .
s

116

(4.43)

Figure 4.7: Non-Abelian pinched vertex-correction diagrams for the q q → C amplitude at
¯
one-loop. Particle notation is as defined in Fig. 4.2.

In order to obtain Eq. (4.42) we have used the equations of motion for the external spinors,
together with the relations

2
2 cot(2θc ) rL = −1 + rL ,

2
2 cot(2θc ) rR = −1 + rR ,

(4.44)

which are true for any charge assignment of the quarks. As anticipated, iQ + iVAbelian +
iVnon-Abelian is free of UV divergences, as manifestly shown by adding together Eqs. (4.33),
(4.35) and (4.42). This part of the amplitude is, however, IR divergent in the limit of zero
gluon mass. Setting mg = 0 and < 0 gives

iQ + iVAbelian + iVnon-Abelian
2
3 + 2i
αs
C2 (r) − −
=
2
4π

+ C2 (G)

iπ

(0)
iMq q →C + finite .
¯

(4.45)

µνρ
Of course we still need to include the contribution from ΓP (k, p, p) (of Eq. (4.40)) in
¯
the full non-Abelian vertex correction. This contains the pinching momenta: the action of p
and p on the external spinors gives zero, and the remaining piece cancels the internal fermion
¯
117

propagator in the diagram. Thus, the internal fermion line in each diagram is pinched away,
leaving an effective diagram with a four-point coupling between fermions and gauge bosons
as shown in Fig. 4.7. The UV divergences of the pinched diagrams have the same groupand momentum-structure as those of the VPAs, and can be absorbed in the counterterms
for the gauge field propagators. In order to see this clearly, we will now consider the form of
the “true” propagator corrections to the q q → C amplitude in the following subsection.
¯

4.4.4

Form of the Vacuum Polarization Amplitudes

The NLO corrections to the q q → C amplitude due to the VPAs are found, from Eqs. (4.26)
¯
and (4.27), to have the form
δZC
(ˆ)
s
(0)
(0) Π
iP = iMq q →C
+ iMq q →C GC
,
¯
¯
2
s
ˆ

(4.46)

2
δZC = ΠCC (MC ) ,

(4.47)

(0)
iMq q →C = gs v r (¯) iγ µ ta us (p) εaλ∗ (r) .
¯ p
µ
¯

(4.48)

where

and

aµ
In order to obtain the second term of Eq. (4.46), we have replaced ΓqqG with its LO com2
2
ponent iγ µ ta . Notice also that at this order we can swap MCphys for MC .
At one-loop, ΠCC (q 2 ) is given by the diagrams of Figs. 4.8 and 4.10, in which the
gluon ghost is represented by dotted lines, the coloron ghost by a sequence of filled circles, and the eaten Goldstone bosons are represented by dashed lines. There are poles at
2
d = 2 proportional to both q 2 and MC . The latter correspond to quadratic divergences
118

Figure 4.8: Coloron-coloron vacuum polarization amplitude at one-loop. A gluon field is,
as usual, represented by a coiling line; a coloron field is represented by a zigzag line. The
coloron ghost is represented by a sequence of filled circles, and the eaten Goldstone bosons
are represented by dashed lines.

Figure 4.9: Gluon-coloron mixing amplitude at one-loop. Particle notation is as defined in
Fig. 4.8.

Figure 4.10: Fermion contributions (solid lines) to coloron-coloron vacuum polarization amplitude and gluon-coloron mixing amplitude. Particle notation is as defined in Fig. 4.8.

119

(renormalizing the coloron mass scale f ), whereas the former can only be logarithmic by
dimensional analysis (renormalizing the coloron field).12 The momentum-dependent part of
the full coloron-coloron VPA is not transverse, as the coefficients of the q 2 and q µ q ν terms
are different. However, we have explicitly verified that the infinite part is transverse: this
is necessary, because the corresponding Lagrangian counterterms are transverse. For small
values of

we obtain

αs −1
ΠCC (q 2 )g µν + q µ q ν –terms
4π
1
µ2
dx
2 1 + 4x(1 − x) E + 2(1 − 2x)2 g µν q 2 − q µ q ν
= C2 (G)
∆GC
0
µ2
+
1 − x(4 − 3x) E − x(1 − x) g µν q 2
∆GC
µ2
2
+
2x E + 3 − 5x g µν MC
(4.49)
∆GC
1
µ2
dx
+ 4 cot2 (2θc )C2 (G)
1 + 4x(1 − x) E + (1 − 2x)2 g µν q 2 − q µ q ν
∆CC
0
x(1 − x)
x(1 − x) µν 2
(1 − 2x)2
µ2
µ2
E−
E qµqν
+ −
g q +
∆CC
4
4
∆CC
8
5
1
µ2
2
E + g µν MC
+
∆CC
4
4
1
µ2
2
2
dx
+ (rL + rR )Nf
− 2x(1 − x) E g µν q 2 − q µ q ν ,
∆qq
0
where our results depend only on the coefficient of g µν , the quantity Nf is the number of

12 This situation parallels the renormalization of the electroweak chiral lagrangian [105,
106].
120

quark flavors in the loop (see Fig. 4.10),

E≡

1

− γ + log 4π ,

(4.50)

and γ is the Euler-Mascheroni constant. The ∆ functions in Eq. (4.49) are
2
∆GC ≡ x m2 + (1 − x)MC − x(1 − x)q 2 − iη ,
g
2
∆CC ≡ MC − x(1 − x)q 2 − iη ,

(4.51)

∆qq ≡ − x(1 − x)q 2 − iη .
Notice that the coloron-coloron VPA of Eq. (4.49) is not IR divergent in the limit mg → 0,
since there are no contributions with only massless (gluon) states. However, what enters
in Eq. (4.46) is the derivative of ΠCC (see Eq. (4.47)), which is IR divergent in the limit
mg → 0.
The momentum-dependent part of the gluon-coloron mixing amplitude (Figs. 4.9 and
4.10) is found to be transverse, both in the infinite and the finite parts. For small values of
we find
1
αs −1
ΠGC (q 2 )g µν + q µ q ν –terms = 2 cot(2θc )C2 (G)
dx
4π
0
×

3
+ 5x(1 − x) E + (1 − 2x)2
4

+ (rL + rR )Nf

1
0

dx

µ2
∆qq

µ2
∆CC

2
q 2 g µν − q µ q ν + E MC

− 2x(1 − x)

There are no potential IR divergences hidden in ΠGC .

121

q 2 g µν − q µ q ν

(4.52)
.

4.4.5

Non-Abelian Vertex Corrections a la Pinch-Technique: Pinched
Diagrams

The pinched diagrams of Fig. 4.7 are obtained from the diagrams of Fig. 4.5 by replacing the full non-Abelian vertex momentum structure Γµνρ (k, p, p) from Eq. (4.37), with
¯
µνρ
ΓP (k, p, p) from Eq. (4.40). This leads to the amplitude
¯

iPpinched =

1
αs
dx 2
C2 (G)
4π
0

µ2
∆GC

+ 4 cot2 (2θc )

1
αs
dx
+
2 cot(2θc )C2 (G)
4π
0

µ2
∆CC

µ2
∆CC

(0)
E Mq q →C
¯

(4.53)

(0)
E Mq q →C ,
¯

where we have used Eq. (4.44) to rewrite the fermion couplings in terms of θc . This contribution to the amplitude has the form of a VPA correction, like that in Eq. (4.46). In fact,
we can write
Π
(M 2 )
(ˆ)
s
(0) Π
(0)
,
iPpinched = iMq q →C CC C + iMq q →C GC
¯
¯
2
s
ˆ

(4.54)

where
1
αs −1
ΠCC (q 2 ) = C2 (G)
dx
4π
0

µ2
∆GC

+ 4 cot2 (2θc )C2 (G)

2
4(q 2 − MC )E
1
0

dx

µ2
∆CC

(4.55)

2
2(q 2 − MC )E ,

and
1
αs −1
ΠGC (q 2 ) = 2 cot(2θc )C2 (G)
dx
4π
0
122

µ2
∆CC

E q2 .

(4.56)

1PI
1PI
1PI

+

+

+ ...

1PI
1PI
1PI

+

+

+

+ ...

Figure 4.11: The relevant contributions to the coloron Dyson series; as before, the zigzag
lines represent colorons. The first row is the sum of the coloron VPA diagrams in the
propagator, while the second row represents the sum of the VPA diagrams on top of the
one-loop contribution from the pinched vertex correction (the double curly line illustrates
generically all the allowed gauge bosons in the original non-Abelian vertices). The overall
pinched amplitude factors out, and has no effect on the coloron pole mass.

4.4.6

Full Propagator Correction

We have just seen that, due to the pinch technique, the coloron-coloron and gluon-coloron
VPAs receive an additional contribution from the pinched non-Abelian vertex corrections.
Combining the VPAs, the UV divergences can be removed by two wavefunction renormalization counterterms (which arise from renormalizing the gauge eigenstates G1µ and G2µ ) and
one mass counterterm (which arises from renormalizing the vacuum expectation value f ), in
123

the usual way. In the MS scheme we obtain
αs −1
ΠCC (q 2 ) + ΠCC (q 2 )
4π
1
µ2
dx
= C2 (G)
− 1 2 3 + 4x(1 − x) E + 2(1 − 2x)2 q 2
∆GC
0
µ2
+
1 − x(4 − 3x) E − x(1 − x) q 2
∆GC
µ2
2
+ −
(4.57)
− 1 2(2 − x)E + 3 − 5x MC
∆GC
1
µ2
+ 4 cot2 (2θc )C2 (G)
dx
−1
3 + 4x(1 − x) E + (1 − 2x)2 q 2
∆CC
0
µ2
1
x(1 − x) 2
µ2
3
2
−
−1 E+1
q + −
−1
E + MC
∆CC
4
∆CC
4
4
1
µ2
2
2
−1
− 2x(1 − x) E q 2 ,
+ (rL + rR )Nf
dx
∆qq
0
and
1
αs −1
ΠGC (q 2 ) + ΠGC (q 2 ) = 2 cot(2θc )C2 (G)
dx
4π
0
µ2
7
2
−1
+ 5x(1 − x) q 2 + MC E + (1 − 2x)2 q 2
∆CC
4
1
µ2
−1
− 2x(1 − x) E q 2 .
+ (rL + rR )Nf
dx
∆qq
0

(4.58)

The overall UV-finite propagator correction to the q q → C amplitude can be found by
¯
insering these expressions in
2
2
ΠCC (MC ) + ΠCC (MC )
(0)
iP + iPpinched = iMq q →C
¯
2
(ˆ) + ΠGC (ˆ)
s
s
(0) Π
+ iMq q →C GC
.
¯
s
ˆ
124

(4.59)

Letting mg → 0, we find that P + Ppinched becomes IR divergent, with the divergence
arising from ΠCC . Setting mg = 0 and < 0 gives

iP + iPpinched =

αs
1
C2 (G) −
4π

(0)
iMq q →C + finite .
¯

(4.60)

We have seen that the pinched diagrams contribute to the full propagators of the gluoncoloron system. This might seem in conflict with the expectation that the mass poles should
be a property of freely propagating particles, and should not depend on any initial and/or
final state. However, when we sum the Dyson series to obtain the full propagator, the pinched
diagrams always appear as an overall prefactor, as pictorially shown in Fig. 4.11. This has
an overall effect on the full propagators, which depend on the initial and final states, but
has no effect on the propagator poles. Thus when we compute physical masses, we can do so
by employing the true propagators in the computation, without the contribution from the
pinched diagrams.

4.4.7

Cross Section at One-Loop

Adding up the tree-level contribution and the NLO contributions from iQ + iVAbelian +
iVnon-Abelian , and iP + iPpinched , gives a q q → C amplitude of the form
¯
(0)
iMq q →C = iMq q →C + iQ + iVAbelian + iVnon-Abelian + iP + iPpinched
¯
¯
αs
(0)
(0)
(0)
(0)
≡ iMq q →C +
T iMq q →C + T iMq q →C + T iMq q →C ,
¯
¯
¯
¯
4π
where expressions for the real parts of T , T , and T

(4.61)

are given below. Averaging the

squared amplitude over initial spins and colors, summing over final polarization states, and
125

integrating over the phase space, gives the NLO result of the form

(0)
(1)
σvirt ≡ σq q →C + σq q →C
ˆ
ˆ ¯
ˆ ¯
2
2
4
4
αs A(rL + rR )
rL + rR
r + rR
αs
δ(1 − χ) 1 +
Re T +
Re T + L
Re T
=
2
2
2
2
s
ˆ
2π
rL + rR
rL + rR

(4.62)
.

2
At s = MC it is possible to integrate over the Feynman parameter space in the expressions
ˆ
for iQ + iVAbelian + iVnon-Abelian , and iP + iPpinched . As we have seen, the UV infinities
cancel in iQ + iVAbelian + iVnon-Abelian and are absorbed by propagator conterterms in
iP + iPpinched . Thus for mg = 0 the overall amplitude is finite. Taking the mg → 0 limit
leads to IR divergences in Re T , which are parametrized by taking
negative values of

Re T =

4πµ2
2
MC

< 0. For small and

we obtain
2
4π 2
3
Γ(1 + ) −
C2 (r)
− −8+
2
3

(4.63)

M2
5π
π2 8
61
− √ −
− log C C2 (G)
9
3
3
2 3
µ2
2
2
MC
7π
29
77
2 (2θ )C (G) + − 1 + 1 log MC (r2 + r2 )N ,
4 cot
+
− √ −
log
c 2
L
R f
48 16 3 16
9 6
µ2
µ2
11 2π 2
5π
2π 2
√ −
Re T = −
+
C2 (r) + 1 +
C2 (G) ,
2
3
3
2 3
√
2
2
MC
MC
95 7 3π 43
5 1
Re T =
−
−
log
2 cot(2θc )C2 (G) + − + log
(rL + rR )Nf .
9
4
12
9 3
µ2
µ2
+

−E+

In the next section we will compute the corrections to the tree-level cross section due to
the emission of soft and collinear gluons. We will show that the real emission cross section
has IR divergences which exactly cancel the IR divergences contained in σvirt (Eq. (4.62)),
ˆ
leading to a total cross section free of both UV and IR divergences.
126

(a)

(b)

(c)

Figure 4.12: Diagrams contributing to the real emission processes. A gluon field is, as usual,
represented by a coiling line; a coloron field is represented by a zigzag line. (a) Gluon
emission. (b) Quark emission. (c) Antiquark emission.

4.5

NLO Coloron Production: Real Corrections

The real emission corrections, at NLO, are given by the diagrams of Fig 4.12. We first
consider the diagrams with real emission of a gluon, shown in Fig 4.12(a). The squared
amplitude, averaged over initial colors and spins, and summed over final colors and polarizations, is found to be, in d = 2(2 − ) dimensions,
4 2
2
C2 (r) gs (rL + rR ) 2
(1)
|Mq q →gC |2 =
µ (1 − )
¯
dim(r)
−1
×
C (r) + C2 (G)
ω(1 − ω) 2
127

1 + χ2
−
+ 2ω (1 − ω)
(1 − χ)2

(4.64)
,

where
ω≡

1 − cos θ
.
2

(4.65)

χ was defined in Eq. (4.22), and θ is the angle between the emitted gluon and the colliding
quarks. The cross section for the real gluon emission is

1
(1)
σq q →gC =
ˆ ¯
2ˆ
s

(1)
dΠ2 |Mq q →gC |2 ,
¯

(4.66)

where the integral is over the two-body Lorentz-invariant phase space in parton CM. In
d = 2(2 − ) dimensions,
−
2
MC (1 − χ)2
1 1−χ
dΠ2 =
8π Γ(1 − )
4πχ

1
0

dω [ω (1 − ω)]− .

(4.67)

This leads to the partonic cross section
2
2
αs (rL + rR )A αs
(1)
σq q →gC =
ˆ ¯
s
ˆ
2π
×

4πµ2
Γ(1 − )
2
Γ(1 − 2 )
MC
2 χ (1 + χ2 )
2 χ (1 + χ + χ2 )
− C2 (r)
− C2 (G)
3 (1 − χ)1+2
(1 − χ)1+2

(4.68)
.

Now χ is no longer constrained to be equal to one. Instead we must have χ ≤ 1, or else
no on-shell coloron can be produced. The term proportional to C2 (r) features a collinear
singularity, parametrized by , and a soft singularity, parametrized by 1 − χ. The term
proportional to C2 (G) only features a soft singularity. The integral over χ in Eq. (4.68) is
finite for

< 0, in spite of the singularity of the integrands. For small and negative values

128

of

we can rewrite the χ-dependence as follows

χ (1 + χ2 )
1
1 + χ2
= − δ(1 − χ) +
− 2(1 + χ2 )
(1 − χ)+
(1 − χ)1+2
χ (1 + χ + χ2 )
3
1 + χ + χ2
= − δ(1 − χ) +
,
2
(1 − χ)+
(1 − χ)1+2

1 + χ2
log(1 − χ)
−
log χ
1−χ
1−χ
+

,

(4.69)

where, as conventional, the “+” distributions are defined by
1

dχ

f (χ)
≡
(1 − χ)+

0
1
log(1 − χ)
dχ f (χ)
≡
1−χ
0
+

1
0
1

dχ

f (χ) − f (1)
,
1−χ

log(1 − χ)
dχ [f (χ) − f (1)]
.
1−χ
0

(4.70)

The coefficients of the delta functions are found by integrating both sides of the equations.
The partonic cross section becomes
2
2
αs A(rL + rR ) αs
(1)
σq q →gC =
ˆ ¯
δ(1 − χ) R + R
s
ˆ
2π

,

(4.71)

where, using Eq. (4.50), and expanding for small values of ,

R=

4πµ2
2
MC

Γ(1 − )
2
3
1
C2 (r)
+
+ C2 (G)
2
Γ(1 − 2 )

R = −2 E − log

,

2
MC
Pq→q (χ)
µ2

+ C2 (r) 4(1 + χ2 )

(4.72)

log(1 − χ)
1+χ
2 1 + χ + χ2
−2
log χ + C2 (G)
.
1−χ
1−χ
3 (1 − χ)+
+

In the second equation Pq→q (χ) is the Altarelli-Parisi splitting function for an on-shell
quark to evolve into a virtual quark and a real gluon

129

Pq→q (χ) = C2 (r)

1 + χ2
3
+ δ(1 − χ) .
(1 − χ)+ 2

(4.73)

(1)
Adding together σvirt , given by Eqs. (4.62) and (4.63), and σq q →gC , given by Eqs. (4.71)
ˆ
ˆ ¯
and (4.72), shows that the IR divergences proportional to δ(1 − χ) cancel. There is still
a collinear singularity in R , proportional to the Altarelli-Parisi evolution Pq→q (χ). This
singularity arises from integrating over all collinear initial-state gluons. As we will see in the
next section, these collinear IR divergences will be absorbed through renormalization of the
PDFs.

The real quark and antiquark emission diagrams are shown in Figs. 4.12(b) and 4.12(c),
respectively. The corresponding summed-averaged squared amplitudes in d = 2(2 − ) are
2
4 2
C (r) gs (rL + rR ) 2
(1)
2= 2
|Mqg→qC |
µ
dim(G)

C2 (r) + C2 (G)

(1 − χ)(1 − ω)
(1 − (1 − χ)(1 − ω))2

1 − − 2χ(1 − χ)
× 2 ( + χ) +
+ (1 − )(1 − χ) (1 − ω)
(1 − χ) (1 − ω)

(4.74)

,

and
4 2
2
C (r) gs (rL + rR ) 2
(1)
2= 2
|Mq g→¯C |
µ
¯ q
dim(G)

C2 (r) + C2 (G)

(1 − χ) ω
(1 − (1 − χ) ω)2

1 − − 2χ(1 − χ)
× 2 ( + χ) +
+ (1 − )(1 − χ) ω
(1 − χ) ω

(4.75)

,

where dim(G) ≡ 8 is the dimension of the adjoint representation. Note that the amplitudes
for quark and antiquark emission are related by crossing, i.e. ω ↔ (1 − ω). The integration
over the two-body Lorentz-invariant phase space proceeds as in the gluon emission case,
yielding
130

2
2
αs A(rL + rR ) αs
(1)
(1)
σqg→qC = σq g→¯C =
ˆ
ˆ¯ q
R ,
s
ˆ
2π

(4.76)

where

R =

dim(r)
3 + 2χ − 3χ2
(1 − χ)(2 + χ + 2χ2 )
C2 (r)
+ C2 (G)
+ 2(1 + χ) log χ
dim(G)
2
χ
2
MC
(1 − χ)2
− E − log
− log
+ 1 Pg→q (χ) .
χ
µ2

(4.77)

Here Pg→q (χ) is the Altarelli-Parisi splitting function for an on-shell gluon to evolve to a
virtual-real quark pair,

Pg→q (χ) =

C2 (r) · dim(r) 2
χ + (1 − χ)2
dim(G)

,

(4.78)

(1)
(1)
where C2 (r) · dim(r)/dim(G) = 1/2. There is no soft singularity in σqg→qC ≡ σq g→¯C ,
ˆ
ˆ¯ q
only a collinear singularity proportional to the Altarelli-Parisi evolution Pg→q (χ). As noted
(1)
above regarding σq q →gC , this singularity will be canceled by renormalization of the PDFs
ˆ ¯
when we compute the total hadronic cross section.

4.6

NLO Cross Section

Our calculations in the previous sections have produced all of the relevant partonic cross
sections at NLO and demonstrated them to be both UV and IR finite.13
13 Note that the gg → C process vanishes at tree-level [107] and the one-loop contributions
are small, less than of order 0.1% of the q q -initiated leading order contribution [108]; we,
¯
therefore, do not include this process in this work.
131

The full NLO cross section for coloron production at the LHC is

σ NLO =

dx1

dx2

0
0
0
0
fq (x1 )fq (x2 ) + fq (x1 )fq (x2 )
¯
¯

q
+
q

(0)
(1)
(1)
σq q →C + σq q →C + σq q →gC
ˆ ¯
ˆ ¯
ˆ ¯

(4.79)

(1)
0
0
0
0
0
0
0
0
fq (x1 )fg (x2 ) + fg (x1 )fq (x2 ) + fq (x1 )fg (x2 ) + fg (x1 )fq (x2 ) σqg→qC
ˆ
¯
¯

,

where the partonic cross sections σ are given in Eqs. (4.20), (4.62), (4.71), and (4.76), and
ˆ
where the superscript “0” in the PDFs will be clear in a moment. We saw that all IR
divergences contained in σ cancel, except for a couple of collinear singularities proportional
to Altarelli-Parisi evolutions. Such singularities arise because we integrated over all collinear
quarks and gluons, even those which we should have included in the PDFs. Therefore, the
corresponding IR singularities are absorbed by renormalizing the bare PDFs in Eq. (4.79).
In the MS scheme,
0
fi (x, µF ) = fi (x)
2
g3
−
8π 2

1

µ2
− γ + log(4π) − log F
µ2

dχ
χ

j

0
fj

x
χ

(4.80)
Pj→i (χ) ,

where i, j = q, g, and µF is the factorization scale. Exchanging the bare PDFs for the
renormalized ones replaces E with log µ2 /µ2 in Eqs. (4.72) and (4.77). The hadronic cross
F

132

section becomes

σ NLO =

αs A H1 (θc )
s

dx1
x1

dx2
x2

(4.81)

fq (x1 , µF )fq (x2 , µF ) + fq (x1 , µF )fq (x2 , µF )
¯
¯

q
+
q

δ(1 − χ) +

αs qq
F (χ)
2π

fq (x1 , µF )fg (x2 , µF ) + fg (x1 , µF )fq (x2 , µF ) + (fq → fq )
¯

αs qg
F (χ)
2π

,

where the function H1 (θc ) is defined below, in Eq. (4.85), A is defined in Eq. (4.21), and the
partonic CM energy s has been traded for the hadronic one, as in Eq. (4.25). Notice that
ˆ
since the integrand is now finite, we can ignore the 1 − factor in A. The functions F qq (χ)
and F qg (χ) are
2
qq (χ) = 2 log MC P
F
q→q (χ) + Dq (χ) ,
µ2
F
2
M
F qg (χ) = log C Pg→q (χ) + Dg (χ) ,
µ2
F

(4.82)

where
log(1 − χ)
1+χ
−2
log χ
1−χ
1−χ
+
2 1 + χ + χ2
+ C2 (G)
+ δ(1 − χ) Q ,
3 (1 − χ)+
(1 − χ)2
dim(r)
Dg (χ) =
C2 (r) χ2 + (1 − χ)2
log
−1
dim(G)
χ
Dq (χ) = C2 (r) 4(1 + χ2 )

+ C2 (G)

(1 − χ)(2 + χ + 2χ2 )
+ 2(1 + χ) log χ
χ

133

3
3
+ + χ − χ2
2
2
,

(4.83)

and

Q = Nf

2
MC
1 1
− + log
9 6
µ2

+ C2 (r) − 8 +
+ C2 (G)
+

2π 2
+
3

H1 (θc ) +
−

2
MC
5 1
− + log
9 3
µ2

11 2π 2
+
H3 (θc )
2
3

H2 (θc )
(4.84)

M2
5π
61
π 2 11
− √ −
−
log C +
9
3
3
2 3
µ2

2π 2
5π
H3 (θc ) +
1+ √ −
3
2 3

M2
7π
77
29
− √ −
log C cot2 (2θc )
12 4 3
4
µ2
√
2
MC
190 7 3π 43
cot(2θc )H4 (θc ) .
−
−
log
9
2
6
µ2

The functions Hi (θc ) are determined by the chiral couplings of the quarks to the colorons (which depend on the charges of the quarks under the full SU (3)1C × SU (3)2C
symmetry)

134

H1 (θc ) =

H2 (θc ) =

H3 (θc ) =

H4 (θc ) =




2 tan2 θ

rL = rR = − tan θc

c




2
2
,
tan θc + cot θc rL = rR






2 cot2 θc

rL = rR = cot θc



2

rL = rR = − tan θc





2(1 + cos(4θc ))
,
rL = rR
 3 + cos(4θc ))






2

rL = rR = cot θc


 2
tan θ

rL = rR = − tan θc
c





tan4 θc + cot4 θc
,
rL = rR
 tan2 θ + cot2 θ

c
c





cot2 θ

rL = rR = cot θc
c




− cot θc
rL = rR = − tan θc





sin(4θc )
.
 3 + cos(4θc ) rL = rR







tan θc
rL = rR = cot θc

(4.85)

At NLO the µ dependence is removed by trading the MS couplings g1s and g2s , or gs and
θc , for the corresponding running couplings. Since θc is a free parameter, we simply set
µ ≡ MC , and express the cross section as a function of the MS couplings. At the same
time, the NLO µF dependence weakens once the renormalized PDFs are employed, as σ in
Eq. (4.79) is independent of µF to this order in perturbation theory. From these results we
may also compute the transverse momentum distribution of the produced coloron, which is

135

given by

dσ
=
dpT

dx1
+
q

dx2

fq (x1 , µF )fq (x2 , µF ) + fq (x1 , µF )fq (x2 , µF )
¯
¯

q

dˆq q →gC
σ ¯
dpT

fq (x1 , µF )fg (x2 , µF ) + fg (x1 , µF )fq (x2 , µF )

+ fq (x1 , µF )fg (x2 , µF ) + fg (x1 , µF )fq (x2 , µF )
¯
¯

(4.86)
dˆqg→qC
σ
dpT

,

where
dˆq q →gC
σ ¯
dpT

dˆqg→qC
σ
dpT

=

1
2 (1 − χ2 )
4πˆ
s

=

1
4πˆ2 (1 − χ2 )
s

×

pT

(1)
× 2|Mq q →gC |2 ,
¯

4p2
T
1−
s(1 − χ)2
ˆ

(4.87)

pT

1−

4p2
T

s(1 − χ)2
ˆ

(1)
(1)
|Mqg→qC |2 + |Mqg→qC |2 ω→1−ω

,

(4.88)

and ω (Eq. (4.65)) is given by

ω=

1−

1−

4p2
T

s(1 − χ)2
ˆ

2

.

(4.89)

Note that this is the leading order prediction for dσ/dpT , and, therefore, this distribution is
strongly µF -dependent.

136

20

Q

0

20

40

60

0.0

0.2

0.4

0.6

0.8

1.0

sin2 Θc
Figure 4.13: Behavior of the Q function defined in Eq. (4.84), for µ = MC : this gives the
contribution from the virtual corrections to the NLO cross section for coloron production.
The upper curve is for the rL = rR scenario, whereas the almost identical lower curves are
for rL = rR = − tan θc , and rL = rR = cot θc . Note that Q, and, therefore, the NLO
corrections, become very large when sin2 θc is either too small or too large.

4.7

Conclusion & Discussion

We now illustrate14 our results for the NLO coloron production cross section in Figs. 4.134.16. In each figure we consider the three possible flavor-universal scenarios for quark charge
assignment: rL = rR = − tan θc , rL = rR , and rL = rR = cot θc . All of the plots refer to
√
coloron production at the LHC with s = 7 TeV.
Notice that the perturbative expansion is only meaningful as long as sin θc is neither
14 For the purposes of illustration we use the Mathematica package for CTEQ5 [109] to
evaluate the relevant parton distribution functions.
137

too close to zero (where g2s

g1s ) nor too close to one (where g1s

g2s ). This is clear

from Fig. 4.13, in which we plot the quantity Q defined in Eq. (4.84), for µ = MC : the
contribution from the virtual corrections to the NLO cross section. The upper curve is for
the rL = rR scenario, whereas the almost identical lower curves are for rL = rR = − tan θc ,
and rL = rR = cot θc . For sin2 θc

0.05 and sin2 θc

0.95 the virtual corrections become

large, and the perturbative expansion in αs breaks down. Since αs
these boundaries correspond to g2s

2.7 and g1s

0.118 at the Z pole,

2.7, respectively.

In Fig. 4.14 we plot the µF dependence of the LO and NLO production cross sections
of a 2.0 TeV coloron (with sin2 θc |µ=2.0 TeV =0.25). The scale-dependence of the LO cross
section is of order 30% while, as expected, the NLO cross section has a much weaker scaledependence, only of the order of 2%.
In Fig. 4.15 we plot the cross section times branching ratio to quark jets as a function of
MC , allowing µF to vary from MC /2 to 2MC . Here, in order to compare to the experimental
results of [4] (shown as the solid line in the figures), we correct for the acceptance of the
detector by multiplying our partonic-level NLO production cross section by the factor

R=

σ(pp → C) · B · A

CMS
axigluon

σ LO (pp → C)axigluon

.

(4.90)

CMS
is the CMS (LO) prediction for axigluon
axigluon
production cross section, times dijet branching ratio, times acceptance15 reported in [4],
In this expression, σ(pp → C) · B · A

and σ LO (pp → C)axigluon is the leading order cross section in Eq. (4.23) in the case of an
15 The CMS acceptance for isotropic decays is of order 0.6, independent of resonance mass
[4].
138

tan Θc

rL rR
0.56
0.54

NLO

pb

0.52
0.50
0.48
LO

0.46
0.44
1.0

1.5

2.0

2.5

3.0

3.5

4.0

3.5

4.0

3.5

4.0

ΜF TeV

rL rR
3.2
NLO

3.0

pb

2.8
2.6
2.4

LO

2.2
1.0

1.5

2.0

2.5

3.0

ΜF TeV

rL rR cot Θc
5.0
NLO

4.8

pb

4.6
4.4
4.2

LO

4.0
1.0

1.5

2.0

2.5

3.0

ΜF TeV

√
Figure 4.14: Dependence of LO and NLO cross sections at the LHC ( s = 7 TeV), as a
function of factorization scale µF for MC = 2.0 TeV, sin2 θc |µ=2.0 TeV = 0.25, and the
three possible flavor-universal scenarios for the quark charge assignments. As expected, the
NLO cross section has a much weaker (formally, two-loop) residual scale-dependence.
139

tan Θc

rL rR
10

Σ B A pb

1
0.1
0.01
0.001
10

4

10

5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

3.0

3.5

4.0

3.5

4.0

MC TeV

rL rR
100

Σ B A pb

10
1
0.1
0.01
0.001
1.0

1.5

2.0

2.5
MC TeV

rL

rR

cot Θc

Σ B A pb

100

1

0.01

1.0

1.5

2.0

2.5

3.0

MC TeV

Figure 4.15: NLO cross section times branching ratio to quarks for on-shell coloron produc√
tion at the LHC ( s = 7 TeV), corrected for acceptance as described in the text. We consider
the three possible flavor-universal scenarios for the quark charge assignments, take the renormalization scale µ to be equal to MC , and plot σ for sin2 θc |µ=M = 0.05 (dashed), 0.25
C
(dot-dashed), and 0.5 (dotted). We plot these cross sections for µF ranging from MC /2
to 2 MC and, reflecting the weak dependence of the NLO cross section on the factorization
scale, the resulting bands for each sin2 θc are very narrow. To give a sense of current experimental reach, we plot the CMS [4] upper limit (solid line) on the cross section times dijet
branching ratio for a narrow resonance.
140

axigluon (i.e. rL = −rR = 1), assuming the branching ratio to quarks B(C → q q ) = 1.16
¯
The three sets of thin bands correspond to sin2 θc |µ=M = 0.05 (dashed), 0.25 (dot-dashed),
C
and 0.5 (dotted). Here, the weak residual µF dependence is shown by the narrowness of the
bands. To give a sense of current experimental reach, we also show the 1 fb−1 CMS upper
bounds on the cross section times dijet branching ratio for a narrow resonance [4]. Note that
the bound on the axigluon [5] corresponds to the rL = rR plot with sin2 θc = 0.5 — and,
hence, a narrow axigluon resonance is constrained to have a mass of order 2.6 TeV or higher.
The enhancement of the axigluon cross section at NLO is responsible for the increase in the
bound from of order 2.5 TeV as reported in [4].
Next, we compute the “K-factor” for coloron production,
σ NLO (MC , sin θc |µ=M , µF = MC )
C
,
K(MC , sin θc |µ=M , µF = MC ) ≡
LO (M , sin θc |
C
σ
, µF = MC )
C
µ=MC

(4.91)

shown in Fig. 4.16 for sin2 θc = 0.05 (dashed), 0.25 (dot-dashed) and 0.50 (dotted). Again,
we see that the NLO corrections are of order 30%. In Appendix we report the numerical
values of the K-factors corresponding to Fig. 4.16, as well as those corresponding to the
ATLAS KK-gluon search reported in [6].
At leading order, the coloron is produced with zero transverse momentum. We may use
our results to compute the pT spectrum in coloron production to leading nontrivial order
from Eq. (4.86). Using these formulae, we may compute the fraction of colorons produced

16 It is worth noting that there are examples of models with colorons which do not decay
primarily to dijets, e.g. [110].
141

rL r R

tan Θc , Μ ΜF MC

1.1

K

1.
0.9

0.8

0.5

1.0

1.5
MC TeV

2.0

2.5

2.0

2.5

rL rR , Μ ΜF MC
1.3
1.25

K

1.2
1.15
1.1
1.05
1.
0.5

1.0

1.5
MC TeV

rL rR cot Θc , Μ ΜF MC

1.1

K

1.

0.9

0.8
0.5

1.0

1.5
MC TeV

2.0

2.5

Figure 4.16: “K-factor”, the ratio of the NLO to LO cross section for coloron production
√
at the LHC ( s = 7 TeV), plotted as a function of MC for sin2 θc = 0.05 (dashed), 0.25
(dot-dashed) and 0.50 (dotted), µF = MC , and the three different quark charge assignments.
142

above a momentum pT min

P(pT ≥ pT min , MC , sin θc |µ=M , µF = MC )
C
pT max
1
dσ
,
≡
dpT
NLO (M , sin θc |
dpT
σ
C
µ=MC , µF = MC ) pT min

(4.92)

where pT max is the kinematic maximum transverse momentum (which depends on the
coloron mass). For illustration, we plot this fraction for vectorial colorons (rL = rR =
− tan θc , with sin2 θc = 0.05) with masses of 1.2, 2.0, and 3.0 TeV in Fig. 4.17. Note that
of order 30% of the colorons in this model and mass range are produced with pT ≥ 200
GeV. Below a pT of 200 GeV the corrections become larger than 30%, terms proportional
2
to log(MC /p2 min ) become large, and this fixed-order calculation becomes unreliable.
T
In conclusion, we have reported the first complete calculation of QCD corrections to
the production of a massive color-octet vector boson. Our next-to-leading order calculation
includes both virtual corrections as well as corrections arising from the emission of gluons
and light quarks, and we have demonstrated the reduction in factorization scale-dependence
relative to the leading order approximation used in previous hadron collider studies. In
particular, we have shown that the QCD NLO corrections to coloron production are as
large as 30%, and that the residual factorization scale-dependence is reduced to of order 2%.
We have also calculated the K-factor and the pT spectrum for coloron production, since
these are valuable for comparison with experiment. Our computation applies directly to
the production of the massive color-octet vector bosons in axigluon, topcolor, and coloron
models, and approximately to the production of KK gluons in extra-dimensional models or
colored technivector mesons in technicolor models. We look forward to future results from
the LHC, and the possible discovery of colorons.
143

% events

100.0
50.0
10.0
5.0
1.0
0.5
0.1

200

400

pT HGeVL
600

800

1000

Figure 4.17: Fraction of colorons produced with a pT greater than pT min , as a function of
pT min . The curves are for MC = 1.2 (highest, thin blue line), 2.0 (middle, medium purple
line), and 3.0 TeV (lowest, thick green line), for the vectorial case rL = rR = − tan θc and
sin2 θc = 0.05. Note that of order 30% of the colorons in this mass range are produced
with pT ≥ 200 GeV. As denoted by the red shaded region, below a pT of 200 GeV the
2
corrections become larger than 30%, terms proportional to log(MC /p2 min ) become large,
T
and this fixed-order calculation becomes unreliable.

144

Chapter 5
CONCLUSION & DISCUSSION

“Physics is like sex: sure, it may give some practical results, but that’s not why
we do it! ”
—Richard Feynman (1918 – 1988)

In the present Thesis, we have analyzed two separate extensions of the Standard Model
of particle physics (SM). As explained in Chapter , the SM rests on the foundations of an
SU (3)C ×SU (2)L ×U (1)Y gauge group, accounting for all three microscopically relevant interactions; namely, the electromagnetism, the strong and the weak forces. On one hand, the
field theoretical descriptions of the electromagnetic and the weak interactions are interconnected, forming the electroweak sector of the SM, and are described by the SU (2)L × U (1)Y
subgroup. On the other hand, the strong force is contained within the quantum chromodynamics (QCD) SU (3)C subgroup.
We have seen that the SU (2)L × U (1)Y subgroup is spontaneously broken to U (1)EM by
means of the Higgs mechanism, which in the SM is facilitated by employing a fundamental
145

scalar doublet. This is the mechanism by which the weak gauge bosons and the fermions
acquire mass. Introduction of this scalar doublet predicts the existence of a real degree of
freedom, called the Higgs boson, the mass of which appears to be highly sensitive to the high
energy behavior of the theory through quadratically divergent quantum corrections. While
the actual mass of the Higgs boson, if it exists, needs to lie naturally around the electroweak
symmetry breaking energy scale, its sensitivity to ultra high energies necessitates a finetuning and causes the Hierarchy problem. Various beyond the Standard Model theories
(BSM) have been proposed in order to cancel the quadratic divergences of the Higgs mass
in a natural way and to solve the Hierarchy problem.
In Part I of the Thesis, we explored various formal and phenomenological aspects of one of
the proposed solutions to the Hierarchy problem, the Lee-Wick Standard Model (LW SM). In
this BSM theory, inspired by the Pauli-Villars regulatory scheme, a set of (heavy) auxiliary
fields are introduced, which form the Lee-Wick (LW) partners of the usual SM particles. Unlike the SM fields, however, these LW partners carry an overall negative sign as part of their
description. It is this extra negative sign which induces a cancellation of the quadratically
divergent quantum corrections among the contributions originating from the SM particles
and their LW partners. Therefore the LW SM provides a natural and economical solution
to the Hierarchy problem.
We have seen that the SM, as a renormalizable theory, employs a few experimentally
determined observables as input in order to make robust predictions regarding the outcome
of many other experiments. In the electroweak sector, the SM predictions have been tested
to an impressive accuracy through the electroweak precision data. In order to facilitate
this comparison between theory and experiment, convenient parametrizations have been
146

introduced. The relatively small deviations of the experimental values of these parameters
from their SM predictions can then be utilized as a powerful tool to place tight constraints
on the BSM variables possibly influencing those parameters.
Since the LW particles have not been observed so far in colliders, they must, if they exist,
be heavy with masses presumably beyond the previous colliders’ production threshold. In
Chapter we analyzed the phenomenological consequences of the LW SM, using the available
electroweak precision data in order to set lower bounds on the masses of the LW particles.
We found a lower bound of several TeV at 95% C.L. for the masses of these hypothetical
LW particles to be consistent with experimental data.
The LW SM with the auxiliary partners may, equivalently, be expressed in a higherderivative formulation, in which, instead of introducing new LW degrees of freedom, one
adds higher-derivative terms to various sectors of the original SM Lagrangian. Addition of
these higher-derivative terms might raise concerns about the overall renormalizability of the
LW theories and their consideration as reliable BSM alternatives with an arbitrary accuracy
in predictions. Previous power counting arguments have exhibited the renormalizability
of the LW theories in the higher-derivative formulation; however, exact translation of this
property to the auxiliary-field formulation remained unexplored. In Chapter we investigated
the global symmetries and renormalizability of the auxiliary-field formalism, by considering
a LW scalar QED theory as an Abelian toy model. We were able to identify a global SO(1, 1)
symmetry, which, together with the U (1) gauge invariance and an SO(1, 1) invariant gaugefixing condition, allowed us to prove the renormalizability of this class of theories and to
clarify the physics involved.
Part II of the Thesis was dedicated to a separate class of BSM theories, forming an
147

extension to the strong sector of the SM. These strong sector extensions arise naturally,
for example as an integral feature of theories in which the Higgs boson is a composite
scalar rather than a fundamental degree of freedom, with its constituents held together by
a new strong interaction. The SM strong sector gauge group is extended to an SU (3)1C ×
SU (3)2C structure, which is spontaneously broken to the ordinary QCD’s SU (3)C group.
This introduces, in addition to the usual massless gluon-octet, a massive vector color-octet
of states, called colorons.
As is the case in ordinary QCD, a perturbative expansion of the theory in terms of the
strong coupling, αs , is appropriate at high energies, given its asymptotically free nature. To
date, colorons have been studied only to leading order (LO) in perturbation theory, as the
colored nature of this final state massive vector boson makes its higher-order non-Abelian
analysis rather nontrivial. In Chapter the first complete and consistent calculation of
coloron production at next-to-leading order (NLO) in perturbation theory was presented. We
provided a finite expression for the production cross section at the LHC, and demonstrated
that the NLO effect is as large as 30%; thereby, dramatically improving upon previous LO
results. In addition, we constructed coloron kinematic variables, such as its transversemomentum distribution, which make a direct comparison with experiment possible.
The research outlined in the present Thesis can be summarized as formal and phenomenological explorations of extensions to two separate sectors of the SM: the electroweak sector
and the strong sector. Motivations for going beyond the SM in these sectors, as explained
above, are different as they address different issues within the SM. The presented analyses,
however, by no means mark an end of theoretical investigations of Lee-Wick and coloron
theories, and there is much room left for extending the research in these areas in future
148

studies. Establishing the renormalizability of the non-Abelian LW theories in the auxiliaryfield formalism remains, for example, unexplored and poorly understood, mainly since the
discussed SO(1, 1) symmetry is violated by the gauge interactions. A thorough non-Abelian
investigation of LW theories in this formulation promises to be a formidable task, but remains necessary in order to extend the demonstrated Abelian results. The coloron NLO
analysis may also be enhanced by taking into account the coloron’s finite lifetime and its
subsequent decay into quarks and gluons through various channels. This allows for achieving
an even higher theoretical accuracy in order to compare with future experimental results.
Furthermore, using the NLO results, one may exploit the available experimental data to
improve the previously determined theoretical LO lower bounds on coloron masses.

As one might appreciate, we are currently progressing through a particularly exciting
era in particle physics research, specifically due to the experiments conducted at the LHC.
These experiments might discover the existence of the elusive SM Higgs boson, in which
case the SM and its symmetry breaking mechanism would be confirmed, with its last missing piece finally in place. Moreover, in light of a solution to the Hierarchy problem, the
LHC may find distinctive signals related to any of the proposed BSM theories, including the
LW SM; thereby, defining the direction of the future theoretical research. A more exciting
scenario, however, would involve a physical exclusion of the SM Higgs boson by the LHC
data. In that case, theoretical focus will be shifted towards alternative electroweak symmetry
breaking mechanisms, such as the strong interaction theories, including the coloron theory.
Additionally, one might imagine the discovery of signals not anticipated previously by the
proposed BSM theories, which would pave the way for fresh ideas and more advanced theoretical developments. In any case, the LHC findings promise to open the gates to a wealth
149

of knowledge concerning the mysteries of the microscopic world — a world whose precise
small-scale exploration might, perhaps not so surprisingly, be connected to the large-scale
properties of the macroscopic universe, answering some of the big questions regarding its
future, present, and past!

150

‫ﻟﺐ ﺑﺮ ﻟﺐ ﮐﻮزه ﺑﺮدم از ﻏﺎ$ﺖ آز‬
‫ﺗﺎ زو ﻃﻠﺒﻢ واﺳﻄﻪ ﻋﻤﺮ دراز‬
‫ﻟﺐ ﺑﺮ ﻟﺐ ﻣﻦ ﻧﻬﺎد و ﻣﯽ ﮔﻔﺖ ﺑﻪ راز‬
‫ﻣﯽ ﺧﻮر ﮐﻪ ﺑﺪ$ﻦ ﺟﻬﺎن ﳕﯽ آ$ﯽ ﺑﺎز‬
‫ّﺎم‬J‫ﺧ‬

Then to this earthen Bowl did I adjourn
My Lip the secret Well of Life to learn:
And Lip to Lip it murmur’d -“While you live,
Drink! - for once dead you never shall return!”
-Omar Khayyám (1048-1131)
Persian polymath, philosopher, mathematician, astronomer, and poet
Translation by Edward FitzGerald (1809-1883)

151

APPENDICES

152

Appendix A

EVALUATION OF THE φ0 → bL¯L
b
AMPLITUDE

The triangle diagram of Fig. 2.10 can be easily evaluated once the mass matrix has been
diagonalized and the Yukawa couplings have been computed. For an arbitrary theory with
heavy replicas of the third generation quarks, and neglecting the bottom Yukawa sector, the
interactions with the Nambu-Goldstone bosons eaten by the W and Z boson read

i,j

y
¯
¯
¯
−i √t φ0 αij ti PR tj − αji ti PL tj − i yt βij φ−¯i PR tj − φ+ tj PL bi
b
2

,

(A.1)

where t0 and b0 are the SM top and bottom, respectively, and where the remaining ones
are heavy replicas. From this expression one may extract the Feynman rules. Shifting the
momentum of the ¯L to zero, and omitting the external fermion wavefunctions, the amplitude
b
153

reads

iM =

/ /
i(k + p + mt )
y
d4 k
i
√t
(yt β0i PR )
4
2 − m2 + i
2
(2π)
(k + p)
ti
i,j
/
i(k + mt )
i
j
(−yt β0j PL )
,
× (αij PR − αji PL )
k2 + i
k 2 − m2 + i
tj
N
(−) ij

(A.2)

where Nij is the number of LW fermions in the i, j pair. Combining the denominators into
a single one, and shifting the loop momentum in the usual way, leads to
3
iyt
iM = − √ pPL
/
2

1
1−x
N
(−) ij β0i β0j αji mt
dx
dy
j 0
0
i,j

d4 l 2(1 − x)
(2π)4 (l2 − ∆)3

iy 3
− √t pPL
/
2

1
1−x
N
(−) ij β0i β0j αij mt
dx
dy
i 0
0
i,j

d4 l
2x
,
(2π)4 (l2 − ∆)3

(A.3)

where
∆ ≡ −x(1 − x)p2 + x m2 + y m2 .
ti
tj

(A.4)

Evaluating the integrals in the p2 → 0 limit gives
3
1 yt
√ pPL
iM = −
/
16π 2 2
+
i=j

2
β0i αii
mt
i
i

N
(−) ij β0i β0j αji mt
j

−

3m2 − m2
m2
ti
tj
ti
1
+
log
2 (m2 − m2 )2
− m2
m2
tj
ti
tj
tj
1

m2
ti

154

(A.5)
.

Comparing this expression with Eqs. (2.31) and (2.32) gives
3
b¯ = 1 y√v
b
t
δgL
16π 2 2 2
+
i=j

2
β0i αii
mt
i
i

N
(−) ij β0i β0j αji mt
j

−

m2
ti

3m2 − m2
m2
ti
tj
ti
1
1
+
log
2 (m2 − m2 )2
− m2
m2
tj
tj
ti
tj

to leading order in the weak gauge coupling.

155

(A.6)
.

Appendix B
FEYNMAN RULES OF THE
COLORON THEORY
The Feynman rules1 for the trilinear and quartic vertices are shown in Figs. B.1-B.4. The
coloron is represented by a zigzag line, the coloron ghost by a sequence of small circles, and
the eaten Goldstone bosons by dashed lines. All other particles are denoted as in QCD
standard notation. Note that a coupling between the eaten Goldstone boson and quarks is
absent in the zero quark mass limit.

1 The Feynman rules discussed here are equivalent to those in [97], aside from those for
the triple-coloron vertex which is not specified in that reference.
156

a, µ

a, µ

c, ⇢

r
p

q
a, µ

a, µ

= gs f abc

q
a, µ

µ a

t

= igs

[g µ⌫ (r

b, ⌫
c, ⇢

r

p)⇢ + g ⇢⌫ (p

q)µ + g µ⇢ (q

r)⌫ ]

= gs f abc

p

a, µ

[g µ⌫ (r

b, ⌫
c

p)⇢ + g ⇢⌫ (p

q)µ + g µ⇢ (q

r)⌫ ]

= gs f abc pµ

p

p

a, µ

b
c

a, µ

p

a, µ

= gs f abc pµ

a, µ

b
b

a, µ

= gs f abc (q

q

p)µ

c
b, ⌫

a, µ

=

igs MC f abc g µ⌫

c
Figure B.1: Feynman rules for the trilinear vertices. In each diagram the momenta are
toward the vertex. A gluon field is, as usual, represented by a coiling line; a coloron field is
represented by a zigzag line. The coloron ghost is represented by a sequence of filled circles,
and the eaten Goldstone bosons are represented by dashed lines.
157

a, µ

r

µ⇢

µ⇢

a, µ

=i

q
a, µ
(q

r)⌫ ]

r

= 2 cot(2✓) gs f abc

p

r)⌫ ]

(gL PL + gR PR ) ta

c, ⇢

b, ⌫
c

[g µ⌫ (r

p)⇢ + g ⇢⌫ (p

q)µ + g µ⇢ (q

= gs f abc pµ

a, µ
(q

µ

p
p

b
b
= gs f abc pµ

a, µ

c
c
a, µ

= 2 cot(2✓) gs f abc pµ

p
p

b
b

a, µ

= cot(2✓) gs f abc (q

q

p)µ

c

Figure B.2: Feynman rules for the trilinear vertices (continued).

158

r)⌫ ]

a, µ

b, ⌫
⇥
2
= igs f abe f cde (g µ⇢ g ⌫

g µ g ⌫⇢ )

+ f ace f bde (g µ⌫ g ⇢
+ f ade f bce (g µ⌫ g ⇢

c, ⇢

d,

a, µ

g µ g ⌫⇢ )
g µ⇢ g ⌫ )

b, ⌫
⇥
2
= igs f abe f cde (g µ⇢ g ⌫

g µ g ⌫⇢ )

+ f ace f bde (g µ⌫ g ⇢

d,

a, µ

g µ g ⌫⇢ )

+ f ade f bce (g µ⌫ g ⇢

c, ⇢

⇤

g µ⇢ g ⌫ )

b, ⌫
⇥
2
= 2i cot(2✓) gs f abe f cde (g µ⇢ g ⌫

⇤

g µ g ⌫⇢ )

+f ace f bde (g µ⌫ g ⇢
+f ade f bce (g µ⌫ g ⇢

c, ⇢

d,

a, µ

g µ g ⌫⇢ )
g µ⇢ g ⌫ )

b, ⌫
=

c, ⇢

d,

⇤

⇥
i 2
2
csc (2✓)(5 + 3 cos(4✓)) gs f abe f cde (g µ⇢ g ⌫
2
+ f ace f bde (g µ⌫ g ⇢
g µ g ⌫⇢ ) + f ade f bce (g µ⌫ g ⇢

g µ g ⌫⇢ )
g µ⇢ g ⌫ )

⇤

Figure B.3: Feynman rules for the quartic vertices. A gluon field is, as usual, represented
by a coiling line; a coloron field is represented by a zigzag line.

159

a, µ

b, ⌫
a, µ

c

d
c

⇥
⇤
2
= igs f ace f bde + f ade f bce g µ⌫

=

b, ⌫

d

a, µ

⇥
⇤
2
igs f ace f bde + f ade f bce g µ⌫

c

b, ⌫

d

⇥
⇤
2
= i cot(2✓) gs f ace f bde + f ade f bce g µ⌫

Figure B.4: Feynman rules for the quartic vertices (continued).

160

Appendix C
NUMERICAL VALUES OF THE
COLORON K-FACTOR
The numerical values of the K-factors for various values of the coloron mass and the three
patterns of coloron coupling are shown in Tabs. C.1-C.3. Finally, the values of the K-factor
corresponding to the KK-gluons of [7], corresponding to the experimental search reported in
[6], are shown in Tab. C.4.

161

MC (GeV) sin2 (θc ) = 0.05 sin2 (θc ) = 0.25 sin2 (θc ) = 0.50
1000
0.780
0.980
0.990
1200
0.800
1.00
1.01
1400
0.820
1.02
1.04
1600
0.840
1.04
1.06
1800
0.870
1.06
1.08
2000
0.890
1.09
1.11
2200
0.920
1.12
1.13
2400
0.950
1.14
1.16
2600
0.970
1.17
1.19
2800
1.00
1.20
1.22
3000
1.04
1.23
1.25
3200
1.07
1.27
1.28
3400
1.10
1.30
1.32
3600
1.14
1.34
1.36
3800
1.18
1.38
1.39
4000
1.22
1.42
1.43
Table C.1: K-factors for colorons of various masses, and rL = rR = − tan θc .
MC (GeV) sin2 (θc ) = 0.05 sin2 (θc ) = 0.25 sin2 (θc ) = 0.50
1000
1.11
1.11
1.12
1200
1.13
1.13
1.14
1400
1.15
1.15
1.16
1600
1.17
1.17
1.18
1800
1.20
1.20
1.21
2000
1.22
1.22
1.23
2200
1.25
1.25
1.26
2400
1.28
1.27
1.29
2600
1.30
1.30
1.32
2800
1.33
1.33
1.35
3000
1.37
1.36
1.38
3200
1.40
1.40
1.41
3400
1.43
1.43
1.44
3600
1.47
1.47
1.48
3800
1.51
1.51
1.52
4000
1.55
1.55
1.56
Table C.2: K-factors for colorons of various masses, and rL = rR . The classic “axigluon”
[5] corresponds to sin2 θc = 0.50.
162

MC (GeV) sin2 (θc ) = 0.05 sin2 (θc ) = 0.25 sin2 (θc ) = 0.50
1000
0.800
0.980
0.990
1200
0.820
1.00
1.01
1400
0.840
1.02
1.04
1600
0.870
1.04
1.06
1800
0.890
1.07
1.08
2000
0.920
1.09
1.11
2200
0.940
1.12
1.13
2400
0.970
1.15
1.16
2600
1.00
1.18
1.19
2800
1.03
1.21
1.22
3000
1.06
1.24
1.25
3200
1.09
1.27
1.28
3400
1.13
1.31
1.32
3600
1.17
1.34
1.36
3800
1.20
1.38
1.39
4000
1.24
1.42
1.43
Table C.3: K-factors for colorons of various masses, and rL = rR = cot θc .

163

MC (GeV) −0.20gs
500
0.660
600
0.670
700
0.670
800
0.680
900
0.690
1000
0.700
1100
0.710
1200
0.720
1300
0.730
1400
0.740
1500
0.760
1600
0.770
1700
0.780
1800
0.790
1900
0.800
2000
0.820
2100
0.830
2200
0.840
2300
0.860
2400
0.870
2500
0.880
2600
0.900
2700
0.910
2800
0.930
2900
0.940
3000
0.960

−0.25gs
0.770
0.780
0.790
0.800
0.810
0.810
0.820
0.840
0.850
0.860
0.870
0.880
0.890
0.900
0.920
0.930
0.940
0.960
0.970
0.980
1.00
1.01
1.03
1.04
1.06
1.07

−0.30gs
0.830
0.840
0.850
0.860
0.870
0.880
0.890
0.900
0.910
0.920
0.930
0.940
0.950
0.970
0.980
0.990
1.00
1.02
1.03
1.04
1.06
1.07
1.09
1.10
1.12
1.13

−0.35gs
0.870
0.880
0.890
0.890
0.900
0.910
0.920
0.930
0.940
0.960
0.970
0.980
0.990
1.00
1.01
1.03
1.04
1.05
1.07
1.08
1.09
1.11
1.12
1.14
1.15
1.17

−0.40gs
0.890
0.900
0.910
0.920
0.930
0.940
0.950
0.960
0.970
0.980
0.990
1.00
1.01
1.03
1.04
1.05
1.06
1.08
1.09
1.10
1.12
1.13
1.15
1.16
1.18
1.19

Table C.4: K-factors for KK-gluons of various masses considered in [6]. This calculation
is based on the theoretical framework of [7], with the KK-gluon coupling (specified in the
column heading) varying between −0.20gs and −0.40gs .

164

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