STRONG DYNAMICS AT THE LHC
By
Pawin Ittisamai

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics-Doctor of Philosophy
2014

ABSTRACT
STRONG DYNAMICS AT THE LHC
By
Pawin Ittisamai
The limitations of the Standard Model of particle physics, despite its being a wellestablished theory, have prompted various proposals for new physics capable of addressing
its shortcomings. The particular issue to be explored here is the mechanism of electroweak
symmetry breaking, the probing of which lies within the TeV-scale physics accessible to
the Large Hadron Collider (LHC). This thesis focuses on the phenomenology of a class of
models featuring a dynamical breaking of the electroweak symmetry via strong dynamics.
Consequences of recent experiments and aspects of near-future experiments are presented.
We study the implications of the LHC Higgs searches available at the time the related
journal article was written for technicolor models that feature colored technifermions. Then
we discuss the properties of a technicolor model featuring strong-top dynamics that is viable
for explaining the recently discovered boson of mass 126 GeV. We introduce a novel method
of characterizing the color structure of a new massive vector boson, often predicted in various
new physics models, using information that will be promptly available if it is discovered in
the near-future experiments at the LHC. We generalize the idea for more realistic models
where a vector boson has flavor non-universal couplings to quarks. Finally, we discuss the
possibilities of probing the chiral structure of a new color-octet vector boson.

ACKNOWLEDGMENTS

The road to the end of the first step of a long adventure in high-energy physics has been
positively challenging and my journey has been full of pleasure and joy. While only my name
appears on the end product, this thesis, it is actually a result of contributions from a great
many people. Without influences and supports from others, either directly or indirectly, this
would not have been possible. So I would like to mention some of them here.
A journey is a journey only if one could start. For this, I am greatly thankful to my dear
parents, Wasin and Passanee, for their devotions and supports to make sure that I could
follow the path of my interests.
Without the following medical doctors, several physical challenges might have derailed
me along the way during the last decade. I am able to live upright thanks to Prof. Wicharn
Yingsakmongkol who replaced a broken part of my spine. I am able to hear stereo thanks
to Doctor Andrea Palmieri who helped me recover from not-so-partial hearing loss. I am
able to use my right arm again thanks to Doctor Andrew Schorfhaar who repaired my torn
shoulder tendon.
Now, in terms of education, past scientific advisors greatly contributed to many pivotal
points. My deep gratitude goes to Prof. Ahpisit Ungkitchanukit and Prof. Goran Senjanovi´c.
I also thank all the professors who helped me appreciate different layers of beauty in
physics. At Michigan State University (MSU), I am particularly thankful to Prof. Mark
I. Dykman, Prof. Carl R. Schmidt, and Prof. Vladimir G. Zelevinsky, with whom I have
multiple wonderful experience in their classes. I also would like thank the first two professors,
along with Prof. Reinhard H. Schwienhorst, for serving as committee members of the thesis.
A great amount of physics appreciations happened outside classrooms. I will not forget a
iii

series of “dinner discussions” back there in Thailand led by Prof. Rujikorn Dhanawittayapol
and “lunch discussions” at MSU led by Prof. Chien-Peng “C.-P.” Yuan.
I would like to extend my gratitude to past and current members of the theory group
for interesting and friendly discussions: Dr. Arsham Farzinnia, Dr. Chiu Man Ho, Joshua
Isaacson, Dr. Vikram Rentala, Dr. Natascia Vignaroli, Dr. Devin Walker, Dr. Jiang-Hao
Yu. Many thanks also go to Josh for his feedbacks on the introduction of the thesis.
This thesis is based on a set of collaborative works. For that, I am deeply grateful to
all my collaborators, particularly Dr. Anupama Atre and Dr. Jing Ren for their significant
contributions to both the works and my education.
Financial support played an important role during decisive moments. I was fully supported during the first five years at MSU from a government-funded program called the
Development and Promotion of Science and Technology talents project (DPST), Thailand.
Thus, I am deeply grateful to the people of Thailand for giving me this rare opportunity.
My life in the Department of Physics has been good. I am grateful to Brenda Wenzlick,
Debby Barratt and Kim Crossland for their kind assistances throughout the years.
My life outside the Department has also been good. I am indebted to a great atmosphere
my friends in the Thai community have created.
My life everywhere inside this planet has always been wonderful once it is shared extensively with Jaruswan Warakanont years ago.
Above all, I very deeply appreciate a complete package of opportunity, support, and care
that I have been receiving from my advisors, Prof. R. Sekhar Chivukula and Prof. Elizabeth
H. Simmons, who have also taught me well what “infinity” meant through their patience.
It has been a great privilege working with both of you.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 1 Introduction . . . . . . . . . . . . . . . . . .
1.1 Particle Physics and the Standard Model . . . . . .
1.2 Going Dynamically Beyond the Standard Model . .
1.3 Phenomenology, the LHC, and this Thesis . . . . .
1.3.1 Part I: Phenomenology of Strong Dynamics
1.3.2 Part II: Coloron Phenomenology . . . . . . .

Part I

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Phenomenology of Strong Dynamics . . . . . . . . . . . . . 31

Chapter 2 Technipion Limits from LHC Higgs Searches . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Technicolor and Technipions . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Models with colored technifermions and a topcolor mechanism . . . . . . .
2.3.1 LHC Limits on Models with Light Technipions . . . . . . . . . . .
2.3.2 LHC Limits on Heavier Technipions Decaying to Tau-Lepton Pairs
2.4 Models with colored technifermions and a top mass generated by ETC . .
2.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3 Discovering Strong Top Dynamics at the LHC . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Scalar Spectrum and Properties . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The Triangle Moose Model . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 The Triangle Moose Potential: Scalar Spectrum and Isospin Violation
3.2.3 Scalar Couplings to Fermions . . . . . . . . . . . . . . . . . . . . . .
3.3 Neutral top-pion phenomenology . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Couplings and decays . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Production cross-section . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Current and Prospective Limits from the Diphoton channel . . . . . .
3.3.4 Decays to ZZ, Zγ and W W . . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Limits from the Ditau Channel . . . . . . . . . . . . . . . . . . . . .
3.4 Charged top-pion phenomenology . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Branching ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Limits from t → Π+ b . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Top-Higgs phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II

Coloron Phenomenology . . . . . . . . . . . . . . . . . . . . . . 97

Chapter 4 Distinguishing Color-Octet and Color-Singlet
the Large Hadron Collider . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 General Parameterization . . . . . . . . . . . . . . . . . .
4.2.1 Flavor Universal Scenario . . . . . . . . . . . . . .
4.2.2 An Illustrative Flavor Non-universal Scenario . . .
4.3 Color Discriminant Variable . . . . . . . . . . . . . . . . .
4.4 Parameter Space in di-jet Channel . . . . . . . . . . . . .
4.5 Sensitivity at the LHC . . . . . . . . . . . . . . . . . . . .
4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Flavor Universal Scenario . . . . . . . . . . . . . .
4.6.2 Flavor Non-universal Scenario . . . . . . . . . . . .
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

Resonances
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Chapter 5 Distinguishing Flavor Non-universal Colorons from Z Bosons
with Heavy Flavor Measurements . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 General Parameterization and Assumptions . . . . . . . . . . . . . . . . . .
5.3 Defining Color Discriminant Variables in Flavor Non-universal Models . . . .
5.3.1 General prescription for using Dcol in flavor-nonuniversal scenarios . .
5.4 Accessible Dijet Resonances at the 14 TeV LHC . . . . . . . . . . . . . . . .
5.5 Systematic Uncertainties at the 14 TeV LHC . . . . . . . . . . . . . . . . . .
5.6 Applying the Color Discriminant Variable to Flavor-non-universal Models at
the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Demonstration that C and Z lie in different regions of coupling ratio
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Using heavy flavor measurements to tell C from Z . . . . . . . . . .
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 6 Probing Color Octet Couplings at the Large Hadron Collider
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 General Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Collider Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Charged Current Channel with W -Decays to Leptons . . . . . . .
6.3.3 Neutral Current Channel with Z-Decays to Charged Leptons . . .
6.4 Sensitivity at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
APPENDICES . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . 189

Appendix A
Appendix B
Appendix C
Appendix D

The Standard Model . . . . . .
Strong Top Dynamics . . . . . .
Uncertainty of Intrinsic Width
Curves with Constant Dcol . . .

BIBLIOGRAPHY

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Measurement
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vii

LIST OF TABLES

Table 1.1

Table 1.2

Table 2.1

Table 2.2

Table 2.3

Fermions of the Standard Model are categorized into two classes:
quarks and leptons (top and bottom entries, respectively). Both
classes are subdivided into 3 generations (columns labeled I, II, and
III) which are identical apart from the different masses of the members of each generation. Properties of the particles are taken from
[17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Bosons of the Standard Model: the gauge bosons (top panel) and the
Higgs (bottom panel). Properties of the particles are taken from [17].

5

Properties of the lightest relevant PNGB (technipion) in representative technicolor models with colored technifermions. In each case, we
show the name and technifermion content of the state (in the notation
of the original paper), the ratio of the weak scale to the technipion decay constant, the anomaly factors for the two-gluon and two-photon
couplings of the technipion, and the technipion’s couplings to leptons
and quarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Branching ratios for phenomenologically important modes (in percent) for technipions of mass 130 GeV for NT C = 2, 4 and for a
standard model Higgs [159] of the same mass. . . . . . . . . . . . . .

41

Branching ratios for phenomenologically important modes (in percent) for technipions of mass 350 GeV for NT C = 2, 4 and for a
standard model Higgs [159] of the same mass. . . . . . . . . . . . . .

42

Table 4.1

Sources of systematic uncertainty contributing to uncertainties in
measurement of the cross section, mass and width of a resonance
in the di-jet channel at various experiments and c.m. energies. The
values used in this analysis are indicated by an asterisk. . . . . . . 119

Table 6.1

Representative cross sections at parton level in fb for√the CW and
CZ signal modes and backgrounds at the LHC with s = 14 TeV.
The signal is shown for mC = 2.5 to 4.5 TeV and for a few sample
coupling values that correspond to ΓC /mC = 0.05, 0.10, 0.20 and
0.30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Table 6.2

Selection efficiencies (in percent) for signal ( mC ) and background
( BG ) samples for the case of the CW channel with leptonic decays
of W into electrons and muons. . . . . . . . . . . . . . . . . . . . . 171

viii

Table 6.3

Same as Table 6.2, but for mc = 4.0 and 4.5 TeV. . . . . . . . . . . . 172

Table 6.4

Selection efficiencies (in percent) for signal ( mC ) and background
( BG ) samples for the case of the CZ channel with leptonic decays
of Z into electron and muon pairs. The couplings and widths for the
mass points shown are the same as those in Table 6.1. . . . . . . . 173

Table A.1

Hypercharge assignments for fermions. . . . . . . . . . . . . . . . . . 195

Table B.1

Lower bound on M (Π+
t ) from b → sγ assuming the fermion couplings
in Eq. (B.43). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

ix

LIST OF FIGURES

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Feynman diagrams for single technipion production at LHC. The
shaded circle in diagrams (b) and (c) represents an ETC coupling
between the ordinary quarks and techniquarks. For interpretation
of the references to color in this and all other figures, the reader is
referred to the electronic version of this thesis. . . . . . . . . . . . .

36

Comparison of experimental limits and technicolor model predictions
for production of a new scalar decaying to photon pairs. In each pane,
the shaded region (above the solid line) is excluded by the combined
95% CL upper limits on σh Bγγ normalized to the SM expectation as
observed by CMS [142] and ATLAS [141]. Each pane also displays (as
open symbols) the theoretical prediction from one of our representative technicolor models with colored technifermions, as a function of
technipion mass and for several values of NT C . Values of mass and
NT C for a given model that are not excluded by the data are shown
as solid (green) symbols. Continued next page. . . . . . . . . . . . .

45

Comparison of experimental limits and technicolor model predictions
for production of a new scalar decaying to tau lepton pairs. In each
pane, the shaded region (above the solid line) is excluded by the
combined 95% CL upper limits on σh Bτ + τ − normalized to the SM
expectation as observed by CMS [145] and ATLAS [143]. Each pane
also displays (as open symbols) the theoretical prediction from one of
our representative technicolor models with colored technifermions, as
a function of technipion mass and for several values of NT C . Values
of MP and NT C for a given model that are not excluded by the data
are shown as solid (green) symbols. . . . . . . . . . . . . . . . . . .

47

Comparison of experimental limits and technicolor model predictions
for production of a new scalar decaying to tau lepton pairs for scalar
masses in the mass range 110 - 600 GeV . In each pane, the shaded
region (above the solid line) is excluded by the 95% CL upper limits
on σh Bτ + τ − from ATLAS [144]. Each pane also displays (as open
symbols) the theoretical prediction from one of our representative
technicolor models with colored technifermions, as a function of technipion mass and for several values of NT C . Values of MP and NT C
for a given model that are not excluded by this data are shown as
solid (green) symbols; nearly all such values at low technipion masses
are excluded by the data shown in Fig. 2.3.1. Continued next page. .

50

x

Figure 2.5

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Comparison of data and theory for production of a new scalar of
mass 150 - 350 GeV that decays to tau lepton pairs; here, technipion production through techniquark loops is potentially modified by
including production via top quark loops assuming extended technicolor generates most of the top quark’s mass. Continued next page. .

53

The gauge structure of the model in Moose notation. g and g are approximately the Standard Model SU (2) and hypercharge gauge couplings while g˜ represents the ‘bulk’ gauge coupling. The left (right)
handed light fermions are mostly localized at site 0 (2) while their
heavy counterparts are mostly at site 1. The links connecting sites
0 and 1 and sites 1 and 2 are nonlinear sigma model fields while the
one connecting sites 0 and 2 is a linear sigma field. Site 2 is dotted
to indicate that only the τ3 component is gauged. . . . . . . . . . .

67

Branching ratios of the Π0t into its dominant decay modes for sin ω =
0.3 (top), 0.5 (middle), and 0.7 (bottom). The order of the curves
in the key (from top to bottom) reflects the order of the curves at
M (Π0t ) = 300 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Total decay width (top) and branching ratio for Π0t → γγ (bottom)
of a 125 GeV Π0t as a function of sin ω. . . . . . . . . . . . . . . . .

79

Top: Cross section times branching ratio for gg → Π0t → γγ for a
125 GeV top-pion at the 7 and 8 TeV LHC. Bottom: 95% confidence
level exclusion limits in the MΠ0 vs. sin ω plane, extrapolated from
the LHC SM Higgs search limits in the γγ channel with 4.8–4.9 fb−1
at 7 TeV from Refs. [206, 207]. . . . . . . . . . . . . . . . . . . . .

82

Branching ratios of the charged top-pion to the dominant final states,
for sin ω = 0.3, 0.5, and 0.7 (top to bottom). We include off-shell
decays to t∗ b and also off-shell decays to Π0t W + assuming M (Π0t ) =
125 GeV. These branching ratios were computing using a modified
version of HDECAY [160]. . . . . . . . . . . . . . . . . . . . . . . . . .

87

+
Top: Contours of BR(Π+
t → τ ν) as a function of M (Πt ) and sin ω.
+
Bottom: Contours of BR(t → Π+
t b) × BR(Πt → τ ν) as a function
+
of M (Π+
t ) and sin ω. We interpret the ATLAS t → H b search [208]
+
to exclude B ≡ BR(t → Π+
t b) × BR(Πt → τ ν) > 0.01. . . . . . . .
+
∗
Top: Contours of BR(t → Π+
t b) × BR(Π → t b) as a function of
+
M (Π+
t ) and sin ω. Bottom: comparison of the ATLAS t → H b
exclusion and the ATLAS (solid) and CMS (dashed) Π0t → γγ limits
[206, 207], assuming degenerate Π0t and Π+
t . . . . . . . . . . . . . .

xi

89

91

Figure 3.8

Figure 3.9

Figure 4.1

Figure 4.2

The CMS exclusion contours for MHt from searches for the SM Higgs
in W W and ZZ final states [209], as a function of M (Π+
t ) for the
0
special case M (Πt ) = 125 GeV. The horizontal (red) lines show our
+
lower bound on M (Π+
t ) from the ATLAS t → H b search [208] for
the same four sin ω values. . . . . . . . . . . . . . . . . . . . . . . .

93

The production of a neutral top-pion and a Z from an s-channel
top-Higgs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

(a) Top left: 5σ discovery reach for a flavor universal coloron in the
plane of the mass of the √
coloron (in TeV) and the square of the
couplings at the LHC with s = 14 TeV. See text for further details.
(b) top right: same as (a) but for a leptophobic Z and the discovery
reach is shown in varying shades of green. (c) bottom left: same as (a)
but for the flavor non-universal coloron where g t (CL,R ) = 3g q (CL,R ).
(d) bottom right: same as (b) but for the flavor non-universal Z with
g t (Z L,R ) = 3g q (Z L,R ). . . . . . . . . . . . . . . . . . . . . . . . . 112
√
(a) Top left: Sensitivity at the LHC with s = 14 TeV and integrated luminosity of 30 fb−1 for distinguishing a coloron from a leptophobic Z in the plane of the log of the color discriminant variable
Dcol = M 3 σjj /Γ and mass (in TeV) for the flavor universal scneario. (b) Top right: Same as (a) but for an integrated luminosity of
100 fb−1 . (c) Bottom left: Same as (a) but for an integrated luminosity of 300 fb−1 (d) Bottom right: Same as (a) but for an integrated
luminosity of 1000 fb−1 . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 4.3

Same as Fig. 4.2, but with the systematic uncertainties in M , Γ and
σjj increased by a factor of 1.5. . . . . . . . . . . . . . . . . . . . . 124

Figure 4.4

(a) Top left: same as Fig. 4.2(c), but for the flavor non-universal
case with g t (L, R) = 2g q (L, R). (b) Top right: same as (a) but for
integrated luminosity of 1000 fb−1 . (c) Bottom left: same as (a)
but for g t (L, R) = 3g q (L, R). (d) Bottom right: same as (c) but for
integrated luminosity of 1000 fb−1 . . . . . . . . . . . . . . . . . . . 126

Figure 4.5

(a) Top left: Sensitivity to√distinguish color-octet and color-singlet
scenarios at the LHC with s = 14 TeV and integrated luminosity of
300 fb−1 for a resonance of mass 3 TeV for different values of ξ. (b)
Top right: same as (a) but for integrated luminosity of 1000 fb−1 . (c)
Bottom left: same as (a) but for a resonance with mass 4 TeV. (d)
Bottom right: same as (c) but for integrated luminosity of 1000 fb−1 .
The different points marked a, b, c, d correspond to different parameter points used as examples in the text. . . . . . . . . . . . . . . . . 128
xii

Figure 5.1

Ratios of parton density functions as a function of the mass of the produced vector resonance. This figures shows two such ratios: the relative contribution of parton density functions, defined in Eq. (4.20), for
the down-type quarks of the first two generations (Wd + Ws )/(Wu + Wc )
(top curve, in brown) and for bottom quarks Wb /(Wu + Wc ) (bottom
curve, in orange) relative to the up-type quarks for the first two generations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 5.2

2
Region of parameter space (scaled coupling strength (gu2 + gd2 )/gQCD
vs. coloron mass M ) where
√ the color discriminant variable analysis
applies at the LHC with s = 14 TeV. The configurations of coupling
ratios vary as follows: ρu ≡ gu2 /(gu2 + gd2 ) increasing from 0 to 1 (from
left to right panels), and ρt +ρb ≡ (gt2 + gb2 )/(gu2 + gd2 ) increasing from
0.25 to 4 (from top to bottom panels). Continued next page. . . . . 146

Figure 5.3

Same as Fig. 5.4 but for a flavor-nonuniversal Z . Notice that while a
leptophobic Z having relatively large couplings to quarks in the third
generation is still within the reach of the future LHC, its total width
would typically be too large to be included in analyses for narrow
resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure 5.4

Illustration that the proposed measurements suffice to identify the
color structure of a new dijet resonance. These plots show regions
of the 3-d parameter space gu2 /(gu2 + gd2 ) vs. σtt¯/σjj vs. σb¯b /σjj ,
at fixed values of mass (3 TeV on the top panels and 4 TeV on the
bottom panels) where dijet cross section and Dcol fall within a certain
range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Figure 5.5

Further illustration that measuring σtt¯ and σb¯b will show whether a
resonance of a given mass, dijet cross-section, and Dcol is a coloron
or Z . We display regions of parameter space that correspond to a
dijet cross section within 1σ uncertainty
of the value required for
√
a discovery at the LHC with s = 14 TeV at L = 1000 fb−1 (the
same values used in Fig.5.4), and that also have the illustrative color
discriminant variable within a range of 50% around value of Dcol =
3 × 10−3 , for resonances with masses 3 TeV (top panel) and 4 TeV
(bottom panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 6.1

A cartoon illustration of the form of prospective constraints on chiral couplings from the di-jet channel (dashed black circle with red
band), the channel with associated production of a W boson (solid
black parallel lines with green band) and the channel with associated
production of a Z boson (dotted black ellipse with blue band). . . . 160

xiii

Figure 6.2

Representative Feynman diagrams for associated production of a W, Z
gauge boson with a color-octet resonance, C. Both s and t channel
diagrams along with initial or final state radiation of the associated
gauge boson are shown. We assume that the weak gauge boson decays
leptonically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Figure 6.3

(a) Top: transverse momentum of the hardest jet (pT1 ) for a coloroctet with mass mC = 3 TeV and two different widths ΓC /mC = 0.05
and 0.20. (b) Bottom: same as (a) but for the second hardest jet
j
(pT2 ). The relatively broad and flat transverse momentum distributions of the signal events contrast with the distributions for the
backgrounds, which all peak sharply at lower transverse momenta. . 170

Figure 6.4

(a) Top left: sensitivity plot for a color-octet of mass mC = 2.5 TeV
produced in association with a W gauge boson, in the√plane of the
q q
couplings gL , gR at the LHC with 10 fb−1 of data and s = 14 TeV;
(b) top right: same as (a) but for 100 fb−1 of data; (c) bottom left:
same as (a) but for a color-octet produced in association with a Z
boson; (d) bottom right: same as (c) but for 100 fb−1 of data. . . . 175

Figure 6.5

Same as Fig. 6.4 but for mC = 3.5 TeV.

Figure 6.6

Same as Fig. 6.4 but for mC = 4 TeV.

Figure 6.7

(a) Left: same as Fig. 6.4(a) but for mC = 4.5 TeV; (b) right: same
as (a) but for 100 fb−1 of data. The figures for the corresponding CZ
channels are not presented as there is not sufficient sensitivity at this
mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Figure 6.8

(a) Left: same as Fig. 6.4(a) but for mC = 3 TeV. Here we have combined the sensitivity in the associated production channel from both
W and Z bosons. The couplings allowed by the current measurement
tt¯ at the Tevatron [288, 289] as interpreted in Ref. [217] are
of AF
B
shown by the translucent yellow dotted region; (b) right: same as (a)
but for 100 fb−1 of data; . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 6.9

Reconstructed invariant mass (mrec
C ) for a 3 TeV color-octet resonance with ΓC /mC = 0.05 and 0.30 after acceptance cuts. . . . . . 183

Figure B.1

Diagrammatic representation of “fermion bubble” approximation yielding the kinetic energy and mass (left) and self-couplings (right) of the
composite Φ field. The two-point function is resumed to generate the
kinetic energy term for the composite scalar field. . . . . . . . . . . 211

j

xiv

. . . . . . . . . . . . . . . 177
. . . . . . . . . . . . . . . . 178

Figure B.2

Diagrams corresponding to the two leading contributions to α∆T in
the TC2 model. The diagram on the left gives rise to the operator
shown in Eq. (B.29). The diagram on the right arises from the fourtechnifermion operator shown in Eq. (B.32). The small black circles
in these diagrams represent the dynamical technifermion mass arising
from technicolor chiral symmetry breaking, as parameterized by the
field Σ in the chiral Lagrangian of Eq. (B.14). . . . . . . . . . . . . 218

Figure B.3

Top: Approximate constraints on MΠ and sin ω in the TC2 model
in the NJL approximation coming from bounds on α|∆T |. The
constraints shown arise from taking |∆T | < 0.5 and assuming that
c1 /2 = cT = cT = 1; the shaded pink region is excluded. The red
solid line shows the bound arising from the operator in Eq. (B.31)
(red line); the blue long-dashed line shows the bound from Eq. (B.34)
(blue dashed line). The dotted purple curves on the top pane depict
contours of constant η from Eq.(B.4); the dashed purple curves in the
bottom pane are contours of constant top-Higgs mass from Eq. (B.28).221

Figure B.4

Loop corrections to δgZbb and b → sγ arising from exchange of
charged top-pions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Figure D.1

The region of parameter space for the colorons (in blue) and Z bosons
(in green) that is consistent with measurements of the √
ratios σtt¯ and
σb¯b , the mass, cross section, and Dcol at the LHC with s = 14 TeV.
The value of Dcol in a given plot is listed at upper right. The three
blue (green) lines in a given plot correspond to colorons (Z bosons)
with three different values of the experimentally inaccessible coupling
ratio (the gu2 /(gu2 +gd2 )). These plots illustrate the measurement precision in σtt¯ and σb¯b that is required to distinguish between the coloron
and the leptophobic Z . The set of plots for resonances with a mass
of 3.0 TeV (3.5 TeV) is in the top (bottom) panel. . . . . . . . . . . 231

Figure D.2

Same as Fig. D.1, but for 4.0 TeV (top panel) and 4.5 TeV (bottom
panel) resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

xv

Chapter 1
Introduction
The Standard Model (SM) is a well-established theory describing particle physics. Simple
and beautiful as it is, the model is considered phenomenologically incomplete and theoretically unsatisfactory. Various types of “new physics” beyond the Standard Model have been
proposed. This thesis was written during the time when the Large Hadron Collider (LHC),
hosting experiments that could potentially confirm or exclude various new physics models,
had finished its first phase of operation. It focuses on two phenomenological aspects of new
physics: implications of the data from the LHC on models explaining the electroweak symmetry breaking via strong dynamics, and the future-LHC prospects for characterizing a new
massive vector boson which is a common prediction of new physics models.
In section 1.1, I give an overview of the Standard Model and its problems, focusing on
the aspects related to electroweak symmetry breaking. Then, in Section 1.2, I present an
overview of the theories based on strong dynamics that are relevant to this thesis. I briefly
discuss the issues these theories address and the features they add. Finally, Section 1.3
illustrates where the works [1, 3, 4, 6, 7] to which my collaborators and I contributed stand
in the field.

1

1.1

Particle Physics and the Standard Model

The ultimate goal of particle physics is to understand, at the most fundamental level, particles and their interactions describing the observed picture of the universe with a set of,
hopefully, simple laws. The modern picture of the subject has been developed immensely
during the past century, thanks to the healthy interplays between advances in various aspects
of theoretical ideas and experiments. Let us begin this section with the theoretical idea that
has become established as a well-tested theory.
Successful theoretical development has been achieved via a prescription of quantum field
theory, where an elementary particle is regarded as an excited state of a quantum field. This
follows the pioneering interacting theory of this sort: Quantum Electrodynamics (QED)
[8, 9, 10, 11, 12], where the electromagnetic interaction between charged particles is explained
by the quantum version of the so-called gauge theory. In a gauge theory, one requires that
physical phenomena are equally well described by any member of a class of matter fields
that are related by a continuous group of spacetime-dependent (i.e., local) transformations
called a gauge transformation. Doing so not only implies the existence of a gauge field, but
also constrains its interactions to be determined by the Lie group describing the symmetry
transformations. The gauge field’s quanta are called gauge bosons. A quantum gauge theory
(in four dimensions) possesses a feature of making consistent predictions to arbitrarily small
length scales in terms of a finite number of parameters as long as the theory remains perturbative. In other words, a finite number of measurements of its parameters at a particular
length scale uniquely determines its predictions at any other scale. This property, called
“renormalizability”, while not being an absolute necessity, is a much-desired feature in a
model attempting to be the ultimate theory.

2

The current experimentally-accepted theory of particle physics that has been developed
along these lines is referred to as the “Standard Model”1 [13, 14, 15]. It is a renormalizable
gauge theory [16] describing physical processes, governed by all known fundamental interactions (except gravity) relevant below subatomic length scales. In the Standard Model,
matter particles — the quarks and the leptons — are grouped according to their interactions
into three generations that are identical in most aspects apart from their masses. The matter
contents are illustrated in Table 1.1. The interactions between the particles are classified
into two sectors: strong and electroweak.
The strong sector utilizes a non-Abelian gauge invariance under an SU (3) group [18]
to describe the strong nuclear interaction. The underlying theory is known as quantum
chromodynamics (QCD) [19, 20]. A striking feature that sets it apart from QED2 is that
the non-Abelian nature of QCD implies that the charge of the interaction, called color, is
carried by both the matter particles (the quarks) and the gauge mediators, called gluons.
Its properties are shown in Table 1.2. Therefore, unlike the photon, the gluons also interact
directly among themselves. This leads to several interesting phenomena. First, in contrast to
what happens in QED, the gluon-gluon interactions turn out to cause the coupling strength
to be smaller when probing to smaller length scales. This feature, known as asymptotic
freedom [21, 22], allows the theory to sensibly describe strong interactions at small distances
as a perturbation theory. Second, this means at a certain finite length scale — around
O(10−15 m) or equivalently the energy scale ΛQCD ∼ O(200 MeV) — the coupling becomes
so large that the perturbative method completely stops making sense. In other words,
QCD has a dimensional scale built in [23]! Third, this leads to an experimentally verified
1 A partial review of the Standard Model is presented in Appendix A.
2 In QED, photons do not interact among themselves. This leads to the phenomenon that strength of
QED interaction increases with decreasing distance.

3

Table 1.1: Fermions of the Standard Model are categorized into two classes: quarks and
leptons (top and bottom entries, respectively). Both classes are subdivided into 3 generations
(columns labeled I, II, and III) which are identical apart from the different masses of the
members of each generation. Properties of the particles are taken from [17].

Quarks

Leptons

name
symbol
charge
mass

I

II

III

up
u
+2/3
2.3 MeV

charm
c
+2/3
1.28 GeV

top
t
+2/3
173 GeV

down
d
−1/3
4.8 MeV

strange
s
−1/3
95 MeV

bottom
b
−1/3
4.18 GeV

electron
e
−1
0.511 MeV

muon
µ
−1
105.7 MeV

tau
τ
−1
1.77 GeV

electron
neutrino
νe
0
O(eV)

muon
neutrino
νµ
0
O(eV)

tau
neutrino
ντ
0
O(eV)

4

Table 1.2: Bosons of the Standard Model: the gauge bosons (top panel) and the Higgs
(bottom panel). Properties of the particles are taken from [17].

Gauge Bosons

symbol
charge
mass

photon (γ)
γ
< 10−35
< 10−18 eV

gluon
g
0
0

Z boson
Z0
0
91.19 GeV

W boson
W±
±1
80.4 GeV

Higgs Boson
h or H
0
126 GeV

conjecture that at length scales greater than ΛQCD , only a “colorless” object is observed.
This phenomenon is called confinement. These phenomena of QCD serve as principal ideas
underpinning models considered in this thesis.
The electroweak sector uses a product gauge group SU (2) × U (1) to describe processes
involving the electromagnetic and the weak nuclear interactions, with the typical length
scale around O(10−18 m), the electroweak scale. Under the weak interaction, the two states
describing fermions, the left- and right-handed chirality Dirac fermions, interact in a different
fashion. Only a left-handed fermion carries the weak charge and can transform into another
fermion belonging to the same representation of the SU (2) symmetry group (i.e., the same
“generation”, as shown in Table 1.1). This so-called charged current interaction is explained
by the charged gauge bosons W ± . Together with the Z boson and the photon that govern
neutral current phenomena, these four gauge bosons constitute the gauge mediators of the
SU (2) × U (1) gauge theory.
The weak interaction exhibits several interesting phenomena. First, unlike the electromagnetic interaction whose range spans infinitely, its subatomically short range means that
5

the gauge bosons are massive. However, a prescription for describing a massive vector boson
is not manifestly gauge invariant. Second, only the symmetry of electrodynamics remains
exact at large length scales. The electroweak symmetry SU (2) × U (1) corresponding to the
unified interaction has to be hidden or broken. A spontaneously hidden symmetry would
have implied an unobserved massless scalar particle called a Nambu-Goldstone boson (NGB)
remaining in the physical spectrum if the symmetry were global [24, 25, 26]. Third, non-zero
mass of fermions also implies that the electroweak symmetry is hidden at length scales above
the electroweak scale. This is because the mass of a fermion field is explained by a coupling
between the left- and right-handed chirality fields, which carry different gauge charges; hence
violating the gauge invariance.
The massiveness of a gauge field and the spontaneous hidden of a local symmetry turn
out to be related to each other under a prescription known as the “BEH mechanism” — for
Brout, Englert, and Higgs (the firsts, among many, of its pioneers) [27, 28, 29, 30, 31]. This
can be understood as follows. When the hidden symmetry is local, the NGBs are no longer
physical and can be gauge transformed away from the spectrum: becoming the longitudinal
degrees of freedom of the gauge field and making the gauge field massive. Roughly speaking,
the vacuum of quantum field theory is not empty, but is filled with an agent that has a nonzero averaged field structure preserving only a subset of the full symmetry group describing
the interaction. Particles interacting with this field then have “effective mass”. This agent
can be either an additional elementary scalar field or a composite object consisting of other
elementary particles that interact strongly at smaller length scales. The Standard Model
is constructed using the fundamental scalar field option. This has an additional benefit of
explaining the fermion masses via an interaction between the fermion bilinear and the scalar
field with the right quantum number. I will focus on reviewing the Standard Model for now
6

and will discuss the latter approach in Section 1.2.
In the SM, electroweak symmetry breaking is realized via the BEH mechanism featuring
additional fundamental scalar fields in a doublet representation of SU (2) [14]. This doublet is
usually referred to as the Higgs field. It is assumed that the field develops a non-zero vacuum
expectation value that respects only the electromagnetic gauge group. As mentioned above,
the doublet structure of the Higgs is also convenient for the fermion sector. The mass of a
fermion can be introduced into the Lagrangian via the Yukawa interaction of the left- and
right-handed chiralities of the fermion with the Higgs doublet. Constructed in this manner,
the Standard Model is renormalizable [16].
The implementation of the fundamental scalar doublet also predicts an additional elementary scalar particle, a particle whose existence has no previous example in nature, as an
excitation of the Higgs field. This particle is known as the Higgs boson. It leads to unique
experimental features of the Standard Model. First, while the Higgs is produced from and
decays to various states, the characteristics of its production and decay (production cross
section, and decay branching ratios) depend only on the mass of the Higgs! Second, the
doublet structure of the Higgs field implies a particular ratio between the masses of W and
Z [32] (or the ratio between isospin-triplet charged and neutral current interactions at zero
momentum). This ratio is generally not guaranteed in other models but the one predicted
by the Standard Model agrees well with experiments [17]. It is explained by an accidental
global SU (2) symmetry, called the custodial symmetry, respected by the interaction of the
Higgs doublet. The custodial symmetry is broken by unequal electromagnetic charges and
the unequal masses of the fermions within an SU (2) doublet. This leads to small deviations
of the ratio from the predicted value at leading order. These unique features, along with
others, provide definite experimental tests of the model.
7

Experiments have been playing vital roles in developing and verifying the Standard
Model, particularly to see the origin of electroweak symmetry breaking (or at least where to
look further). Of particular importance are particle accelerators that collide well-controlled
beams with one another; they are called colliders. Unlike accelerators that direct a beam
into a fixed target, colliders allow all of the beam energy to be used for production of
elementary particles. Various kinds of colliding beams have been used: from the electronpositron collider at the “Organisation europ´eenne pour la recherche nucl´eaire”3 (CERN)
known as “Large Electron-Positron collider” (LEP), to the Fermi National Laboratory (Fermilab) proton-antiproton collider named the Tevatron, to the current CERN’s proton-proton
collider called the “Large Hadron Collider” (LHC). The latter started its operation in 2009,
achieved the goal for the first run of center-of-mass energy of 7 TeV in 2011, and 8 TeV in
2012, and eventually finished the run in 2012. It will resume operation at 13 TeV in 2015
and will eventually be operating at its designed center-of-mass energy of 14 TeV. Analyses
of existing data have been taken care of by various collaborations including the ATLAS (A
Toroidal LHC Apparatus) [33, 34] and CMS (Compact Muon Solenoid) [35, 36].
These accelerators and the collaborations working with them have contributed to notable
recent discoveries in particle physics. They include the weak gauge bosons in 1983 [37, 38,
39], the unusually heavy top quark in 1995 [40, 41], and eventually the the τ neutrino in
2000 [42]. The latter completely established the physical existence of the Standard Model’s
fermion and gauge contents. In addition to making the discoveries, some of these experiments
also contributed to measuring certain observables to unprecedented precision to uncover
any deviations from the Standard Model’s predictions. They are referred to as “precision
electroweak tests” (See, for example, [43]). While the Standard Model has been surviving
3 Previously, “Conseil Europ´een pour la Recherche Nucl´eaire”.

8

well enough under these tests, there have also been some indications of room for small
deviations.
Along with the aforementioned discoveries and measurements, one of the primary goals
for pushing towards “the next accelerator” during the past three decades had been to look
for the Higgs boson. The Higgs hunting lasted until the 4th of July 2012, the day the ATLAS
and CMS collaborations announced the observation of a particle having properties that are
consistent with the Higgs boson of the Standard Model [44, 45]. Following the discovery, a
natural next step is to inspect the new state in detail. Is it actually a scalar or a pseudoscalar?
Are the couplings (hence branching fraction of its decays to various channels) in agreement
with the Standard Model predictions? Does it have any partner states not predicted by the
Standard Model such as a charged one? So far, to the present precision, it appears to be
“the Higgs” of the SM [46, 47, 48, 49, 50, 51, 52, 53]. However, there are reasons suggesting
that this does not need to be the case when the Higgs is probed to better precision.
The question whether the recently discovered particle actually is the Higgs boson of the
Standard Model stems from the reason that the Standard Model itself is regarded as incomplete. The needs for a new physics model can be illustrated from both phenomenological
and theoretical points of view.
Phenomenologically, the need for a new physics model is clear. The Standard Model is
unable to cope with several experimentally established facts. As the ideas and the evidence of
massive neutrinos [54, 55, 56] were not established when the Standard Model was proposed,
neutrinos were assumed to be massless — we know this assumption is no longer valid. The
Standard Model does not account for the existence of the enigmatic constituents inferred from
cosmology known as Dark Matter and Dark Energy (see, for example, [57, 58, 59, 60, 61])
— these two are not unsubstantial!. In addition, while matter and anti-matter feature in
9

the SM, the model does not explain why the majority of the observable universe consists of
matter. Last but not least, what about gravity?
Theoretically, the Standard Model accounts for several descriptions in an unsatisfying
manner. For example, electroweak symmetry breaking is assumed without any explanation
of underlying dynamics. In addition, the renormalizability of the model itself could cause an
issue. It allows one to extrapolate the model to arbitrary smaller length scales inaccessible
by current experiments; i.e., going across the “desert” between the electroweak symmetry
breaking scale O(10−18 m) down to the Planck scale O(10−35 m), for gravity, if one assumes
no new physics showing up in between. This poses a problem, known as “naturalness”
problem, for a theory with an elementary scalar.
A theory is considered natural if macroscopic phenomena are not explained by a set of
excessively carefully adjusted parameters of microscopic physics working at a much smaller
length scale. In particle physics, the consensus is that the smallness of a parameter at a
certain scale is acceptable only if its smallness is viewed as a result of an enhanced exact
symmetry had the parameter vanished4 . For example, fermion masses of O(eV − GeV) or
gauge boson masses of O(GeV) can be naturally small (compared to the Planck scale) because
their disappearances would have led to a chiral symmetry and a gauge symmetry, respectively.
These symmetries also do protect the masses from being large in the calculations including
quantum effects. A theory with a fundamental scalar field, however, does not typically have
this feature of “naturalness” [62].
There is no symmetry restored upon having a scalar’s mass set to zero. Then quantum
corrections in that particular theory would drive its mass to the largest mass scale of the
4 Note that this naturalness argument is for a model construction point of view. On a physical point of
view, one does not say that the mass of a fermion is small because of chiral symmetry. But one rather says,
the smallness mass of a fermion implies that chiral symmetry is only approximate.

10

theory unless unnaturally fine cancellations between the theory’s parameters occurred. For
the Standard Model [63], that mass scale is the Planck scale and the cancellations would
have to be of O(10−32 ) if the particle at 126 GeV is the fundamental scalar predicted by the
model. In other words, according to this naturalness guideline, the Standard Model is not a
natural theory describing the recently-discovered scalar particle with mass 126 GeV [64, 65].
If the argument is taken seriously, although it is important to stress that this need not be
so, models addressing the naturalness issue usually predict new physics appearing around a
TeV energy scale. Coincidentally, there is another clue leading to the TeV scale as the arena
one expects to learn more about the electroweak symmetry breaking or new physics model.
The TeV energy scale is actually a “built-in” feature of the SM. In a standard-modellike theory without a scalar field that couples like the Higgs, the tree-level amplitudes for
longitudinal W + W − scattering (W W → W W ) violate unitarity around ∼ 1 TeV [66]. This
means either the W and Z interactions become strongly coupled or a sub-TeV particle
capable of canceling out the dangerous effect shows up — or both. Now the sub-TeV particle
has recently been discovered. In the case that the particle is actually the Standard Model
Higgs boson, new physics at the TeV scale is no longer guaranteed unless one takes the
naturalness argument seriously.
So far, apart from having discovered a particle that might behave like the Higgs of the
Standard Model, the LHC has not yet found a clue to what kind of new physics is preferred
by nature. But after all, we expect that a new physics model is out there somewhere. So
while we wait (and hope) for the near-future experiments to shed light on what the model
viable for explaining new physics looks like, it is worthwhile to see what the current data
“tell” us about some of those models, and what we could do if something is found from the
near-future data.
11

1.2

Going Dynamically Beyond the Standard Model

From the aspect of the agent of electroweak symmetry breaking and the naturalness issue, one
can classify the paths beyond the Standard Model to two major directions: more fundamental
scalars and no fundamental scalar. Each has its own merits. The “more fundamental scalars”
path includes Supersymmetry (SUSY), where an additional mechanism is introduced to
stabilize the hierarchy. Among many benefits, it could provide a unified picture of strong
and electroweak interactions [67, 68, 69] and stands as an essential part of other branches of
new physics such as string theory (see, for example, [70, 71, 72, 73]). The “no fundamental
scalars” path is based on theories with symmetries broken dynamically, where a composite
particle plays the role of the Higgs boson. This picture is motivated by the superconductor
theory in condensed matter physics — by Bardeen, Coopers, and Schrieffer (BCS) [74, 75]
— and has been brought to particle physics by Nambu and Jona-Lasinio [76, 77]. In the
BCS theory, a symmetry is broken dynamically by the formation of a condensate due to
interactions among electrons, making the photon massive. In this thesis, we focus on models
transplanting this idea into the beyond-the-Standard-Model physics. To introduce the idea,
let us begin by considering a well-studied example of dynamical symmetry breaking in the
regime of low-energy phenomena of QCD. This will illustrate how the electroweak gauge
bosons could become massive without the existence of a fundamental scalar in the theory.
Dynamical symmetry breaking is implemented to QCD under the context of chiral symmetry breaking in the presence of electroweak interaction. Without the electroweak interaction, the chiral symmetry breaking works as follows. The intrinsic energy scale of QCD,
ΛQCD ∼ O(200 MeV) [23], is large enough to consider the up and down quarks as approximately massless. This means their left- and right-handed degrees of freedom transform

12

independently under unitary transformations. In the exact massless limit, this exhibits an
SU (2)L ×SU (2)R global chiral symmetry. Interestingly, this symmetry is hidden even in this
massless quark limit. At energy scales lower (or length scales larger) than the confinement
scale ΛQCD , the attractive interaction for color singlet states leads to a Lorentz-invariant
bound state u¯L uR of a quark and an antiquark with opposite chiralities. It has non-zero
vacuum expectation value of this bound state 0|¯
uL uR |0 = 0|d¯L dR |0 ∼ Λ3QCD (where |0
is a hadronless ground state), called a condensate. The condensate is no longer invariant
under the chiral transformation, while still it respects the vectorial symmetry SU (2)L+R
(where both left- and right-handed fields transform in the same way). So there exists an
SU (2)-triplet of massless Nambu-Goldstone bosons of the chiral symmetry breaking and
these are interpreted as the pions of QCD5 .
What is interesting for our purposes is that the left- and right-handed chirality quarks that
form the condensate carry different charges under the electroweak group. The left-handed
ones “feel” the SU (2)L (now the weak gauge group), while the right-handed “feel” the U (1)Y
hypercharge subgroup of the SU (2)R . So the condensate’s formation breaks the electroweak
symmetry, and the pattern of electroweak symmetry breaking is indeed SU (2)L × U (1)Y →
U (1)em . The Nambu-Goldstone bosons interact with the gauge currents of electroweak
interactions and become the longitudinal degrees of freedom of the W and Z, making them
massive. Electroweak symmetry breaking by condensates provides additional benefits. The
ratio of the W and Z masses also turns out to be correct thanks to the SU (2)L+R global
symmetry which is still respected by the condensates. The unitarity behavior at energy
large (energy) scales of the model (for processes such as the W W scattering) is taken care
5 Pions are not exactly massless, but have small masses compared to the hadronic scale. This means
that the chiral symmetry is only approximate. They are referred to as Pseudo-Nambu-Goldstone Bosons
(PNGB).

13

of by composite resonances. Perhaps most interestingly, the intrinsic energy scale of QCD,
which could be far below from the scale above which new physics appear, means there is no
naturalness problem.
QCD, nevertheless, is not a suitable candidate for fully explaining electroweak symmetry
breaking as it accounts for only a small fraction of the mass of the gauge bosons; i.e.,
O(ΛQCD ) ∼ 200 MeV — many orders of magnitude too small! This is expected, as QCD
becomes strong at energy scales much lower than the electroweak symmetry breaking scale.
A new source of strong interaction with much higher intrinsic energy scale is needed.
A model describing electroweak symmetry breaking via strong dynamics utilizes new
QCD-like interactions to explain the O(100 GeV) masses of the electroweak gauge bosons.
This class of models is known as Technicolor (TC) [78, 79, 80]. Its earlier versions are
essentially a scaled-up version of QCD where the new strong interaction governed by a
SU (NTC ) gauge theory are experienced by a set of NT f flavors of new massless fermions
called technifermions, T = {U, D, . . .}. Similar to what happened in QCD, the simplest
realization requires a doublet of technifermions so the SU (2) chiral symmetry is exhibited,
and then hidden spontaneously by the presence of a condensate 0|U¯L UR |0 (and similarly
for other technifermions). The resulting Nambu-Goldstone bosons are called the technipions.
Once coupled with the electroweak interaction, they will be the main source6 of the W and Z
masses provided that the energy scale of technicolor ΛTC is large enough. The structure of a
model determines how large this scale has to be. A larger number of technifermion doublets,
say ND , leads to a greater contribution to the gauge bosons’ masses (by a factor of

√

ND ).

A larger gauge group for Technicolor interaction (larger NTC ) also means an enhancement
to the gauge bosons’ masses (also by a factor of

√

NTC ). This means the scale of Technicolor

6 The quark condensate of QCD still provides masses of the gauge boson, but to a much less extent.

14

could be ∼ O(TeV). As simple and elegant as it may appear, this basic Technicolor scenario
does not explain the origins of the masses of the fermions.
The masses and the mixings of quarks and leptons require an additional mechanism to
transfer the effect of the technifermion condensate to the Standard Model fermion fields. At
energies far below the scale above which new physics responsible for the interaction takes
over, the non-renormalizable effective interaction appears as an apparent “four-fermion”
interaction (suppressed by the mass scale of that interaction). A class of models that handles
this by coupling these fields to the Technicolor currents via an additional interaction at higher
energies is called Extended Technicolor (ETC) [81, 82].
ETC is a gauge theory having a gauge group GETC that contains technicolor and flavor
as sub-groups. The Standard Model fermions and technifermions are in the same irreducible
representation of this group so they interact via an ETC gauge-boson exchange. The group
GETC is assumed to be spontaneously hidden at high energies, the size of the mass METC
of the ETC-gauge bosons. At energies low compared to METC , this interaction manifests as
2
. A technifermion condensate formed via techa “four-fermion” interaction (¯
q T )(T¯q)/METC
2
nicolor interactions then induces the operator q¯q T¯T ETC /METC
. Masses of the fermions

(and other observables) at low energies are obtained by renormalization group running of
the interaction down from METC , where the value of the condensate is evaluated. Given
that Technicolor behaves as a scaled-up version of QCD where the coupling decreases very
quickly as energy increases, there is approximately no enhancement of the value of the condensate when running from ΛTC to METC . This means T¯T ETC ∼ T¯T TC ∼ Λ3TC ; hence,
2
mf ∼ Λ3TC /METC
. In addition, each fermion requires its own ETC scale. Heavy fermions

such as the tau, charm, bottom, and especially the top demands low scale METC — the
top with mt = 175 GeV implies a very small METC ∼ mt as ΛTC ∼ v ∼ mt . This is a
15

defining feature of the model: unlike the Standard Model where flavors are treated as free
parameters, ETC models deal with flavor physics dynamically.
There is a cost for a model attempting to explain both mass generation and flavor physics.
The fermion-technifermion interaction that is introduced to give mass to the Standard Model
fermions must also allow a generation-changing interaction between fermions. In ETC models, various scales of the METC are required for the theory to account for the wide range of
2
¯ ds)/M
¯
fermion masses. This is problematic as an interaction such as (ds)(
ETC which leads

to the highly constrained flavor-changing neutral current (FCNC) phenomena is also present
in ETC models [83]. The values of METC that account for the correct fermion masses could
give rise to FCNC effects larger than what have been observed in experiments. Current
bounds from FCNC observables such as those from neutral kaons (KL − KS ) mass difference
are very stringent [17] and require that METC > O(10 − 100 TeV), a range of values much
larger than what is required to generate even the mass of the strange and charm quarks!
The flavor problem of ETC models arises because the theory becomes asymptotically
free with increasing energy above TC scale quickly the way QCD does. The evolution
of couplings eventually affects how the fermion condensate changes via what is called the
anomalous dimension for the technifermion mass operator. To deal with this problem, a class
of models where the coupling decreases very slowly with increasing energy called Walking
Technicolor (WTC) have been proposed [84, 85, 86, 87, 88, 89]. The slow evolution of
the coupling is achieved by having more technifermions to enhance the “screening effect”.
This in turn enhances the value of the technifermion condensate evaluated at the METC
scale. A larger fermion mass is then possible before the flavor problem re-emerges. In short,
viable extended Technicolor models cannot be a scaled-up version of QCD and have to have
structure leading to walking. Unfortunately, the idea of walking is still not sufficient to deal
16

with the third generation [90].
The third generation is special. On the one hand, it contains the fermion that couples to
the electroweak symmetry breaking much more strongly than the others: the top quark. It
requires an (walking) ETC scale so low that it is even too close to the TC scale for comfort.
This would invalidate the picture of “low-energy” contact interaction that is used to explain
the mass of the other fermions. On the other hand, as the bottom quark’s mass is ∼ 4 GeV,
the mechanism explaining the mass of the top must also exhibits larger weak isospin violation
compared to the other generations [91]. As the violation of weak isospin is constrained by
precision electroweak experiments, the isospin violating effect must be suppressed by a high
ETC scale. But then that scale is too high to produces the top quark mass. So, should the
top quark be treated in the same manner as other quarks after all?
The intimate relationship between the top quark and electroweak symmetry breaking
suggests that perhaps the top plays a special role [92, 93, 94, 95, 96, 97]. In other words,
the top, rather than the technifermions, could form a condensate itself via a new strong
isospin-symmetric interaction in an extended strong sector. To prevent the bottom from
also forming a condensate, another isospin-violation interaction is introduced, which could
be either weak or strong. The new symmetries corresponding to both the extended strong
sector and the isospin-violation sector do not manifest at low energies, implying the existence
of new massive gauge bosons. A class of models describing the mechanism for this condensate
to form is known as “Topcolor”.
In the minimal topcolor model [98], one introduces two isospin-symmetric strong interaction groups and two “hypercharge” groups. The third-generation quarks are charged under
one SU (3) group where the rest are charged under the other SU (3) group. Similar assignments apply for the two U (1) groups with specific conditions ensuring the bottoms do not
17

form a condensate. The pattern of symmetry breaking SU (3)1 × SU (3)2 × U (1)1 × U (1)2 →
SU (3)C × U (1)Y means there will be nine new massive gauge bosons. One set is color octet,
which is referred to collectively as colorons (or topgluons in some of the literature). The
other is a color singlet to which we refer generically as a Z . Still, Topcolor is not a viable
model by itself. It was found that Topcolor implemented this way either requires a mass
for the top quark ∼ O(500 − 1000 GeV), or requires a significant amount of fine tuning
Λt /mt

1, which could also mean it provides a weak scale (hence masses of the weak gauge

bosons) that is slightly too low [99]. These issues can be addressed in two ways. One is
incorporating Extended Technicolor to supply the missing part of the weak scale. Another
is introducing an electroweak iso-singlet quark to help the top “supply” the EWSB and still
have the observed mass, so that we might not need technicolor at all.
Topcolor-assisted Technicolor (TC2 or TC2) is a model proposed to utilize the good
parts of both technicolor and topcolor [99, 100, 101, 102, 103, 104]. Electroweak symmetry is hidden mainly by the technifermion condensate while the third generation is mainly
taken care of by Topcolor in the ways introduced in the previous paragraph. TC2 is rich
in phenomenology without being too complicated at low energies. Having two sources of
dynamical electroweak symmetry breaking means there are two sets of iso-triplet NambuGoldstone Bosons. One set of the triplets becomes the longitudinal parts of the W and Z
while the orthogonal set survives in the spectrum. The members of the latter, known as the
top-pions, couple only (i.e., at leading order) to the quarks in the third generation. There
could also be a composite scalar degree of freedom orthogonal to the neutral top-pion state,
known as the top-Higgs.
The alternative top-condensate model does not need Technicolor. Top-Seesaw [105, 106,
107] is a model where the electroweak symmetry is broken by a top condensate, formed via
18

Topcolor interaction, in the presence of an electroweak isosinglet vectorlike quark (usually
denoted as χ) that mixes with the top. The mixing is arranged in such a way that the
heaviness of this extra quark results in a mass for the top that agrees with experiment.
Being an isospin singlet, the χ can generally be introduced in a manner that does not spoil
various aspects of results from precision electroweak experiments. In this case, the top-Higgs
is a bound state of the top and the even heavier χ.
In the next section of this chapter, I will discuss the task of realizing the neutral top-pion
or the top-Higgs in either of the top-condensate models as the recently-discovered 126 GeV
particle, which is a chapter in this thesis on its own. Until then, I present an overview of
how quantitative predictions from models with strong dynamics are carried out.
The models introduced so far present qualitative pictures of strong dynamics while actual calculations (and predictions!) could be anything but trivial. Knowledge of the nonperturbative dynamics required for doing such calculations is gradually being established
through lattice gauge theory calculations. Nevertheless, quantitative predictions can be
achieved in approximate models with sensible assumptions motivated by QCD and experiments. Typically, these models are constructed as a low-energy effective theory, which
captures mainly the low-energy degrees of freedom of interest at energy scales accessible to
experiment while systematically including only the leading indirect impacts of the majority
of the degrees of freedom that are too massive to actually be produced at those energy scales.
In the context of dynamical symmetry breaking, one uses an effective theory that focuses
on the scalar sector of the model. It is characterized mainly by parameters relevant to the
strong-top sector; namely, the masses of the top-pions and top-Higgs, and the fraction of the
contribution of the strong-top dynamics to the mechanism of electroweak symmetry breaking. The rest of this section discusses an approximate model for describing the generation
19

of the condensate dynamically.
The low-energy effective model describing the top quark sector in the presence of a
separate mechanism responsible for generating electroweak symmetry breaking is constructed
based on a minimal model called the Higgsless model [108, 109, 110, 111, 112, 113, 114, 115,
116]. The Higgsless model utilizes the AdS-CFT correspondence [73, 72, 117] between a
strongly-interacting four-dimensional theory (parts of which we do not know how to calculate
perturbatively) to a weakly-interacting five-dimensional gauge theory (where calculations
can be done). The presence of the extra dimension is suppressed as it decouples from lowenergy physics via compactification. While the details on extra-dimension inspired theories
are beyond the scope of this thesis, here are some typical features of this type of extra
dimension models as follow: First, electroweak symmetry is broken by boundary conditions
in the extra-dimension. Second, the onset of unitarity violation within W and Z scattering
is delayed by contributions arising from exchange of tower of Kaluza-Klein (KK) gauge
bosons. Third, electroweak precision constraints can be satisfied by feeding the fermions
their electroweak properties through multiple gauge groups located at different “sites” in
the extra dimension. Chivukula et al. [118] found that the minimal model featuring these
properties combines an extended electroweak gauge group SU (2) × SU (2) × U (1) with the
Standard Model color group. The symmetry breaking of the extended electroweak group
to U (1)EM is governed by two non-linear sigma model fields

SU (2)×SU (2)
;
SU (2)

each connects

the gauge group at the “middle site” to one at an “outer site”7 on the boundary of the
extra dimension. The dual picture of this “three-site” Higgsless model is a model describing
dynamical symmetry breaking.
Based on the three-site model, a model called the “top-triangle moose” (TTM) was intro7 So the non-linear sigma model is SU (2) × SU (2) × SU (2).

20

duced by Chivukula et al. [119] to consistently focus on the low-energy behavior of models
featuring separate sectors for top mass generation and electroweak symmetry breaking (i.e.,
mass generation of other particles). The low-energy phenomenology is still captured by a
three-site gauge group conventionally arranged as SU (2)0 × SU (2)1 × U (1)2 . The symmetry
breaking patterns SU (2)0 × SU (2)1 → SU (2) and SU (2)1 × U (1)2 → U (1) are governed by
the non-linear sigma fields presented in the three-site model. The TTM extends the scope
of the three-site model to incorporate strong top dynamics by introducing a top-Higgs field
connecting the “outer sites” SU (2)0 and U (1)2 . This top-Higgs field is an SU (2) doublet
field corresponding to the low-energy picture of the bound-states of t and b. The electroweak
symmetry is broken by both the non-linear sigma model and the top-Higgs fields. The toppions arise as a triplet orthogonal to the would-be Nambu-Goldstone bosons “eaten” by the
Standard Model W and Z, while the top-Higgs state remains in the spectrum.
The most interesting feature of the top-triangle moose model is the observation that
different points in its parameter space correspond to different strong-top dynamics models.
It serves as a low-energy effective theory for a TC2 model in the region where the contribution
from strong-top dynamics to electroweak symmetry breaking and the mass of the top-Higgs
are both relatively small. It also serves as a low-energy effective theory for top-seesaw
assisted technicolor when the top-dynamics contribution to electroweak symmetry breaking
is relatively large (due to the seesaw partner of top) and the top-Higgs are massive, O(2mt )
as seen in the previous paragraphs. This property makes the top-triangle moose model
versatile in describing the scalar sector of low-energy strong-dynamics models where only the
top-pions and the top-Higgs are present. An overview of the implications of LHC constraints
on models with strong-top dynamics obtained by studying this model will be discussed in
the next section.
21

1.3

Phenomenology, the LHC, and this Thesis

The LHC has done much more than discovering the 126 GeV-particle having properties
resemble the “standard Higgs”. Higgs boson property measurements that have been carried
out by the collaborations at the LHC [46, 47, 48, 49, 50, 51, 52, 53, 120] can be used as a guide
to directly and indirectly probe for viable new physics models. After all, the experiments
at the LHC can be considered just started, with the 8 TeV program ended in 2012. The
physics program resumes at 13 TeV in 2015. Then, the LHC will eventually go through a
long-run period of high luminosity at 14 TeV. That is the period one can safely expect a
large abundance of TeV-scale related events and sees whether they are pointing to any new
physics.
This thesis is organized into two parts which are roughly summarized as follow:
• What are the implications from the data available at the time of writing on theories
with strong dynamics?
• What can one do if a new vector resonance, one of the key predictions of these kinds
of model or others, is discovered in the near future?
In short, Part I of this thesis is dedicated to phenomenological implications from the LHC
for both technicolor models and models with strong top dynamics. Part II provides the
possibilities of characterizing color and chiral properties of a new massive vector resonances,
which will typically appear in most new physics models, if they are discovered during the
future run of the LHC at 14 TeV. The rest of this chapter provides an introduction to each
part.

22

1.3.1

Part I: Phenomenology of Strong Dynamics

Technicolor models feature technipions as a result of global chiral symmetry breaking. While
three of them are engineered to become the longitudinal degrees of freedom of the W and Z
bosons through the BEH mechanism, models constructed with more than one technifermion
doublets will result in a number of technipions’ remaining in the spectrum. Various phenomenological reasons could lead to a model’s having color-singlet, colored, or electroweak
charged technipions. The electrically neutral technipion behaves like the Higgs in many
ways. So analyses from the Higgs searches at colliders could be used to provide constraints
on corresponding strong dynamics models.
The general strategy is that the current lack of evidence for the Higgs boson (or similar)
of a particular mass in a particular decay channel will place an upper bound on the cross
section for that mass and channel [121, 122, 123]. The bound can be translated directly for
constraining the neutral technipion which shares the same kinematical properties as it is also
a spin-0 particle. A variant of model that predicts much greater rates than the Standard
Model Higgs will be greatly constrained. While analyses in this fashion have been carried
out since the time of the LEP e+ e− collider, the LHC has great advantages over LEP for
studying several Technicolor models.
The LHC, being a hadron collider, could produce a copious amount of particles that have
strong interactions; i.e., carry color charges. This means that if there exists a technipion
in a Technicolor model featuring colored technifermions, it will be produced in abundance,
particularly by a gluon fusion via a colored-technifermion loop. As noted in a previous
study on extracting the information from searches for a Higgs decaying to τ + τ − and γγ
by Alexander Belyaev, Alexander Blum, R. Sekhar Chivukula, and Elizabeth H. Simmons

23

in [124] in 2005, the signal rates for these channels are expected to be enhanced in models
featuring colored technifermions.
In 2011, about a year after the LHC began its Higgs hunting operation, R. Sekhar
Chivukula, Elizabeth H. Simmons, Jing Ren, and I used data from ATLAS and CMS available at the time to provide a follow-up to that strategy in [1]. We found that the constraints
on a large class of Technicolor predicting light technipions, in the mass range of 110−350 GeV,
are rather tight if the model has at least one of the following properties: (a) includes colored
technifermions (b) s significant contributions from topcolor dynamics and (c) has Technicolor groups with three or more Technicolors (NT C ≥ 3). Thus, the work provides a set of
guidelines for building a viable Technicolor model with colored technifermions that are not
affected significantly by these constraints. Chapter 2 of Part I is based on this work.
Now, let us turn the attention to models featuring strong top dynamics in general — ones
that might or might not involve colored technifermions. This is where the low-energy effective
theory that captures all the essential properties of strong top dynamics — the top-triangle
moose — could play a role. Recall that the scalar particles in this theory include both the
scalar top-Higgs and the pseudoscalar top-pions. Key phenomena in the scalar sector are
determined by masses of the top-Higgs and the top-pions, and the fraction of the contribution
of top-color dynamics (along with technicolor) to the electroweak symmetry breaking. That
fraction is usually denoted as the “mixing” sin ω. On the one hand, this fraction enhances the
top-Higgs and top quark coupling; i.e., by 1/ sin ω (hence the production from gluon-fusion
and the decays to gluon pairs by 1/ sin2 ω). On the other hand, it suppresses the production
of top-Higgs and the Standard Model vector-boson pairs, W W and ZZ; i.e., by sin ω (hence
the production from vector boson fusion and the decays to vector boson pairs by sin2 ω).
The neutral top-pion, being a pseudoscalar, does not even have tree-level couplings to the
24

W W and the ZZ states. The couplings are possible only at the quantum level; e.g., via a
top quark loop.
The properties and the LHC search strategies of the scalar sector of various models with
strong top dynamics via the top-triangle moose model was discussed by R. Sekhar Chivukula,
Elizabeth H. Simmons, Baradhwaj Coleppa, Heather E. Logan, and Adam Martin in [125].
The authors also presented the implications of data from the Higgs searches, in the W W
and ZZ decay channels, for the top-Higgs in [126]. It was found that the searches in these
final states could pose a stringent bound on the top-Higgs, particularly when the top-pion
is heavier than the top quark or when the strong top dynamics contributes substantially
to EWSB. Specifically, a 150 GeV-top-pion would imply the exclusion of a top-Higgs with
mass below 300 GeV. The top-pions themselves cannot be too light, particularly if they
are degenerate, or the charged top-pions would have been the final states of the top quarks
decays — and those processes are also well-constrained [127, 128]. In short, the date from
2011 already showed that it would be very challenging to explain a light Higgs-like particle
with the top-Higgs state. The remaining candidate was then the neutral top-pion.
Near the mid-2012, there was an indication of a “Higgs-like” signature with mass 126 GeV
observed at the LHC by the ATLAS and CMS collaborations. At the time, the evidence for
the new state was larger than what to be expected from the Standard Model Higgs in the
di-photon and ZZ ∗ (to four charged leptons) channels, and moderate for the W W ∗ channel.
The data for the di-tau and bottom-pair final states were inconclusive. My colleagues —
Sekhar Chivukula, Baradhwaj Coleppa, Heather Logan, Adam Martin, Jing Ren, Elizabeth
Simmons — and I used the top-triangle moose model to find the range of model parameters
such that the Higgs-like state could be explained by the neutral top-pion [3]. We found that
the top-pion, lacking tree-level couplings to the ZZ and W W , could not explain the excess
25

in the data for those corresponding final states. In addition, while it is possible to explain
the 126 GeV data in the di-photon final state with the neutral top-pion, this requires large
mass splitting from the charged top-pions which are constrained by the top quark decays.
This splitting requires larger isospin violation than exists in TC2 models. Further details on
this study and its results can be found in Chapter 3, which is based on this work.
Now we have seen the implications from the recent experiments at the LHC for the two
classes of model built on strong dynamics: technicolor and strong top dynamics. The LHC
still has a long-term high-luminosity plan to follow at 14 TeV. That will open windows to
both new physics at high-energy scales or physics leaving traces that can be uncovered by
having large number of events. So we will now focus on a class of particles that features in
most new physics models, not just those with strong dynamics: new massive vector bosons.
We would like to know what one can do shortly after the particle has been discovered. This
is the main focus of the Part II of the thesis.

1.3.2

Part II: Coloron Phenomenology

New massive vector bosons are often predicted in theories of physics beyond the standard
model, particularly if they contain an extra local symmetry group that is hidden spontaneously. Those bosons having couplings with the Standard Model’s fermions will have a
chance to be produced directly at a collider that has a sufficiently high beam energy. The
new vector bosons can be either colored or color-singlet. The colored particles that transform as color octets naturally arise from models with extended color sectors including strong
dynamics, as well as models built on extra dimensions. Color-octet bosons are the main
focus in this thesis. Color-singlet vector bosons are a very common prediction of models
having an extended electroweak gauge group (see, for example, [129] for a review of models).
26

We generically refer to the color-octet and the color-singlet vector bosons as a coloron and
a Z , respectively.
The possibility of having the ‘’coloron and the Z coupled to hadrons provides a twofold advantage for searches at a hadron collider such as the LHC [121, 130]. First, those
resonances within the reach of the LHC can be produced copiously. Second, both particles
decay to so-called “di-jet” final states of simple topology consisting of two groups of hadrons
called “jets”, while the Z can also decay to the two-lepton final state. The searches for new
vector resonances have been ongoing from the 1980’s (see, for example, [131] for a review) up
to the current LHC era. While none of the new resonances have been discovered, the LHC
still has years left in its program and the potential of future colliders has been under serious
discussions [132, 133]. There is room for a new vector resonance to be discovered within the
foreseeable future.
The question one might ask is then: suppose a vector resonance is discovered, how does
one quickly characterize its properties? Some of the properties that could allow identification
of the origin of the vector boson include the color structure and the chiral structure of its
couplings to the Standard Model fermions. While there have been several proposals of
ways to characterize the resonance (see, for example, [130, 134, 135, 136, 137, 138, 139]),
several require detailed knowledge from observables that could take time to measure. It is
then helpful to have some means to characterize a particle using the information after its
discovery such as the cross section from the discovery channel, its mass, and its decay width
together with as few complementary measurements of other observables as possible.
The knowledge of the color structure of the new gauge boson could allow identifications
or exclusion of a number of new physics models. Both the colorons and Z that could be
produced copiously at the LHC are likely to have the dijet final state as the discovery channel.
27

The Z can even have the dilepton final states. A “bump” in both di-jet and di-lepton final
states having the same mass and the same total decay width would immediately imply that
the resonance could not be a color-octet. However, a bump that is present in the dijet
final state but not in the dilepton final state could come from either a color-octet vector
resonance or a color-singlet one that does not have leptonic couplings. The latter is called
a “leptophobic Z ”. How can we then distinguish between the coloron and the leptophobic
Z?
In 2012, my colleagues — Anupama Atre, Sekhar Chivukula, Elizabeth Simmons — and
I addressed this question of how one can still quickly tell a coloron apart from a leptophobic
Z [4, 5]. The goal is to construct a variable from observables that allows one to do so as
model-independent a manner as possible; i.e., without having to consider different points in
a parameter space of the model’s couplings. We found that their different color structures
lead the coloron and Z’ to lie in different ranges of value of a variable constructed from mass,
width and dijet cross section corresponding to each resonance. The variable is referred to
as a “color discriminant”. We illustrated the use of this method in the scenarios where the
resonance couples universally to all quarks, or have different couplings to quarks in the third
family. The work is presented in Chapter 4.
The next step is to implement the method based on the color discriminant variable
within a realistic scenario where the new vector boson couplings are not necessarily flavoruniversal. More realistic models usually predict a Z carrying SU (2)L charges which makes
it couple differently to up- and down-type quarks. The right-handed couplings are typically
not constrained from the theory point of view. At first sight, this would have suggested that
a large number of complementary measurements are needed to identify the color structure
of the resonance. However, in 2013, Sekhar Chivukula, Elizabeth Simmons and I found that
28

this is not the case [7] for the following reasons. First, the differences in couplings to quarks
in the first two generations, which are inaccessible in the dijet final state, were found to
be practically irrelevant to our method in most reasonable cases. Second, the difference in
chiral couplings would play no role as long as the variables considered are not sensitive to
it (fermion-antifermion cross sections, mass, and width). Supplementary measurements on
heavy-flavor decays will determine the top and bottom couplings. So we extended the scope
of the method to a flavor non-universal scenario, where in most cases the measurements
for the di-top and di-bottom final states could supplement to the information on the color
structure. Chapter 5 is based on this work.
Another important aspect of the new vector boson’s properties that could eventually lead
to uncovering its origin is the chiral structure of its couplings to the Standard Model fermions.
Measuring the cross-section for the boson’s production followed by decay into di-fermion
channels, either dijet or di-leptons, allow one to probe the magnitude of the gauge couplings
but not the separate couplings to left- and right-handed fermions. However, the associated
production of the new resonance with the Standard Model W bosons (sensitive only to lefthanded couplings) and Z boson (sensitive asymmetrically to left- and right-handed couplings)
can provide the additional information. In 1992, Cveti˘c and Langacker [140] discussed the
associated productions for the new electroweak gauge bosons, the W and Z , with the
ordinary bosons: the W, Z and γ. The dijet cross section is to be supplemented by cross
sections for the rare process pp → V → f¯f V where V = (W , Z ) and f, f = SM fermions,
and the observable proportional to the difference between the numbers of charged leptons
that go into the forward and backward directions. The accuracy of this method relies on the
assumption that the new gauge bosons have leptonic decay channels, which are considered
“clean”. The situation for probing the chiral structure of the gauge boson that do not couple
29

to leptons will be substantially different, both in terms of the backgrounds (now the “dirty”
QCD backgrounds) and the decay final states (such as the dijets).
Back in 2012, inspired by a question raised by Devin G. Walker8 , my colleagues —
Anupama Atre, Sekhar Chivukula, Elizabeth Simmons, Jiang-Hao Yu — and I proposed a
new channel, the associated production of W , Z gauge bosons and color octet resonances,
to help determine the chiral structure of the couplings [4, 5]. This is a counting experiment
where a combination between cross sections of the two modes of associated productions along
with a dijet cross section allows one to determine the chiral structure of the vector boson
with flavor-universal couplings to quarks. We found that the early run of the 14 TeV-LHC
(i.e., 10 fb−1 to 100 fb−1 ) can probe a large region of the parameter space down to very small
couplings. This work is presented in Chapter 6 of this thesis.
A summary for each set of results is presented at the end of each chapter. In Chapter 7,
I conclude the thesis, giving an overview summary and a discussion of the aspects one can
explore in the future.

8 Private communication.

30

Part I
Phenomenology of Strong Dynamics

31

Chapter 2
Technipion Limits from LHC Higgs
Searches
— This chapter is based on a work in collaboration with R. Sekhar
Chivukula, Elizabeth H. Simmons, and Jing Ren which has appeared in [1]. The contents also have appeared in parts in the
conference proceeding [2].

2.1

Introduction

Before the Higgs-like signal was observed on July 4th 2012, experiments at the Large Hadron
Collider were striving to discover the agent of electroweak symmetry breaking. The negative
findings were generally phrased in terms of placing constraints on the properties of the
fundamental scalar Higgs boson state (hSM ) predicted to exist in the standard model. In
2011, both the ATLAS and CMS collaborations at the CERN LHC reported the results from
searches for the standard model Higgs in the two-photon [141, 142] and τ + τ − [143, 144, 145]
decay channels. They placed upper bounds on the cross-section times branching ratio (σ · B)
for each channel over the approximate mass range 110 GeV ≤ mh ≤ 145 GeV, finding that
σ · B cannot exceed the standard model prediction by more than a factor of a few. In
addition, ATLAS independently constrained the production of a heavy neutral scalar SM

32

Higgs boson with mass up to 600 GeV and decaying to τ + τ − .
This chapter presents the work [1] where we applied these limits to the neutral “technipion” (ΠT ) states predicted to exist in technicolor models that include colored technifermions.
Because both the technipion production rates and their branching fractions to γγ or τ τ can
greatly exceed the values for a standard model Higgs, the LHC results place strong constraints on technicolor models. This strategy was first suggested as a possible for hadron
supercolliders over fifteen years ago in Refs. [121, 122, 123].
As we have introduced in Sec. 1.2, which will be reviewed below (see also [99]), many
technicolor models, including those with walking and topcolor dynamics, feature technipion
states, pseudo-scalar bosons that are remnants of electroweak symmetry breaking in models
with more than one weak doublet of technifermions. Production of light technipion states
at lepton colliders has been studied by a variety of authors [146, 147, 148, 149, 150, 151];
the most comprehensive analysis [151] used LEP I and LEP II data to constrain the anomalous couplings of technipions to neutral electroweak gauge bosons and derived limits on the
size of the technicolor gauge group and the number of technifermion doublets in various
representative technicolor models. Subsequently, the authors of [124] considered technipion phenomenology at hadron colliders; they demonstrated both that technipions can be
produced at a greater rate than the standard model Higgs, because the technipion decay
constant is smaller than the electroweak scale, and also that the technipions can also have
higher branching fractions to γγ or τ τ final states. As a result, the technipions are predicted
to produce larger signals in these two channels at LHC than the hSM would [124].
In [1], we showed that the ATLAS [141, 143, 144] and CMS [142, 145] searches for the
standard model Higgs exclude, at 95% CL, technipions of masses from 110 GeV to nearly 2mt
in technicolor models that (a) include colored technifermions (b) feature topcolor dynamics
33

and (c) have technicolor groups with three or more technicolors (NT C ≥ 3). For certain
models of this kind, the limits also apply out to higher technipion masses or down to the
minimum number of technicolors (NT C = 2). We also showed how the limits may be modified
in models in which extended technicolor plays a significant role in producing the mass of the
top quark; in some cases, this makes little difference, while in other cases the limit is softened
somewhat. Overall, we found that the ATLAS and CMS significantly constrain technicolor
models. Moreover, as the LHC collaborations collect additional data on these di-tau and
di-photon final states and extend the di-photon analyses to higher mass ranges, they should
be able to quickly expand their reach in technicolor parameter space.
The rest of this chapter is organized as follows. In Sec. 2.2, we provide a brief review of
Technicolor models and properties of a neutral technipion relevant to the Higgs production
and decays to the final states of interest. Then, in Sec. 2.3, we show how the LHC data
constraints technipions composed of colored technifermions and discuss the constraints on
theories where mass of the top quark is mainly generated by topcolor dynamics. We then
discuss the implications of the constraints in theories where the top quark’s mass receives a
substantial contribution from ETC in Sec. 2.4. Finally, the conclusions and discussions are
presented in Sec. 2.5.

2.2

Technicolor and Technipions

Dynamical theories of electroweak symmetry breaking embody the possibility that the scalar
states involved in electroweak symmetry breaking could be manifestly composite at scales not
much above the weak scale v ≈ 246 GeV. In technicolor theories [78, 79, 80], a new asymptotically free strong gauge interaction breaks the chiral symmetries of massless fermions T

34

at a scale Λ ∼ 1 TeV. If the fermions carry appropriate electroweak quantum numbers (e.g.
left-hand (LH) weak doublets and right-hand (RH) weak singlets), the resulting condensate
T¯L TR = 0 breaks the electroweak symmetry correctly to its electromagnetic subgroup.
Three of the Nambu-Goldstone Bosons of the chiral symmetry breaking become the longitudinal modes of the W and Z, making those gauge bosons massive. The hierarchy and
triviality problems plaguing the standard model are absent: the logarithmic running of the
strong gauge coupling renders the low value of the electroweak scale natural, while the absence of fundamental scalars obviates concerns about triviality.
In so-called minimal technicolor models, there are no composite scalars left in the spectrum. However, many dynamical symmetry-breaking models include more than the minimal
two flavors of technifermions needed to break the electroweak symmetry. In that case, there
will exist light pseudo Nambu-Goldstone bosons known as technipions, which could potentially be accessible to a standard Higgs search. Technipions that are bound states of colored
technifermions can be produced through quark or gluon scattering at a hadron collider, like
the LHC, through the diagrams in Fig. 2.1. In the models with topcolor dynamics, where
ETC interactions (represented by the shaded circle) contribute no more than a few GeV to
the mass of any quark, there is only a small ETC-mediated coupling between the technipion
and ordinary quarks in diagrams 2.1(b) and 2.1(c). Combining that information with the
large size of the gluon PDF at the LHC and the NT C enhancement factor in the techniquark
loop at left, we expect that the diagram in Fig. 2.1(a) will dominate technipion production
in these theories, which we study here and in Section 2.3. Technipions in models without
strong top dynamics could, in contrast, have a large top-technipion coupling, making diagram 1(c) potentially important; we will consider that scenario in Section 2.4. Technipions
that are bound states of non-colored technifermions would be produced at hadron colliders
35

only through diagrams 2.1(b) and 2.1(c), which would generally yield a significantly lower
production rate; we comment on these models in the discussion (Section 2.5).
g

P

Q
g

g

Q

Q

(a) gluon-gluon fusion through
techniquark loop

t

b

P

t
P
g

¯b

(b) b¯b annihilation

t

(c) gluon-gluon fusion through
top quark loop

Figure 2.1: Feynman diagrams for single technipion production at LHC. The shaded circle
in diagrams (b) and (c) represents an ETC coupling between the ordinary quarks and techniquarks. For interpretation of the references to color in this and all other figures, the reader
is referred to the electronic version of this thesis.
No single technicolor model has been singled out as a benchmark; rather, different classes
of models have been proposed to address the challenges of dynamically generating mass
while complying with precision electroweak and flavor constraints. We will study the general
constraints that the current LHC data can place a variety of theories with colored technifermions and light technipions. Following [124, 151], the specific models we examine are:
1) the original one-family model of Farhi and Susskind [152] with a full family of techniquarks and technileptons, 2) a variant on the one-family model [150] in which the lightest
technipion contains only down-type technifermions and is significantly lighter than the other
pseudo Nambu-Goldstone bosons, 3) a multiscale walking technicolor model [153] designed
to reduce flavor-changing neutral currents, 4) a low-scale technicolor model (the Technicolor Straw Man – TCSM – model) [154] with many weak doublets of technifermions and
5) a one-family models with weak-isotriplet technifermions [146]. Properties of the lightest
electrically-neutral technipion in each model that couples to gluons (and can therefore be
readily produced at LHC) are shown in Table 2.1. For completeness, we show the name
and technifermion content of each state in the notation of the original paper proposing its
36

existence; while each paper has its own conventions, all technifermion names including “Q”
or “D” refer to color-triplets (a.k.a. techniquarks) while those including “L” or “E” refer
to color-singlets (a.k.a. technileptons).1 In the TCSM low-scale model, the second-lightest
technipion is the state relevant for our study (the lightest, being composed of technileptons,
lacks an anomalous coupling to gluons); in the other models the lightest technipion is the
relevant one. For simplicity the lightest relevant neutral technipion of each model will be
generically denoted P . Furthermore, we will assume that the lightest technipion state is
significantly lighter than other neutral (pseudo)scalar technipions in the spectrum, in order
to facilitate the comparison to the standard model Higgs boson.2
Single production of a technipion can occur through the axial-vector anomaly which
couples the technipion to pairs of gauge bosons. For an SU (NT C ) technicolor group with
technipion decay constant FP , the anomalous coupling between the technipion and a pair of
gauge bosons is given, in direct analogy with the coupling of a QCD pion to photons,3 by
[155, 156, 157]
NT C AV1 V2

g1 g2
µ ν λ σ
µνλσ k1 k2 1 2
2
8π FP

(2.1)

where
AV1 V2 ≡ T r [T a (T1 T2 + T2 T1 )L + T a (T1 T2 + T2 T1 )R ]

(2.2)

is the anomaly factor, T a is the generator of the axial vector current associated with the
techipion, subscripts L and R denote the left- and right-handed technifermion components of
1 Note that the LR multiscale model [153] incorporates six technileptons, which we denote L .
2 The detailed spectrum of any technicolor model depends on multiple factors, particularly the parameters
describing the “extended technicolor” [81, 82] interaction that transmits electroweak symmetry breaking to
the ordinary quarks and leptons. Models in which several light neutral PNGBs are nearly degenerate could
produce even larger signals than those discussed here.
3 Note that the normalization used here is identical to that in [124] and differs from that used in [151] by
a factor of 4.

37

the technipion, the Ti and gi are the generators and couplings associated with gauge bosons
Vi , and the ki and i are the four-momenta and polarizations of the gauge bosons. The value
of the anomaly factor Agg for the lightest PNGB of each model that is capable of coupling
to gluons appears in Table 2.1, along with the anomaly factor Aγγ coupling the PNGB to
photons. Also shown in the table is the value of the technipion decay constant, FP for each
model.4
Examining the technipion wave functions in Table 2.1 we note that the PNGB’s do not
decay to W boson pairs, since the W + W − analog of Fig. 2.1 vanishes due to a cancellation
between techniquarks and technileptons. The corresponding ZZ diagrams will not vanish
but, again due to a cancellation between techniquarks and technileptons, will instead yield
small couplings for the technipion to ZZ (and Zγ) proportional to the technifermion hypercharge couplings [151]. The small coupling and phase space suppression yield much smaller
branching ratios for the PNGB’s to decay to ZZ or Zγ, and hence these modes are irrelevant
to our limits.
The rate of single technipion production via glue-glue fusion and a techniquark loop
(Fig.2.1(a)) is proportional to the technipion’s decay width to gluons through that same
techniquark loop
Γ(P → gg) =

m3P
8π

αs NT C Agg 2
.
2πFP

(2.3)

In the SM, the equivalent expression (for Higgs decay through a top quark loop) looks like
[158]
m3h
Γ(hSM → gg) =
8π

2
αs 2 3τ
(1 + (1 − τ )f (τ )) ,
3πv
2

(2.4)

4 In the multi-scale model [model 3], various technicondensates form at different scales; we set F (3) =
P
v in keeping with [153] and to ensure that the technipion mass will be in the range to which the standard
4
Higgs searches are sensitive.

38

Table 2.1: Properties of the lightest relevant PNGB (technipion) in representative technicolor
models with colored technifermions. In each case, we show the name and technifermion
content of the state (in the notation of the original paper), the ratio of the weak scale to the
technipion decay constant, the anomaly factors for the two-gluon and two-photon couplings
of the technipion, and the technipion’s couplings to leptons and quarks. The symbols“Q”
or “D” refer to color-triplets (a.k.a. techniquarks) while those including “L” or “E” refer to
color-singlets (a.k.a. technileptons). The multiscale model incorporates six technileptons,
which we denote by L . For the TCSM low-scale model, ND refers to the number of weakdoublet technifermions contributing to electroweak symmetry breaking; this varies with the
size of the technicolor group. The parameter y in the isotriplet model is the hypercharge
assigned to the technifermions.

TC models
FS one family[152]

PNGB and content
1 (3Lγ
¯ 5 L − Qγ
¯ 5 Q)
√
P1

Variant one family[150]

P0

LR multiscale[153]

P0

TCSM low scale[154]

πT0

MR Isotriplet [146]

P1

4 3
1 (3Eγ
¯ 5 E − Dγ
¯ 5 D)
√
2 6
1 (L
¯ γ5 L − 2Qγ
¯ 5 Q)
√
6 2
1 (3Lγ
¯ 5 L − Qγ
¯ 5 Q)
√
4 3
1 (3Lγ
¯ 5 L − Qγ
¯ 5 Q)
√
6 2

v/FP
2
1
4
√
ND
4

Agg
− √1

3
1
−√
√6
2 2
− 3
− √1
3
− √1
2

Aγγ
4
√

3 3
16
√
3 √6
8 2
9
100
√
27
√3

24 2y 2

λl
1
√
6

λf
1

1

1

1

1

1

1

2
3

where τ ≡ (4m2t /m2h ) and

f (τ ) =



1 2


 sin−1 (τ − 2 )

√



− 1 log 1+√1−τ
4
1− 1−τ

if τ ≥ 1
− iπ

2

(2.5)

if τ < 1.

so that the expression in square brackets in Eq. (2.4) approaches 1 in the limit where the
top quark is heavy (τ >> 1). Therefore, the rate at which P is produced from gg fusion
exceeds that for a standard Higgs of the same mass by a factor
v2
Γ(P → gg)
9
κgg prod =
= NT2 C A2gg 2
Γ(hSM → gg)
4
FP

39

−2
3τ
(1 + (1 − τ )f (τ ))
2

(2.6)

where, again, the factor in square brackets is 1 for scalars much lighter than 2mt . A large
technicolor group and a small technipion decay constant can produce a significant enhancement factor.
Technipions can also be produced at hadron colliders via b¯b annihilation (as in Fig. 2.1(b)),
because the ETC interactions coupling quarks to techniquarks afford the technipion a decay mode into fermion/anti-fermion pairs. The rate is proportional to the technipion decay
width into fermions:

Γ(P → f f ) =

NC λ2f m2f mP
8π FP2

1−

s
2

4m2f

(2.7)

m2P

where NC is 3 for quarks and 1 for leptons. The phase space exponent, s, is 3 for scalars
and 1 for pseudoscalars; the lightest PNGB in our technicolor models is a pseudoscalar. For
the technipion masses considered here, the value of the phase space factor in (2.7) is so close
to one that the value of s makes no practical difference. The factors λf are non-standard
Yukawa couplings distinguishing leptons from quarks. The variant one-family model has
√
2 and λ
6; the multiscale model also includes a similar factor, but
λquark =
lepton =
3
with average value 1; λf = 1 in the other models. For comparison, the decay width of the
SM Higgs into b-quarks is:
3 m2b mh
Γ(hSM → bb) =
8π v 2

1−

4m2b
m2h

3
2

(2.8)

Thus, the rate at which P is produced from b¯b annihilation exceeds that for a standard Higgs

40

Table 2.2: Branching ratios for phenomenologically important modes (in percent) for technipions of mass 130 GeV for NT C = 2, 4 and for a standard model Higgs [159] of the same
mass.

One
Variant
Decay
Family
one family
Channel NT C NT C NT C NT C
=2
=4
=2
=4
b¯b
77
56
61
50
c¯
c
7
5.1
0
0
+
−
τ τ
4.5
3.3
32
26
gg
12
35
7
23
γγ
0.011 0.033 0.11 0.35
+
−
W W
0
0
0
0

Multiscale
NT C
=2
64
5.8
3.8
26
0.025
0

NT C
=4
36
3.2
2.1
59
0.056
0

TCSM
low-scale
NT C NT C
=2
=4
77
56
7
5.1
4.5
3.3
12
35
0.088 0.26
0
0

Isotriplet
NT C
=2
60
5.4
3.5
14
17
0

NT C
=4
31
2.8
1.8
29
36
0

SM
Higgs
49
2.3
5.5
7.9
0.23
31

of the same mass by
λ2 v 2
Γ(P → bb)
κbb prod =
= b2
FP
Γ(hSM → bb)

1−

4m2b

s−3
2

m2h

(2.9)

The enhancement is smaller than that in Eq. (2.6) because there is no loop-derived factor
of NT C .
For completeness, we note that the branching fraction for a technipion into a photon pair
via a techniquark loop is:
m3P
Γ(P → γγ) =
64π

αs NT C Aγγ 2
.
2πFP

(2.10)

as compared with the result for the standard model Higgs boson (through a top quark loop)
[158]
Γ(hSM → γγ) =

m3h
9π

2
α 2 3τ
(1 + (1 − τ )f (τ )) ,
3πv
2

41

(2.11)

Table 2.3: Branching ratios for phenomenologically important modes (in percent) for technipions of mass 350 GeV for NT C = 2, 4 and for a standard model Higgs [159] of the same
mass.

One
Variant
Decay
Family
one family
Channel NT C NT C NT C NT C
=2
=4
=2
=4
b¯b
44
18
42
20
c¯
c
4
1.6
0
0
+
−
τ τ
2.6
1
22
11
gg
49
79
35
68
γγ
0.047 0.076 0.54
1
+
−
W W
0
0
0
0

Multiscale
NT C
=2
24
2.2
1.4
72
0.069
0

TCSM
low-scale
NT C NT C NT C
=4
=2
=4
7.7
44
18
0.69
4
1.6
0.45
2.6
1
91
49
79
0.087 0.36 0.58
0
0
0

Isotriplet
NT C
=2
20
1.8
1.2
34
42
0

NT C
=4
6.2
0.56
0.36
41
51
0

SM
Higgs
0.036
0.0017
0.0048
0.085
∼0
68

From these decay widths, we can now calculate the technipion branching ratios to all of
the significant two-body final states, taking NT C = 2 and NT C = 4 by way of example. In
the TCSM low-scale model we set ND = 5 (10) for NT C = 2 (4) to make the technicolor
coupling walk; in the Isotriplet model, we set the technifermion hypercharge to the value
y = 1. We find that the branching ratio values are nearly independent of the size of MP
within the range 110 GeV - 145 GeV and also show little variation once MP > 2mt ; to give
a sense of the patterns, the branching fractions for MP = 130 GeV are shown in Table II
and those for MP = 350 GeV are shown in Table III. The branching ratios for the SM Higgs
at NLO are given for comparison; these were obtained from the Handbook of LHC Higgs
Cross Sections [159]. The primary differences are the absence of a W W decay for technipions
and the enhancement of the two-gluon coupling (implying increased gg → P production);
the di-photon and di-tau decay widths can also vary moderately from the standard model
values.

42

Pulling this information together, and noting that the PNGBs are narrow resonances, we
may define an enhancement factor for the full production-and-decay process yy → P → xx
as the ratio of the products of the width of the (exclusive) production mechanism and the
branching ratio for the decay:

κP
yy/xx =

Γ(P → yy) × BR(P → xx)
≡ κyy prod κxx decay .
Γ(hSM → yy) × BR(hSM → xx)

(2.12)

And to include both the gluon fusion and b-quark annihilation production channels when
looking for a technipion in the specific decay channel P → xx, we define a combined enhancement factor

κP
total/xx =

σ(gg → P → xx) + σ(bb → P → xx)
σ(gg → hSM → xx) + σ(bb → hSM → xx)

κP
+ σ(bb → P → xx)/σ(gg → hSM → xx)
gg/xx
=
1 + σ(bb → hSM → xx)/σ(gg → hSM → xx)
=

κP
+ κP
σ(bb → hSM → xx)/σ(gg → hSM → xx)
gg/xx
bb/xx
1 + σ(bb → hSM → xx)/σ(gg → hSM → xx)

P
≡ [κP
gg/xx + κbb/xx Rbb:gg ]/[1 + Rbb:gg ].

(2.13)

Here Rbb:gg is the ratio of b¯b and gg initiated Higgs boson production in the Standard Model,
which can be calculated using the HDECAY program [160]. In practice, as noted in [124],
the contribution from b-quark annihilation is far smaller than that from gluon fusion for
colored technifermions.

43

2.3

Models with colored technifermions and a topcolor
mechanism

We will now show how the LHC data constrains technipions composed of colored technifermions in theories where the top-quark’s mass is generated by new strong “topcolor”
dynamics [100] preferentially coupled to third-generation quarks. In such models, the ETC
coupling between ordinary quarks and technifermions (or technipions) is very small, so that
gluon fusion through a top-quark loop will be negligible by comparison with gluon fusion
through a technifermion loop, as a source of technipion production.

2.3.1

LHC Limits on Models with Light Technipions

Here we report our results for technipions in the 110 - 145 GeV mass range where direct
comparison with Higgs production is possible. We consider final states with pairs of photons
or tau leptons, since the LHC experiments have reported limits on the standard model Higgs
boson in both channels.
First, we show the limits derived from the CMS and ATLAS searches for a standard
model Higgs boson decaying to γγ in Fig. 2.3.1. The multiscale [153], TCSM low-scale [154],
and isotriplet [146] models predict rates of technipion production and decay to diphotons
that exceed the experimental limits in this mass range even for the smallest possible size of
the technicolor gauge group (larger NT C produces a higher rate). Note that we took the
value of the technifermion hypercharge parameter y in the isotriplet model to have the value
y = 1 for purposes of illustration; choosing y ∼ 1/7 could make this model consistent with
the di-photon data for NT C = 2, but that would not affect the limits from the di-tau channel
discussed below. For the original [152] and variant [150] one-family models, the data still
44

allow NT C = 2 over the whole mass range, and NT C = 3 is possible for 115 GeV < MP <
120 GeV; even 135 < MP < 145 GeV is marginally consistent with the data for NT C = 3 in
the original one-family model.

101

102

γγ channel

One Family (Farhi-Susskind)
-1
-1
CMS (1.66 fb )+ATLAS (1.08 fb )
NTC=4
NTC=3
NTC=2

(σ x BR)P / (σ x BR)SM

(σ x BR)P / (σ x BR)SM

102

100

10-1
110

115

120

125
130
MP [GeV]

135

140

101

100

10-1
110

145

(a) Original one-family model [152].
105

(σ x BR)P / (σ x BR)SM

104

103

γγ channel

Variant One Family (Casalbuoni et al)
-1
-1
CMS (1.66 fb )+ATLAS (1.08 fb )
NTC=4
NTC=3
NTC=2

115

120

125
130
MP [GeV]

135

140

145

(b) Variant one-family model [150].
γγ channel

Multiscale (Lane-Ramana)
-1
-1
CMS (1.66 fb )+ATLAS (1.08 fb )
NTC=6
NTC=4
NTC=3
NTC=2

102

1

10

100
110

115

120

125
130
MP [GeV]

135

140

145

(c) Multiscale walking technicolor model [153].

Figure 2.2: Comparison of experimental limits and technicolor model predictions for production of a new scalar decaying to photon pairs. In each pane, the shaded region (above the
solid line) is excluded by the combined 95% CL upper limits on σh Bγγ normalized to the
SM expectation as observed by CMS [142] and ATLAS [141]. Each pane also displays (as
open symbols) the theoretical prediction from one of our representative technicolor models
with colored technifermions, as a function of technipion mass and for several values of NT C .
Values of mass and NT C for a given model that are not excluded by the data are shown as
solid (green) symbols. Continued next page.

The limits from the the CMS and ATLAS searches for a standard model Higgs boson
decaying to τ + τ − in the same mass range need extra considerations. The results from the

45

Figure 2.2 (cont’d)

103

106

γγ channel

TCSM Low-Scale (Lane)
-1
-1
CMS (1.66 fb )+ATLAS (1.08 fb )
NTC=4, ND=10
NTC=3, ND=8
NTC=2, ND=5

γγ channel

Isotriplet (Manohar-Randall)

5

10
(σ x BR)P / (σ x BR)SM

(σ x BR)P / (σ x BR)SM

104

102

101

104
103
102

CMS (1.66 fb-1)+ATLAS (1.08 fb-1)
NTC=4
NTC=3
NTC=2

101
100
110

115

120

125
130
MP [GeV]

135

140

100
110

145

(d) TCSM Low-scale technciolor model (the Technicolor Straw Man model) [154].

115

120

125
130
MP [GeV]

135

140

145

(e) Isotriplet model [146]. The magnitude of the
technifermion hypercharge variable y has been set to
1 for illustration.

analyses [161, 144, 162], arose in large part from the production of a standard model Higgs
boson through vector-boson fusion or in association with a W bosonproduction mechanisms
that are absent for technipions. Thus, the updated results on searches for the Higgs in the
τ τ decay mode from the ATLAS experiment [163] are used instead.
The search for the standard model Higgs boson in the τ τ decay mode reported in [163]
is based on the combination of searches for three different decay channels of the final state
τ particles: τhad τhad νν, lτhad 3ν, and ll4ν. The results of the individual analysis in each
channel I is expressed as a 95% C.L. upper limit on

µI (mH ) =

σ(pp → H)BR(H → τ τ → I)
,
σSM (pp → H)BRSM (H → τ τ → I)

(2.14)

as a function of the Higgs boson mass mH . The individual channels are then combined

46

102

One Family (Farhi-Susskind)
ATLAS (4.7 fb-1) / Scaled for Pseudoscalar
NTC=4
NTC=3
NTC=2

103

ττ channel

(σ x BR)P / (σ x BR)SM

(σ x BR)P / (σ x BR)SM

103

101

100
110

ATLAS 2012
115

120

125
130
MP [GeV]

135

140

102

(σ x BR)P / (σ x BR)SM

102

ATLAS 2012

100
110

145

ττ channel

101

(a) Original one-family model [152].
103

Variant One Family (Casalbuoni et al)
ATLAS (4.7 fb-1) / Scaled for Pseudoscalar
NTC=4
NTC=3
NTC=2

115

120

125
130
MP [GeV]

135

140

145

(b) Variant one-family model [150].

TCSM Low-Scale (Lane)
ATLAS (4.7 fb-1) / Scaled for Pseudoscalar
NTC=4, ND=10
NTC=3, ND=8
NTC=2, ND=5

ττ channel

101

100
110

ATLAS 2012
115

120

125
130
MP [GeV]

135

140

145

(c) TCSM Low-scale technciolor model (the Technicolor Straw Man model) [154].

Figure 2.3: Comparison of experimental limits and technicolor model predictions for production of a new scalar decaying to tau lepton pairs. In each pane, the shaded region (above the
solid line) is excluded by the combined 95% CL upper limits on σh Bτ + τ − normalized to the
SM expectation as observed by CMS [145] and ATLAS [143]. Each pane also displays (as
open symbols) the theoretical prediction from one of our representative technicolor models
with colored technifermions, as a function of technipion mass and for several values of NT C .
Values of MP and NT C for a given model that are not excluded by the data are shown as
solid (green) symbols.

47

using5



µ(mH ) = 

N
l

−(1/2)

1

µ2I (mH )

.

(2.15)

to find an overall bound on the standard model Higgs boson in the τ τ decay channel.
In the Higgs search, each τ τ decay channel receives contributions from three different
production modes for the Higgs boson: gluon fusion (gg), vector-boson fusion (VBF), and
associated production (VH). In order to correctly scale these results to find limits on technipions, we need the limits on µI that would arise solely from gluon fusion. This can be done by
scaling µI (mH ) by using the fraction of expected signal events (including both production
cross section and detection efficiency) arising from gluon fusion, through

µ
˜I (mH ) =

nIgg (mH )
nIgg (mH ) + nIVBF (mH ) + nIVH (mH )

−1

µI ,

(2.16)

where nIP (mH ) is the number of signal events expected for a standard model Higgs boson
produced via mechanism P and detected in τ τ decay channel I. In [163] the expected number
of signal events for each production mode and decay channel, including both cross section
and detection efficiency, is listed for a 120 GeV standard model Higgs boson. Assuming that
the detection efficiency is roughly constant over the mass range of interest (100 GeV < mH <
145 GeV), we estimate the number of expected signal events by

nIP (mH ) =

σP (mH )
nI (120 GeV) ≡ RP (mH ) · nIP (120 GeV) .
σP (120 GeV) P

(2.17)

We take the ratio of production cross sections, RP (mH ), from [7].
5 This equation is only approximately correct since the limit on each µ arises from a finite number of
I
events in the data sample in channel I, and this formula does not account for fluctuations in the number
of events observed. For a recent analysis of the validity of this approximation, see [164] — on the basis of
which we expect this approximation to be good to 20%.

48

Using (2.17) and (2.16), we scale the data from [163] to obtain the upper bound µ
˜(mH ) on
σ(pp → H → τ τ → I)/σSM (pp → H → τ τ → I) arising solely from gluon fusion. We then
combine these different τ τ decay channels using (2.15). This result is then used to bound
the production cross section for technipions in the three models in Refs. [152, 150, 154]. Our
results are shown in Fig. 2.3.
The constraints are even more stringent, as shown in Fig, 2.3. Only the original [152], the
variant [150] one-family model, and the TCSM low-scale [154] with NT C = 2 is marginally
consistent with data. Forthcoming LHC data on τ τ final states should provide further insight
on these two models for NT C = 2.

2.3.2

LHC Limits on Heavier Technipions Decaying to Tau-Lepton
Pairs

We now consider technipions that are too heavy to be directly compared with a Higgs in the
LHC data, but which can be directly constrained by looking at data from final states with
tau-lepton pairs. ATLAS has obtained [144] limits on the product of the production cross
section with the branching ratio to tau pairs at 95% confidence level for a generic scalar
boson in the mass range 100 − 600 GeV. We use this limit to constrain technicolor models
as follows. The production cross section σ(gg → P ) for technicolor models can be estimated
by scaling from the standard model6 using the production enhancement factor calculated for
each technicolor model [124]. And the branching fraction of the technipions into tau pairs
6 The standard model production cross section σ(gg → h
SM ) at several values of the Higgs mass can be
obtained from the Handbook [159].

49

102

102

ττ channel

One Family (Farhi-Susskind)

ATLAS (1.06 fb )
NTC=4
NTC=3
NTC=2

101
σgg x BR(ττ) [pb]

σgg x BR(ττ) [pb]

-1

ATLAS (1.06 fb )
NTC=4
NTC=3
NTC=2

101

ττ channel

Variant One Family (Casalbuoni et al)

-1

100

10-1

100

10-1

10-2
110 150

200

250

300

2mt
400
MP [GeV]

450

500

550

10-2
110 150

600

(a) Original one-family model [152].
103

250

300

2mt
400
MP [GeV]

450

500

550

600

(b) Variant one-family model [150].
ττ channel

Multiscale (Lane-Ramana)

ATLAS (1.06 fb-1)
NTC=6
NTC=4
NTC=3
NTC=2

102
σgg x BR(ττ) [pb]

200

1

10

100

10-1
110 150

200

250

300

2mt
400
MP [GeV]

450

500

550

600

(c) Multiscale walking technicolor model [153].

Figure 2.4: Comparison of experimental limits and technicolor model predictions for production of a new scalar decaying to tau lepton pairs for scalar masses in the mass range 110
- 600 GeV . In each pane, the shaded region (above the solid line) is excluded by the 95%
CL upper limits on σh Bτ + τ − from ATLAS [144]. Each pane also displays (as open symbols)
the theoretical prediction from one of our representative technicolor models with colored
technifermions, as a function of technipion mass and for several values of NT C . Values of
MP and NT C for a given model that are not excluded by this data are shown as solid (green)
symbols; nearly all such values at low technipion masses are excluded by the data shown
in Fig. 2.3.1. As discussed in the text, limits to the right of the vertical bar at MP = 2mt
apply only when a topcolor sector, rather than extended technicolor, generates most of the
top quark’s mass. Continued next page.

50

Figure 2.4 (cont’d)
103

103

ττ channel

TCSM Low-Scale (Lane)

10

101

100

10-1

10-2
110 150

ATLAS (1.06 fb )
NTC=4
NTC=3
NTC=2

2

σgg x BR(ττ) [pb]

10
σgg x BR(ττ) [pb]

-1

ATLAS (1.06 fb )
NTC=4, ND=10
NTC=3, ND=8
NTC=2, ND=5

2

ττ channel

Isotriplet (Manohar-Randall)

-1

101

100

10-1

200

250

300

2mt
400
MP [GeV]

450

500

550

10-2
110 150

600

(d) TCSM Low-scale technicolor model (the Technicolor Straw Man model) [154].

200

250

300

2mt
400
MP [GeV]

450

500

550

600

(e) Isotriplet model [146].The magnitude of the technifermion hypercharge variable y has been set to 1 for
illustration

is shown in Table II, above. Therefore,

σ(gg → P )BR(P → τ τ ) = κgg prod σ(gg → hSM )BR(P → τ τ ) .

(2.18)

Our comparison of the experimental limits with the model predictions is shown in Fig. 2.4.
In the region of the figures to the left of the vertical bar, we see that the data excludes
technipions in the mass range from 145 GeV up to nearly 2mt in all models for NT C ≥ 3.
For the multiscale and isotriplet models, NT C = 2 is excluded as well in this mass range;
for the TCSM low-scale model, NT C = 2 is excluded up to nearly 300 GeV (the few points
that are allowed at low mass on this plot are excluded by the data shown in Figs. 2.3.1 and
2.3; while for the original and variant one-family models, NT C = 2 can be consistent with
data at these higher masses. Again, further LHC data on di-tau final states will be valuable
for discerning whether the models with only two technicolors remain viable. At present,
technicolor models with colored technifermions are strongly constrained even if their lightest
technipion is just below the threshold at which it can decay to top-quark pairs.

51

Moreover, as the region of the figures to the right of the vertical bar demonstrates, the
data also impacts technipions in the mass range above 2mt in some cases: MP ≤ 450 GeV
(375 GeV) is excluded for any size technicolor group in the multiscale (isotriplet) model and
MP ≤ 375 GeV is excluded for NT C ≥ 3 in the TCSM low-scale model. Note that these
limits apply only in cases where the technipion has a very small branching fraction into top
quarks, and the branching fraction to di-taus just varies smoothly with the increasing mass
of the technipion. As we shall discuss shortly, such limits on technipions heavier than 2mt
would not hold in models where the extended technicolor dynamically generates the bulk of
the top quark mass and the technipion has an appreciable top-quark branching fraction.

2.4

Models with colored technifermions and a top mass
generated by ETC

We will now illustrate how the above constraints are modified in theories where the topquark’s mass includes a substantial contribution from extended technicolor. In such models,
the ETC coupling between the top quark and technipion can be relatively large, which has
several consequences.
First, it means that for technipions heavy enough to decay to top-quark pairs that channel
will dominate, so that the branching fractions to τ + τ − and γγ become negligible. So these
models can be constrained by the LHC data discussed in this work only for MP < 2mt .
Second, it implies that charged technipions P + that are lighter than the top quark can open
a new top-quark decay path: t → P + b. Existing bounds on this decay rate preclude charged
technipions lighter than about 160 GeV; for simplicity, we will take this to be an effective
lower bound on the mass of our neutral technipions in our discussion here – though, in
52

εt=0.5

ATLAS (1.06 fb-1)
NTC=4
NTC=3
NTC=2
Top-loop (NTC=2)

2

10
σgg x BR(ττ) [pb]

103

ττ channel

One Family (Farhi-Susskind)

101

Variant One Family (Casalbuoni et al)

100

10-1

ττ channel
εt=0.5

ATLAS (1.06 fb-1)
NTC=4
NTC=3
NTC=2
Top-loop (NTC=2)

2

10
σgg x BR(ττ) [pb]

103

101

100

10-1

10-2
150

200

250
MP [GeV]

300

10-2
150

350

(a) Original one-family model [152].
103

200

250
MP [GeV]

300

350

(b) Variant one-family model [150].

Multiscale (Lane-Ramana)

ττ channel
εt=0.5
-1

ATLAS (1.06 fb )
NTC=6
NTC=4
NTC=3
NTC=2
Top-loop (NTC=2)

σgg x BR(ττ) [pb]

102

101

100

10-1
150

200

250
MP [GeV]

300

350

(c) Multiscale walking technicolor model [153].

Figure 2.5: Comparison of data and theory for production of a new scalar of mass 150 350 GeV that decays to tau lepton pairs; here, technipion production through techniquark
loops is potentially modified by including production via top quark loops assuming extended
technicolor generates most of the top quark’s mass. In each pane, the shaded region (above
the solid line) is excluded by the 95% CL upper limits on σh Bτ + τ − from ATLAS [144]. As in
Fig. 2.4, each pane displays the theoretical prediction (including techniquark loops only) from
one technicolor model with colored technifermions, as a function of technipion mass and for
several values of NT C . Values of MP and NT C for a given model that are not excluded by this
data are shown as solid (green) symbols. The hatched region indicates (for NT C = 2) how
including the contributions of top-quark loops could impact the model prediction, assuming
t = 0.5. If the top and techniquark loop contributions interfere constructively, the model
prediction moves to the top of the hatched region; if they interfere destructively, the model
prediction moves to the bottom of the hatched region. Continued next page.

53

Figure 2.5 (cont’d)
εt=0.5

ATLAS (1.06 fb-1)
NTC=4, ND=10
NTC=3, ND=8
NTC=2, ND=5
Top-loop (NTC=2)

2

10
σgg x BR(ττ) [pb]

103

ττ channel

TCSM Low-Scale (Lane)

101

10-1

10-2
150

εt=0.5

ATLAS (1.06 fb-1)
NTC=4
NTC=3
NTC=2
Top-loop (NTC=2)

2

100

ττ channel

Isotriplet (Manohar-Randall)

10
σgg x BR(ττ) [pb]

103

101

100

10-1

200

250
MP [GeV]

300

10-2
150

350

(d) TCSM Low-scale technciolor model (the Technicolor Straw Man model) [154].

200

250
MP [GeV]

300

350

(e) Isotriplet model [146].

principle, it is possible for the neutral technipion to be lighter than its charged counterpart.
Based on these considerations, we will be considering possible LHC bounds on technipions
with substantial coupling to top quarks and lying in the mass range 160 GeV < mP < 2mt ;
at present only data on di-tau final states exists for this mass range.
Within this mass range, the presence of a large top-technipion coupling allows gluon
fusion through a top-quark loop (as in Fig.2.1(c)) to become a significant source of technipion
production. Extrapolating from the expressions for decay of a pseudoscalar boson in [158],
one finds that the decay of technipion P to gluons through a top-quark loop has the rate:

Γtop (P → gg) =

m3P
8π

αs t 2
[τ f (τ )]2
2πFP

(2.19)

where t is the ETC-mediated top-quark coupling to technipions, τ and f (τ ) are as defined
in Eqs. (2.4) and (2.5), and the expression [τ f (τ )] → 1 in the limit of large top-quark mass.
Comparing this with Eqn. (2.3), we see that the ratio
Γtop (P → gg)
=
Γ(P → gg)

t [τ f (τ )]

NT C Agg
54

2

≡ Rloops

2

(2.20)

can be substantial if t ≈ 1 and NT C is small.
The relative sign of the techniquark loop and top-quark loop contributions depends on the
structure of the ETC sector of the theory. In models where this sign is positive, the top-quark
and techniquark amplitudes will add constructively and the limits derived in the previous
section will be strengthened. However, in models where the relative sign is negative, the
diagrams in Fig. 2.1(a) and 2.1(c) will interfere destructively, reducing the rate of technipion
production calculated in the previous section by a factor of

1 − Rloops

2

(2.21)

That has the potential to weaken the bounds from the LHC data.
Moreover, in a technicolor model where both NT C and Agg are relatively small, for light
technipion masses where τ f (τ ) ≈ 1, the ratio Rloops can be greater than one, meaning that
the top-quark loop can contribute more to technipion production than the techniquark loop.
For heavier technipion masses, the relative importance of the top-quark loop declines, and
the two contributions interfere strongly, so that the production rate declines and the limits
from LHC data become much weaker. For still heavier technipion masses, the techniquark
loop begins to dominate again and the interference loses its impact on the strength of the
bounds.
This behavior is visible in Fig. 2.5, which shows how the limits on the NT C = 2 version
of each model would be affected by the presence of top-quark loops with t = 0.5. The data,
shaded region, and model prediction curve are as in Fig. 2.4, for the mass range 160 GeV
< MP < 2mt . Also shown here is a hatched region that illustrates how the model curve
would move upwards (downwards) in the presence of constructive (destructive) interference

55

between the top and technifermion loops. The destructive interference would have little
impact on the constraint the LHC data places upon the multiscale model, and progressively
greater impact on the viability of the N = 2 versions of the isotriplet, TCSM low-scale, and
original one-family models. In the variant one-family model, we see that the contribution
from the top loop would, as discussed above, dominate at lower mP , cancel the techniquark
loop contribution at mP ≈ 300 GeV (so that the expected cross-section would vanish), and
then diminish in size for larger mP .
We have also explored the impact of top loop contributions with t = 0.5 on the NT C = 4
versions of the models, where the value of Rloops would be smaller by a factor of two.
We find that destructive interference from top loops would leave the LHC data’s exclusion
of technipions intact across the range 160 GeV < mP < 2mt in the multiscale model,
would bring the upper end of the excluded range down to 325 GeV (300 GeV, 250 GeV) in
the isotriplet (TCSM low-scale, variant one-family) model from the value of 2mt shown in
Fig. 2.4, and bring the upper range of the excluded range down to about 250 GeV from the
previous 325 GeV (per Fig. 2.4) in the original one-family model. The impact on models
with even larger values of NT C would be proportionately smaller.
Finally, we note that if data were available for di-photon final states in the applicable
mass range, it would be possible to discern the impact of destructive interference between
top and technifermion loops on the data’s ability to constrain the models. In this case, one
would need to include effects of top-quark loops both on technipion production from gluon
fusion and also on technipion decay to two photons.

56

2.5

Discussion and Conclusions

In [1] we have used the LHC limits on the γγ [141, 142] and τ + τ − [143, 144, 145] decay
modes of a standard model Higgs boson to constrain the technipion states predicted in
technicolor models with colored technifermions. As discussed in [124], the technipions tend
to produce larger signals in both channels than hSM would, so that this is an effective
way of constraining such technicolor models. Because the technipions are spinless, just like
the standard model Higgs boson, the di-photon and di-tau final states resulting from decay
of the produced boson would have the same kinematic properties, so there should be no
change in the efficiencies and acceptances. Hence, it is possible to adapt the limits quoted
by the collaborations for the Higgs searches very directly to technicolor models with colored
technifermions.
We have found that the combined limits on Higgs bosons decaying to di-photon or di-tau
final states from the ATLAS and CMS collaborations exclude at 95% CL the presence of
technipions in the mass range from 110 GeV nearly up to 2mt for any of the representative
models considered here for NT C ≥ 3. Even if one takes NT C = 2 to make the production
rate as small as possible, the multiscale [153] and isotriplet [146] models are excluded up
to 2mt ; the TCSM low-scale [154] is excluded for technipion masses up to nearly 300 GeV;
and the original [152] and variant [150] one-family model are only marginally consistent with
data. The implication for technicolor model building is clear: models with light technipions
and colored technifermions are not allowed by the LHC data, except possibly in a few models
with NT C = 2. Model-builders will need to consider scenarios with heavier pseudo NambuGoldstone bosons or theories in which the technifermions are color-neutral.
Moreover, we have also seen that the ATLAS limits on a scalar decaying to τ + τ − con-

57

strain the presence of technipions in the mass range 2mt < MP < 450 GeV if the technipion
decays only negligibly to top quarks – as in models where the top quark’s mass is being
generated by a topcolor [98] sector instead of by extended technicolor. The excluded mass
range extends to 450 GeV (375 GeV) for a multi scale (isotriplet) technicolor sector for any
value of NT C and reaches 375 GeV for a TCSM low-scale technicolor sector with NT C ≥ 3.
Hence, starting from these technicolor sectors, building a topcolor-assisted technicolor [100]
model would now require ensuring that the technipions have masses above 375 - 450 GeV.
This complements recent LHC searches for H → W W, ZZ that exclude the top-Higgs state
of TC2 models for masses below 300 GeV if the associated top-pion has a mass of 150 GeV
(the lower bound rises to 380 GeV if the top-pion mass is at least 400 GeV) [126].
In principle, there are several ways to construct technicolor models that could reduce
the scope of these limits. As discussed earlier, one possibility is to arrange for the extended
technicolor sector to provide a large fraction of the top quark’s mass (though it would be
necessary to find a new way to evade bounds on FCNC and weak isospin violation). In this
case, gluon fusion through a top-quark loop (as in Fig. 2.1(c)) could provide an alternative
production mechanism for the technipions. If the ETC structure of the model caused the
top-quark and techniquark loop amplitudes to interfere constructively, our bounds would be
strengthened; but, as illustrated in Fig. 2.5, in a model where the interference was destructive,
our limits on the technipion mass could be weakened, at least for small values of NT C .
Another possibility is to build a technicolor model that includes technipions but not
colored technifermions7 . In order for extended technicolor to provide mass to the quarks,
color must then be embedded in the ETC group alongside technicolor, and some ETC gauge
7 One example is the “minimal walking technicolor” model in [165] with technifermions in the symmetric
tensor representation and NT C = 2; various aspects of its collider phenomenology have been predicted, for
instance, in [166, 167]

58

bosons will carry color charge. It would be more difficult to use the LHC data discussed here
to set broadly-applicable limits on technipions appearing in such models. The production
mechanism contributing most strongly to the rate for the states we studied would not be
operative; that is, without colored technifermions, the process illustrated in Fig. 2.1(a) would
be absent. The analogous process with top quarks instead of colored technifermions in the
loop (as in Fig. 2.1(c)) could, in principle, contribute, but there will be no loop-derived
enhancement by NT C as in the diagram of Fig. 2.1(a). If the coupling of the top quarks to
the technipion were large, that could provide an enhancement to replace the missing NT C
factor – but, as we have seen, the coupling is highly model-dependent. And, as mentioned
earlier, building a model where ETC provides most of the top quark mass (and the toptechnipion coupling is large) remains an open challenge, because it is hard to accomplish
this without contravening experimental limits on flavor-changing neutral currents [81, 82] or
isospin violation [91].
A third option would be to base a model around a technicolor sector devoid of technipions,
such as the original one-doublet model of [78, 79, 80] or a modern “next-to-minimal” walking
technicolor model with technifermions in the symmetric tensor representation of technicolor
and NT C = 3 [165]. Of course, these models come with their own complexities and challenges.
This first set of LHC data has excluded a large class of technicolor and topcolor-assisted
technicolor models that include colored technifermions – unless the technipions states can be
made relatively heavy or the extended technicolor sector can be arranged to cause interference
between top-quark and techniquark loops. Model builders will need to either identify specific
technicolor theories able to withstand the limits discussed here, while generating the top
quark mass without excessive weak isospin violation or FCNC, or else seek new directions
for a dynamical explanation of the origin of mass. Finally, we would like to stress that
59

additional LHC data that gives greater sensitivity to new scalars decaying to τ + τ − or that
addresses scalars with masses over 145 GeV decaying to γγ could quickly probe models down
to the minimum number of technicolors and up to higher technipion masses.

60

Chapter 3
Discovering Strong Top Dynamics at
the LHC
— This chapter is based on a work in collaboration with R. Sekhar
Chivukula, Elizabeth H. Simmons, Baradhwaj Coleppa, Heather
E. Logan, Adam Martin, and Jing Ren, which has appeared in
[3].

3.1

Introduction

In Chapter 2, we saw how the searches for the Higgs placed restrictions on new physics
models describing the electroweak symmetry breaking via strong dynamics. The “4th of
July 2012 discovery” of the Higgs-like boson with mass of approximately 125 GeV was able
to provide much stronger constraints on models of this kind. In this chapter, we focus on
models where the top-quark plays a special role in electroweak symmetry breaking; i.e.,
models with strong top dynamics.
As discussed in Section 1.2 of Chapter 1 phenomenology of the scalar sector of a model
with separate sectors for mass generation of the top quark and of the electroweak bosons can
be conveniently described by the Top Triangle Moose model [119]. In section, 1.3, we noted
that a light top-Higgs has already been tightly constrained [125, 126] by negative findings

61

from the Higgs searches. The work on which this chapter is based then presents an update
of that result in light of new data from the LHC and considers bounds on the top-pions that
are also present. In particular, it explores the possibility that the new boson with a mass of
approximately 125 GeV [168, 45, 161, 44] observed at the LHC is consistent with a neutral
pseudoscalar top-pion state.1 We demonstrate that a neutral pseudoscalar top-pion can
generate the diphoton signal at the observed rate. However, the region of model parameter
space where this is the case does not correspond to classic topcolor-assisted technicolor
scenarios with degenerate charged and neutral top-pions and a top-Higgs mass of order 2mt ;
rather, additional isospin violation would need to be present and the top dynamics would be
more akin to that in top seesaw models [105, 106, 107]. Moreover, the interpretation of the
new state as a top-pion can be sustained only if the ZZ (four-lepton) and W W (two-lepton
plus missing energy) signatures initially observed at the 3σ level decline in significance as
additional data is accrued.2
The LHC evidence for a new boson is composed of several components, based on separate
event samples optimized to be sensitive to the production of the new boson via gluon-fusion,
via vector-boson fusion, or in association with an electroweak boson or a top-quark pair,
and the subsequent decay of the boson to two photons, two massive electroweak bosons, or
pairs of tau-leptons or bottom-quarks [45, 44]. While the totality of evidence including all
subchannels provides convincing evidence of a new bosonic state – one consistent with a
Standard Model (SM) Higgs – the statistical significance of the different subchannels varies,
and it is not yet certain that the object discovered is the Higgs boson. With the data
1 The possibility that the boson observed at the LHC is a pseudo-scalar has been considered by a number
of authors recently [169, 170, 171, 172, 173, 174, 175].
2 Currently, the production and decay properties of the observed 126 GeV boson in the ZZ and W W final
states are consistent with the expectations for the Higgs boson of the Standard Model [46, 52, 53, 120].

62

available at the time of the work the evidence for the new boson is strongest in the diphoton
channel, with a local p-value showing that the “background-only” hypothesis is excluded
at more than the 4σ level by both experiments (a level which is larger than would have
been expected with the current data set for the SM Higgs). The evidence in the next most
sensitive decay channel, ZZ ∗ subsequently decaying to four charged-leptons (e or µ), is also
strong – with a local p-value rejecting the background-only hypothesis at the 3σ level. The
search for the W W ∗ decay mode, in which the W -bosons subsequently decay to e or µ and
corresponding neutrino, is less constraining since it is not possible to measure the diboson
invariant mass – though the background only hypothesis is disfavored by 2-3 σ. Finally, the
evidence for the decay of the new boson to fermions, either tau-leptons or bottom-quarks, is
so far inconclusive.
Our goal in this work [3] is to further the phenomenological investigation of the top-pions
and top-Higgs at the LHC that was started in [119, 125, 126]. We begin in Sec. 3.2 by setting
out the relevant details of the Top Triangle Moose model. Sections 3.3 – 3.5 contain the
bulk of our phenomenological results. In Sec. 3.3 we first consider the possibility that the
diphoton signal observed at the LHC arises from the neutral pseudoscalar top-pion and find
the range of model parameters consistent with these experimental results. Since this object
is a psuedoscalar, it lacks tree-level couplings to ZZ and W W [173, 176, 177]. While the toppion can decay to ZZ or W W through a top-quark loop, we show that these effects would
be too small to be observable in the current data. In Sec. 3.4 we demonstrate that, for the
value of model parameters such that the neutral top-pion can account for the observed LHC
diphoton signal, the properties of top-quark decay imply that the corresponding charged
top-pions would have to be heavier than 150 GeV. As reviewed in Appendix B.1, however,
this implies that the model would need to include more isospin violation than is the minimum
63

required to produce a heavy top-quark – i.e., more isospin violation than is usually assumed
to exist in these models. In Sec. 3.5 we review and update the constraints previously derived
in [126], in the case that the 125 GeV object is associated with the neutral top-pion. We
summarize our findings and discuss their implications in Sec. 3.6.

3.2

The Scalar Spectrum and Properties

On one level, the Top Triangle Moose model is an example of a deconstructed Higgsless model
of electroweak symmetry breaking. Inspired by the possibility of maintaining perturbative
unitarity in extra-dimensional models through heavy vector resonance exchanges in lieu of
a Higgs [108, 109, 110], Higgsless models were initially introduced in an extra-dimensional
context as SU (2) × SU (2) × U (1) gauge theories living in a slice of AdS5 , with symmetry
breaking codified in the boundary condition of the gauge fields [111, 112, 113, 114, 115, 116].
The low energy dynamics of these extra-dimensional models can be understood in terms of
a collection of 4-D theories, using the principle of “deconstruction” [178, 179]. Essentially,
this involves latticizing the extra dimension, associating a 4-D gauge group with each lattice
point and connecting them to one another by means of nonlinear sigma models; the picture
that emerges is called a “Moose” diagram [180]. The five dimensional gauge field is now
spread in this theory as four dimensional gauge fields residing at each lattice point, and the
fifth scalar component residing as the eaten pion in the sigma fields.
The key features of these models [181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192]
that are relevant to our discussion are as follows: Spin-1 resonances created by the strong
dynamics underlying the sigma fields are described as massive gauge bosons, following the
Hidden-Local-Symmetry scenario originally used for QCD [193, 194, 195, 196, 197] and also

64

the BESS [181, 182] models. The phenomenology of those resonances in the Top Triangle
Moose have been discussed in Refs. [118, 119]. Standard model (SM) fermions reside primarily on the exterior sites – the sites approximately corresponding to SU (2)w and U (1)Y gauge
groups; these fermions become massive through mixing with massive, vector-like fermions
located on the interior, ‘hidden’ sites. The phenomenology of these fermions has previously
been discussed in [119, 125]. Precision electroweak parameters [198], are accommodated by
adjusting the SM fermion’s distribution across sites [191] to match the gauge boson distribution, a process called “ideal delocalization” [184]. This is identical to the solution used in
extra-dimensional Higgsless models, where the spreading of a fermion among sites becomes
a continuous distribution, or profile, in the extra dimension [114].
The AdS/CF T correspondence suggests that these weakly-coupled Higgsless models can
be understood to be dual to the strongly coupled models of electroweak symmetry breaking. Indeed the Top Triangle Moose is a deconstructed analog of topcolor-assisted technicolor (TC2) [98, 100, 101, 102, 103, 99, 104], a scenario of dynamical electroweak symmetry
breaking in which the new strong dynamics is partitioned into two different sectors. The
technicolor sector [79, 80] is responsible for the bulk of electroweak symmetry breaking,
through condensation of a technifermion bilinear, and is therefore characterized by a scale
F ∼ v, where v = 246 GeV is the EWSB scale. Consequently, technicolor dynamics is responsible for the majority of the weak gauge boson masses and, more indirectly [82, 81], the
masses of the light fermions. The second strong sector, the topcolor sector [98, 100], communicates directly with the top quark. Its purpose is to generate a large mass for the top
quark through new strong dynamics that cause top quark condensation [92, 93, 94, 95, 96].
In generating a top-quark mass, this second sector also helps to break the electroweak symmetry. If the characteristic scale of the topcolor sector is low, f
65

F , it plays only a minor

role in electroweak breaking, but can still generate a sufficiently large top-quark mass given
a strong enough top-topcolor coupling. Because electroweak symmetry is effectively broken
twice in this scenario, there are two sets of Goldstone bosons. One linear combination of
the weak-triplet Goldstone bosons (the combination primarily composed of technifermions)
is eaten to become the longitudinal modes of the W ± /Z 0 , while the orthogonal triplet and
accompanying weak singlet state remain in the spectrum. These remaining states, typically
referred to as the top-pions and the top-Higgs, are the focus of this chapter.
Probing the dynamics of topcolor assisted technicolor will involve discovering the topHiggs and top-pions which are associated with the generation of the large top-quark mass,
and measuring their properties. In this section we describe briefly our expectations for the
properties of these states, and summarize the model-dependence of their couplings.

3.2.1

The Triangle Moose Model

The Top Triangle Moose model [119] is shown in Moose notation in Fig. 3.1. The circles
represent global SU(2) symmetry groups; the full SU(2) at sites 0 and 1 are gauged with
gauge couplings g and g˜, respectively, while the τ 3 generator of the global SU(2) at site 2 is
gauged with U(1) gauge coupling g . The lines represent spin-zero link fields which transform
as a fundamental (anti-fundamental) representation of the group at the tail (head) of the
link. Σ01 and Σ12 are nonlinear sigma model fields, describing the technicolor/three-site
[118] sector of the theory, while Φ (the top-Higgs doublet) is a linear sigma model field
arising from top-color [98, 100].
The kinetic energy terms of the link fields corresponding to these charge assignments are:

Lgauge =

F2
F2
Tr[(Dµ Σ01 )† Dµ Σ01 ] +
Tr[(Dµ Σ12 )† Dµ Σ12 ] + (Dµ Φ)† Dµ Φ,
4
4
66

(3.1)

0
g
Σ01
g˜

Φ

g

�

1

Σ12

2
Figure 3.1: The gauge structure of the model in Moose notation. g and g are approximately
the Standard Model SU (2) and hypercharge gauge couplings while g˜ represents the ‘bulk’
gauge coupling. The left (right) handed light fermions are mostly localized at site 0 (2) while
their heavy counterparts are mostly at site 1. The links connecting sites 0 and 1 and sites
1 and 2 are nonlinear sigma model fields while the one connecting sites 0 and 2 is a linear
sigma field. Site 2 is dotted to indicate that only the τ3 component is gauged.

where the covariant derivatives are:

Dµ Σ01 = ∂µ Σ01 + igW0µ Σ01 − i˜
g Σ01 W1µ ,
Dµ Σ12 = ∂µ Σ12 + i˜
g W1µ Σ12 − ig Σ12 τ 3 Bµ ,
Dµ Φ = ∂µ Φ + igW0µ Φ −

ig
Bµ Φ.
2

(3.2)

a τ a and W
a a
Here the gauge fields are represented 3 by the matrices W0µ = W0µ
1µ = W1µ τ ,

where τ a = σ a /2 are the generators of SU(2). The nonlinear sigma model fields Σ01 and
Σ12 are 2×2 special unitary matrix fields. To mimic the symmetry breaking caused by
underlying technicolor and topcolor dynamics, we assume all link fields develop vacuum
3 Here the subscripts appearing in the fields will refer to the “site” numbers and the superscripts will be
reserved for SU (2) indices.

67

expectation values (vevs):

√
 f/ 2 


Φ =
.


0


Σ01 = Σ12 = 12×2 ,

(3.3)

In order to obtain the correct amplitude for muon decay, we parameterize the vevs in terms
of a new parameter ω,

F =

√
2 v cos ω,

f = v sin ω,

(3.4)

where v = 246 GeV is the weak scale. We will explore the parameter range4 0.2 ≤ sin ω ≤ 0.8,
in which the Top Triangle Moose acts as a low-energy effective theory for a variety of models
with strong top dynamics [126]. As a consequence of the vacuum expectation values, the
gauge symmetry is broken all the way down to electromagnetism and we are left with massive
gauge bosons (analogous to techni-resonances), top-pions and a top-Higgs. To keep track
of how the degrees of freedom are partitioned after we impose the symmetry breaking, we
expand Σ01 , Σ12 and Φ around their vevs. The coset degrees of freedom in the bi-fundamental
link fields Σ01 and Σ12 can be described by nonlinear sigma fields:

Σ01 = exp

2iπ0a τ a
F

,

Σ12 = exp

2iπ1a τ a
F

,

(3.5)

4 The extreme case in which sin ω → 1 would have a rather different phenomenology, as the properties
of the top-Higgs boson would approach those of the Standard Model Higgs boson, the top-Higgs could
potentially be light, and the top-pions would become heavier.

68

while the degrees of freedom in Φ fill out a linear representation,




Φ=



√
(f + Ht + iπt0 )/ 2 

.

iπt−

(3.6)

The gauge-kinetic terms in Eq. (3.1) yield mass matrices for the charged and neutral
gauge bosons. The photon remains massless and is given by the exact expression

e 3
e 3
e
+ W1µ
+ Bµ ,
Aµ = W0µ
g
g˜
g

(3.7)

where e is the electromagnetic coupling. Normalizing the photon eigenvector, we get the
relation between the coupling constants:

1
1
1
1
= 2 + 2 + 2.
2
e
g
g˜
g

(3.8)

This invites us to conveniently parametrize the gauge couplings in terms of e by

g=

e
g
= 0 ,
sin θ cos φ
cos φ

We will take g˜

g˜ =

e
g
= 0 ,
sin θ sin φ
sin φ

g =

g, which implies that tan φ ≡ x is a small parameter.

69

e
.
cos θ

(3.9)

3.2.2

The Triangle Moose Potential: Scalar Spectrum and Isospin
Violation

Counting the number of degrees of freedom, we see that there are six scalar degrees of
freedom on the technicolor side (Σ01 , Σ12 ) and four on the topcolor side (Φ). Six of these
will be eaten to form the longitudinal components of the W ± , Z 0 , W ± , and Z 0 . This leaves
one isospin triplet of scalars, the top-pions Πat , and the top-Higgs Ht as physical states in
the spectrum. While the interactions in Eq. (3.1) are sufficient to give mass to the gauge
bosons, the top-pions and top-Higgs remain massless at tree level. Quantum corrections
will give the top-pions a mass, however this loop-level mass is far too small to be consistent
with experimental constraints. To generate phenomenologically acceptable masses for the
top-pions and top-Higgs, we add three5 additional interactions:

LM = −λ Tr
+

M †M

f2
−
2

2

2
f
− κf 2 Tr M − √ Σ01 Σ12
2

f 2 Tr M † Σ01 Σ12 τ 3

2

+ h.c.

,

(3.10)

where the first of these interactions arises from topcolor interactions, the second from ETClike interactions [82, 81], and the third is an example of possible isospin-violating interactions
in the top-color sector. Here λ, κ, and are three new dimensionless parameters that depend
on the details of the top-color dynamics, f is the same vacuum expectation value appearing
˜
in Eq. (3.4), and M is the Φ field expressed as a matrix6 , schematically given by M = (Φ, Φ)
5 In [119] the possibility of isospin violation, and hence the last term in Eq. (3.10), was neglected. As we
show in Appendix B.1, isospin violation in the top color sector is usually assumed to be small, and hence
the size of the dimensionless parameter , is small. We introduce it here to explore the phenomenology that
would arise from non-degenerate top-pions.
6 This corrects the expression in [119].

70

˜ = −iσ2 Φ∗ :
with Φ


√
 (f + Ht + iπt0 )/ 2
iπt+

M =

√
iπt−
(f + Ht − iπt0 )/ 2





,


(3.11)

where πt+ = (πt− )∗ . The first term in Eq. (3.10) depends only on the modulus of M , and
therefore contributes only to the mass of the top-Higgs. The second and third terms give
mass to both the top-Higgs and the physical (uneaten) combination of pion fields, as we will
show shortly. Because these masses depend on three parameters, λ, κ, and , we can treat
the mass of the top-Higgs and the masses of the uneaten charged and neutral top-pions as
three independent parameters. In addition to generating masses, the potential in Eq. (3.10)
also induces interactions between the top-Higgs and top-pions which are important in our
analysis.
The next step towards understanding top-pion phenomenology is to identify the combination of degrees of freedom which make up the physical (uneaten) top-pions. While the
top-Higgs Ht remains a mass eigenstate, the pions π0a , π1a and πta mix. We can identify
the physical top-pions as the linear combination of states that cannot be gauged away. We
do this by isolating the Goldstone boson states that participate in interactions of the form
Vµ ∂ µ π in the Lagrangian. We start by expanding the nonlinear sigma fields to first order in
π/F ,
2iπ0a τ a
+O
Σ01 = 1 +
F
2iπ1a τ a
Σ12 = 1 +
+O
F

71

π2
F2
π2
F2

,

(3.12)

.

(3.13)

Plugging this in Eq. (3.1), we can read off the various interaction terms. The gauge-Goldstone
mixing terms are of the form:

g˜ aµ
g µ
g aµ
Lmixing = W0 ∂µ [F π0a + f πta ] + W1 ∂µ [F π1a − F π0a ] − B2 ∂µ F π13 + f πt3 . (3.14)
2
2
2
Note that the pion combination in the third term can be written as a linear combination of
those appearing in the first two terms:

F π13 + f πt3 = [F π03 + f πt3 ] + [F π13 − F π03 ].

(3.15)

The two eaten triplets of pions span the linear combinations that appear in the first two terms
of Eq. (3.14), leaving the third linear combination as the remaining physical top-pions, which
we will denote Πat :
Πat = − sin ω

π0a + π1a
√
2

+ cos ω πta ,

(3.16)

where we have normalized the state properly using the definitions of F and f in Eq. (3.4).
The physical top-pions can also be identified by expanding the top-Higgs potential given
in Eq. (3.10) and collecting the mass terms. The physical masses of the top-Higgs and
top-pions are

M 2 ± = 2κv 2 tan2 ω
Πt

M 2 0 = 2(κ − )v 2 tan2 ω
Πt

2
MH

t

= 2(4λ + κ)v 2 sin2 ω = 8λv 2 sin2 ω + M 2 + cos2 ω.
Πt

(3.17)

while the other two linear combinations of pions are massless, as true Goldstone bosons

72

should be. Equation (3.10) also contains trilinear couplings between Ht and two top-pions;
−
0 0
the Feynman rules for the Ht Π+
t Πt and Ht Πt Πt interactions are given by

−
Ht Π+
t Πt :

−2iv sin ω 4λ cos2 ω + κ

sin4 ω
cos2 ω

−i
2 cos2 ω − M 2 + 2M 2 sin2 ω
MH
t
Π+
Π+
v sin ω
t
t
4
sin ω
sin2 ω
−2iv sin ω 4λ cos2 ω + κ 2 − 2
cos ω
cos2 ω
−i
2 cos2 ω − M 2 + 2M 2 sin2 ω .
=
MH
t
Π0t
Π+
v sin ω
t

=
Ht Π0t Π0t :

(3.18)

These couplings are important for top-Higgs decays when MHt > 2MΠt .
For the purposes of our phenomenological analysis we will take the masses of the topHiggs, and of the charged and neutral top-pions as independent parameters. To give a sense
of what might be expected from TC2 dynamics, we have looked at the expectations for these
parameters in a Nambu– Jona-Lasinio (NJL) [76] approximation for the topcolor dynamics;
our NJL calculation is summarized in Appendix B.1. From the NJL analysis we find:
• The top-Higgs mass satisfies MHt = O(2mt ) [76]. This result is known to change once
subleading interactions are taken into account [96], and hence we take this result as
only indicative that the top-Higgs should have a mass of order 200 - 700 GeV.
• The mass splitting between the charged and neutral top-pions is relatively small – with
∆MΠ /MΠ less than about 10%. We therefore conclude that the minimum amount of
isospin violation required in topcolor (the amount necessary to yield the top-quark
mass) need not produce a large mass splitting between the top-pions.
• The analysis also confirms that the form of the potential in Eq. (3.10), with

0,

correctly summarizes the non-derivative interactions yielding the top-pion and top73

Higgs masses and interactions. We therefore typically expect MΠt

MHt , c.f. Eq.

(3.17) for small sin ω.
Based on these considerations, in what follows we explore the possibility that the new
state at a mass of approximately 125 GeV observed at the LHC is consistent with a neutral
pseudoscalar top-pion state. We consider two representative cases: (1) assuming degenerate
charged and neutral top-pion masses, MΠ± = MΠ0 , and (2) fixing MΠ0 ≈ 125 GeV and
t

t

t

allowing the charged top-pion mass to vary. As discussed above, the first case is that generically expected in top color models, and the second allows us to illustrate how these results
would change if the top-color dynamics includes additional sources of isospin violation.

3.2.3

Scalar Couplings to Fermions

The couplings of the top-pion and top-Higgs to fermions are model dependent. Unlike in
the standard model, the presence of two different sources for the quark masses (topcolor and
technicolor) implies that the top-pion and top-Higgs couplings depend on the individual lefthanded and right-handed rotations in the separate up- and down-quark sectors that relate
the topcolor gauge eigenstates (in which the top-pion and top-Higgs couplings are simple) to
the mass-eigenstates [100, 199, 200].
For our analysis, we make the following assumptions:
• Following [100, 199, 200], we assume that the top- and bottom-quarks both receive most
of their mass as a result of topcolor (which would naturally explain why Vtb

1), while

the other quarks and the leptons receive their masses from the (extended) technicolor
sector. That is, if we were to “turn off” technicolor electroweak symmetry breaking
(F → 0 or cos ω → 0) the top and bottom quarks would have masses close to their
74

observed values, but all other quarks and the leptons would be massless.
• The usual CKM angles are related to the difference between the left-handed up- and
down-quark rotations which are required. Since the observed CKM matrix is nontrivial, it is not possible that both of the left-handed up- and down-quark rotations
are trivial. As we show in appendix B.2, however, if the observed CKM angles arise
predominantly from rotations in the left-handed down-quark sector, charged top-pion
exchange will lead to unacceptably large contributions to the process b → sγ. We
therefore assume that CKM mixing arises from the rotations in the left-handed upquark sector.
• The rotations in the right-handed sector are, a priori, unconstrained. However, if
¯0
present, they have the potential to lead to unacceptably large contributions to Bd0 − B
d
¯ 0 meson mixing. We therefore assume that there is no mixing in the
[199] and D0 − D
right-handed sector.
With these assumptions, to leading order, the flavor-diagonal couplings of the neutral
top-pions to the third generation fermions7 are
i Π0t
mt cot ω t¯L tR + mb cot ω ¯bL bR + mτ tan ω τ¯L τR + h.c. .
v

(3.19)

The mixing in the left-handed up-quark sector will necessarily lead to flavor-changing decays
of the neutral top-pion [201, 202] of the form
i Π0t
CKM c¯ t + V CKM u
mt cot ω Vcb
¯L tR
L R
ub
v

.

(3.20)

7 Couplings to the light quarks and leptons would follow the same pattern as for the τ lepton, but will
not be needed in what follows.

75

The couplings of the top-Higgs to fermions are the scalar analogs of the pseudo-scalar couplings of the Π0t listed in Eqs. (3.19) and (3.20) above.
Similarly, the corresponding charged-pion couplings are of the form8
√
i 2Π+
t [m cot ω t¯ b + m cot ω t¯ b
t
R L
L R
b
v
CKM c¯ b + m tan ω ν
+mb cot ω Vcb
¯τ L τR + mc tan ω Rcs c¯R sL + h.c. , (3.21)
τ
L R

where Rcs is an unknown mixing parameter which, for the purposes of illustration, we take
equal to its maximum value Rcs

cos θC

1.9

The relation between the assumptions made here and the simpler form of the fermion
couplings used in [119] is presented in Appendix B.2.

3.3

Neutral top-pion phenomenology

In this section, we will discuss the phenomenology of the neutral top-pion assuming it has a
mass of 125 GeV. We start by reviewing the couplings and decays, examine the production
cross-section, and then discuss various decay modes in light of the LHC data. More details
about the model can be found in Ref. [125].

3.3.1

Couplings and decays

The couplings of the neutral top-pion that are most relevant to our analysis are those to
gg, γγ, b¯b, and τ τ¯. The couplings to gluon pairs or photon pairs arise from top quark loops
8 The coupling of Π+ to t¯ b gives a potentially large contribution to the process Z → b¯b [203], which
R L
t
must be compensated for by adjusting the properties of the top-quark [125]. See the discussion in appendix
B.2.
9 If this coefficient were smaller, this would increase the branching ratio BR(Π+ → τ¯ν ) which would
τ
t
strengthen the limits in section 3.4.

76

(contributions from loops containing heavy top-quark partners would be suppressed by powers of the heavy quark mass). Those to fermions arise from top color (for t and b) and/or
extended technicolor dynamics (especially for lighter fermions). Being a pseudoscalar, the
top-pion lacks tree-level couplings to W W and ZZ, and the loop induced couplings to these
massive gauge bosons are small compared to the dominant ones listed above. These decays
do occur through a top-quark loop, and are discussed separately.
We have calculated the branching ratios of Π0t using the MSSM pseudoscalar decay
routines in HDECAY version 3.531 [160], modified to take into account the different fermion
coupling structure of Eqs. (3.19)–(3.20) and the absence of superpartners. The resulting
branching ratios are illustrated in Fig. 3.2 for sin ω = 0.3, 0.5 and 0.7. Decays to b¯b dominate
at low Π0t mass, with the gg and tc channels becoming important only once MΠ0
t

Decays to tt¯ turn on at MΠt

2mt

200 GeV.

350 GeV and completely dominate above this mass.

Note that our calculation using HDECAY includes decays to off-shell tt¯ below threshold. As
these plots indicate, for a 125 GeV top-pion, only the decay branching ratios to gg, γγ, b¯b
and τ τ¯ are significant.
The total width of Π0t is shown in the top panel of Fig. 3.3 as a function of sin ω, with
MΠ0 = 125 GeV. Because its mass is well below the tt¯ threshold, the Π0t remains a narrow
t

resonance with width below 1 GeV for all values of sin ω ≥ 0.2.
In the bottom panel of Fig. 3.3 we display the branching ratio for Π0t → γγ as a function
of sin ω with MΠ0 = 125 GeV. This branching ratio reaches at most 0.5 parts per mil and is
t

roughly five times smaller than the SM Higgs branching ratio into photons.

77

1

Branching ratio

0.1

0.01

0.001

0.0001

1e-05
100

sin ω = 0.3
gg
tc
bb
tt
tu
γγ
ττ
150

200
250
MΠ 0 (GeV)

300

350

200
250
MΠ 0 (GeV)

300

350

200
250
MΠ 0 (GeV)

300

350

t

1

Branching ratio

0.1

0.01

0.001

0.0001

1e-05
100

sin ω = 0.5
gg
tc
bb
tt
tu
ττ
γγ
150

t

1

Branching ratio

0.1

0.01

0.001

0.0001

1e-05
100

sin ω = 0.7
gg
tc
bb
tt
ττ
tu
γγ
150

t

Figure 3.2: Branching ratios of the Π0t into its dominant decay modes for sin ω = 0.3 (top),
0.5 (middle), and 0.7 (bottom). The order of the curves in the key (from top to bottom)
reflects the order of the curves at MΠ0 = 300 GeV.
t

78

0.1

MΠ 0 = 125 GeV

Total width (GeV)

t

0.01

0.001
0.2

0.3

0.4

0.5
sin ω

0.6

0.7

0.8

0.0006
MΠ 0 = 125 GeV
t

BR(Πt0 → γγ)

0.00055
0.0005
0.00045
0.0004
0.00035
0.0003

0.2

0.3

0.4

0.5
sin ω

0.6

0.7

0.8

Figure 3.3: Total decay width (top) and branching ratio for Π0t → γγ (bottom) of a 125
GeV Π0t as a function of sin ω.

79

3.3.2

Production cross-section

Here, we calculate the production cross-section of the neutral top-pion; in subsequent subsections we will compare this prediction to various ATLAS and CMS results to analyze the
current and future LHC sensitivity to neutral top-pions.
The neutral top-pion is produced at the LHC almost exclusively via gluon fusion. We
calculate the cross section for Π0t production in gluon fusion according to

σ(gg → Π0t ) =

A
f αf F1/2 (τf )
H
f F1/2 (τf )

2

2

× σ(gg → HSM ),

(3.22)

where in the sum over fermions we include10 t, b and c; also αt = αb = cot ω and αc =
H (τ ) and F A (τ ), for scalars and pseudoscalars
tan ω. Here the fermion loop functions F1/2
1/2

respectively, are given by [158]:

H
F1/2
= −2τ [1 + (1 − τ )f (τ )] ,
A
F1/2
= −2τ f (τ ),

(3.23)

2 and
where τf = 4m2f /MΠ

f (τ ) =

with η± = (1 ±

√





 sin−1

1/τ

2




− 41 [ln(η+ /η− ) − iπ]2

if τ ≥ 1

(3.24)

if τ < 1,

H → −4/3 and
1 − τ ). In the limit of a heavy fermion in the loop, F1/2

10 Technifermion loops do not contribute to top-pion production because the SU (2)
2
weak × [SU (3)]
anomaly vanishes for any realistic technicolor theory.

80

A → −2.
F1/2

We take the SM gluon-fusion Higgs production cross section σ(gg → HSM ) from Ref. [204]
for the 7 TeV LHC. This SM Higgs cross section includes the state-of-the-art radiative
corrections, which boost the cross section by a substantial factor ∼ 2. Our cross section
in Eq. (3.22) relies on the equality of the k-factors for pseudoscalar production and scalar
production. In fact, because most of the QCD k-factor comes from real radiation, this
equality has been shown to hold to within 20%, as illustrated in [205].

3.3.3

Current and Prospective Limits from the Diphoton channel

The diphoton decay channel has played a leading role in LHC searches for the Standard
Model Higgs boson. Although this is not the dominant decay mode for the neutral toppion, it would certainly be highly visible in the LHC detectors. We have calculated σ(gg →
Π0t )×BR(Π0t → γγ) and our results are shown as a function of sin ω (fixing MΠ0 = 125 GeV)
t

in the top pane of Fig. 3.4. The signal rate is largest for small sin ω, due to the enhancement
of both the Π0t production cross section and the branching ratio to γγ at small sin ω.
The LHC SM Higgs searches in Refs. [206, 207] have exclusion sensitivity to γγ resonances
with a cross section of order 50 fb for resonance masses between 110 and 150 GeV. We find
that this excludes a neutral top-pion in this mass range with sin ω

0.4 − 0.5. We show

the excluded region in the right plot in Fig. 3.7, based on the 95% confidence level limit on
σ/σSM in the γγ channel alone for the SM Higgs from Refs. [206, 207]; those limits are based
on 4.9 fb−1 (ATLAS) and 4.8 fb−1 (CMS) at 7 TeV. We translated the LHC results into
bounds on our model by comparing the CMS and ATLAS limits on σ/σSM with
σ
σSM

=

σ(gg → Π0t ) × BR(Π0t → γγ)
,
[σ(gg → HSM ) + σ(VBF → HSM )] × BR(HSM → γγ)
81

(3.25)

1

MΠ 0 = 125 GeV

σ(gg → Πt0) * BR(Πt0 → γγ) (pb)

t

8 TeV

0.1

7 TeV

0.01

0.001
0.2

0.3

0.4

0.5

0.6

0.7

0.8

sin ω

0.8
0.7

sin ω

0.6
0.5
0.4
0.3
0.2
110

Excluded
ATLAS
CMS
115

120

125

130
135
MΠ 0 (GeV)

140

145

150

t

Figure 3.4: Top: Cross section times branching ratio for gg → Π0t → γγ for a 125 GeV
top-pion at the 7 and 8 TeV LHC. Bottom: 95% confidence level exclusion limits in the
MΠ0 vs. sin ω plane, extrapolated from the LHC SM Higgs search limits in the γγ channel
with 4.8–4.9 fb−1 at 7 TeV from Refs. [206, 207].

82

where σ(gg → Π0t ) is obtained from Eq. (3.22) and σ(VBF → HSM ) is the SM Higgs
production cross section via vector boson fusion (VBF). Note that the CMS analysis includes
a contribution from a dedicated VBF search topology channel, which would not be present
for the top-pion. The ATLAS analysis does not include a dedicated VBF channel and is
thus more directly applicable to the top-pion. However, the inclusion of the dedicated VBF
search channel by CMS does not appear to significantly affect our results: the limits are
consistent with each other in excluding low values of sin ω.
Both CMS and ATLAS observe a new state with a mass of about 125 GeV decaying
to diphotons whose properties appear to be consistent with those of a SM Higgs boson.
However, the observed diphoton rate is nearly twice that expected for a SM Higgs [168, 161],
which also makes the excess consistent with a neutral top-pion with sin ω

0.5, as shown

in the bottom pane of Fig. 3.4.

3.3.4

Decays to ZZ, Zγ and W W

It is interesting to consider how one would be able to distinguish a neutral top-pion from a
SM Higgs boson once more data is in hand. The SM Higgs has tree-level couplings to W + W −
and ZZ, while couplings to γγ arise only at one loop. In contrast, being a pseudoscalar, Π0t
does not have tree-level couplings to W + W − or ZZ [173, 176, 177]. It can, however, have
couplings to W + W − , ZZ, and Zγ at one loop. Ref. [177] considered the possibility that the
loop-induced pseudoscalar coupling to the SU (2) and hypercharge gauge bosons can account
for the observed γγ and 4 signal. Essentially, the strategy consisted in adjusting the relative
value of the SU (2) and hypercharge gauge couplings so that the equation
Γc (H → ZZ ∗ → 4e)
Γc (φ → 4e)
=
.
Γ(H → γγ)
Γ(φ → γγ)
83

(3.26)

is satisfied. Here, φ refers to the pseudoscalar and the superscript c means the quantities are
computed with the experimental cuts imposed. It was shown that the Zγ ∗ contribution to
the 4 signal completely dominates the ZZ ∗ contribution, in direct contrast to the SM case
where the Zγ ∗ contribution is negligible. Fixing the ratio of the coupling strengths this way
leads to a well-defined prediction [177]:

φ

RZγ/γγ ≡

Γ(φ → Zγ)
= 121.
Γ(φ → γγ)

(3.27)

Thus, in order to see if the top-pion can generate the experimentally required signal
strength, it suffices to compute the ratio of the partial widths to Zγ and γγ and compare
with the number in Eq. (3.27). This number turns out to be ≈ 0.02 for the top-pion.11
Thus, we conclude that the top-pion cannot generate the observed ratio of the 4 to γγ
rates. Though this might seem to be a problem for models with strong top-quark dynamics
in general, we point out that the 4 signal involves very few events and conclusions about the
viability of our model based on this observation should be postponed until higher-statistics
results are available from the ATLAS and CMS collaborations.

3.3.5

Limits from the Ditau Channel

Searching for light neutral top-pions decaying to τ τ is difficult because of the large Drell-Yan
background. For scalars that are produced in part by vector boson fusion, the sensitivity
can be enhanced by implementing cuts that preferentially select the VBF channel, but unfortunately this option is not available for pseudoscalars like the top-pion.
11 We note that this is independent of sin ω, which cancels out in the ratio. sin ω in our model is analogous
to the parameter c in [177] - one that can be tuned to adjust the production cross-section to the proper
value.

84

Looking specifically at the case where the neutral top-pion is responsible for the diphoton
excess at 125 GeV (MΠ0 = 125 GeV and sin ω
t

0.5, corresponding to an enhancement in

the γγ channel by about a factor of 2 compared to the SM Higgs prediction), then we expect
a τ τ signal rate, from the gluon fusion channel, approximately equal to that of the SM Higgs.
This is about a factor of 3 below the sensitivity of the HSM → τ τ search from ATLAS [163]
(4.7 fb−1 at 7 TeV). However, that ATLAS analysis includes events in a “Higgs plus two
jet” event category, corresponding to VBF production, to improve the sensitivity to the SM
Higgs, so this limit does not directly apply to the neutral pseudoscalar top-pion of our model.
The papers [168, 45, 161, 44] reporting the discovery of a new scalar in the diphoton channel
do analyze data from the ditau channel, but neither finds conclusive evidence that the new
state decays to tau lepton pairs. In the future, perhaps a dedicated search focused on the
gluon fusion production channel would be sensitive to the Π0t .

3.4

Charged top-pion phenomenology

Charged top-pions would be pair-produced via electroweak processes at LHC and their dominant decay channels are hadronic. Therefore a direct search for Π+
t would be hampered by
a combination of low cross-section and high backgrounds. The main constraints on these
states presently come from top quark decays.

3.4.1

Branching ratios

We plot the branching ratios of the charged top-pion as a function of MΠ± in Fig. 3.5,
t

assuming that the mass of the neutral top-pion is fixed at 125 GeV. For top-pion masses
below mt , the dominant decays are into τ ν and offshell t∗ b; their relative rates depend on
85

the top-pion mass and sin ω. The decay to cs has a branching ratio a little less than half
that of τ ν; the rate for bc is many times smaller when sin ω

0.5. For masses above mt ,

decays to tb overwhelmingly dominate.
0 +∗ becomes possible. As this
If MΠ+ > MΠ0 then the off-shell decay Π+
t → Πt W
t

t

branching ratio never exceeds 5%, it is phenomenologically unimportant for our purposes.
Since the charged top-pion seldom decays to the neutral top-pion even when kinematically
allowed to do so, Fig. 3.5 also gives a good sense of the branching ratios of the charged toppion for the case in which the top-pions are degenerate.

3.4.2

Limits from t → Π+ b

The ATLAS collaboration has searched for evidence of charged scalars in top quark decays
using 4.6 fb−1 of data gathered at 7 TeV [208]. Because this was motivated as a search
for the charged Higgs of the MSSM, which decays almost exclusively to τ ν for large tan β,
their search assumed that the charged scalar would decay only to τ ν. Specifically, they set
a limit on B ≡ BR(t → H + b), assuming that BR(H + → τ ν) = 1. The latter assumption
is built directly into their analysis in that they scale the simulated cross section for SM tt¯
background, in which t → W b, by (1 − B)2 .
The conclusions of the ATLAS t → H + b analysis cannot be directly applied to the
charged top-pion because BR(Π+
t → τ ν) = 1, as can be seen in Fig. 3.5. In fact, as also
illustrated in the left pane of Fig. 3.6, the value of BR(Π+
t → τ ν) ranges from a maximum
of about 0.7 for a relatively light Π+
t and large sin ω, to close to zero for a heavier top-pion
and lower sin ω (due to the competing t∗ b decay).
Nevertheless, we can adapt the ATLAS t → H + b limits to extract information about
the charged top-pion. The charged top-pion signal is the same as that for the charged Higgs
86

1

sin ω = 0.3
τν
cs
bc
tb

0.9
0.8
Branching ratio

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100

110

120

130

140
150
MΠ + (GeV)

160

170

180

140
150
MΠ + (GeV)

160

170

180

140
150
MΠ + (GeV)

160

170

180

t

1
0.9
0.8

sin ω = 0.5
τν
cs
bc
tb

Branching ratio

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100

110

120

130

t

1
0.9
0.8

sin ω = 0.7
τν
cs
bc
tb

Branching ratio

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100

110

120

130

t

Figure 3.5: Branching ratios of the charged top-pion to the dominant final states, for
sin ω = 0.3, 0.5, and 0.7 (top to bottom). We include off-shell decays to t∗ b and also off-shell
decays to Π0t W + assuming MΠ0 = 125 GeV. These branching ratios were computing using
t

a modified version of HDECAY [160].
87

+
studied in Ref. [208], provided that the parameter B is replaced by BR(t → Π+
t b)×BR(Πt →

τ ν). We calculated the top quark decay branching ratio at tree level neglecting the bottom
quark mass, using

BR(t →

Π+
t b)

=

cot2 ω(1 − M 2 + /m2t )2
Π

2 /m2 )(1 − M 2 /m2 )2 + cot2 ω(1 − M 2 /m2 )2
(1 + 2MW
+
t
t
t
W

.

(3.28)

Π

We have calculated the Π+
t decay branching ratios using a modified version of HDECAY [160]
as discussed before. Combining these branching fractions, we show contours of BR(t →
+
Π+
t b) × BR(Πt → τ ν) in the MΠ+ vs. sin ω plane in the right-hand pane of Fig. 3.6.
t

However, the “SM-like” top-pair events to which the signal events are compared in setting
a limit on exotic top decays will no longer include only t → W b events. This sample will now
potentially contain events in which a top-pion decays to bt∗ , yielding t → Π+ b → W + b¯bb,
where the W + b comes from the off-shell top quark. While the kinematic features of these
top decays will differ from those of SM decays, the events may be similar enough to be picked
up in the SM top quark sample. To see how common these events are, we show contours
+
∗
of BR(t → Π+
t b) × BR(Π → t b) in the plane of MΠ+ and sin ω in the left-hand panel of
t

Fig. 3.7. The product of branching ratios can be significant: it lies above 0.3 for sin ω < 0.45
and MΠ+ ∼ 140 GeV. In this case more than half of all tt¯ events would contain at least one
t

±
∗
± ¯
top quark decaying to Π±
t b followed by Πt → t b → W bb; we suspect that this could distort

kinematic distributions and b-tag rates in the tt¯ sample enough to be noticed. Similarly, for
sin ω = 0.5 and MΠ+
t

+
∗
145 GeV, we find BR(t → Π+
t b) × BR(Π → t b)

0.2, leading

to about 40% of tt¯ events containing at least one top quark decaying to Π±
t b followed by
∗
± ¯
Π±
t → t b → W bb.

While deliberately distinguishing these t → Π+ b → W + b¯bb events from SM top quark
88

0.8
0.7

sin ω

0.6
B = 0.01

0.5

BR(Πt+ → τ ν)
0.6
0.5
0.4
0.3
0.2
0.1

0.4
0.3
0.2
100

110

120

130

140
150
MΠ + (GeV)

160

170

180

t

0.8
0.7

sin ω

0.6
0.5
BR(t → Πt+ b)*BR(Πt+ → τ ν)

0.4

0.2
0.1
0.05
0.02
0.01

0.3
0.2
100

110

120

130
MΠ + (GeV)

140

150

160

t

Figure 3.6:

Top: Contours of BR(Π+
t → τ ν) as a function of MΠ+ and sin ω. Bottom:
t

+
Contours of BR(t → Π+
t b) × BR(Πt → τ ν) as a function of MΠ+ and sin ω. We interpret
t

+
the ATLAS t → H + b search [208] to exclude B ≡ BR(t → Π+
t b) × BR(Πt → τ ν) > 0.01.

89

decays would require a dedicated analysis, in the meantime, we can make the conservative
assumption that all of these events will be included in the “SM-like” sample. When this is
the case, the comparison between exotic and SM-like events gives a conservative upper limit
+
on BR(t → Π+
t b) × BR(Πt → τ ν). When some of these events are not picked up in the

SM-like sample, the true upper bound on the product of branching fractions is actually even
stronger.
We are now ready to determine the constraints on our model. Reference [208] sets
an upper bound on B ≡ BR(t → H + b) (with BR(H + → τ ν) = 1) of B
MH + = 90 GeV, falling to B

0.05 for

0.01 for MH + = 120–160 GeV. Therefore we can take

the right-most contour in the left-hand pane of Fig. 3.7 as the rough exclusion limit on Π+
t
from this search channel. This excludes charged top-pion masses below about 118, 140,
149, and 153 GeV for sin ω = 0.2, 0.4, 0.6, and 0.8, respectively. We have overlaid this
exclusion curve on the plots in Fig. 3.6 to make it easier to see what values of BR(Π+
t → τ ν)
+
∗
and BR(t → Π+
t b) × BR(Π → t b) are still allowed in our model. Note, for instance,

that, for sin ω

0.6, the region of parameter space where the B ≤ 0.01 limit falls has

BR(Π+
t → τ ν) < 0.1.
Finally, examining the right-hand pane of Fig. 3.7, we see that if the new state observed in
diphotons at around 125 GeV is to be interpreted as a Π0t (with the event rate yielding sin ω =
0.5), then we would interpret the combination of the diphoton data from Refs. [206, 207]
and ATLAS search [208] for t → H + b with H + → τ ν as jointly constraining the Π+
t to be
heavier than about 145 GeV.
Therefore, the only phenomenologically viable case involves non-degenerate top-pions. As
discussed in detail in Appendix B.1, however, this differs from the standard expectation in
top color models and implies that new sources of isospin violation would have to be present.
90

0.8

+

+

BR(t → Πt b)*BR(Πt → t b)
0.02
0.05
0.1
0.2
0.3

0.7

sin ω

0.6
0.5
0.4
B = 0.01

0.3
0.2
100

110

120

130

140
150
MΠ + (GeV)

160

170

180

t

0.8
Excluded by top decay, B > 0.01
0.7

sin ω

0.6
0.5
0.4
Excluded by ATLAS/CMS Πt0 → γγ

0.3
0.2
100

110

120
130
140
MΠ 0 = MΠ + (GeV)
t

150

160

t

+
∗
Top: Contours of BR(t → Π+
t b) × BR(Π → t b) as a function of MΠ+ and
t
sin ω. On each plot, only the region of parameter space to the right of the red dot-dashed
curve labeled “B=0.01” is still allowed by data on top-quark decays, as shown in Fig. 3.6.
Bottom: comparison of the ATLAS t → H + b exclusion and the ATLAS (solid) and CMS
(dashed) Π0t → γγ limits [206, 207], assuming degenerate Π0t and Π+
t .

Figure 3.7:

91

3.5

Top-Higgs phenomenology

In addition to the top-pion states discussed above, models in which the top-quark plays a
direct role in electroweak symmetry breaking contain a “top-Higgs” state. Such a state is
expected to have a mass greater than about 200 GeV, and we have previously demonstrated
[126] that such a top-Higgs state would produce ZZ and W W signals much larger than
those characteristic of a SM Higgs of the same mass when decays to pairs of top-pions are
kinematically forbidden. In this section we consider the constraints on the top-Higgs state
assuming that the neutral top-pion is the new boson discovered at the LHC.
The couplings of the top-Higgs, along with its decay widths to the most relevant channels
+ −
∓
0
0 0
W W , ZZ, tt¯, Π±
t W , Πt Z, Πt Πt , and Πt Πt , are given in detail in Ref. [126]. For

completeness, we reproduce the formulas for the key decay widths in Appendix B.3, along
with the ratio between the LHC production cross-sections for the top-Higgs and the SM
Higgs. We will first establish the current mass limits on the top-Higgs based on data from
the ATLAS and CMS experiments. We then comment on the discovery prospects for the
top-Higgs in the channel Ht → Π0t Z at the 14 TeV LHC.
Reference [126] used the combined SM Higgs limits from the LHC to determine the
excluded range of top-Higgs masses as a function of sin ω, for various values of the toppion mass. In the mass range of interest, the LHC limits come entirely from the SM Higgs
decays into W W and ZZ, and so are directly applicable to the top-Higgs after rescaling by
the appropriate ratios of production cross section and decay branching ratios. The limits
of Ref. [126] used ATLAS results with 1.0–2.3 fb−1 and CMS results with 1.1–1.7 fb−1 of
integrated luminosity at 7 TeV. Here we update the limits using the more recent CMS SM
Higgs search results based on 4.6–4.8 fb−1 at 7 TeV and also consider how the limits translate

92

to the case where the charged and neutral top-pions are not degenerate.
400
sin ω = 0.40
0.47
0.59
0.70

350

MΠ 0 = 125 GeV
t

Excluded by SM Higgs search
250

t

MΠ + (GeV)

300

200
150
Excluded by top decay, B > 0.01
100
250

300

350

400
450
MH (GeV)

500

550

600

t

Figure 3.8: The CMS exclusion contours for MHt from searches for the SM Higgs in W W
and ZZ final states [209], as a function of MΠ+ for the special case MΠ0 = 125 GeV. We
t

t

show sin ω values of 0.40 (solid lines), 0.47 (long-dashed lines), 0.59 (short-dashed lines), and
0.70 (dotted lines), which correspond to a rate for the 125 GeV Π0t in the γγ channel relative
to that of the SM Higgs of σ/σSM = 3.0, 2.0, 1.0, and 0.5, respectively. The horizontal (red)
lines show our lower bound on MΠ+ from the ATLAS t → H + b search [208] for the same
four sin ω values.

t

Fig. 3.8 shows how the top-Higgs exclusion curves behave for a variety of sin ω values.
Given that light charged top-pions are excluded by the ATLAS search [208] for t → H + b,
the top-Higgs cannot have a mass lower than about 250-300 GeV.
There is also a theoretical bound to bear in mind. For small values of sin ω, the top-Higgs
couplings violate perturbativity for sufficiently high Ht masses, when the decay channels to
two tops and two top-pions open up; roughly speaking this occurs when the top-Higgs width
exceeds its mass. For sin ω ≈ 0.4 we find this constrains the top-Higgs mass to lie below
about 600 GeV, while for sin ω ≥ 0.5, perturbativity considerations do not constrain the
region of interest.
Many of the decay channels that are available to a heavy top-Higgs result in hadronic

93

t
Ht

Z

Figure 3.9: The production of a neutral top-pion and a Z from an s-channel top-Higgs.
final states with large SM backgrounds. A potential exception is Ht → ZΠ0t →

γγ as

shown in Fig. 3.9. Assuming the state discovered at 125 GeV is the neutral top-pion, one
can then take advantage of the Ht Π0t Z coupling, and look for the top-Higgs in the process
pp → ZΠ0t →

γγ by using an invariant mass cut on the diphotons to cull background.

We find that discovery in the allowed parameter space (see Fig. 3.8) is not possible for
sin ω values of 0.7 and above in this channel. Even for lower values of sin ω, the minimum
integrated luminosity required for a 5σ discovery at the 14 TeV LHC in this mode is 100
f b−1 and a luminosity several times greater is required in most of the MHt vs MΠ+ plane.
t
Therefore, we conclude that this will not be a realistic discovery mode for the top-Higgs in
the case of a light neutral top-pion. The most promising search channels for the top-Higgs
therefore remain the W W and ZZ final states as used in the SM Higgs search.

3.6

Conclusions

In this chapter, we have analyzed the phenomenology of the top-pion and top-Higgs states
in models with strong top dynamics, and have translated the present LHC constraints on
the SM Higgs into bounds on these scalar states.
We have seen that it is possible for the observed excess in the HSM → γγ search channel
to correspond to a neutral top-pion of mass MΠ0 = 125 GeV. Based on the size of the crosst

94

section observed [168, 45, 161, 44], the corresponding value of sin ω would be approximately
0.5. Because Π0t is a pseudo scalar, however, models of strong top dynamics do not predict
a visible signal in the ZZ → 4 channel or the W W channel, nor a diphoton signal in
the vector boson fusion production channel, nor any associated production of the 125 GeV
object with a W or Z. Therefore, as additional data is accumulated, we would expect the
diphoton resonance to continue to grow in significance, the initial signals in the ZZ → 4
and W W channels to fade away, and the dijet-tagged diphoton signal to persist only at
a level consistent with dijet-tagged gg → Π0t rather than dijet-tagged vector boson fusion
events. Moreover, in the context of these models, we would also expect that a signal in the
ditau decay channel would be present but less visible for the Π0t than for the SM Higgs.
For the range of model parameters where the neutral top-pion can account for the LHC
diphoton signal, searches for non-standard top-quark decays to charged scalar plus bottom
quark exclude charged top-pions with masses up to about 145 GeV (as in the left-hand panel
of Fig. 3.7). These searches continue to become more sensitive as the decay properties of
the top-quark are measured more accurately. As a result, if the neutral top-pion has a mass
of 125 GeV, it cannot be degenerate with the charged top-pion, as one would more typically
expect in models of strong top dynamics. Instead, the model must contain substantial isospin
violation to produce this top-pion mass splitting.
We have also updated limits on the top-Higgs. Our results show that current LHC
searches for the SM Higgs in W W and ZZ exclude the existence of a top-Higgs state up to
masses of order 300 GeV, with some dependence on the charged top-pion mass and sin ω as
shown in Fig. 3.8.
The implication is that current searches at the LHC strongly constrain theories with
strong top-dynamics. The top triangle moose model interpolates [126] between a variety
95

of strong top dynamics models as the value of sin ω varies between about 0.2 and 0.8,
the range studied in this chapter. In the context of strong top-dynamics, the new boson
observed at the LHC is too light to be the top-Higgs [126]. Instead, the diphoton signal can
be produced by a neutral top-pion of the appropriate mass and couplings, assuming that
one constructs a theory including additional isospin violation, but in this case we would not
expect a significant signal in the ZZ → 4 channel. This last stipulation is problematic since
both LHC experiments report a 3σ signal in the four-lepton channel with the current data
set. Moreover, if the diphoton signal corresponds to a neutral top-pion, then the theoretical
context cannot be the most familiar part of the top-triangle-moose parameter space in which
0.2 ≤ sin ω ≤ 0.5, the top-pions are degenerate, and the top-Higgs has a mass of order 2mt :
i.e. the portion of the parameter space corresponding to classic TC2 models. Rather, the
context would be the less-explored region in which sin ω is of order 0.5 or greater, the toppions have a substantial mass splitting, and the top-Higgs is heavier: i.e. a model in which
the strong top dynamics are of the top seesaw form.
We anticipate that additional LHC data will provide further clarity about the nature of
the diphoton resonance and its possible connection to strong top dynamics.

96

Part II
Coloron Phenomenology

97

Chapter 4
Distinguishing Color-Octet and
Color-Singlet Resonances at the Large
Hadron Collider
— This chapter is based on a work in collaboration with Anupama
Atre, R.Sekhar Chivukula, and Elizabeth H. Simmons which has
appeared in [6]. The contents have also appeared in the conference
proceeding [5].

4.1

Introduction

We have seen in Part I how the searches for and the discovery of the Higgs boson place
constraints on new physics models describing electroweak symmetry breaking via strong
dynamics. Now, in Part II, let us switch gears slightly to a common feature among a larger
class of new physics models: a vector resonance. In particular, we focus on the resonances
that could be discovered at a hadron collider.
Hadron colliders are a rich source for the production of new resonances with strong
coupling due to colored particles in the initial state. A particularly simple and powerful
probe of new colored resonances is the di-jet channel where the resonance decays to two
98

partons. Each successive hadron collider with an increase in center of mass energy and
integrated luminosity has been able to probe for di-jet resonances with higher masses. Many
well motivated theories of physics beyond the Standard Model (SM) predict new particles
that give rise to signatures in the di-jet channel. These new particles can have different spin
and color structure and a sample of such possibilities is listed below.
The color-octet vector boson arises as a result of extending the gauge group of the strong
sector. The chiral structure of the couplings between quarks and the color-octet varies
and the couplings can either be flavor universal or flavor non-universal. Examples of flavor
universal scenarios are the axigluon [210, 211] and coloron [212, 213] where all quarks are
charged under the same SU (3) group. Flavor non-universal scenarios appear in the case of
the topgluon where the third generation quarks are assigned to one SU (3) group and the
light quarks to the other [98, 100] and the axigluon where different chiralities of the same
quark can be charged under different groups [214, 215, 216, 217, 218, 219] . Other examples
include Kaluza-Klein (KK) gluons which are excited gluons in extra-dimensional models
[220], technirhos which are composite colored vector mesons found in technicolor [152, 99,
221], models that include colored technifermions and low-scale string resonances [222].
The electrically neutral color-singlet vector boson, collectively called a Z , also appears
in many beyond SM physics scenarios and can originiate from extending the electroweak
U (1) or SU (2) gauge group. The Z can also have flavor universal [223, 224, 225] or flavor
non-universal couplings to fermions [226, 227, 228]. For reviews of Z models, see Refs. [129,
134, 229] and the references therein. Other examples of color-singlet states probed in the
di-jet channel include the spin-2 gravitons in Randall-Sundrum models [230, 231] which are
KK excitations of the gravitational field [232, 233] and the electrically charged color-singlet
vector boson, namely the W [121].
99

Searches for new resonances in the di-jet channel have been performed at the CERN
S¯
ppS [234, 235], Tevatron [236, 237, 238, 239, 240] and the Large Hadron Collider (LHC) [241,
242, 243, 244, 245, 246, 247, 248, 249]. Once a resonance has been discovered the next step
is to measure the properties of the resonance. The di-jet invariant mass mjj and the angular
distributions of energetic jets relative to the beam axis are sensitive observables to determine
the mass and spin of the resonance. In this work [6], we explore the question of determining
the color structure of the resonance produced in the di-jet channel; in particular we explored
whether the resonance is a color-octet or a color-singlet state.
To distinguish a color-octet and a color-singlet resonance produced in the di-jet channel
we introduce a new variable called the color discriminant variable. Assuming that the new
resonance decays only to quarks, this variable reflects the color structure of the resonance and
is constructed from measurements which can be made in the di-jet discovery channel, namely
the di-jet cross section, mass and width of the resonance. We demonstrate the utility of this
variable in distinguishing color-octet and color-singlet resonances using the simple flavor
universal example of a coloron and a leptophobic Z . We also demonstrate the robustness
of this method using a simplified flavor non-universal scenario1 . We study the sensitivity
of the LHC with center of mass (c.m.) energy of 14 TeV and integrated luminosities of
30, 100, 300 and 1000 fb−1 to distinguish color-octet and color-singlet resonances. Motivated
by current constraints on di-jet resonances and future prospects for discovery, we probe
masses ranging from 2.5 − 6 TeV with various couplings and widths.
The rest of the chapter is organized as follows. In Sec. 4.2 we describe phenomenological
models for a coloron and a leptophobic Z used as illustrative examples in this chapter. We
introduce a color discriminant variable to distinguish color-octet and color-singlet resonances
1 The general flavor non-universal scenarios are studied in Chapter 5.

100

in Sec. 4.3. We discuss the current constraints from collider searches and the allowed regions in parameter space in Sec. 4.4 and the uncertainties involved in measuring the color
discriminant variable at the LHC in Sec. 4.5. We present our results in Sec. 4.6 and conclusions in Sec. 4.7. A discussion about uncertainties relevant to our analysis is presented in
Appendix C.

4.2

General Parameterization

Color-octet and color-singlet resonances of interest to our study may be motivated in many
beyond the SM physics scenarios as described in the introduction. Therefore we study
a phenomenological model of color-octet and color-singlet resonances independent of the
underlying theory to keep our study widely applicable. We will assume that there are no
additional colored states into which the resonance can decay. If new light states are present,
studying the properties of the decays of the coloron or Z into these new particles will be
instructive (see Refs. [250, 251] and references therein). The couplings of the color-octet
and color-singlet resonances to SM quarks can all be the same in the simple flavor universal
scenario and can all be independent in the most general flavor non-universal case. In this
section, we present details about the parameterization of the interactions of color-octet and
color-singlet resonances for two cases, the flavor universal scenario and an illustrative flavor
non-universal scenario.
The interaction of a color-octet resonance Cµ with the SM quarks qi has the form

LC = igs

i P + gi P
q¯i γ µ gC
L
R qi C µ ,
C
i

L

101

R

(4.1)

i
i
where Cµ = Cµa ta with ta an SU(3) generator, gC
and gC
denote left and right chiral
L
R

coupling strengths of the color-octet to the SM quarks relative to the QCD coupling gs ,
the projection operators have the form PL,R = (1 ∓ γ5 )/2 and the quark flavors run over
i = u, d, c, s, t, b. We will denote the color-octet resonance by C and its chiral couplings to
q

t
light quarks by gC
and to the third generation by gC
and use the terms color-octet
L,R
L,R

and coloron interchangeably.
The color-octet resonance with the interactions as in Eq. (4.1) decays primarily to two
jets or a top pair and its decay width is given by

ΓC = αs
+

MC
q 2
q 2
t 2 + gt 2
4 gC + gC
+ gC
CR
L
L
R
12

t 2 + g t 2 )(1 − µ ) + 6g t g t µ
(gC
t
CL CR t
CR
L

1 − 4µt ,

(4.2)

where MC and ΓC are the mass and intrinsic width of the color-octet respectively. Decays
to top quarks are modified by the kinematic factors involving µt = m2t /MC2 with mt the
top quark mass. Strictly speaking, the bottom quark’s contribution to the width is modified
by factors involving µb = m2b /MC2 but we ignore these factors since m2b << MC2 . For a
color-octet that is heavy compared to the top quark, the expression for the total width of a
color-octet in Eq. (4.2) simplifies to

ΓC = αs

MC
q2
t2
4gC + 2gC
12

(4.3)

where we have defined
i 2 ≡ gi 2 + gi 2
gC
CL
CR

for i = q (light quarks u, d, c, s) and i = t (quarks in the third generation t, b).
102

(4.4)

We parameterize the interaction of a color-singlet similarly to that of the color-octet. If
the color-singlet has tree level couplings to SM leptons also, it can be easily distinguished from
a color-octet as the octet has no decays to leptons. Hence we consider only the leptophobic
variant of the color-singlet, as this can mimic a color-octet resonance in the di-jet channel.
We will call such a resonance a leptophobic Z henceforth. The interactions of a leptophobic
Z with the SM quarks are given by

LZ = igw
where g i

ZL

and g i

ZR

i

i P + gi
q¯i γ µ gZ
P qZ ,
L
ZR R i µ
L

(4.5)

denote left and right chiral coupling strengths of the leptophobic Z to

the SM quarks relative to the weak coupling gw = e/ sin θW and the quark flavors run over
q
Z L,R

i = u, d, c, s, t, b. Again the chiral couplings of the Z to light quarks are denoted by g
and to the third generation by g t

Z L,R

.

The leptophobic Z with the interactions as in Eq. (4.5) decays primarily to two jets or
a top pair and its decay width is given by

ΓZ

= αw
+

MZ
q 2
q 2
4 g
+g
Z
Z
2
L
R

t 2 + gt 2
+ gZ
ZR
L

t 2 + g t 2 )(1 − µ ) + 6g t g t
(gZ
µ
t
ZR
ZL ZR t
L

1 − 4µt ,

(4.6)

where MZ and ΓZ are the mass and intrinsic width of the leptophobic Z respectively.
Similar to the case of the color-octet, we can neglect the kinematic factors involving the
mass of the top and bottom quarks when the Z is much heavier than the SM quarks. Hence

103

the expression for the decay width of a Z as given in Eq. (4.6) simplifies to

ΓZ = αw

MZ
q 2
t2 ,
4g + 2gZ
Z
2

(4.7)

where g i 2 ≡ g i 2 + g i 2 for i = q, t. In the rest of the chapter, and also the next chapter,
ZL

Z

ZR

we will only consider color-octet and color-singlet resonances that are much heavier than the
top quark.

4.2.1

Flavor Universal Scenario

In the flavor universal scenario all SM quarks have the same coupling to the color-octet
q

resonance, i.e. gC

L,R

t
= gC

L,R

= gC

L,R

. The expression for the decay width of a color-octet

as given in Eq. (4.3) simplifies further to

ΓC =

αs
M g2 ,
2 C C

(4.8)

and the branching fraction for the color-octet resonance to decay to jets obeys the simple
relation
BR(C → jj) = 5/6,

where j = u, d, c, s, b.

(4.9)

Similarly, the decay width of a Z with flavor universal couplings to SM quarks simplifies
to
2 ,
ΓZ = 3αw MZ gZ
q

(4.10)

2 /4π and g
where αw = gw
= gt
= gZ
. The branching fraction for a Z to decay
Z L,R
Z L,R
L,R

104

to jets obeys the simple relation

BR(Z → jj) = 5/6,

where j = u, d, c, s, b

(4.11)

Note that although the width of the coloron is proportional to the strong coupling
(αs (mZ )
(αw

0.12) and that of the leptophobic Z is proportional to the weak coupling

2 + g2
0.04), the two resonances will have comparable widths when the couplings gL
R

are the same. This is due to the difference in the color factors for the two resonances.

4.2.2

An Illustrative Flavor Non-universal Scenario

The couplings of a color-octet and color-singlet resonance to SM quarks can all be independent in the most general flavor non-universal scenario. While it is desirable to study
the most general case, it is computationally cumbersome and will be postponed to Chapter
5. Instead we consider an intermediate scenario where the couplings of the color-octet and
the color-singlet to quarks in the third generation are different from the couplings of the
quarks in the first two generations. An interesting example of such a scenario is that of
a color-octet resonance (described in Refs. [214, 216, 215, 217]) that can enhance top-pair
forward-backward asymmetry observed at the Tevatron [252, 253, 254, 255, 256]. For the
Z , models related to strong dynamics typically feature a Z that couples with an enhanced
strength to quarks in the third family (see Ref. [129] for a review of such cases).
The couplings of a color-octet or color-singlet resonance to SM quarks in the flavor nonuniversal scenario we consider can be parametrized by

t
gC/Z

L,R

≡ ξg
105

q
C/Z L,R

,

(4.12)

where t = t, b and q = u, d, c, s. The change in the total decay width of a color-octet and
color-singlet in the flavor non-universal scenario compared to the width in the flavor-universal
case (given by Eq. (4.8) and Eq. (4.10) respectively) is given by

Γnon−universal = Γuniversal

4 + 2ξ 2
6

.

(4.13)

The branching fraction of the color-octet and color-singlet to jets, where the jets are defined
to include j = u, d, c, s, b changes from

Br(C/Z → jj) = 5/6

(4.14)

in the flavor universal case to

Br(C/Z → jj) =

4 + ξ2
4 + 2ξ 2

(4.15)

in the flavor non-universal case. Hence the branching fraction to jets in the flavor nonuniversal case decreases to 1/2 from 5/6 in the flavor universal case as ξ becomes large.
This tendency towards smaller branching fraction in the flavor non-universal case tends to
decrease the overall di-jet cross section for the resonance. The production rate stays the
same, as it is dominated by the contribution from the first (and to a small extent from the
second) generation quarks; the bottom quark with negligible parton luminosity contributes
very little. However, the branching fraction reduces compared to the flavor universal scenario
and hence the total di-jet cross section decreases.
Finally we note that in the narrow-width approximation the quantity ξ can be determined
by a measurement of the ratio of the cross sections where the resonance decays to top quark
106

pairs or to jets:
σ(pp → C/Z → tt¯)
σ(pp → C/Z → jj)

Br(C/Z → tt¯)
ξ2
=
.
Br(C/Z → jj)
4 + ξ2

(4.16)

We introduce next a new variable to help distinguish color-octet and color-singlet states,
described in this section, in the di-jet channel.

4.3

Color Discriminant Variable

In the previous section, we introduced generic color-octet and leptophobic color-singlet states
that couple to SM quarks with differing strengths and color structures. A resonance of either
type will be produced copiously at a hadron collider and will decay into two jets or top quark
pairs. Decays in the top quark channel have the advantage of possible leptons in the final
state and hence better reconstruction efficiencies compared to the SM QCD background.
However they suffer from a smaller cross section (due to smaller branching fraction to top
pairs), overall smaller efficiency due to the large number of final state particles and still a
relatively large background. On the other hand the simple topology of the decay into two
highly energetic central jets with a larger branching ratio and higher efficiency of jet selection
makes the di-jet channel a discovery mode despite very large QCD backgrounds. Such
resonances will be discovered in the di-jet channel as simple “bumps” (in the narrow width
approximation) over an exponentially falling QCD di-jet background. The channel with
decays to top quark pairs, while suffering from a smaller rate, is nonetheless an important
one as it helps determine flavor universal from flavor non-universal scenarios as described in
Sec. 4.6.2.
Once such a resonance has been discovered in the di-jet channel, it is possible to obtain a
measurement of its cross section, mass and width. The question then arises about the nature
107

of the resonance - is it a color-octet or a color-singlet resonance? An estimate of the couplings
2 + g 2 ) for each resonance can be obtained from the cross section measurement [257]. This
(gL
R

can possibly be used to eliminate some scenarios where the deduced couplings are either not
motivated theoretically or excluded by other experiments. However, there will be large
regions in parameter space where both color-octet and leptophobic color-singlet scenarios
survive. To keep our study widely applicable we propose a model-independent approach to
distinguish color-octet and color-singlet resonances that depends mainly on the kinematics
of the process.
The cross section for the production and decay of a resonance in the narrow-width approximation can be written as

V

V ≡ σ(pp −→ jj)
σjj

σ(pp → V )Br(V → jj),

(4.17)

where V is a generic resonance, σ(pp → V ) is the cross section for producing the resonance
and Br(V → jj) is the branching fraction for the resonance to decay to jets. Using this
approximation and factoring out the couplings, color factors and mass dependence explicitly
we can write the cross section for a color-octet as

4
2 1
αs gC
Wq (MC )Br(C → jj)
9
MC2 q
8 ΓC
=
Wq (MC )Br(C → jj),
9 MC3 q

σ(pp → C → jj) =

(4.18)

where the expression for the width of the flavor universal coloron in Eq. (4.8) has been used
to obtain the final form of Eq. (4.18). Similarly, we can express the cross section for the

108

leptophobic Z as

σ(pp → Z → jj) =

1
1
2
αw gZ
3
M2

Z

1 ΓZ
=
9 M3

Z

q

q

Wq (MZ )Br(Z → jj)

Wq (MZ )Br(Z → jj),

(4.19)

where the expression for the width of the flavor universal leptophobic Z in Eq. (4.10) has
been used to obtain the final form of Eq. (4.19). Here, Wq (MV ) (V = C, Z ) is dependent
only on the parton distribution functions (PDFs), kinematics, and phase space factors and
is defined by

Wq (MV ) = 2π 2

MV2
s

1

dx
fq x, µ2F fq¯
2
x
MV /s

MV2 2
,µ
sx F

+ fq¯ x, µ2F fq

MV2 2
,µ
sx F

,
(4.20)

where fq x, µ2F is the parton distribution function at the factorization scale µ2F . Throughout this and the next chapter, we set µ2F = MV2 .
Suppose a new di-jet resonance is discovered with a particular cross section and mass; it
is important to determine whether it is a coloron described by Eq. (4.18) or a Z described
by Eq. (4.19). Comparing Eqs. (4.18) and (4.19) for equal σ and equal M and noting that Σ
and di-jet branching ratios are also equal in the two cases, we find the following relationship
between the widths:
Γ∗Z = 8Γ∗C ,

(4.21)

where the asterisk signifies that we are comparing bosons with equal production cross-sections
(σ). This implies that if a resonance is discovered in the di-jet channel, a measurement of
the width can point to the color-structure of the resonance discovered.

109

We get a similar expression for the flavor non-universal case by replacing the expressions
for the width and the branching fraction in Eqs. (4.18) and (4.19) by those in Eqs. (4.13)
and (4.15). For the flavor non-universal coloron we have

1
4
2
αs gC
Wq (MC )Br(C → jj)
R M2
9
C q
8 ΓC
6
4 + ξ2
=
W
(M
)
q
C
9 MC3 4 + 2ξ 2
4 + 2ξ 2
q

σ(pp → C → jj) =

,

(4.22)

where ΓC is the width of the coloron in the flavor non-universal case. For the flavor nonuniversal leptophobic Z we have

σ(pp → Z → jj) =

1
1
2
αw gZ
3
M2

1 ΓZ
=
9 M3

Z

Wq (MZ )Br(Z → jj)

q

Z

6
4 + 2ξ

2

q

Wq (MZ )

4+ξ
4 + 2ξ

2
2

,

(4.23)

where ΓZ is the width of the leptophobic Z in the flavor non-universal case. For a given ξ,
which is determined by a measurement of the ratio of the cross sections as given in Eq. (4.16),
we see that, for resonances with a mass M and yielding equal total di-jet cross sections, the
relation between the width of the coloron and the leptophobic Z in the flavor non-universal
case remains the same as before:
Γ∗Z = 8Γ∗C .

(4.24)

We use this relation to introduce a new variable to distinguish color-octet and color-singlet
resonances and parameterize it in a model-independent way.
As discussed earlier, a discovery in the di-jet channel will inspire three immediate measurements - cross section, mass and width, which in turn depend on model dependent pa110

rameters such as couplings, color-structure and mass of the resonance. Note that the cross
section is proportional to the color structure, square of the couplings and M −2 while the
width is proportional to square of the couplings and M. Denoting by σjj the cross section
for producing a resonance in the di-jet channel, we define

Dcol ≡

M3
σ ,
Γ jj

(4.25)

as a color discriminant variable that is dimensionless by construction. This variable depends
only on the color structure of the resonance being considered for a given ξ. For example,
any two points in the parameter space of a coloron will lead to the same Dcol for a given ξ
where as points in parameter space of a coloron and a leptophobic Z will lead to different
values of Dcol for the same ξ. Thus one can distinguish a color-octet and a color-singlet state
in a relatively model-independent fashion i.e. without analyzing each point in parameter
space separately. Next we discuss the constraints on the parameter space and the discovery
potential of color-octet and color-singlet states at the LHC.

4.4

Parameter Space in di-jet Channel

In this section we describe the region of parameter space in which using the color discriminant
variable is applicable. First we discuss the current constraints on the parameter space of
coloron and leptophobic Z models. Next we discuss the discovery prospect for colorons and
leptophobic Z s at the LHC with c.m. energy of 14 TeV, since the question of distinguishing
the color structure of a resonance will arise only after the resonance has been discovered.
The discovery and exclusion regions for the coloron and leptophobic Z are presented in

111

Fig. 4.1(a) and (b), respectively, for the flavor universal scenario and in Fig. 4.1(c) and (d),
respectively, for the flavor non-universal scneario.

3.0

3.5

fb −1
100
0
5σ fb −1
sen
siti
vity

fb −1

300

no
4.0

4.5
M [TeV]

5.0

5.5

6.0

b −1
sen
siti
vity

fb −1

no

5σ

0f

0.25

Γ Mres

Γ Mres
2.5

3.5

g = 3g

gLq 2 + gRq 2 /gw2

30 f −
b 1
100
fb −1
300
fb −1
100
no 0 fb −1
5σ
sen
siti
vity

LHC ex
clusion

gLq 2 + gRq 2 /gs2

0.5

3.0

Non-universal
Leptophobic
Z
t
q
Γ ≥0.15M

gt = 3gq

Γ ≥0.15M

30 fb −1

30 fb −1
0.5

Non-universal Coloron
1.0

LHC e
xclusio
n

gLq 2 + gRq 2 /gw2

2.5

100

3.5

Γ Mres

300

3.0

fb −1

fb −1
30

Γ Mres

0.5

fb −1

2.5

fb
300 fb−1
1000itivity
sens
no 5σ
4.0 4.5 5.0 5.5 6.0
M [TeV]

1.0

100

1.0

1

−

100

2.0

LHC e
xclusio
n

gLq 2 + gRq 2 /gs2

3.0

Leptophobic Z

1.5 Γ ≥0.15M

100

Coloron

4.0 Γ ≥0.15M

4.0

4.5
M [TeV]

5.0

5.5

2.5

6.0

3.0

3.5

4.0

4.5
M [TeV]

5.0

5.5

6.0

Figure 4.1: (a) Top left: 5σ discovery reach for a flavor universal coloron in the
√ plane of the
mass of the coloron (in TeV) and the square of the couplings at the LHC with s = 14 TeV.
The discovery reach is shown in varying shades of blue for different luminosities ranging
from 30 fb−1 to 1000 fb−1 . The area marked “no 5σ sensitivity” corresponds to no discovery
reach at 1000 fb−1 but may have some reach at higher luminosities. The area marked
“LHC exclusion” in gray corresponds to the exclusion from 8 TeV LHC [249]. The region
above the dashed line marked Γ ≥ 0.15M corresponds to the region where the narrow-width
approximation used in di-jet resonance searches is not valid [258, 259, 131]. The region
below the horizontal dashed line marked Γ ≤ Mres corresponds to the region where the
experimental mass resolution is larger than the intrinsic width [249]. See text for further
details. (b) top right: same as (a) but for a leptophobic Z and the discovery reach is shown in
varying shades of green. (c) bottom left: same as (a) but for the flavor non-universal coloron
q
t
where gC
= 3gC
. (d) bottom right: same as (b) but for the flavor non-universal Z
with

L,R
t
g
=
Z L,R

L,R
q
3g
.
Z L,R

112

Searches for di-jet resonances where the width of the resonance is small compared to the
mass have been carried out at the LHC by both CMS and ATLAS collaborations2 . They
have found no evidence of such resonances and set exclusion limits on the product of cross
section, branching ratio and acceptance for the 8 TeV LHC run [246, 248, 249]. They also
present the theoretical estimate of the product of cross section times branching fraction for
various sample models, including colorons and sequential Z s. The acceptance for each model
can then be estimated as a function of the mass without doing a full detector simulation.
Using this estimated acceptance we translate the exclusion limits from Ref. [249] to obtain
excluded regions in the plane of mass and coupling for the case of a coloron. The di-jet
analysis [249] presents results only for a sequential Z and hence we apply the acceptance
for a sequential Z to the leptophobic Z as well. This is reasonable as the acceptance is
dependent mainly on kinematics and not on the couplings and their structure at leading
order. Note that in estimating the excluded regions we have used the most stringent results
which come from CMS [249]. The results of this exercise are presented as gray regions labeled
“LHC exclusion” in Fig. 4.1 for the coloron and leptophobic Z . Note that a larger region
of parameter space is excluded for the coloron compared to the leptophobic Z due to the
stronger coupling strength of the coloron to the SM quarks. On the other hand the excluded
region for the flavor universal scenario is larger than that of the flavor non-universal scenario
due to the smaller total cross section in the latter case as discussed earlier.
A resonance in the di-jet channel will be observed as a fluctuation in the exponentially
falling QCD background. An estimation of the QCD di-jet background is notoriously difficult
due to the reliance on leading order cross sections, uncertainties in the estimation of jet energy
2 For a recent compilation of bounds on di-jet resonances, and their interpretation in terms of resonance
couplings, see Ref. [257]. The results we present here, obtained from the experimental bounds cited, are
consistent with the results of Ref. [257].

113

scale and other systematic uncertainties. Hence data driven methods are often employed to
normalize the QCD background in a region away from the signal. In the absence of real data,
a fit is performed to samples from full detector simulation to estimate the experimental
sensitivity. As a full estimation of the QCD background and the sensitivity in the di-jet
channel is beyond the scope (and focus) of this article, we use the results presented in
Ref. [260] to estimate the discovery potential of a coloron and a leptophobic Z in the di-jet
channel at the LHC with

√
s = 14 TeV.

Similar to the case of current CMS studies, the authors of Ref. [260] present the minimum
cross section that can be observed at the LHC with

√
s = 14 TeV and for luminosities up to 10

fb−1 . They present their discovery potential results as a product of cross section, branching
ratio and acceptance for different masses of the resonance after taking into account statistical
uncertainties (from background fluctuation) and systematic uncertainties. The systematic
uncertainties include various sources such as jet energy scale, jet energy resolution, radiation
and low mass resonance tails and luminosity. In addition they also present the theoretical
estimate of the product of cross section times branching fraction for various sample models,
including colorons and sequential Z s. As before, the acceptance for each model can then
be estimated as a function of mass and we translate the 5σ discovery reach to regions in the
plane of mass and coupling. The discovery reach from 10 fb−1 is then scaled appropriately
to obtain the discovery reach for other values of integrated luminosities, L = 30, 100, 300
and 1000 fb−1 . The regions that can be probed at the 5σ level for the case of the flavor
universal and flavor non-universal coloron are presented as regions of varying shades of blue
in Fig. 4.1(a) and Fig. 4.1(c) respectively for the different luminosities listed above. For
the leptophobic Z , we use the acceptance for a sequential Z as explained earlier and the
discovery reach is presented in Fig. 4.1(b) and Fig. 4.1(d) as different shades of green for
114

the flavor universal and flavor non-universal case respectively. The region shown in red and
labeled “no 5σ sensitivity” in Fig. 4.1 corresponds to the case where the resonance will not
be discovered at 5σ with 1000 fb−1 . Owing to the stronger coupling strength of the coloron
to SM quarks the discovery reach for a coloron extends to larger masses while the reach
for the leptophobic Z is limited to lower masses due to the weak coupling to SM quarks.
Moreover, the discovery region for the flavor non-universal scenario shrinks compared to the
flavor universal scenario due to the smaller total di-jet cross section in the former case as
discussed earlier.
Note that the experimental search for resonances in the di-jet channel applies in the
region where the narrow-width approximation is valid. The authors of Refs. [258, 259, 131]
point out that this approximation is valid only up to Γ/M ≤ 0.15. The area above the top
dashed line in Fig. 4.1 indicates the region where the narrow-width approximation is not
valid. Similarly, the measurement of the width of a resonance is limited by the experimental
mass resolution Mres and intrinsic widths smaller than Mres cannot be distinguished. The
area below the bottom dashed line in Fig. 4.1 indicates the region where the intrinsic width
is smaller than the experimental resolution and cannot be distinguished. The estimate of the
experimental mass resolution varies with mass and has been obtained from Ref. [249]. Hence
the region where our analysis is applicable is between the two dashed lines and where there
is discovery potential indicated by blue (green) colored regions for a coloron (leptophobic
Z ).
Finally we present some details about our simulation of signal samples. The production
cross section was calculated using MadGraph5 [261] and CTEQ6L1 PDFs [262] were used.
The factorization and renormalization scales were set to be equal to the mass of the resonance.
Next we discuss the sensitivity of the LHC to measure the color discriminant variable to
115

distinguish color-octet and color-singlet states.

4.5

Sensitivity at the LHC

The color discriminant variable (Dcol ) is a function of the mass (M ) and intrinsic width (Γ)
of the resonance as well as the cross section for producing the resonance in the di-jet channel
(σjj ). Hence statistical and systematic uncertainties in the measurement of the di-jet cross
section, mass and intrinsic width of the resonance play a key role in the measurement of
the color discriminant variable and hence in distinguishing a color-octet from a color-singlet
resonance. In this section we discuss the uncertainties in the measurement of the mass,
intrinsic width and the cross section for producing the resonance in the di-jet channel and
their effect on the uncertainties in the measurement of Dcol at the LHC with

√
s = 14 TeV.

Motivated by current constraints and future prospects described in Sec. 4.4 we consider
resonance masses in the range 2.5 − 6 TeV.
The uncertainty in the measurement of the di-jet cross section can be written as
∆σjj
1
= √ ⊕ εσ sys ,
σjj
N

(4.26)

where N is the number of signal events, εσ sys is the fractional systematic uncertainty and
⊕ indicates that the uncertainties are added in quadrature. The discovery of a resonance
in the di-jet channel is a pre-requisite for measuring Dcol . Hence in the rest of the article
N indicates the number of signal events required (above background fluctuation) to obtain
a 5σ discovery and has been obtained from Ref. [260] as described in Sec. 4.4. The sources
of systematic uncertainties in measuring the di-jet cross section include jet energy scale, jet

116

energy resolution, radiation and low mass resonance tail and luminosity [36]. The effect of
all these systematic uncertainties was estimated in Ref. [260] and presented as a fractional
uncertainty (as a function of the mass) normalized to the di-jet cross section required to
obtain 5σ discovery (above background fluctuation) at the LHC with

√

s = 14 TeV. The

fractional uncertainty (εσ sys ) varies from 0.28 to 0.41 in the mass range of interest and is
listed in Table 4.1.
The uncertainty in the measurement of the di-jet mass is given by

∆M
1
σΓ Mres
=√
⊕
⊕
M
M
N M

∆M
,
M JES

(4.27)

where σΓ is the standard deviation corresponding to the intrinsic width of the resonance (Γ
2.35σΓ assuming a Gaussian distribution), Mres is the experimental di-jet mass resolution
and (∆M/M )JES is the uncertainty in the mass measurement due to uncertainty in the
jet energy scale. The various components of systematic uncertainties contributing to the
uncertainty in the mass measurement depend on each experiment and detector and their
estimate for different experiments and c.m. energies are listed in Table 4.1. The specific
values used in our analysis are indicated by an asterisk (*).
The uncertainty in the measurement of the intrinsic width is given by

∆Γ
=
Γ

1
1+
2(N − 1)

2
Mres 2
+
σΓ

Mres 4
σΓ

∆Mres 2
,
Mres

(4.28)

where ∆Mres is uncertainty in the di-jet mass resolution due to uncertainty in the jet energy
resolution. Again, the estimate of the various components contributing to the uncertainty in
the width measurement are listed in Table 4.1 for various experiments and the values used

117

in the analysis are indicated by an asterisk(*). See Appendix C for details on calculating
the expression for uncertainty in the intrinsic width given in Eq. (4.28).
The estimation of systematic uncertainties depends on detector details and the energy
of the collider and an accurate estimate of any systematic uncertainty can be done only
after the machine is operational and has been calibrated. However as the LHC is yet to
run at

√
s = 14 TeV, the energy for which we present our results, we have the choice

of using the systematic uncertainties estimated using full detector simulation (but no real
data) at
√

√
s = 14 TeV or of using the systematic uncertainties from real LHC data but for

s = 8 TeV. We will use the estimate for systematic uncertainties from actual LHC data

where available and assume that any future LHC run will be able to reach at least the current
level of uncertainties, if not better. This is a reasonable assumption as experiments tend to
make improvements in their estimation of errors and efficiencies with real data, compared
to original estimates from simulated data, due to improved experimental techniques. For
example, the uncertainty in mass due to the uncertainty in jet energy scale at the LHC
with

√
s = 14 TeV was expected to be about 5% [36] while the current analyses at 8 TeV

show that an uncertainty of 1.25% is achievable [249]. Note that we have modeled all
systematic uncertainties to be Gaussian and hence added them in quadrature. We have also
not included any correlation between the uncertainties. Most of the systematic uncertainties
were estimated for resonance masses up to 5 TeV for the different experiments and we have
extrapolated this estimate to resonance masses up to 6 TeV. To account for the possibility
that the systematic uncertainties are larger than the ones we use in the analysis we also
present our results for the case where all the systematic uncertainties used in evaluating
Dcol are increased by a factor of 1.5. We believe this rather conservative estimate will be
able to cover reasonable fluctuations in systematic uncertainties due to higher energy, larger
118

luminosity and other effects not included in our study.
Table 4.1: Sources of systematic uncertainty contributing to uncertainties in measurement of
the cross section, mass and width of a resonance in the di-jet channel at various experiments
and c.m. energies. The values used in this analysis are indicated by an asterisk.

Systematic Uncertainty
Di-jet cross section
uncertainty (fractional)

Notation

Value

Mass Range

√
s

Experiment

εσ sys

0.28 − 0.41∗

2.5 − 6 TeV

14 TeV

LHC [260]

0.045 − 0.035∗
0.045 − 0.031
0.071 − 0.062
0.1∗
0.1
0.013∗
0.028
0.035

2.5 − 6 TeV
2.5 − 6 TeV
2.5 − 6 TeV
any
any
any
any
any

8 TeV
8 TeV
14 TeV
8 TeV
14 TeV
8 TeV
8 TeV
14 TeV

CMS [249]
ATLAS [247]
LHC [260]
CMS [248]
LHC [260]
CMS [248]
ATLAS [247]
LHC [260]

Mass resolution

Mres
M

Mass resolution
uncertainty
Mass uncertainty
from jet energy
scale (JES)

∆Mres
M

4.6

∆M
M JES

Results

In this section we describe the sensitivity of the LHC to distinguish a color-octet resonance
from a color-singlet resonance using the color discriminant variable introduced in Sec. 4.3.
We estimate the sensitivity at the LHC by evaluating Dcol and include the uncertainties
in estimating Dcol as described in Sec. 4.5. Motivated by current constraints on di-jet
resonances and future sensitivity of the LHC to discover di-jet resonances as described in
Sec. 4.4, we consider the mass range of 2.5 − 6 TeV. We will present our results for the
flavor universal and the flavor non-universal case at the LHC with
integrated luminosities, namely, L = 30, 100, 300 and 1000 fb−1 .

119

√
s = 14 TeV for varying

4.6.1

Flavor Universal Scenario

The sensitivity of the LHC with c.m. energy of 14 TeV to distinguish color-octet and colorsinglet resonances in the flavor universal scenario is shown in Fig. 4.2(a) - (d) for varying
luminosities, L = 30, 100, 300 and 1000 fb−1 . The sensitivity is presented in the plane of the
mass of the resonance (M ) in TeV and the log of the color discriminant variable (Dcol ). In
each panel, there are two separate bands, corresponding to a coloron and a leptophobic Z .
For each resonance, the uncertainty in the measurement of Dcol due to uncertainties in the
measurement of the cross section, mass and width of the resonance is indicated by gray bands
around the central value of Dcol represented as a black dashed line. The outer (dark gray)
band corresponds to the uncertainty in Dcol when the width is equal to the mass resolution,
i.e. Γ = Mres . A determination of the intrinsic width is not possible when Γ < Mres . The
inner (light gray) band corresponds to the case where the width Γ = 0.15M . The narrow
width approximation used in di-jet searches is not valid when Γ > 0.15M . Resonances with
width Mres ≤ Γ ≤ 0.15M will have bands that extend between the outer and inner gray
bands. The blue (green) colored region indicates the region in parameter space of the coloron
(leptophobic Z ) that has not been excluded by current searches [249] and has the potential
to be discovered at the LHC at a 5σ level as described in Sec. 4.4.
The results in Fig. 4.2 illustrate several features. First, note that the bands for coloron
and leptophobic Z are well separated vertically. This implies that the color discriminant
variable is able to clearly distinguish between a color-octet and a color-singlet at the LHC
after all uncertainties have been taken into account. The mass range (from 2.5 − 6 TeV) can
be roughly divided into three regions: the low mass region where the Z band is green but
the coloron band is grayed out; the high mass region where at least one band is grayed out,

120

LHC 14TeV, 30 fb−1

Colo
−1

Lep

2.5

3.0

toph

obic

3.5

log σM3/Γ

ron

−2

4.0

4.5
M [TeV]

5.0

5.5

6.0

ron

−1

2.5

3.0

2.5

bic Z

4.0

4.5
M [TeV]

3.0

5.0

5.5

opho

3.5

bic Z
4.0

4.5
M [TeV]

6.0

5.0

5.5

6.0

LHC 14TeV, 1000 fb−1
Colo

ron

−1

Lept

−2

opho

3.5

Lept

0

Colo

Lept

−1

Z

LHC 14TeV, 300 fb−1

−2

Colo
ron

−2

0
log σM3/Γ

0

log σM3/Γ

log σM3/Γ

0

LHC 14TeV, 100 fb−1

2.5

3.0

opho
bic Z

3.5

4.0

4.5
M [TeV]

5.0

5.5

6.0

√
Figure 4.2: (a) Top left: Sensitivity at the LHC with s = 14 TeV and integrated luminosity
of 30 fb−1 for distinguishing a coloron from a leptophobic Z in the plane of the log of the color
3

discriminant variable Dcol = MΓ σjj and mass (in TeV) for the flavor universal scneario.
The central value of Dcol for each particle is shown as a black dashed line. The uncertainty
in the measurement of Dcol due to the uncertainties in the measurement of the cross section,
mass and width of the resonance is indicated by gray bands. The outer (darker gray) band
corresponds to the uncertainty in Dcol when the width is equal to the experimental mass
resolution i.e. Γ = Mres . The inner (lighter gray) band corresponds to the case where
the width Γ = 0.15M . Resonances with width Mres ≤ Γ ≤ 0.15M will have bands that
extend between the outer and inner gray bands. The blue (green) colored region indicates
the region in parameter space of the coloron (leptophobic Z ) that has not been excluded by
current
searches [249] and has the potential to be discovered at a 5σ level at the LHC with
√
s = 14 TeV after statistical and systematic uncertainties are taken in to account. (b) Top
right: Same as (a) but for an integrated luminosity of 100 fb−1 . (c) Bottom left: Same as
(a) but for an integrated luminosity of 300 fb−1 (d) Bottom right: Same as (a) but for an
integrated luminosity of 1000 fb−1 . Note that the colored regions in all panels correspond
to the same colored regions in the mass and coupling plane used in Fig. 4.1 for different
luminosities.
121

and an intermediate region where the coloron band is blue and the Z band is green. The
color discriminant variable can be used to distinguish colorons and leptophobic Z s in the
intermediate region where both resonances are allowed, discoverable at the LHC, and have
widths in the appropriate range (Mres ≤ Γ ≤ 0.15M ). Note that this overlap region expands
with increasing luminosity.
Our analysis is not useful in the low mass region because a coloron there is either already
excluded by LHC searches or is too narrow. For example, consider a future discovery of a
resonance in the di-jet channel at a mass of 3.0 TeV and 100 fb−1 of integrated luminosity.
This could certainly correspond to a Z lying within the green band of Fig. 4.2(b). However,
from the lower left corner of Fig. 4.1(a) we also see that it could correspond to a very narrow
coloron, with a width less than the detector resolution. Being unable to measure the width
of such a coloron accurately, the uncertainty in the measurement of our discriminant variable
Dcol would be very large and hence we would not be able to distinguish between a coloron
and a leptophobic Z .
Now contrast this with the high mass region where the coloron has the potential to
be discovered while the leptophobic Z band is grayed out. In this case if a resonance is
discovered in the di-jet channel with a mass of 5.0 TeV with 100 fb−1 integrated luminosity,
then we would have confidence that it is a coloron and not a leptophobic Z . A discovery in
the di-jet channel (with the current analyses) requires the width to be relatively narrow (Γ ≤
0.15M ), while the width of a corresponding Z would be very broad as seen in Fig. 4.1(b).
Note that there are no current experimental strategies to discover very broad resonances and
it is also beyond the scope of our analysis.
The color discriminant variable is a dimensionless quantity that depends only on the
color factors for a given ξ. All points in the parameter space of a coloron and leptophobic
122

C and D Z respectively and D C = D Z for a given ξ. However
Z give the same value of Dcol
col
col
col

the results in Fig. 4.2 show a clear dependence of Dcol on the mass of the resonance (M ).
This dependence on the mass is an artifact of the implicit dependence on mass coming from
PDFs. A resonance with a large (small) mass corresponds to parton luminosity at large
(small) x and hence small (large) cross section. Note that at a fixed collision energy (at
the parton level) the value of Dcol for a given resonance will be universal and have no mass
dependence.
The uncertainties in the estimation of Dcol are large for higher mass resonances compared
to the ones at low mass. This is easily understood as high mass resonances have smaller
cross sections and hence fewer signal events leading to large uncertainties. As the luminosity
increases from 30 fb−1 to 1000 fb−1 the uncertainty at higher masses decreases due to the
larger number of signal events and the width of the uncertainty bands becomes uniform. This
is because systematic uncertainty will become progressively dominant when the number of
events is sufficiently large that a 5σ discovery is possible.
The uncertainty in the color discriminant variable is smaller for the case of larger width
(inner gray band) and is larger for the case of smaller width (outer gray band) resonances.
This is because smaller width corresponds to smaller couplings and hence smaller cross
section which leads to larger statistical uncertainty. In addition the uncertainty in the
intrinsic width is inversely proportional to Γ/Mres and hence leads to larger systematic
uncertainty for smaller widths. This relation between the uncertainty in the width and the
mass resolution is described in Appendix C. In addition, the size of the colored band (which
corresponds to the discovery reach) is smaller for larger masses. This is because the couplings
corresponding to a width of Γ = Mres (outer region of the colored band) are small leading
to a smaller cross section and hence not enough signal events for a 5σ-discovery at a given
123

LHC 14TeV, 30 fb−1

0

0
Colo

ron

log σM3/Γ

log σM3/Γ

Colo
−1
−2

2.5

Lep
3.0

toph
3.5

LHC 14TeV, 100 fb−1
ron

−1
−2

Lep

obic

Z
4.0 4.5
M [TeV]

5.0

5.5

6.0

2.5

LHC 14TeV, 300 fb−1

0

3.0

toph

log σM3/Γ

log σM3/Γ

2.5

3.0

3.5

6.0

ron

−1
−2

obic

Z
4.0 4.5
M [TeV]

5.5

Colo

−1

toph

5.0

LHC 14TeV, 1000 fb−1

0

ron

Lep

Z
4.0 4.5
M [TeV]

3.5

Colo

−2

obic

5.0

5.5

6.0

2.5

Lept

3.0

opho
b

3.5

ic Z

4.0

4.5
M [TeV]

5.0

5.5

6.0

Figure 4.3: Same as Fig. 4.2, but with the systematic uncertainties in M , Γ and σjj increased
by a factor of 1.5.

124

mass. The larger couplings which correspond to larger widths (inner region of the colored
band) and larger cross sections have the potential for a 5σ discovery.
In Fig. 4.3, we display the results for the scenario where systematic uncertainties in the
measurement of mass, width and cross section of the resonance in the di-jet channel listed in
Sec. 4.5 were increased by a factor of 1.5. Note that our results remain robust even with this
very conservative estimate of systematic uncertainties and the color-octet and color-singlet
states can still be distinguished from each other.

4.6.2

Flavor Non-universal Scenario

In this section we discuss the sensitivity of the LHC with c.m. energy of 14 TeV to distinguish
between color-octet and color-singlet resonances in the flavor non-universal scenario. As
illustrative examples we present the results for two different flavor non-universal scenarios in
Fig. 4.4 for the mass range 2.5 − 6 TeV and for various couplings and widths. In Fig. 4.4(a)
and Fig. 4.4(b) we present the results in the plane of mass of the resonance and the log
q

t
of the color discriminant variable for gL,R
= 2gL,R for integrated luminosities of 300 fb−1

and 1000 fb−1 respectively. In Fig. 4.4(c) and Fig. 4.4(d) we present similar results for
q

t
= 3gL,R for integrated luminosities of 300 fb−1 and 1000 fb−1 respectively.
gL,R

The results for the flavor non-universal case in Fig. 4.4 illustrate several features. As
discussed in Sec. 4.3, the ratio between the widths of a color-octet and color-singlet depends
only on the color factors and hence the color discriminant variable can be used to distinguish
color-octet and color-singlet resonances in the flavor non-universal scenario for a given ξ. This
is demonstrated by the separation between the bands for colorons and leptophobic Z s in
Fig. 4.4, for the cases ξ = 2 and ξ = 3. Several features present in the flavor universal case
are preserved. The reach in mass increases with increasing luminosity; the uncertainty in the
125

LHC 14TeV, 300 fb−1
gt = 2gq
ξ= 2

Colo

ron

−1
−2

Lept

bic Z

3.0

3.5

4.0

4.5
M [TeV]

0

5.5

6.0

ron

4.5
M [TeV]

5.0

5.5

opho

bic Z

3.0

3.5

4.0

4.5
M [TeV]

6.0

5.0

5.5

6.0

LHC 14TeV, 1000 fb−1
gt = 3gq
ξ= 3
Colo

ron

−1
−2

Lept

4.0

Lept

0

Colo

opho
bic Z
2.5 3.0 3.5

ron

−1

2.5

log σM3/Γ

log σM3/Γ

5.0

LHC 14TeV, 300 fb−1
gt = 3gq
ξ= 3

−1

Colo

−2

opho

2.5

−2

0
log σM3/Γ

log σM3/Γ

0

LHC 14TeV, 1000 fb−1
gt = 2gq
ξ= 2

Lept

2.5

opho

bic Z
3.0 3.5

4.0

4.5
M [TeV]

5.0

5.5

6.0

Figure 4.4: (a) Top left: same as Fig. 4.2(c), but for the flavor non-universal case with
q
t
gL,R
= 2gL,R . (b) Top right: same as (a) but for integrated luminosity of 1000 fb−1 . (c)
q

t
Bottom left: same as (a) but for gL,R
= 3gL,R . (d) Bottom right: same as (c) but for
−1
integrated luminosity of 1000 fb .

126

estimation of Dcol is large for larger masses and is dominated by systematic uncertainties
when the number of events is large and the uncertainties for resonances with small (large)
widths is large (small). There are also some new features specific to the flavor non-universal
case. The reach in mass decreases with increasing ξ. As discussed in Sec. 4.4, as ξ changes
the production cross section remains the same while the branching fraction to jets decreases
leading to a smaller di-jet cross section and hence fewer signal events. Note that the central
value of Dcol decreases with increasing ξ due to the dependance of the di-jet cross section
on ξ 2 as shown in Eqs. (4.22) and (4.23).
So far we have presented results for distinguishing a color-octet and a color-singlet state
in the flavor universal case and in the flavor non-universal case for a given ξ. In Fig. 4.5
we present the sensitivity of the color discriminant variable for varying ξ at the LHC with
√

s = 14 TeV. In Fig. 4.5(a) and (b) we present results for M = 3 TeV and integrated

luminosities of 300 fb−1 and 1000 fb−1 respectively while Fig. 4.5(c) and (d) are for M =
4 TeV and integrated luminosities of 300 fb−1 and 1000 fb−1 respectively. As before the
central values are indicated by dashed (black) lines and the reach for colorons (leptophobic
Z s) are denoted by blue (green) regions. The different points marked a, b, c, d correspond
to different parameter points used as examples below.
With the discovery of a resonance in the di-jet channel a measurement of the mass, width
and cross section and hence of the color discriminant variable is possible. Several scenarios
may be allowed for a given measured value of mass and Dcol . For example, in Fig. 4.5(a), for
log(Dcol ) = −1.5 a few of the allowed possibilities include a leptophobic Z with ξ = 1 and a
coloron with ξ = 3.5 or 4.5 and are marked by the points labeled a, b, c respectively. Similar
possibilities are shown for other masses and luminosities in Fig. 4.5(b) - (d) for different
measured values of Dcol and are labeled as a, b, c, d. With just the measurements from the
127

LHC 14 TeV, 300 fb−1
M =3 TeV

−2

0

Coloron
a

b

1

2

ξ

log σM3/Γ

3

4

5

n

Lept
1

b

c

0

−2

bic Z
ξ

a

3

4

b

Lept
1

c

5

0

d

opho

bic Z

2

ξ

3

4

5

LHC 14 TeV, 1000 fb−1
M =4 TeV

Coloro

n

−1

opho

2

n

0

Coloro
a

−1

Coloro

−2

LHC 14 TeV, 300 fb−1
M =4 TeV

−1

0

c

Lepto
phobi
cZ

0

−2

log σM3/Γ

−1

LHC 14 TeV, 1000 fb−1
M =3 TeV

0

log σM3/Γ

log σM3/Γ

0

a

Lepto
ph
1

b
c

d

obic Z
2

ξ

3

4

5

Figure 4.5: (a)√
Top left: Sensitivity to distinguish color-octet and color-singlet scenarios at
the LHC with s = 14 TeV and integrated luminosity of 300 fb−1 for a resonance of mass
3 TeV for different values of ξ. (b) Top right: same as (a) but for integrated luminosity of
1000 fb−1 . (c) Bottom left: same as (a) but for a resonance with mass 4 TeV. (d) Bottom
right: same as (c) but for integrated luminosity of 1000 fb−1 . The different points marked
a, b, c, d correspond to different parameter points used as examples in the text.

128

di-jet discovery channel it would not be possible to distinguish which of these scenarios is
being realized in nature. Additional information is required, for instance, by measuring ξ
from the ratio of the cross sections when the resonance decays to top pairs vs jets as given
in Eq. (4.16). With these two measurements, i.e. Dcol and ξ, one can distinguish between
color-octet and color-singlet resonances and their flavor structure.
Finally, we note that distinguishing between color-octet and color-singlet resonances of
various flavor structures depends on the reach of ξ as well as the precision with which ξ can
be measured at the LHC. This in turn depends on the measurements of the cross section in
the di-jet channel and the top pair channel and the associated uncertainties. For example,
a resonance may be discovered at a given mass in the di-jet channel and a measurement of
Dcol may be made. However a measurement of the cross section in the top pair channel
may not be possible or may have large uncertainties due to limited statistics (because of
small branching to top pairs). In such a case a measurement of ξ is not possible or has very
large errors and the ability to distinguish various flavor scenarios is reduced. A detailed
exploration of the reach and precision with which ξ can be measured is beyond the scope
and focus of this article. Instead we refer the interested reader to experimental studies of
resonance searches in the top pair [263, 264, 265, 266] and di-jet channels [260] and the
results of such studies can provide an estimate of ξ and the uncertainties in measuring ξ.

4.7

Summary

Di-jet resonance searches are simple but powerful model-independent probes for discovering
new particles motivated in many new physics scenarios. Once a resonance has been discovered in the di-jet channel it will be very important to measure its properties. The di-jet

129

(discovery) channel can provide information about the mass and spin of the resonance as
well as constrain the coupling strength. It does not provide information about the chiral
structure; that can be obtained by including information from the channel where the resonance is produced in association with a SM electroweak gauge boson. The next question that
remains to be resolved is the color structure of the new resonance; in particular, whether it
is a color-octet or color-singlet resonance.
In the work on which this chapter is based, we proposed a new variable called the color
discriminant variable (Dcol ) to distinguish color-octet from color-singlet resonances. This
variable is relatively model-independent and is sensitive to the color and flavor structure. In
order to make our study as widely applicable as possible, we studied phenomenological models
of color-octet and color-singlet resonances without being tied down to a specific theory. As
illustrative examples we studied colorons and leptophobic Z s in the flavor universal case
as well as an illustrative flavor non-universal scenario. We analyzed the current constraints
on the parameter space and the discovery potential of coloron and leptophobic Z models
at the future LHC. We sampled a wide range of masses, couplings and values of ξ, which
parameterizes the relative coupling strength of the third generation and first generation SM
quarks to the new resonance.
We studied the sensitivity to distinguish color-octet and color-singlet resonances in the
flavor universal as well as flavor non-universal scenarios and presented our results for the
LHC with c.m. energy of 14 TeV for varying luminosities, L = 30, 100, 300 and 1000 fb−1 ,
after including all uncertainties. We found that our method has a wide reach in mass and
couplings as well as ξ. A measurement of Dcol alone can distinguish between color-octet
and color-singlet states for a given flavor scenario. However to distinguish between different
flavor scenarios we will need additional information which comes from a measurement of ξ.
130

Together, Dcol and ξ can help distinguish a color-octet from a color-singlet state as well as
establish the nature of the couplings in the flavor sector. We find that the LHC will be able
to provide information about the color and flavor structure of a new di-jet resonance for a
wide range of couplings and masses and hence point us in the direction of the underlying
theoretical structure.

131

Chapter 5
Distinguishing Flavor Non-universal
Colorons from Z Bosons with Heavy
Flavor Measurements
— This chapter is based on a work in collaboration with R. Sekhar
Chivukula and Elizabeth H. Simmons which has appeared in [7].

5.1

Introduction

In the previous chapter, we have established the possibility of distinguishing whether a vector
resonance is either a leptophobic color-singlet or a color-octet, using a variable available
with the discovery of the resonance: the “color discriminant variable”, Dcol . The variable is
constructed from the dijet cross-section for the resonance (σjj ), its mass (M ), and its total
decay width (Γ), observables that will be available from the dijet channel measurements of
the resonance:
Dcol ≡

M3
σ ,
Γ jj

(5.1)

For a narrow-width resonance, the color discriminant variable is independent of the resonance’s overall coupling strength. We also illustrated applications of the color discriminant
132

variable technique for two simple cases. The first was a flavor universal model with identical
couplings to all quarks — see Sec. 4.2.1. In the second, the overall strength of couplings
to quarks in the third generation was allowed to be different from those in the first two
(couplings to top and bottom were kept equal) — see Sec. 4.2.2. Combining Dcol with information about the tt¯ cross-section1 for the resonance (pp → V → tt¯) still enabled us to
distinguish C from Z in the second case. While these two scenarios clearly illustrate the
application of the method, they did not encompass the features of a typical Z , whose upand down-type couplings are usually also different from one another.
In this work [7] we demonstrate the application of the Dcol method to more general
scenarios where couplings to quarks within the same generation (e.g., up vs. down, lefthanded vs right-handed) are different, while still allowing couplings to quarks in the third
generation to be different from those in the first two. This general scenario corresponds to
more realistic models, especially those featuring a Z , as in the case where the gauge group
responsible for the existence of the resonance does not commute with the gauge groups of the
Standard Model. We discuss how one could incorporate information from heavy resonance
decays to of a top pair (tt¯) and a bottom pair (b¯b) in order to determine what type of
resonance has been discovered.
The rest of the chapter is organized as follows. We lay out the phenomenological parameters we use, as well as key assumptions imposed on them, in section 5.2. Then, we discuss
the detailed application to flavor-nonuniversal scenarios in section 5.3. The parameter space
to which our method is applicable is then presented in section 5.4. After reprising, in section
5.5, the estimation of uncertainties for the LHC at

√

s = 14 TeV from [6], we present our

1 Studies of the sensitivity of a resonance decaying to tt¯ at the √s = 14 Tev LHC include [267, 268, 269,
270, 271].

133

results in section 5.6 and discuss them in section 5.7. Model-independent plots for various
combinations of resonance couplings and masses are presented in Appendix D to illustrate
how well each relevant observable must be measured to distinguish between the coloron and
the Z .

5.2

General Parameterization and Assumptions

In this section, we introduce the assumptions about properties regarding chiral and generation structures that are used in this chapter. The phenomenological parameterizations of
the coloron and leptophobic Z couplings are the same as what have been used in Chapter
4.
Couplings between the vector boson and the left- and right-handed forms of the upand down-type fermions are not necessarily equal in general. It is true that a color octet
resonance, originating from interactions described by a gauge group commuting with SU (2)L
of the standard model, will have the same couplings to the left-handed up- and down-type
quarks. The same is not generally the case for a Z boson. Moreover, the couplings of either
a coloron or Z to right-handed up- and down-type quarks may differ.
Couplings between the vector boson and the left- and right-handed forms of the upand down-type fermions are not necessarily equal in general. It is true that a color octet
resonance, originating from interactions described by a gauge group commuting with SU (2)L
of the standard model, will have the same couplings to the left-handed up- and down-type
quarks. The same is not generally the case for a Z boson. Moreover, the couplings of either
a coloron or Z to right-handed up- and down-type quarks may differ.
In addition, the couplings can generally be different among the three generations of

134

quarks. The observed suppressions of flavor-changing neutral currents disfavor a TeV-scaled
resonance with non-universal couplings to the first two generations2 . So we will limit our
interests throughout this article to scenarios where couplings for the first two generations
are the same. The third generation is special. In models where top quark plays a unique
role, the couplings to quarks in the third generation are often assumed to be different from
those for the light quark generations.
Therefore, under this assumption, a coloron has 6 free parameters describing its couplings
to quarks:

gC = gC
L
L

u,c

d,s

and

gC , gC
R
R

u,c

d,s

t = gb
gC
CL
L

and

t , gb ,
gC
R CR

(5.2)

while a leptophobic Z has 8:

u,c
d,s
,g
ZL ZL

and

g

t , gb
gZ
L ZL

and

t
gZ

g

u,c
d,s
,g
ZR ZR
R

b
, gZ

R

.

(5.3)

The dependence on these parameters does not fully manifest itself in measurements available
after a discovery; i.e., width and dijet cross section. After all, those observables are not
sensitive to chiral structures of the coupling, as the left- and right-handed couplings enter
symmetrically. So we denote
q2

q2

g q2 ≡ gL + gR
2 See, for example, Table 4 of Ref. [272].

135

(5.4)

and notice that the four relevant parameters for our analysis of colorons are

u 2 = gc 2,
gC
C

d 2 = gs 2,
gC
C

t 2,
gC

b2
gC

(5.5)

d 2 = gs 2,
gZ
Z

t 2,
gZ

b2
gZ

(5.6)

and, similarly, there are four

u 2 = gc 2,
gZ
Z

for a leptophobic Z .
The dijet cross section (σ(pp → V → jj)) plays an important role in evaluating the color
discriminant variable. In this analysis, we make a distinction between quarks from the first
two “light” generations and those from the third. So we will classify what is referred to
as “dijet” resonance accordingly. Not only does this simplify the analysis, as we shall see
later on, but measurements of the cross sections to the tt¯ and b¯b final states (respectively,
σ(pp → V → tt¯) and σ(pp → V → b¯b)) will provide distinct information allowing the
identification of the color structure of the resonance. Throughout the article, quarks that
constitute a jet j are those from the first two generations3 ; namely,

j = u, d, c, s.

(5.7)

With these definitions, we now demonstrate how to construct a color discriminant variable.
3 Notice that the definition of jets is different from that in Chapter 4.

136

5.3

Defining Color Discriminant Variables in Flavor
Non-universal Models

In this section, we extend the definition of the color discriminant variable, introduced in
Section 4.3 of the previous chapter, to general scenarios of a resonance with flavor nonuniversal couplings.
Recall the simple flavor universal scenario from Sec. 4.3. Since the overall coupling
strength can be factored out as a ratio of observables (ΓV /MV ) as shown in (4.18) and
(4.19), the color discriminant variable is defined as

C
Dcol
Z
Dcol

MC3 C
8
σjj =
=
ΓC
9
=

M3

Z

ΓZ

Z =
σjj

for the coloron and Z , respectively.

1
9

Wq (MC )Br(C → jj)

q

q

Wq (MZ )Br(Z → jj)

(5.8)

(5.9)

Here we have used the Wq (M ) function defined

in Eq.4.20. In a flavor non-universal scenario, one cannot always factor out the dependence
of couplings appearing in a production cross section in this manner. Branching fractions of
the decay final states are also not necessarily the same for two resonances. In the following
sections, we will demonstrate that even when this is the case, the color discriminant variable
method remains valuable.
In a flavor non-universal scenario, we will follow the parameterization and assumptions

137

introduced in section 5.2. The production cross section and decay width for the coloron are

σ(pp → C) =

4 αs
d 2 (W + W ) + g b 2 W
g u 2 (Wu + Wc ) + gC
s
d
b
C
9 MC2 C
u2
gC

4 αs
u 2 + gd 2
gC
=
C
2
9 MC

u 2 + gd 2
gC
C

(Wu + Wc )
b2
gC

u2
gC

+ 1 − u2
d2
gC + gC

(Wd + Ws ) + u 2
W ,
d2 b
gC + gC

αs
u 2 + 2g d 2 + g t 2 + g b 2
M 2gC
C
C
C
12 C
t2
b2
gC
gC
αs
u 2 + gd 2
2
+
MC gC
=
+
C
u 2 + gd 2
u 2 + gd 2
12
gC
gC
C
C

(5.10)

ΓC =

,

(5.11)

where they have been written using parametrization that allows some of them to, as we shall
see, correspond to observables. The expressions for leptophobic Z are similar

1 αw
σ(pp → Z ) =
3 M2

u2
gZ

gu 2

d2
+ gZ

Z

gu 2 + gd 2

Z

Z

+ 1−
ΓZ

=

αw
M
2 Z

gu 2
Z
u
2
g + gd 2
Z
Z

(Wu + Wc )

Z

gb 2

(Wd + Ws ) + u 2 Z d 2 Wb ,
g +g
Z

u 2 + gd 2
gZ
Z

Z

gb 2

gt 2

2 + u2 Z d2 + u2 Z d2
g +g
g +g
Z

Z

(5.12)

Z

.

(5.13)

Z

The color discriminant variables are, for the coloron,

C
Dcol
=

gu 2
16
(Wu + Wc ) u 2 C d 2 +
3
gC + gC
gb 2
+ u2 C d2
gC + gC

Wb
Wu + Wc

gu 2
Wd + Ws
1 − u2 C d2
Wu + Wc
g + gC
C













2
×
(5.14)
2

t
2
b
2


gC
gC





 2 + gu 2 +gd 2 + gu 2 +gd 2 

C

138

C

C

C

and for the Z ,

Z
Dcol

gu 2

gu 2
2
(Wu + Wc ) u 2 Z d 2 +
=
3
g +g

Wd + Ws
1 − u2 Z d2
Wu + Wc
g +g
Z
Z
Z
Z












b
2


g
2
Wb
Z
(5.15)
+ u2
×
2

t2
b2
g + g d 2 Wu + Wc


g
g


Z
Z
Z
Z




 2 + gu 2 +gd 2 + gu 2 +gd 2 
Z

Z

Z

Z

where parts related to resonance production are grouped within the square brackets, while

those related to decay are grouped within curly braces. Notice that the appearance of the
factor 2 in the decay part of the expressions is due to our assumption that the first two
generations couple identically to the vector resonance.
The relative strength with which the vector boson couples to the u− and d−type quarks
of the light SM generations, gu2 / gu2 + gd2 , which we will call the “up ratio” for brevity, is
not accessible by experiments available in the dijet channel. However, equivalent information
for quarks in the third generation can be measured by comparing the dijet and heavy flavor
cross sections:
gt2
σ(pp → tt¯)
=
2
σ(pp → jj)
gu2 + gd2

“top ratio”

(5.16)

“bottom ratio”

(5.17)

and
gb2

σ(pp → b¯b)
=
2
.
σ(pp → jj)
gu2 + gd2

Supplementary measurements of these ratios of cross sections will help pinpoint the structure
of couplings of the resonance.
139

While our expressions (5.14 5.15) for Dcol appear to have a complicated dependence on
multiple parameters involving different quark flavors, recalling that the coloron and Z’ are
being produced by collisions of quarks lying inside protons simplifies matters considerably.
First, the contribution of b-quarks to the production part of Dcol [square brackets within
(5.14) and (5.15)] is suppressed significantly by their relative scarcity in the protons, as
W +W

W

b . We
reflected by the range of the ratios of the parton density functions Wd +Ws and W +W
u
c
u
c

plot the values of these functions in Fig. 5.1 for mass range of 2.5−6 TeV at a pp collider with
center-of-mass energy 14 TeV. Here and throughout the chapter, the CT09MCS from the
CTEQ Collaboration is used4 as the parton distribution function set [273]. In that figure,
we allow the factorization scale to vary by a factor of 2 from the mass of the resonance. The
effect of this variation has been illustrated as the width of each band in the plot. From the
plot we conclude that unless the resonance has a considerably stronger coupling to the b than
to quarks in first two generations, the precise strength of the couplings to third-generation
quarks becomes relevant to Dcol only through the decay part of the expressions (5.14) and
(5.15), the part in curly braces.
Second, we see that the experimentally inaccessible parameter that we call the up ratio
appears unlikely to leave us confused as to whether a new dijet resonance is a coloron or
a leptophobic Z . From Fig. 5.1 we see that the ratio (Wd + Ws ) / (Wu + Wc ) is fairly
flat over the range of masses considered — varying between 0.5 and 0.09 as the mass goes
from 2.5 TeV to 6 TeV. The dependence of Dcol on the up ratio is suppressed by a factor of
1 − (Wd + Ws ) / (Wu + Wc ). So the only situation where confusion might arise is if a coloron
had a very small up-ratio and an otherwise similar Z had an up ratio near the maximum
4 The more recent CT09MCS Parton Distribution Function yields results consistent with the CTEQ6L1
set used in the work on which Chapter 4 is based.

140

value of one. This is understood as follows. Consider two resonances having the same top
and bottom ratios. A coloron with negligible up ratio will have

C ∝ 16
Dcol

gu 2
1 − u2 C d2
gC + gC

u2
gC

+
u 2 + gd 2
gC
C

Wd + Ws
Wu + Wc

→ 16

Wd + Ws
Wu + Wc

(5.18)

→ 2.

(5.19)

while a Z with the up ratio close to 1 will have

Z
Dcol

∝2

gu 2
Z

gu 2
Z

+ gd 2
Z

+

gu 2

Wd + Ws
Wu + Wc

1 − u2 Z d2
g +g
Z

Z

C and D Z will be closer to each other
We see that in this particular setting, the values of Dcol
col

at higher masses. We will illustrate this further in Section 5.6.
100

Wd +Ws
Wu+Wc

10−1
10−2
10−3

Wb
Wu+Wc

10−4
10−5
2.5

3.0

3.5

4.0
4.5
M [TeV]

5.0

5.5

6.0

Figure 5.1: Ratios of parton density functions as a function of the mass of the produced
vector resonance. This figures shows two such ratios: the relative contribution of parton
density functions, defined in Eq. (4.20), for the down-type quarks of the first two generations
W

Wd +Ws
Wu +Wc

b
(top curve, in brown) and for bottom quarks W +W
(bottom curve, in orange)
u
c
relative to the up-type quarks for the first two generations. The values have been calculated
using CT09MCS parton distribution functions with factorization scale varied by a factor of 2
away from the mass of the resonance in the range 2.5 − 6.0 TeV. The results of this variation
are illustrated as a band for each function. Note that the upper curve depends only weakly
on the resonance mass and that the lower curve’s values are always ∼ O(10−3 ) or less over
the entire mass range.

141

5.3.1

General prescription for using Dcol in flavor-nonuniversal scenarios

Now that we know the parameters that affect the determination of Dcol , the general prescription of the analysis goes as follows. After a resonance has been discovered, one uses
the measurements of three observables; dijet cross section, mass, and total decay width to
evaluate Dcol . This particular value of Dcol could correspond to various configurations of
flavor-nonuniversal couplings denoted by three coupling ratios; namely, the up ratio, top
ratio, and bottom ratio. The up ratio cannot be experimentally measured, but the top
ratio and bottom ratio are accessible by measuring the tt¯ and b¯b cross sections. In many
circumstances, these two cross section measurements together with Dcol suffice to identify
the color structure of a resonance of given mass and dijet cross section. The illustration of
this method and its limitation are presented in Section 5.6.
First, however, we must determine the region of parameter space where the coloron discriminant variable is relevant: the region where one can discover the resonance and measure
M , Γ and σjj precisely. This is the topic of the next section.

5.4

Accessible Dijet Resonances at the 14 TeV LHC

A resonance discovered in a dijet channel but not in dilepton channel could be either a coloron
or a leptophobic Z . The color discriminant variable allows one to distinguish between the
two particles in a model-independent manner; i.e., without having to analyze each set of
couplings separately. In this section, we describe the region of parameter space to which
this method is applicable5 . In this region the resonance has not already been excluded by
5 A detailed discussion was presented in [6].

142

the current searches, is within the reach of future searches, and has a total width that is
measurable and consistent with the designation “narrow”.
One may deduce the current exclusion limits on resonances in the dijet channel using
the limits on the production cross section times branching ratio (σ × Br(jj)) from the
(null) searches for narrow-width resonances carried out by the ATLAS and CMS collaborations [246, 248, 249] at

√
s = 8 TeV. We use the most stringent constraint, which comes

from CMS [249]. As the exclusion limit is provided in the form of σ × Br(jj) × (Acceptance),
we estimate the acceptance of the detector for each value of the resonance mass by comparing, within the same theoretical model, σ × Br(jj) that we calculated vs. σ × Br(jj) ×
(Acceptance) provided by CMS. The acceptance is a characteristic of properties of the detector and kinematics, the latter being the same for coloron and Z to the leading order;
thus we use throughout our analysis the acceptance obtained from a comparison within a
sequential Z model. The excluded region of parameter space is displayed in gray in Figs.
5.4 and 5.3.
Sensitivity to a dijet resonance in future LHC experiments with

√

s = 14 TeV de-

pends on the knowledge of QCD backgrounds, the measurements of dijet mass distributions, and statistical and systematic uncertainties. CMS [260] has estimated the limits on
σ×Br(jj)×(Acceptance) that will be required in order to attain a 5σ discovery at CMS with
integrated luminosities up to 10 fb−1 , including both statistical and systematic uncertainties.
We obtain the acceptance for CMS at

√

s = 14 TeV in the same manner as described in the

previous paragraph. The sensitivity for the dijet discovery from 10 fb−1 is then scaled to
the integrated luminosities L = 30, 100, 300, 1000 fb−1 considered in our studies (assuming
that the systematic uncertainty scales with the squared root of integrated luminosity). The
predicted discovery reaches for these luminosities are shown in varying shades of blues for
143

coloron and greens for Z in Figs. 5.4 and 5.3. Sensitivity to a dijet resonance in future
LHC experiments with

√
s = 14 TeV depends on the knowledge of QCD backgrounds, the

measurements of dijet mass distributions, and statistical and systematic uncertainties.
Sensitivity to a dijet resonance in future LHC experiments with

√

s = 14 TeV de-

pends on the knowledge of QCD backgrounds, the measurements of dijet mass distributions,
and statistical and systematic uncertainties. Ref. [260] presents estimates of the limits on
σ × Br(jj) × (Acceptance) that will be required in order to attain a 5σ discovery at CMS
with integrated luminosities up to 10 fb−1 with statistical uncertainty alone as well as with
statistical and systematic uncertainties combined. We obtain the acceptance for CMS at
√

s = 14 TeV in the same manner as described in the previous paragraph. The sensitivity

from 10 fb−1 is then scaled to the integrated luminosities L = 30, 100, 300, 1000 fb−1 considered in our studies. The predicted discovery reaches for these luminosities are shown in
varying shades of blue for coloron and green for Z in Figs. 5.4 and 5.3.
The total decay width also constrains the absolute values of the coupling constants. On
the one hand, experimental searches are designed for narrow-width dijet resonances; hence
their exclusion limits are not applicable when the resonance is too broad, which translates
to about Γ/M = 0.15 as the upper limit [258, 259, 131]. On the other hand, the appearance
of (intrinsic) total decay width in the expression for the color discriminant variable requires
that the width be accurately measurable; width values smaller than the experimental dijet
mass resolution, Mres , cannot be distinguished. The region of parameter space that meets
both constraints and is relevant to our analysis is shown in Figs. 5.4 and 5.3 as the region
between the two dashed horizontal curves labeled Γ ≥ 0.15M and Γ ≤ Mres . Regions where
the width is too broad or too narrow are shown with a cloudy overlay to indicate that they
are not accessible via our analysis.
144

In Figs. 5.4 and 5.3, we also show the contours in the region of parameter space along
which the uncertainties in measuring Dcol are 20% and 50% at the LHC with

√

s = 14 TeV

for the integrated luminosity of 1000 fb−1 . We will estimate uncertainties for relevant observables in the accessible region in the next section.

5.5

Systematic Uncertainties at the 14 TeV LHC

Statistical and systematic uncertainties on dijet cross section, mass, and intrinsic width of
the resonance will play a key role in determining how well Dcol can discriminate between
models at the LHC with
the LHC with

√

√

s = 14 TeV. The actual values of the systematic uncertainties at

s = 14 TeV will be obtained only after the experiment has begun. In this

section, we discuss the estimates of the uncertainties that we use in our calculations.
The effect of systematic uncertainties in the jet energy scale, jet energy resolution, radiation and low mass resonance tail and luminosity on the dijet cross section at the 14 TeV
LHC was estimated in Ref. [260]. It is presented there as a fractional uncertainty (as a
function of the mass) normalized to the dijet cross section required to obtain a 5σ discovery
above background fluctuations. The dijet mass resolution, the uncertainty of the dijet mass
resolution, and the uncertainty of the mass itself (due to uncertainty in the jet energy scale),
also affect the determination of both the mass and intrinsic width. Table 4.1 in Chapter 4
lists these estimated uncertainties for
CMS and ATLAS experiments at

√

√

s = 14 TeV together with the values from the actual

s = 8 TeV. Here, we use the estimate for systematic

uncertainties from actual LHC data where available and assume that any future LHC run
will be able to reach at least that level of precision. The values we use are marked in Table
4.1 with asterisks.

145

50%

2 Γ
≤ Mres
0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

2 Γ
≤ Mres

ρu = 1.0

−1

Excluded
−1

30 fb

0.0699

b −1

6
4
2 Γ
≤ Mres

0.0196

Coloron

8 Γ ≥ 0.15M
2
gu2 + gd2 /gQCD

ρt + ρb = 1.0

Dcol
0.2119

30
f

1

fb −
30
−1

fb
fb
300 1000
s
n
e
s
No
0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

−1

b
100 f
ns
se
No
0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]
0.5814

b −1

b −1

4

0.0349

Coloron

4
Γ ≤ Mres

Coloron

8 Γ ≥ 0.15M
6

0.0110

0.1243

Excluded

ded
0.0430

0f

20%

Coloron

8

ρt + ρb = 1.0

Dcol
0.1452

10

4

ρu = 0.5
0.4344

2
gu2 + gd2 /gQCD

6

0.0024

30 f −
b 1
100 −
fb 1
300 −
fb 1
No
1000
sen
s
fb −1

Excluded

2
gu2 + gd2 /gQCD

8 Γ ≥ 0.15M

0.0161

Γ ≥ 0.15M

−1
00 fb −1
100 fb 3100e0nfb
os s
N
0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

ρt + ρb = 1.0

Dcol
0.0785

ρt + ρb = 0.25

Dcol
0.3766

12

−1

Γ ≤ Mres

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]
ρu = 0.0

1.0333

4

30
f

sen
s

Γ ≤ Mres

0.2873

8

ρu = 1.0

0.0195

Coloron

Γ ≥ 0.15M

No

50%

0.0764

Exclu

30 fb −1
100 −
fb 1
300 −
fb 1
1000
fb −1

Excluded

2
gu2 + gd2 /gQCD

20%

4

ρt + ρb = 0.25

Dcol
0.2580

12

Coloron

Γ ≥ 0.15M

8

0.7719

ed

12

ρu = 0.5

0.0042

Exc
lud

0.0286

2
gu2 + gd2 /gQCD

ρt + ρb = 0.25

Dcol
0.1395

2
gu2 + gd2 /gQCD

ρu = 0.0
0.5104

−1

b
3000ffb−1
00ns
1se
o
N
0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]
−1

100 fb

2
vs. coloron
Figure 5.2: Region of parameter space (scaled coupling strength (gu2 + gd2 )/gQCD
√
mass M ) where the color discriminant variable analysis applies at the LHC with s =
14 TeV. The configurations of coupling ratios vary as follows: ρu ≡ gu2 /(gu2 + gd2 ) increasing
from 0 to 1 (from left to right panels), and ρt +ρb ≡ (gt2 + gb2 )/(gu2 + gd2 ) increasing from 0.25
to 4 (from top to bottom panels). The 5σ discovery reach, with statistical and systematic
uncertainties included, is shown in varying shades of blue for different luminosities ranging
from 30 fb−1 to 1000 fb−1 . The red area marked “no 5σ sensitivity” lies beyond the discovery
reach at 1000 fb−1 . The gray area on the left of each plot marked “excluded” has been
excluded by the 8 TeV LHC [249]. In the region above the dashed line marked Γ ≥ 0.15M ,
the narrow-width approximation used in dijet resonance searches is not valid [258, 259, 131].
In the region below the horizontal dashed line marked Γ ≤ Mres , the experimental mass
resolution is larger than the intrinsic width [249], so that one cannot determine Dcol . The
contours marked 20% and 50% indicate the region above which the uncertainty in measuring
Dcol , as estimated in Sec. 5.5, is smaller than 20% and 50%, respectively. Continued next
page.

146

Figure 5.2 (cont’d)

1

50%

0.0175

20%

b −1
0f

b −1
0f
b −1

30

2

30
f

3

0.0049

Coloron

10

Excluded

1

b −1

100

0 fb −

2
gu2 + gd2 /gQCD

20%

ρt + ρb = 4.0

Dcol
0.0530

4 Γ ≥ 0.15M

s

2

Coloron

50%

Γ ≤ Mres

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

5.6

3

0.1456

sen

20%

4 Γ ≥ 0.15M

ρu = 1.0

0.0027

30
0f

Γ ≥ 0.15M

0.0108

fb −1
100
fb −1

Coloron

ρt + ρb = 4.0

Dcol
0.0363

30

0.1088

No

2

ρu = 0.5

0.0006

Exclu
ded

3

0.0040

30 f −
b 1
100
fb −1
300 −
fb 1
1000
−1
f
b
No se
ns

Exclud

2
gu2 + gd2 /gQCD

ed

4

ρt + ρb = 4.0

Dcol
0.0197

2
gu2 + gd2 /gQCD

ρu = 0.0
0.0720

−1

fb
00
10
s
sen

1 Γ
≤ Mres

1 Γ
≤ Mres

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

No

Applying the Color Discriminant Variable to Flavornon-universal Models at the LHC

In this section we illustrate how the color discriminant variable Dcol (as described in Section
5.3) may be used to distinguish whether a newly discovered dijet resonance is a coloron or
a leptophobic Z even if it is flavor non-universal. As previously mentioned, we will focus
on resonances having masses within 2.5 − 6 TeV at the

√

s = 14 TeV LHC with integrated

luminosities up to 1000 fb−1 . The values of Dcol as well as other observables have been
evaluated using the uncertainties estimated in Section 5.5 and the region of parameter space
to which this analysis is applicable was identified in Section 5.4.

5.6.1

Demonstration that C and Z lie in different regions of coupling ratio space

As we have seen, the value of Dcol at a fixed mass and dijet cross section may correspond
g2

to a variety of combinations of values of the three ratios of couplings, the up ratio ( 2 u 2 ),
gu +g
d
g2

g2

the top ratio ( 2 t 2 ), and the bottom ratio ( 2 b 2 ). The last two are directly determined
gu +g
gu +g
d
d
147

0.5

30
0

fb −1

b −1

fb −1
30

ed

gu2 + gd2 /gw2

0.0182

No

fb −1

b −1

100
0

30
0f

0f

0.0022

ρt + ρb = 4.0
0.0006

b −1

Γ ≥ 0.15M

00

30
0f

0f

b −1

1.5

10

1

10

fb −1
s

b −1

00

0f

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

10

30

Dcol
0.0066

Leptophobic Z

20%

0.5
Γ ≤ Mres

b −1

fb −1

ded

Exclu
ρu = 1.0

0.0003

50%

Γ ≤ Mres

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

ρt + ρb = 4.0

Γ ≥ 0.15M

10
20%

0.5

50%

fb −1

1

30

gu2 + gd2 /gw2

s

sen

100

1

0 fb −

1.5

se

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

Leptophobic Z

Γ ≥ 0.15M

No

20%

100

fb −1
300

gu2 + gd2 /gw2

1

0.0013

No

1

Dcol
0.0045

ns

50%

ens

ρu = 0.5
0.0136

0.0025

20%

Γ ≤ Mres

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

0.0001

2

1

50%

ρt + ρb = 4.0

0.0087

3 Γ ≥ 0.15M
gu2 + gd2 /gw2

No

20%

Leptophobic Z
1.5

1

fb −1
sen

s

10
00

fb −1

100

30

Exclu
ded

gu2 + gd2 /gw2

2

ρt + ρb = 1.0

Dcol
0.0265

Leptophobic Z

Γ ≥ 0.15M

sen

0.0005

ρu = 1.0
0.0727

Γ ≤ Mres

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]
Dcol

0 fb −

0.0014

fb −

0.0054

Γ ≤ Mres

0.0025

−1

fb
00
10 ns
se
No

Leptophobic Z

1

ρu = 0.0

b −1

ρt + ρb = 1.0

Dcol
0.0181

3

50%

0.0090

100
s

ρu = 0.5

gu2 + gd2 /gw2

1

fb −1
100
0
No
sens fb −1

fb −1

20%

20%

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

0.0543

Γ ≥ 0.15M

300

1

100

30
fb −

gu2 + gd2 /gw2

2

2

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

fb −1

0.0003

3

1 Γ≤M
res

Leptophobic Z
3

fb −1

1

30
fb −

Exclu
ded

20%

fb − 1

0.0020

0.0044

Leptophobic Z

1 Γ≤M
res

ρt + ρb = 1.0

Dcol
0.0098

0.0155

No
s

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]
ρu = 0.0

2

sen

sen
Γ ≤ Mres

0.0359

3

ρt + ρb = 0.25

Dcol
0.0471

4 Γ ≥ 0.15M

50%

No

1

ρu = 1.0
0.1292

No

20%

50%

4 Γ ≥ 0.15M

0f

Γ ≥ 0.15M

s

2

0.0024

Leptophobic Z
gu2 + gd2 /gw2

30 f −
b 1
100
fb −1
300
fb −1
100
0 fb −1

gu2 + gd2 /gw2

3

0.0096

Exclud

Leptophobic Z
4

ρt + ρb = 0.25

Dcol
0.0323

0f

ρu = 0.5
0.0965

30

0.0005

100

0.0036

10

ρt + ρb = 0.25

Dcol
0.0174

30
0

ρu = 0.0
0.0638

50%

Γ ≤ Mres

0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
M [TeV]

Figure 5.3: Same as Fig. 5.4 but for a flavor-nonuniversal Z . Notice that while a leptophobic
Z having relatively large couplings to quarks in the third generation is still within the reach
of the future LHC, its total width would typically be too large to be included in analyses for
narrow resonances.

148

from the measurements of σtt¯ and σb¯b while Dcol is relatively insensitive to the first ratio,
as mentioned in Section 5.3. The question is, therefore, whether measuring the mass, width,
dijet cross-section, σtt¯ and σb¯b can definitively identify the color charge of a newly discovered
resonance. We find that it can. We will illustrate this finding for resonances of mass 3 TeV
and 4 TeV, as we have seen in Fig. 5.4 and 5.3 that most colorons with lower masses are
excluded by the current experiments and most Z bosons with higher masses are not within
reach of the future LHC run at 1000 fb−1 .
In Fig. 5.4, we show the region of parameter space of the three coupling ratios (using
σ¯

σ

the observables σ tt¯ and σ bb in place of the top and bottom ratios, respectively) in which
jj
jj
coloron or Z models (each displayed as a point) with the same mass lead to a certain range
of dijet cross-section and Dcol . We choose the range for the dijet cross-section to be within
1 standard deviation from the value that allows a 5σ discovery at luminosity 1000 fb−1 . We
selected range of Dcol to be within 50% from Dcol = 3 × 10−3 for this illustration as it
permits the required measurements to be made for either a coloron or Z as discussed in
Section 5.4. Points in the accessible area of parameter space are highlighted in blue if the
discoverable resonance is a C and in green if it is a Z . These points lie in the blue regions
of Fig. 5.4 for colorons or the green regions of Fig. 5.3 for Z .
We now explore the features that Fig. 5.4 exhibits. The 3-dimensional plots in the left
panel show that models of C and leptophobic Z which correspond to measurable observables
σ¯

σ

g2

appear in different region of σ tt¯ vs σ bb vs 2 u 2 three-dimensional parameter space. The
jj
jj
gu +g
d
σ

σ¯

symmetry between the σ tt¯ and σ bb axes illustrates the rarity of having a heavy resonance
jj
jj
produced via b¯b annihilation - the only process in which b quarks could contribute to Dcol

without a corresponding contribution from t quarks. In addition, we see that while the up
ratio is experimentally inaccessible, the top view figures displayed in the right panels show
149

Figure 5.4:
Illustration that the proposed measurements suffice to identify the color
structure of a new dijet resonance. These plots show regions of the 3-d parameter space
gu2 /(gu2 + gd2 ) vs. σtt¯/σjj vs. σb¯b /σjj , at fixed values of mass (3 TeV on the top panels and
4 TeV on the bottom panels) where dijet cross section and Dcol fall within a certain range.
The cross
section is within about 35% the value required for a 5σ discovery at the LHC
√
with s = 14 TeV at L = 1000 fb−1 . The range of 50% from Dcol = 3 × 10−3 is chosen for
illustration. Points in parameter space that are accessible to the LHC are circles highlighted
in blue for C and triangles highlighted in green for Z . Each plot on the left panel is shown
again, as viewed from above, on the right. The two views make clear that colorons and Z’
bosons lie in distinguishably separate regions of parameter space; in particular, our ability
to measure the ratio gu2 /(gu2 + gd2 ) will not prevent us from determining the color structure
of a new vector resonance at the 14 TeV LHC.
150

that this does not typically lead to confusion between a coloron and a Z , just as we have
argued in Sec. 5.3. That is, even having projected the 3D data of all up ratio values onto
the bottom-ratio vs. top-ratio plane, the blue coloron and green Z points still lie in distinct
regions.

5.6.2

Using heavy flavor measurements to tell C from Z

Fig. 5.4 not only shows that the coloron and Z lie in different regions of parameter space,
but also implies that measurements of the top and bottom decays of the new resonance will
almost always enable us to determine its color structure.
Let us assume that a new resonance has been found and that its mass, dijet cross-section,
and Dcol have been measured. For illustration, take the values of these observables to be
those used in Fig. 5.4 (for the same LHC energy, luminosity and estimated uncertainties).
Because there is a gap between the Z and C regions of that figure when the up ratio is 0,
and because the boundaries of the regions are angled rather than vertical, we can see that a
Z with the minimum up ratio value of 0 would not be mistaken for a coloron. On the other
hand, a Z with the maximum up ratio (equal to 1) lies as close as possible to the coloron
region of parameter space. So we will want to compare a Z with an up ratio of 1 (one that
does not couple to d or s quarks) to a coloron with varying values of the up ratio.
This very comparison is presented in Fig. 5.5, which is plotted in the top-ratio vs bottomratio plane. The up ratio for the Z is fixed to be 1; the up ratio for the C is varied from 0
(left panels) to 1 (right panels). We see that as the up ratio for the coloron increases from its
minimum to maximum values, the blue coloron region of parameter space moves out from
the origin, away from the green Z region of parameter space. Correspondingly, if we were
to decrease the up ratio of the Z boson, the green Z region would shift closer to the origin,
151

away from the blue coloron region. In general, the coloron and Z regions do not overlap.
Given the shape and orientation of the regions corresponding to color-singlet and coloroctet resonances in the plots, measuring both the top ratio and bottom ratio would clearly
allow us to distinguish the new resonance’s color structure. Moreover, we see that if either
the top ratio or bottom ratio were measured to be sufficiently large, we would know that the
resonance must be a coloron (because the Z region is already at its maximum distance from
the origin). For example, a measurement of σtt¯/σjj

6 for a 3 TeV resonance or σtt¯/σjj

3

for a 4 TeV resonance, for the values of σjj used in Fig. 5.5, would identify it as a color-octet.
We note that there could still be a rare situation where our inability to measure the up
ratio would prevent us from determining the color structure of a new resonance. The regions
of parameter space corresponding to the extreme cases of a coloron with only down-type
light quark couplings and a Z with only up-type light quark couplings could potentially
overlap. As mentioned in Sec. 5.3, this is more likely to happen for heavier resonances due
to decreasing values of parton distribution functions for down-type light quarks at higher
resonance masses. For example, given our estimations of uncertainties, such an overlap could
occur for a 4 TeV resonance as illustrated in the lower left panel of Fig. 5.5.
The determination of the color structure of a resonance generally requires measurements
of both σtt¯ and σb¯b . As with the dijet cross sections, systematic uncertainties for these
measurements will be obtained after the experiment (at 14 TeV) has started6 . While the
estimation of these uncertainties lies beyond the scope of this article, our result should
illustrate that measuring the tt¯ and b¯b cross sections to an uncertainty of O(1) still provides

6 Systematic uncertainties on the total SM background yields from recent tt¯ resonance searches at √s =
¯
7,
√ 8 TeV [274, 275] are ∼ O(20%). Estimates for the total systematic uncertainty for tt final state at
s = 14 TeV can be extracted, for example, from a study for
√ the fully hadronic channel presented in
reference [276], which is ∼ O(40 − 50 %). The b¯b resonance at√ s = 8 TeV search has been carried out by
CMS [277], while the estimate for systematic uncertainties at s = 14 TeV is currently not available.

152

significant information. For the purpose of comparing models and estimating the required
uncertainties, additional plots illustrating models of coloron and leptophobic Z that lead
to the same value of Dcol , without uncertainties being taken into account, are presented in
Appendix D.
The determination of the color structure of a resonance requires generally requires precise
measurements of both σtt¯ and σb¯b . As with the dijet cross sections, systematic uncertainties
for these measurements will be obtained after the experiment has started. While the estimation of these uncertainties lies beyond the scope of this article, our results in this section
should illustrate the precision required for distinguishing among color structures. For the
purpose of comparing models and estimating the required uncertainties, additional plots illustrating models of coloron and leptophobic Z that lead to the same value of Dcol , without
uncertainties being taken into account, are presented in the Appendix D.

5.7

Discussion

The simple topology and large production rate for a dijet final state not only allows a
Beyond the Standard Model vector boson to be observed via dijet resonance searches, it
also allows the determination of crucial properties of the new resonance. The measurements
are particularly useful in distinguishing between a color-octet vector resonance (C) and a
color-singlet one that couples only to colored particles (leptophobic Z ). In this article, we
showed that the method for distinguishing between the two types of resonances in a modelindependent manner using the color discriminant variable introduced in Ref. [6], can be
extended to more general and realistic scenarios of flavor non-universal couplings in a wide
range of models.

153

LHC

10

2
gu
2 +g 2
gu
d

σb¯b /σjj

8

s = 14 TeV, L =1000f b−1 , M =3.0 TeV, σjj = 0.015±0.0051 pb, Dcol = 0.003 ± 0.0015

= 0(C), 1.0(Z )

8

= 0.5(C), 1.0(Z )

C

0

2

4 6
σtt¯/σjj
√

2
gu
2 +g 2
gu
d

8

10

0

= 1(C), 1.0(Z )

C

6

2

Z
0

2

4 6
σtt¯/σjj

8

10

0

Z
0

2

4 6
σtt¯/σjj

8

10

s = 14 TeV, L =1000f b−1 , M =4.0 TeV, σjj = 0.0073±0.0026 pb, Dcol = 0.003 ± 0.0015

= 0(C), 1.0(Z )

10
8

2
gu
2 +g 2
gu
d

= 0.5(C), 1.0(Z )

10
8

6

6

6

4

4

4

2

2

0

2
gu
2 +g 2
gu
d

4

2

Z

10
8

4

C

LHC

10

2
gu
2 +g 2
gu
d

6

4

0

10
8

6

2

σb¯b /σjj

√

Z C
0 2 4 6
σtt¯/σjj

8

10

0

4 6
σtt¯/σjj

8

10

0

= 1(C), 1.0(Z )

C

2

C
Z
0 2

2
gu
2 +g 2
gu
d

Z
0 2

4 6
σtt¯/σjj

8

10

Figure 5.5: Further illustration that measuring σtt¯ and σb¯b will show whether a resonance of
a given mass, dijet cross-section, and Dcol is a coloron or Z . We display regions of parameter
space that correspond to a dijet cross
section within 1σ uncertainty of the value required
√
for a discovery at the LHC with s = 14 TeV at L = 1000 fb−1 (the same values used in
Fig.5.4), and that also have the illustrative color discriminant variable within a range of 50%
around value of Dcol = 3 × 10−3 , for resonances with masses 3 TeV (top panel) and 4 TeV
(bottom panel). Coloron and Z models that are within reach are displayed in a verticallyhatched blue region and a diagonally-hatched green region, respectively. Note that in nearly
all of the displayed areas, the colorons and Z bosons lie in different regions of parameter
space. However, at the bottom left panel for a 4 TeV resonance, there is a small overlapping
region that could correspond either to a C coupling only to down-type light quarks or to a
Z coupling only to up-type light quarks.

154

The color discriminant variable, Dcol , is constructed from measurements available directly
after the discovery of the resonance via the dijet channel; namely, its mass, its total decay
width, and its dijet cross section. Assuming the new resonance couples identically to quarks
of the first two generations, Dcol depends on three model-specific ratios of coupling constants:
g2

g2

g2

the up ratio ( 2 u 2 ), the top ratio ( 2 t 2 ), and the bottom ratio ( 2 b 2 ). We showed that
gu +g
gu +g
gu +g
d
d
d
the method is generally not dependent on knowing the up ratio, a quantity which is not
presently accessible to experiment. Since Dcol is insensitive to chiral structure, discriminating
between color-singlet and color-octet resonances with flavor non-universal couplings requires
only measurements of the tt¯ and b¯b resonance cross sections. Our results are illustrated in
Figs. 5.4 and 5.5, with further scenarios explored in the Appendix D.
Our analysis assumed that the coloron and Z have only negligible couplings to any nonStandard Model fermions that may exist. It is straightforward to consider an extension to
models where the resonance does couple to new fermions. While the coloron always has only
visible decay channels (as it couples to colored particles), the Z could have non-negligible
decay branching fraction into non-Standard Model invisible particles. In that case, the
leptophobic Z and the coloron would be even easier to distinguish from one another by
the color discriminant variable. Simply put, the Z ’s invisible decays would increase its
total width, which appears in the denominator of the expression for its color discriminant
variable7 . This means the value of the Z ’s color discriminant variable, which is already
smaller than that of the coloron by a factor of 8, would be further reduced due to the
appearance of non-negligible invisible decays. Therefore, a leptophobic Z with invisible
g2

σ

σ¯

decays will correspond to a region in the 2 u 2 vs. σ tt¯ vs. σ bb parameter space (such as
jj
jj
gu +g
d

those presented in Fig. 5.4) that lies even further away from the region occupied by colorons.
7 Cf. the curly braces of Eq. (5.15).

155

To summarize: we have generalized the color discriminant variable for use in determining
the color structure of new bosons that may have flavor non-universal couplings to quarks.
We focused on resonances having masses 2.5 − 6.0 TeV for the LHC with center-of-mass
energy

√
s = 14 TeV and integrated luminosities up to 1000, fb−1 . After having taken

relevant uncertainties and exclusion limits from current experiment and sensitivity for future
experiments into account, we find that the future runs of the LHC can reliably determine
the color structure of a resonance decaying to the dijet final state.

156

Chapter 6
Probing Color Octet Couplings at the
Large Hadron Collider
— This chapter is based on a work in collaboration with Anupama
Atre, R.Sekhar Chivukula, Elizabeth H. Simmons, and Jiang-Hao
Yu which has appeared in [4]. Part of its content have also appeared in the conference proceeding [5].

6.1

Introduction

In addition to the color structure of a new vector resonance, the way the resonance couples
to different chiralities of the known fermions will provide a helpful insight as to the origin of
the particle. At a hadron collider, as the initial states are composed of colored particles, the
LHC will generally produce a resonance coupled to these colored particles at a large rate.
This makes the LHC an ideal machine for exploring new colored resonances associated with
new strong dynamics. This chapter is based on the work [4] studying the chiral structure of
the couplings of a color-octet vector resonance collectively referred to as a coloron.
The high production rate of a colored resonance (due to strong coupling) and the simple
topology of the final state (decay into two jets) makes the search for di-jet resonances one
of the early signatures that are studied at hadron colliders. Once a new colored resonance
157

is discovered, measuring its properties will be the next important task. The di-jet invariant
mass mjj and the angular distributions of energetic jets relative to the beam axis are sensitive
observables to determine the properties, such as mass and spin of the resonance. As I have
shown in Chapters 4 and 5, basic observables available with a dijet resonance discovery
can help distinguish a coloron from a leptophobic Z . The basic observables are the mass,
width, cross section to dijet final states, and cross sections to heavy flavors state of tt¯ and
b¯b. Although one can use these observables to constrain the coupling strength of the colored
resonance to the Standard Model (SM) quarks, they are not sufficient to determine the chiral
structure of the couplings. All these observables receive equal contributions from left- and
right-handed chirality quarks.
Some additional information about the chiral couplings can potentially be gleaned from
¯

the measurement of the forward-backward asymmetry of the top-antitop pair (AtFt B ). Such
a measurement has been made at the Tevatron. According to the Standard Model, QCD
is the interaction responsible for observing more top quarks moving along the direction of
the proton beam than along the anti-proton beam. In QCD, the gluons couple equally to
¯

left- and right-handed quarks. So the non-zero AtFt B could then predicted via next-to-leading
order corrections. The observations in 2011 at the Tevatron [254, 255] were of O(20%), which
were larger than the predictions of O(5%), available at the time this work was completed
in 20121 , by the Standard Model [280, 281]. Motivated by these results, various models
with specific chiral structures of the color-octet particle that give rise to a large asymmetry
have been proposed (see, for example, [216, 218]). Unlike the Tevatron, which uses protonantiproton initial state, the initial proton-proton initial state of the LHC is symmetric. The
1 In November 2014, a next-to-next-to-leading-order (NNLO) calculation combining all known Standard
Model corrections was completed by the authors of Ref. [278]. They found the asymmetry predicted by
the Standard Model to be 9.5 ± 0.7% which is in agreement with the most recent measurement by the
DØ collaboration [279] of 10.6 ± 3%.

158

forward backward asymmetry of the top-antitop pair is diluted at that machine compared
to the large asymmetry at the Tevatron [282, 283].
We proposed a new channel for studying coloron couplings at the LHC: the associated
production of a W or Z gauge boson with the coloron. The chiral couplings of the weak gauge
bosons to the fermions in the associated production channel provides additional information
about the chiral structure of the new strong dynamics. Combining the associated production
channel with the di-jet channel makes it possible to extract the chiral couplings of the
colored resonance because the cross-sections of each channel have a different dependence
on the coloron’s couplings to fermions. The functional form of the dependence of these
2 + g 2 ; in the Wjj channel it
measurements on the chiral couplings in the di-jet channel is gL
R
2 and in the Zjj channel it is ag 2 +bg 2 . A cartoon illustration of these three measurements
is gL
L
R

along with the di-jet measurement is shown in Fig. 6.1. Notice that while combining the
different channels will narrow the allowed range of couplings, there remains an ambiguity in
extracting the sign of the couplings. This method of using the associated production of a
weak gauge boson to illuminate the properties of a new resonance was studied earlier in the
context of the measurement of Z couplings [140, 136].
In this work [4] we studied the sensitivity of the LHC with center of mass (c.m.) energy
of 14 TeV to probe the chiral structure of the couplings for colored resonances with 10 fb−1
and 100 fb−1 integrated luminosity by the method proposed above. We studied colored
resonances with masses in the range 2.5 TeV to 4.5 TeV and various couplings and widths;
this mass range runs from the lightest colorons still allowed by LHC dijet searches to the
heaviest colorons to which the LHC is likely to be sensitive.
The rest of the chapter is organized as follows. In Sec.6.2, we present a simple parameterization for the colored resonances and our notation. In Sec.6.3, we discuss the signal and
159

3

Zjj

2

Dijet

gR

1

0

−1

−3
−3

Wjj

−2
−2

−1

0
gL

1

2

3

Figure 6.1: A cartoon illustration of the form of prospective constraints on chiral couplings
from the di-jet channel (dashed black circle with red band), the channel with associated
production of a W boson (solid black parallel lines with green band) and the channel with
associated production of a Z boson (dotted black ellipse with blue band). Combining the
constraints from different channels will narrow the range of allowed couplings.

160

associated backgrounds, the Monte Carlo simulation details in Sec.6.3.1 and the channels
with charged and neutral gauge bosons in Sec. 6.3.2 and Sec.6.3.3 respectively. We present
a discussion of our results in Sec.6.4 and conclusions in Sec.6.5.

6.2

General Parameterization

The color-octet resonance of interest to our study may be motivated in many BSM scenarios
as noted in the introduction. Hence we explore a phenomenological model of color-octet
resonances independent of the underlying theory. The interaction of the color-octet resonance
C µ with the SM quarks qi has the form

i
i
i γ q = ig q¯ C µ γ
L = igs q¯i C µ γµ gVi + gA
s i
µ g L PL + g R P R q i ,
5 i

(6.1)

i (or g i and g i ) denote vector and
where Cµ = Cµa ta with ta an SU(3) generator, gVi and gA
L
R

axial (or left and right) coupling strengths relative to the QCD coupling gs , the projection
operators have the form PL,R = (1 ∓ γ5 )/2 and the quark flavors run over i = u, c, d, s, b, t.
For simplicity, we will denote the color-octet resonance by C and its chiral couplings to light
q

t .
quarks by gL,R and to the third generation by gL,R
q,t

The various couplings gL,R can all be independent in the most general non-universal
case and can all be the same in simple flavor universal scenarios. A study of the former
case would be desirable but it is computationally cumbersome and the latter case, while
simple, does not include some interesting scenarios. Furthermore the authors of Ref. [259]
found that the flavor universal case is excluded at 99.5% by a global fit to top pair data for
color-octet masses less than 3 TeV. An interesting (yet manageable and general) example is

161

that of a color-octet resonance that can enhance top-pair forward-backward asymmetry in
models allowing non-universal couplings when [215, 284, 217]

q

t < 0.
gA gA

(6.2)

¯

tt measurement at the Tevatron we choose the following non-universal
Motivated by the AF
B

couplings as in Ref. [215] as an example.

q

gVt = gV

and

q

(6.3)

q

(6.4)

t = −g ,
gA
A

or in terms of gL − gR ,
q

t =g ,
gL
R

and

t =g .
gR
L

This example is sufficient to demonstrate the utility of the method of measuring couplings
described in this article. This choice of couplings also allows us to easily compare the reach
¯

of our study with that of Ref. [215] which is consistent with the AtFt B measurement. A more
general study involving fully independent couplings is beyond the scope of this study and
will be addressed in a future publication.
The color-octet resonance with the interactions as in Eq. (6.1) decays primarily to two
jets or a top pair and its decay width is given by

m
q 2
q 2
t 2 + gt 2
ΓC = αs C 4 gL + gR + gL
R
12
+

t 2 + g t 2 )(1 − µ ) + 6g t g t µ
(gL
t
R
L R t

1 − 4µt ,

(6.5)

where the terms in the first line come from decays to light quarks and to the bottom quark,

162

while the terms in the second line come from decays to top quarks. Decays to top quarks are
modified by the kinematic factors involving µt = m2top /m2C with mtop and mC the top quark
and color-octet mass respectively. Strictly speaking, the bottom quark’s contribution to the
width is modified by factors involving µb but we ignore these factors since m2b << m2C . For
an octet that is heavy compared to the top quark and whose couplings are of the form given
in Eq. (6.4) the expression for the decay width simplifies to

ΓC =

αs
2 =α m
2
2
m g 2 + gR
s C gV + gA .
2 C L

(6.6)

In this simplified version, the branching fraction for the color-octet resonance to decay to
any single quark flavor obeys the simple relation

BR(C → qi q¯i ) = 1/6,

6.3

where i = u, d, c, s, t, b.

(6.7)

Collider Phenomenology

In this section we study the collider phenomenology of color-octet states produced in association with a W or Z gauge boson and discuss the signal and associated backgrounds. We
present the the Monte Carlo simulation details in Sec.6.3.1, and in Sec.6.3.2 and Sec.6.3.3
we study the modes of associated production with a W and a Z boson, respectively.
The color-octet states (C) are produced and decay to two jets via the process

C

pp −→ j j.

163

(6.8)

They can also be produced in association with a weak gauge boson via the processes

C

pp −→ j j W ± ,

(6.9)

C

pp −→ j j Z,

(6.10)

where j = u, d, s, c, b. We will refer to the processes in Eq. (6.9) and Eq. (6.10) as the
CW and CZ channels, respectively. Representative diagrams of interest for the associated
production modes, which include s and t channel diagrams with the emission of the gauge
bosons in either the initial or final state, are shown in Fig. 6.2. The final state channels that
we study are
pp →

±E
/T

2j,

+ −

2j,

coming from W ± (→ ± ν) or Z(→ + − ), respectively and

(6.11)
= e, µ. Although the inclusion

of the τ lepton in the final state could increase signal statistics, for simplicity we ignore this
experimentally more challenging channel.
The relevant backgrounds to the signal processes in Eq. (6.11) are
• W + jets, Z+ jets with W, Z leptonic decays;
• top pair production with fully leptonic, semi-leptonic and hadronic decays (where some
final state particles may be missed or mis-identified);
• single top production leading to a W ± b q final state;
• W + W − , W ± Z and ZZ with all possible decay combinations leading to the final state
in Eq. (6.11);
Next, we present some details about the Monte Carlo simulation.
164

l (l+)

W (Z)

q

ν¯ (l−)

j

q

j
l (l+)
ν¯ (l−)

W (Z)

C

C

q¯

j

q¯

j

l (l+)

l (l+)

ν¯ (l−)

W (Z)
q

W (Z)
j

q

C

j

C

j

q¯

ν¯ (l−)

q¯

j

Figure 6.2: Representative Feynman diagrams for associated production of a W, Z gauge
boson with a color-octet resonance, C. Both s and t channel diagrams along with initial or
final state radiation of the associated gauge boson are shown. We assume that the weak
gauge boson decays leptonically.

6.3.1

Monte Carlo Simulation

We have performed a detailed simulation of both the signal and all the relevant backgrounds
using Madgraph/Madevent [285] for event generation at the partonic level, PYTHIA [286]
for parton showering with initial and final state radiation as well as hadronization and
PGS [287] for detector simulation. All the detector simulation parameters were set to default
values that correspond to the LHC Detector in Madgraph/Madevent. The CTEQ6L1 Parton
Distribution Functions (PDFs) [262] were used for both signal and background samples. For
the signal processes we set the factorization and renormalization scales to be µF = µR =
mC . For the background processes the renormalization and factorization scales are set to
µF = µR = Q where Q2 =

j2

(m2i + pTi ).

We simulate signal samples with color-octet masses, mC = 2.5 to 4.5 (4.0) TeV for the

165

q

t ,
CW (CZ) channel. For each mass, we do an exhaustive sampling of couplings gL,R , gL,R
t
where the couplings gL,R
follow the relation in Eq. (6.4). This exhaustive sampling of

couplings gives us color-octets with varying widths, ΓC /mC ∼ 0.025 − 0.50. While we use
the entire sample in estimating the reach of the signal, we will only present details about the
samples with ΓC /mC = 0.05, 0.10, 0.20, 0.30 for simplicity and clarity of presentation. We
show the parton-level cross sections for a few different color-octet masses and widths at the
LHC with

√

s = 14 TeV in Table 6.1 for illustration. Also shown, for comparison, are the

sizes of the standard model backgrounds at parton level.
Table 6.1: Representative cross sections √
at parton level in fb for the CW and CZ signal
modes and backgrounds at the LHC with s = 14 TeV. The signal is shown for mC = 2.5 to
4.5 TeV and for a few sample coupling values that correspond to ΓC /mC = 0.05, 0.10, 0.20
j
j
and 0.30. For the backgrounds a requirement that pT1 > 250 GeV and pT2 > 200 GeV was
j
j
applied in all cases, except for tt¯ where p 1 > 150 GeV and p 2 > 100 GeV was applied.
T

mC (TeV)

Γ/mC

2.5
3.0
3.5
4.0
4.5

0.05
0.10
0.20
0.30
0.30

T

Signal - CW
q
gR
σ (fb)
-0.42
0.82
8.4
0.59
1.2
7.5
0.71
1.7
6.4
1.2
1.8
8.2
-1.2
1.9
7.8
q
gL

Signal - CZ
q
gR
σ (fb)
-0.82 -0.42
3.5
-1.09 -0.71
3.0
-1.49
1.08
3.5
-1.92 -1.17
3.8
q
gL

Background
σ (fb)
(W → ν) + 2 jets
9500
tt¯ semi leptonic
4200
(Z → ) + 2 jets
1000
single top (t → ± νb)
160
Total
15000

The backgrounds for the process of interest are listed in Sec.6.3. We generate all the

166

j

j

backgrounds with the requirement that pT1 > 250 GeV and pT2 > 200 GeV (at the parton
j

j

level) except top-pair production where we used pT1 > 150 GeV and pT2 > 100 GeV. We
verified that requiring large cuts on the pT of the leading jets at the parton level does not
distort the distributions in the region of interest to us (higher pT regions) or affect the
background efficiencies.
For the CW channel the leading background is W + jets with leptonic decays of the W .
There is also a sizable contribution to the background from top pair production with semileptonic decays. Other decay modes of top pair such as the fully leptonic mode where one
lepton is lost and the hadronic mode where there is a fake lepton turn out to be negligible
after the acceptance and optimized cuts are applied. The single top background as well
as those from Z+ jets where one lepton is lost turn out to be relevant at the sub-leading
level. The background from all the diboson channels turns out to be insignificant after the
acceptance and optimized cuts are applied. For the CZ channel the only relevant background
is Z+ jets with leptonic decays of Z. The other backgrounds such as top pair with fully
leptonic decays and diboson channels turn out to be insignificant after the acceptance and
optimized cuts are included. While we analyzed all the backgrounds, we only list the cross
sections for the leading and sub-leading backgrounds in Table 6.1 for illustration.
We summarize below the minimum cuts imposed (after detector simulation) in recon-

167

structing the physics objects for our analysis.

j

pT > 40 GeV,

|ηj | < 2.5,

∆R(jj) > 0.4,

# jets ≥ 2,

pT > 25 GeV,

|η | < 2.5,

∆R(j ) > 0.4,

∆R( ) > 0.2,

E
/T > 25 GeV,

# isolated leptons = 1(2) for CW (CZ) channel,

mZ − 20 GeV < m

< mZ + 20 GeV.

(6.12)

Note that the E
/T cut is only for the CW channel and the ∆R( ) and m

cuts are only for

the CZ channel. All other cuts are applied for both channels.

6.3.2

Charged Current Channel with W -Decays to Leptons

In this section we study the case where the color-octet is produced in association with a W
boson and investigate the prospects of this channel at the LHC with

√

s = 14 TeV. The

color-octet decays to two jets and the W boson decays leptonically W → ν ( = e, µ) giving
us a final state jj E
/T . We probe color-octet masses from 2.5 to 4.5 TeV for a wide range of
q

t
couplings gL,R , gL,R
and decay widths (ΓC ) and describe the analysis in detail below.

Each event is required to contain at least two jets and exactly one lepton isolated from
other leptons and jets, the criteria for which are listed in Eq. (6.12). There are also additional
minimum requirements on the transverse momentum and rapidity of the jets, leptons and
missing energy listed in Eq. (6.12). We will refer to these requirements as acceptance cuts.
As jets from the decay of a heavy resonance are highly energetic, we identify the two jets with
the highest pT as the jets from the decay of the color-octet. In Fig. 6.3(a) and (b) we show
the transverse momentum distributions for these jets normalized to unit area for a coloroctet with mass mC = 3 TeV and two different widths ΓC /mC = 0.05 and 0.20. It is clear
168

from the distributions that the pT of the jets is a good discriminant to separate signal from
the background. The signal distributions are very broad and the background distributions
are sharply peaked at lower transverse momentum. In addition, since associated production
would be studied after the existence and mass of the color-octet have been established by
dijet studies, we can use transverse momentum cuts that are optimized based on the mass
of the color-octet state.
The efficiencies of the acceptance cuts and the optimized transverse momentum cuts
for various signal and background samples (optimized to give 3σ significance) are shown in
Table 6.2 and 6.3 for different mass values of the color-octet at 14 TeV c.m. energy. The
couplings and widths for the mass points shown are the same as in Table 6.1. We can achieve
very good separation of signal from background by just using these simple pT cuts as seen
from Table 6.2 and 6.3. We find that the commonly used kinematic variable HT , which is the
sum of the transverse momenta of final state particles, is dominated by the two hardest jets
and there is no significant gain in efficiency by further optimizing on this variable. Similarly,
we have investigated the possibility of using the reconstruction of the mass peak as another
useful discriminant; as discussed in Sec.6.4, we found that its likely utility is limited.
Finally we mention some details about the signal simulation sample. We study different
color-octet masses from 2.5 TeV to 4.5 TeV. While we show only a few points for illustration
q

t
in Table 6.2 and 6.3, we sample a wide range of couplings gL,R , gL,R
and decay widths (ΓC )

ranging from very narrow widths ∼ 2.5% to upwards of 40% for each mass point. We estimate
√
the significance for each point in parameter space by s/ b, where s (b) are the number of
signal (background) events. The results of this analysis are presented in Sec.6.4.

169

(a)

(b)
j

Figure 6.3: (a) Top: transverse momentum of the hardest jet (pT1 ) for a color-octet with
mass mC = 3 TeV and two different widths ΓC /mC = 0.05 and 0.20. (b) Bottom: same as
j
(a) but for the second hardest jet (pT2 ). The relatively broad and flat transverse momentum
distributions of the signal events contrast with the distributions for the backgrounds, which
all peak sharply at lower transverse momenta.

170

Table 6.2: Selection efficiencies (in percent) for signal ( mC ) and background ( BG ) samples
for the case of the CW channel with leptonic decays of W into electrons and muons. The
couplings and widths for the mass points shown are the same as those in Table 6.1.
Efficiencies (in %)
Cuts for
mC = 2.5 TeV
2.5
W jj
tt¯
Zjj
t

Cuts for
mC = 3.0 TeV
3.0
W jj
tt¯
Zjj
t

Cuts for
mC = 3.5 TeV
3.5
W jj
tt¯
Zjj
t

6.3.3

Acceptance cuts
see Eq. (6.12)
54
54
58
13
56

j

j

pT1 > 825 GeV

pT2 > 775 GeV

Overall

53
1.3
8.9×10−2
1.5
5.0×10−1

63
36
6.1
49
6.1

18
0.25
3.1×10−3
9.2×10−2
1.7×10−2

pT2 > 950 GeV

Overall

60
35
7.5
48
4.1

13
8.8×10−2
7.9×10−4
3.3×10−2
2.5×10−3

pT2 > 1050 GeV

Overall

62
35
1.9
48
4.5

9.8
5.2×10−2
9.5×10−5
2.0×10−2
1.3×10−3

Acceptance cuts j1
pT > 1000 GeV
see Eq. (6.12)
54
41
54
4.6×10−1
58
1.9×10−2
13
5.5×10−1
56
1.1×10−1
Acceptance cuts j1
pT > 1100 GeV
see Eq. (6.12)
51
30
54
2.7×10−1
58
9.0×10−3
13
3.3×10−1
56
5.0×10−2

j

j

Neutral Current Channel with Z-Decays to Charged Leptons

In this section we study the case where the color-octet is produced in association with a Z
boson and investigate the prospects of this channel at the LHC with

√

color-octet decays to two jets and the Z boson decays leptonically Z →
171

s = 14 TeV. The
( = e, µ) giving

Table 6.3: Same as Table 6.2, but for mc = 4.0 and 4.5 TeV.
Efficiencies (in %)
Cuts for
mC = 4.0 TeV
4.0
W jj
tt¯
Zjj
t

Cuts for
mC = 4.5 TeV
4.5
W jj
tt¯
Zjj
t

Acceptance cuts j1
pT > 1200 GeV
see Eq. (6.12)
49
21
54
1.6×10−1
58
6.8×10−3
13
2.1×10−1
56
2.4×10−2
Acceptance cuts j1
pT > 1350 GeV
see Eq. (6.12)
50
16
54
8.0×10−2
58
7.1×10−4
13
9.0×10−2
56
5.4×10−3

j

pT2 > 1150 GeV

Overall

61
34
0
42
1.8

6.5
3.0×10−2
0
1.1×10−2
2.5×10−4

pT2 > 1300 GeV

Overall

59
31
0
35
8.3

4.5
1.4×10−2
0
4.0×10−3
2.5×10−4

j

us a final state jj . All the details of the signal simulation are the same as in Sec.6.3.2,
except that we sample color-octet masses from 2.5 TeV to 4.0 TeV as there is no sensitivity
in this channel for higher mass resonances.
The event is required to contain at least two jets and exactly two leptons isolated from
other leptons and jets in the event, the criteria for which are listed in Eq. (6.12). We
require that the two isolated leptons reconstruct a Z boson with the condition that mZ −
20 GeV < m

< mZ + 20 GeV. There are also additional minimum requirements on the

transverse momentum and rapidity of the jets and leptons listed in Eq. (6.12). We will refer
to these combined requirements as acceptance cuts. The pT of the two hardest jets is a good
discriminant to separate signal from the background as in the case of the CW channel. The
efficiencies of the acceptance cuts and the optimized transverse momentum cuts for various

172

signal and background samples (that give 3σ significance) are shown in Table 6.4 for different
mass values of the color-octet at 14 TeV c.m. energy. The couplings and widths for the mass
points shown are the same as those in Table 6.1. Again, we sample a much broader range of
couplings and widths and show only a few points for illustration.
Table 6.4: Selection efficiencies (in percent) for signal ( mC ) and background ( BG ) samples
for the case of the CZ channel with leptonic decays of Z into electron and muon pairs. The
couplings and widths for the mass points shown are the same as those in Table 6.1.
Efficiencies (in %)
Cuts for
mC = 2.5 TeV
2.5
Zjj

Cuts for
mC = 3.0 TeV
3.0
Zjj

Cuts for
mC = 3.5 TeV
3.5
Zjj

Cuts for
mC = 4.0 TeV
4.0
Zjj

Acceptance cuts
see Eq. (6.12)
34
42
Acceptance cuts
see Eq. (6.12)
34
42

j

j

pT1 > 850 GeV

pT2 > 800 GeV

Overall

47
8.4×10−1

68
43

11
1.5×10−1

pT1 > 975 GeV

pT2 > 925 GeV

Overall

38
3.9×10−1

69
45

8.9
7.3×10−2

pT2 > 1000 GeV

Overall

70
44

6.2
4.6×10−2

pT2 > 1150 GeV

Overall

74
49

4.4
2.2×10−2

j

Acceptance cuts j1
pT > 1050 GeV
see Eq. (6.12)
33
27
42
2.5×10−1
Acceptance cuts j1
pT > 1200 GeV
see Eq. (6.12)
34
17
42
1.1×10−1

j

j

j

It is evident from Table 6.4 that we can achieve very good separation of signal from
background for the CZ channel as well by using just simple pT cuts as in the case of CW
channel. Again, we investigated the possible improvements one could make by using the
invariant mass of the color-octet resonance as a discriminant. The situation for the case
of the associated production of the Z boson with leptonic decays is slightly better, as full
173

information for reconstructing the final state is readily available from the reconstructed
leptons. However, as detailed in Sec.6.4, we determined that it would not add appreciably
to the present analysis. Next, we discuss the results of our analysis and the sensitivity at
the LHC.

6.4

Sensitivity at the LHC

In this section we will describe the sensitivity of the associated production channel at the
LHC, the current constraints from direct searches and future sensitivity at the LHC to
provide a broad view of the reach of the LHC in determining the chiral structure of the
couplings of a color-octet resonance. We have simulated signal and background events for
two different scenarios at the LHC with

√
s = 14 TeV: an early run with an integrated

luminosity of 10 fb−1 and a longer run with 100 fb−1 integrated luminosity. We estimate
the number of signal and background events after optimizing the transverse momentum cuts
√
as described in Sec.6.3.2 and 6.3.3 and the statistical significance is calculated as s/ b where
s (b) is the number of signal (background) events.
We present the results of our analysis for the LHC in Figs. (6.4-6.8) in the plane of the
q

q

couplings gL , gR for different masses, mC , of the color-octet. The sensitivity for the channel
with associated production of a W (Z) gauge boson is presented in the upper (lower) panels
of Figs. (6.4-6.6), while the left (right) panels are for integrated luminosity of 10 (100) fb−1 .
In Figs. 6.7(a) and (b) we show the sensitivity for a 4.5 TeV color-octet resonance for 10
fb−1 and 100 fb−1 respectively in the CW channel. The CZ channel has no sensitivity
at this mass and we do not show that channel in Fig. 6.7. The different colored bands
represent varying significance of signal observation from 2σ to greater than 5σ. The black

174

t S e ns

ij e

Γ/M=0.05

D

-1
fb

0
gL

1

2

−2

−1

0
gL

Dij
e

-1

D

-1
fb

0
gL

t S e ns

Γ/M=0.05

2

5σ
4σ
3σ
2σ

significance

<2σ

Γ/M=0.30

−2

(c)

>5σ

Γ/M=0.10
Γ /M=0 . 15
Γ/M=0.20

−2
1

it y 1 4

ij e

gR

0

−1

Γ/M=0.30

−1

tiv

i

<2σ

-1
fb

i

2σ

Dij
e

D

t S e ns

1

usion 7 TeV
4.8
xcl
tE

ij e

3σ

1 0 0f b

1 0 0f b

gR

4σ

V
Te

V
Te

Γ/M=0.05

usion 7 TeV
4.8
xcl
tE

it y 1 4

5σ

Γ/M=0.10
Γ /M=0 . 15
Γ/M=0.20

−2

2

LHC 14 TeV
∫L = 100 fb-1
MC = 2.5 TeV

-1

1

−2

1

2

>5σ

significance

LHC 14 TeV
∫L = 10 fb-1
MC = 2.5 TeV

−1

2σ

(b)

2

0

4σ
3σ

Γ/M=0.30

(a)

tiv

5σ

<2σ

significance

-1

−1

it y 1 4

Γ/M=0.10
Γ /M=0 . 15
Γ/M=0.20

−2

Γ/M=0.30

−2

0

−1

Γ/M=0.10
Γ /M=0 . 15
Γ/M=0.20

−2

tiv

i

<2σ

gR

i

t S e ns

2σ

Di
j

D

ij e

1

>5σ

lusion 7 TeV
4.8
Exc
et

gR

3σ

1 0 0f b

Γ/M=0.05

−1

4σ

V
Te

1 0 0f b

0

usion 7 TeV
4.8
xcl
tE

it y 1 4
V
Te

tiv

significance

Dij
e

5σ

-1

1

LHC 14 TeV
∫L = 100 fb-1
MC = 2.5 TeV

2

>5σ

-1
fb

LHC 14 TeV
∫L = 10 fb-1
MC = 2.5 TeV

2

−1

0
gL

1

2

(d)

Figure 6.4: (a) Top left: sensitivity plot for a color-octet of mass mC = 2.5 TeV produced
q q
in association with a W gauge boson, in the plane of the couplings gL , gR at the LHC with
√
10 fb−1 of data and s = 14 TeV; (b) top right: same as (a) but for 100 fb−1 of data;
(c) bottom left: same as (a) but for a color-octet produced in association with a Z boson;
(d) bottom right: same as (c) but for 100 fb−1 of data. The various color bands show the
regions with varying significance from 2σ to > 5σ. The inner solid green circle is the limit
from current direct searches for narrow dijet resonances at the LHC [244, 241, 243, 242, 245].
The outer green circle corresponds to a contour with ΓC /mC = 0.15. The region between
the two circles represented as a faded gray region is excluded for narrow resonances while
the region beyond the outer green circle is allowed for broad resonances. The small black
region in the center lies beyond the projected dijet sensitivity at the LHC with 100 fb−1 of
data [260]. The black dotted contours indicate the combinations of couplings that give rise
to varying widths ΓC /mC = 0.05, 0.10, 0.20 and 0.30.
175

dotted curves are contours of constant widths and we show the curves for several values of
ΓC /mC = 0.05, 0.10, 0.20 and 0.30. The small black region in the center lies outside the
projected dijet sensitivity (i.e. couplings within the black region result in a dijet production
rate too small to be observed) at the LHC with 100 fb−1 of data [260].
The Tevatron [240] and the LHC [244, 241, 243, 242, 245] have looked for resonances
in the dijet spectra, and the most stringent constraints come from the LHC as expected.
This data places stronger constraints on low mass resonances and there is essentially no
constraint on color-octets with masses above 3.5 TeV. The inner green solid circle is the
limit from the current non-observation of narrow resonances in the dijet channel at the LHC
[244, 241, 243, 242, 245] and the region outside this inner green circle can potentially be
excluded. However there is a caveat here. The analyses of the LHC dijet searches make
the assumption of a resonance with a narrow width of order 10% - 15%. The authors of
Ref. [258, 259, 131] argue that in the case where the resonance is not narrow (ΓC /mC > 15%)
the constraints from dijet data can be relaxed. We mark the contour of ΓC /mC = 0.15 by the
outer solid green circle. The area between the two green circles which is shown as a faded gray
region (where the narrow-width approximation is valid) is excluded. The region beyond the
outer green circle which corresponds to resonances with broader widths is possibly allowed.
For example, in Fig. 6.4(a) the dijet constraint would be valid for narrow resonances (up to
the outer green circle labeled ΓC /mC = 0.15) and would not be applicable to the regions
outside this curve. As the LHC accumulates more data, the simple dijet analyses would
remain sensitive only to the region inside the ΓC /mC = 0.15 curve. Of course a different
analysis of dijet data without the narrow width assumption could ultimately be sensitive to
the whole region. The region corresponding to broad resonances (beyond the outer green
circle marked Γc /mC = 0.15) has large couplings and hence will be largely accessible at the
176

LHC.

1

i

t S e ns

ij e

gR

−2

−1

0
gL

D

i

t S e ns

0

Γ/M=0.05

−1

Γ/M=0.10
Γ /M=0 . 15
Γ/M=0.20

−2
1

it y 1 4
tiv

ij e

gR

i

2σ
<2σ

-1

D

Γ/M=0.30

0
gL

D

1

-1

t S e ns

3σ

D

ij e

4σ

2

significance

>5σ
5σ
4σ
3σ
2σ
<2σ

Γ/M=0.30

−2

(c)

sion 7 TeV 4
xclu
.8
tE
fb
ije

gR

5σ

1 0 0f b

1 0 0f b

Γ/M=0.05

Γ/M=0.10
Γ /M=0 . 15
Γ/M=0.20

−1

2

LHC 14 TeV
∫L = 100 fb-1
MC = 3.5 TeV

V
Te

V
Te

it y 1 4
tiv

sion 7 TeV 4
xclu
.8
tE
fb
ije

1

−2

1

2

>5σ

significance

LHC 14 TeV
∫L = 10 fb-1
MC = 3.5 TeV

−2

2σ

(b)

2

−1

4σ
3σ

Γ/M=0.30

(a)

0

5σ

<2σ

significance

i

t S e ns

significance

2

>5σ

-1

ij e

0
gL

Γ/M=0.05

-1

D

−1

it y 1 4
tiv

Γ/M=0.10
Γ /M=0 . 15
Γ/M=0.20

−2

Γ/M=0.30

−2

0

−1

1

-1

gR

2σ
<2σ

D

−2

D

1

-1

Γ/M=0.10
Γ /M=0 . 15
Γ/M=0.20

3σ

sion 7 TeV
4.8
xclu
tE
fb
ije

Γ/M=0.05

−1

4σ

1 0 0f b

1 0 0f b

0

5σ

V
Te

V
Te

it y 1 4
tiv

lusion 7 TeV 4.8
Exc
fb jet
Di

1

LHC 14 TeV
∫L = 100 fb-1
MC = 3.5 TeV

2

>5σ

-1

LHC 14 TeV
∫L = 10 fb-1
MC = 3.5 TeV

2

−1

0
gL

1

2

(d)

Figure 6.5: Same as Fig. 6.4 but for mC = 3.5 TeV.

The results in Figs. (6.4-6.8) illustrate several features. Firstly, in the channel with
q

associated production of a W boson, there is no sensitivity in the region near gL = 0 due
to the left-handed couplings of the W boson, and the sensitivity improves as we move away
q

from the gL = 0 axis. The channel with the associated production of a Z boson on the other
hand is sensitive to both left and right-handed couplings and sensitivity in the region close
177

it y 1 4

0
gL

i

t S e ns

Γ/M=0.10
Γ/M=0.20

−2
1

2

Γ/M=0.30

−2

−1

0
gL

(a)

1
it y 1 4

2σ

-1

D

−1

0
gL

it y 1 4

i

t S e ns

ij e

gR

D

Γ/M=0.20

1

2

Γ/M=0.05

Γ/M=0.20

Γ/M=0.30

−2

(c)

−1

0
gL
(d)

Figure 6.6: Same as Fig. 6.4 but for mC = 4 TeV.

178

>5σ
5σ
4σ
3σ
2σ
<2σ

Γ/M=0.10

−2

Γ/M=0.30

−2

0

−1

Γ/M=0.10

−2

tiv

-1

i

1

1 0 0f b

t S e ns

3σ

V
Te

1 0 0f b

ij e

4σ

<2σ

V
Te

gR

5σ

Γ/M=0.05

−1

2

LHC 14 TeV
∫L = 100 fb-1
MC = 4.0 TeV

2

>5σ

significance

LHC 14 TeV
∫L = 10 fb-1
MC = 4.0 TeV

0

1

(b)

2

tiv

2σ
<2σ

significance

D

−1

ij e

gR

i

-1

Γ/M=0.20

Γ/M=0.30

−2

it y 1 4

4σ
3σ

Γ/M=0.05

−1

Γ/M=0.10

−2

0

D

Γ/M=0.05

−1

tiv

-1

t S e ns

2σ
<2σ

5σ

1 0 0f b

ij e

1

>5σ

V
Te

gR

3σ

1 0 0f b

0

4σ

V
Te

tiv

5σ

significance

1

LHC 14 TeV
∫L = 100 fb-1
MC = 4.0 TeV

2

>5σ

significance

LHC 14 TeV
∫L = 10 fb-1
MC = 4.0 TeV

2

1

2

1

2σ

t S e n sit

ij e

gR

0

D

Γ/M=0.20

−1

<2σ

0
gL

Γ/M=0.10
Γ/M=0.20

−2

Γ/M=0.30

−2

2σ

Γ/M=0.05

−1

Γ/M=0.10

−2

4σ
3σ

D

Γ/M=0.05

−1

t y 1 4 Te
ivi

significance

3σ

5σ

0 0fb - 1

t S e n sit

4σ

>5σ

V1

ij e

5σ

<2σ
0 0fb - 1

0

t y 1 4 Te
ivi

V1

gR

1

LHC 14 TeV
∫L = 100 fb-1
MC = 4.5 TeV

2

>5σ

significance

LHC 14 TeV
∫L = 10 fb-1
MC = 4.5 TeV

2

1

2

Γ/M=0.30

−2

(a)

−1

0
gL

1

2

(b)

Figure 6.7: (a) Left: same as Fig. 6.4(a) but for mC = 4.5 TeV; (b) right: same as (a) but
for 100 fb−1 of data. The figures for the corresponding CZ channels are not presented as
there is not sufficient sensitivity at this mass.
q

to gL = 0 is non-zero. The smaller production cross section for this channel along with the
small leptonic branching fractions for Z decays limit the gain in sensitivity. Nonetheless this
channel provides an additional measurement and hence useful information in untangling the
couplings.
We present the results for both 10 fb−1 and 100 fb−1 of data, and as expected, the longer
run with more data has better sensitivity and can probe masses up to 4.5 (4.0) TeV in the
channel with associated production of a W (Z) gauge boson. We also show the projected limit
of sensitivity in the dijet channel for the LHC with 100 fb−1 data as the small black circle in
the center with small couplings. This can be improved even further, essentially allowing one
to get closer to near zero couplings by the following observation. The theoretical analysis
that produced the projected sensitivity [260] selects for a rather narrow region in pseudorapidity of the jets leading to a small acceptance. The analyses of actual LHC data for the
7 TeV dijet spectrum at the LHC [244, 241, 243, 242, 245] include larger regions of pseudo-

179

rapidity giving rise to an acceptance larger than the one in Ref. [260]. It is safe to assume
that the final acceptance will be at least equal to or even better than the current one and
the small black region in the center could shrink even further.
Finally, the dijet reach is much better than that of associated production channel at the
LHC, essentially probing near zero couplings. If the LHC were to discover a resonance with
such small couplings, one would have to find other novel ways of understanding the chiral
structure of couplings as the associated production channel does not have sensitivity in those
regions of parameter space.

−1

1

2

i

0
=0 .10
0 . .1 5
20
0.
30

−2

D

(a)

t S e ns

=0

−2

0
gL

it y 1 4

ij e

gR

i

t S e ns

ij e

-1

gR

−1

.05

ed
w
lo
l
A

0
gL

>5σ

significance

significance

B
AF
y
b

0

tiv

4.8 fb -1
TeV

−1

ed
w
lo
l
A

<2σ

1 0 0f b

−2

2σ

n7

−2

ed
w
lo
l
A

1

3σ

B
AF Dijet Exclu
y
sio
b
V
Te

0
=0 .10
0 . .1 5
20
0.
30

.05

4σ

Γ/M =
Γ/M M
Γ/ =
Γ/M =
Γ/ M

=0

4.8 fb -1
TeV

it y 1 4
1 0 0f b

Γ/M =
Γ/M M
Γ/ =
Γ/M =
Γ/ M

−1

n7

0

tiv

V
Te

ed
w
lo
l
A

5σ

D

1

B
AF Dijet Exclu
y
sio
b

LHC 14 TeV
∫L = 100 fb-1
MC = 3.0 TeV

2

>5σ

-1

LHC 14 TeV
∫L = 10 fb-1
MC = 3.0 TeV

2

5σ
4σ
3σ
2σ
<2σ

B
AF
y
b

1

2

(b)

Figure 6.8: (a) Left: same as Fig. 6.4(a) but for mC = 3 TeV. Here we have combined the
sensitivity in the associated production channel from both W and Z bosons. The couplings
¯
allowed by the current measurement of AtFt B at the Tevatron [288, 289] as interpreted in
Ref. [217] are shown by the translucent yellow dotted region; (b) right: same as (a) but for
100 fb−1 of data;
Separately, we note that the Tevatron has made a measurement of the top-pair forward¯

tt ) [288, 289]. The authors of Ref. [217] have translated this
backward asymmetry (AF
B

measurement into constraints on the couplings of color-octet resonances. We show this
additional constraint for the case of a color-octet with mass mC = 3 TeV in Figs. 6.8(a) and
180

¯

(b) for 10 fb−1 and 100 fb−1 respectively. The region consistent with the AtFt B measurement
is shown in translucent yellow with small dots. Note that for clarity of presentation we have
combined the sensitivity from CW and CZ channels in Figs. 6.8(a) and (b). There are studies
[259] that do global fits of data from top pair production including cross section, angular
measurement, indirect searches and electroweak constraints under certain assumptions. The
interested reader can combine our analysis with that of Ref. [259] to see how those other
constraints (and assumptions) will affect the reach of the LHC for color-octet resonances.
Next, we discuss some possible means for achieving further improvements to the sensitivity. The reconstruction of the mass peak can be a useful discriminant to separate signal from
background provided the mass of the resonance is known from prior measurements. The signal for our process comes from both s and t-channel diagrams as shown in Fig. 6.2. In the case
where the s-channel contribution dominates and the associated W gauge boson is from the
initial state, it is straightforward to form the invariant mass of the resonance mC = m(j1 , j2 )
for the signal and backgrounds. As the jets from the decay of the color-octet are very energetic, there is hard radiation from the jets and we found that using the three hardest jets
reconstructs the invariant mass better and so it is better to use mC = m(j1 , j2 , j3 ) for the
signal and backgrounds. In the case where the associated W gauge boson is produced in the
final state with W → ν decay, we do not have full information to reconstruct the event.
Since we cannot determine which of the two configurations the final state W comes from,
one can make the conservative choice of using the transverse mass:

m2T =

j

j

j

2
2
1
2
3
pW
T + mW + pT + pT + pT

2

j

j

j

− pTW + pT 1 + pT 2 + pT 3

2

.

Our initial investigation found that selecting events based on a transverse mass cut yielded

181

only modest improvement. Even reconstructing events with a leptonically decaying W using
the W rest mass approximation did not help much. However further study may be warranted.
The situation for the case of the associated production of a Z boson with leptonic decays
is slightly better as full information for reconstructing the final state is readily available. In
the case where the s-channel contribution dominates, the invariant mass of the resonance
can be reconstructed from mC = m(j1 , j2 , j3 ) (initial state Z) or mC = m(j1 , j2 , j3 , Z)
(final state Z). Since we cannot determine which of the two configurations the Z boson
comes from, we pick the one closest to the resonance mass and call it mrec
C . In Fig. 6.9
we show the reconstructed invariant mass (mrec
C ) for a 3 TeV color-octet resonance with
width ΓC /mC = 0.05 and 0.30. The distribution for mrec
C is normalized to unit area. While
it is easy to distinguish the mass peak for the case of a very narrow resonance, it becomes
increasingly harder to do so for broader resonances. Furthermore it will be hard to determine
the mass of a broad resonance even in the dijet channel. Since we sample a wide range of
couplings which lead to varying widths from 2.5% to greater than 40% of the mass, using the
invariant mass as a discriminant is not useful for the entire range. Also, in the case where
the t-channel contribution is not negligible the resonance peak is diluted and the utility of
the invariant mass as a discriminant is further reduced. In view of these observations we do
not use the invariant mass as a discriminant to separate signal from backgrounds for either
the CW or CZ channel but mention it here for completeness as a possible improvement one
could make for the appropriate cases (narrow width resonances).
The analysis reported here gives only an estimate of the LHC reach and we discuss some
considerations about uncertainties and systematic effects. Our predictions for the signal and
background are at the leading order (LO) and no k-factors have been taken into account.
Varying the scales by a factor of two around the central value results in a variation of the
182

Figure 6.9: Reconstructed invariant mass (mrec
C ) for a 3 TeV color-octet resonance with
width ΓC /mC = 0.05 and 0.30 after acceptance cuts.
signal cross sections of order 15 − 30% and can be used as an estimate of the uncertainty
of the leading order (LO) cross sections. For the backgrounds one can use the k-factors
where available or they can be normalized to data in regions outside the signal. Multijet
events, which are abundant at the LHC, can fake the signal for the CW channel if jets are
misreconstructed as leptons and if jet transverse energies are poorly measured, leading to a
presence of missing energy in the events. As explained in Ref. [290] these effects will be small
with optimized experimental methods. Also high luminosity measurements have to contend
with the issue of pile up effects which become important for the longer run. Estimating these
effects are beyond the scope of this work and we only mention them as possible sources of
uncertainties that will need to be accounted for in an experimental analysis.

183

6.5

Summary

The LHC has been running successfully and opening up new frontiers at the TeV scale.
Among the many new possibilities for discovery at the LHC are color-octet resonances that
are motivated in many BSM theories. These new colored particles will be produced copiously
at a hadron collider and show up as resonance mass peaks in the dijet spectrum. Once these
particles are discovered it is of paramount importance to measure the properties such as
mass, spin and coupling structure to pinpoint the underlying theory. While some information
(such as mass and spin) can be obtained from the discovery mode, other information vital
to probing the theory such as chiral couplings cannot be obtained from that measurement.
In this chapter we proposed a new channel, namely the associated production of a W
or Z gauge boson with a color-octet resonance to provide information about the chiral
structure. We combined the information from the dijet (discovery) mode with the associated
production to uncover the chiral structure of the couplings of color-octet resonances to SM
quarks. In order to make our study as widely applicable as possible, we have performed a
phenomenological analysis of color-octets without being tied down to a specific theory. We
sampled a wide range of masses, couplings and decay widths of color-octets and optimized the
kinematic cuts to enhance the signal over the SM backgrounds. With this optimized analysis,
we determined the sensitivity for the couplings and masses at the LHC and presented the
results in the couplings plane for each mass after taking into account existing constraints.
For one sample case of mC = 3 TeV we also show the effect of being consistent with the
¯

AtFt B measurement at the Tevatron.
We studied two scenarios at the LHC with c.m. energy of 14 TeV, one with the early
data of 10 fb−1 integrated luminosity and the other with the longer run accumulating 100

184

fb−1 of data. As expected the early run is sensitive to lower masses and larger couplings but
the reach dramatically improves both in terms of mass and couplings for the longer run. In
particular the early run can probe masses up to 4.5 TeV but with relatively large couplings
compared to the strong QCD coupling while the longer run explores a much larger region
in couplings. We find encouraging results that the LHC will be able to provide information
about the chiral structure for a wide range of couplings and masses and hence point us in
the direction of the underlying theoretical structure.

185

Chapter 7
Conclusion
The Standard Model is a well-established theory that has shortcomings in both phenomenological and theoretical aspects. The origin of mass, an aspect that is treated as an input of
the Standard Model, could be addressed by new physics theories featuring dynamical breaking of the electroweak symmetry under the influence of new strong dynamics. The region of
energy in which the physics responsible for electroweak symmetry breaking lies is also within
the coverage of the LHC. Given that the collider still has at least a decade of experiments in
its schedule, we are certainly within one of the most exciting eras of modern particle physics.
This thesis accounts for two questions one generally asks during this data-rich era: what
do the current LHC data tell us about potential extensions of the Standard Model and
what can we do if, in the near-future experiments, a signal from new physics is found? In
particular, my collaborators and I studied how data from the Higgs searches and discovery
constraint models with strong dynamics in Part I. Then we studied ways to characterize a
vector resonance, a common predictions among most new physics models in Part II.
In Part I, we saw in Chapter 2 that even the first set of LHC data in 2011, before the
discovery of the Higgs-like boson, has already put challenges to a large number of technicolor
and topcolor-assisted technicolor models that include colored technifermions. These theories
predicted enhanced rates of the di-tau and di-photon final states in the Standard Model
Higgs searches, which were highly constrained by the data.
The fate of models featuring strong top-dynamics were studied in Chapter 3. We used an
186

effective model focusing on phenomenological aspects of models with strong top-dynamics to
analyze general phenomena of the models. The data in 2012, available with the discovery of
the Higgs-like particle, also highly constrained a large class of strong top-dynamics models.
As a matter of fact, the data on Higgs searches from the first run at the LHC put great
challenges not only on models featuring strong dynamics, but also on most new physics models that account for electroweak symmetry breaking. With the precisions of measurements
achieved so far [17], the data indicate that the discovered particle with mass of 126 GeV is
consistent with properties predicted by the Standard Model. The absence of other states
similar to the Higgs boson also put tight constraints on models inspired by the lack of “naturalness” of the Standard Model. In the scenario that the discovered particle is actually
the Higgs boson, a requirement of a mechanism (be it the Standard Model or new physics)
preventing the unitarity issue at a TeV scale could be considered satisfied. In this scenario,
new physics is no longer “guaranteed” at a TeV scale. Nevertheless, new physics could still
show up as small deviations from the Standard Model predictions. Therefore, high precision
measurements of the Higgs properties will be one of the first priorities of the next run of the
LHC and the next collider. Still, there are other aspects of motivations for new physics and
the chance of having a trace of such new physics showing up at the LHC is everything but
negligible.
In Part II, we studied how one could characterize a common prediction of various new
physics models, a new vector boson. We focused on direct searches in the dijet final state
which provides large rates and simple topology: the final state a large class of new physics
models might show up.
We saw in Chapter 4 and 5 that a color-octet vector boson could be distinguished from a
color-singlet leptophobic one, which has similar decay channels, using simple measurements
187

available with its discovery in a dijet final state. We found that our method for discriminating
the color structure is applicable to a vector boson that could show up during both early and
long-term runs at the 14 TeV LHC.
We also saw in Chapter 6 that the future LHC runs could also help probe the chiral
structure of a new vector boson, allowing us to study its origin. We introduced a method
utilizing associate productions of a color-octet vector boson with the Standard Model’s weak
bosons to determine the chiral couplings. From a phenomenological point of view, a vast
region of non-excluded parameter space is still within reach of the LHC. There are still less
well-explored areas of well-motivated ideas.
With the LHC, the ATLAS and CMS collaborations have pushed particle physics past an
important milestone with the discovery of the particle whose properties consistent with the
Higgs boson of the Standard Model. They have also shaped the directions of developments
towards new physics, or even a new collider, as they observed no deviations from other
Standard Model’s predictions. Within a short period of time, the LHC has already put
great challenges on models describing new physics at a TeV scale — a great benefit of doing
physics in the era with abundant inputs from experiments. We can expect that the future
runs of the LHC, with higher luminosity and higher center of mass energy, will help identify
the behavior of the Higgs and the mechanism electroweak symmetry breaking.

188

APPENDICES

189

Appendix A
The Standard Model
In this appendix, I provide a partial review of the gauge theory for electroweak interactions
and mass generations in the Standard Model.

A.1

Gauge Theory for Electroweak Interaction

A charged current weak interaction involving left-handed leptons and hadrons at energy
scales of beta decay processes ∼ O(MeV) is well-described by a theory pioneered by Fermi
[291] (later generalized by Feynman–Gell-Mann [292] and Sudarshan–Marshak [293]). The
prototypical Hamiltonian describing this class of theory, involving an interaction that changes
a flavor of fermions, is
G
H = √F
2

†

d3 xJ µ (x)Jµ (x)

(A.1)

where GF = 1.16637 × 10−5 GeV−2 is a Fermi’s constant and
Jµ =

¯lγ µ PL νl +
leptons

p¯γ µ (cV + cA γ 5 )n

(A.2)

hadrons

5
with PL = 1−γ
defined as the left-handed projection operator, and cV,A are constants
2

O(1). The coupling dimension (mass)−2 of (A.1) implies this is an effective theory and
is non-renormalizable. This means above a certain energy scale of a few hundred GeVs,
the scattering cross section increases with energy such that unitarity is violated; rendering
190

the theory non-predictive. In other words, one cannot always express formulae in terms of
experimentally-measured variables that remain finite and do not depend on the high-energy
limit of the theory. The behavior at high energy for processes involving four fermions can
be soften by regarding this theory as a low-energy limit of a theory having an electricallycharged intermediate vector bosons W ± , with mass MW ∼ 80 GeV, as a mediator of the
interaction. This is equivalent to writing the Hamiltonian as H ∼

d3 xJ µ (x)W − (x) + h.c..

Still, this does not eliminate the problem with unitarity as the cross section involving the
newly-introduced particle such as e+ e− → W + W − grows as (energy)2 . This can also be
viewed from the point that a momentum propagator for a massive vector field with four
momenta kµ does not have a “safe” high-energy behavior; i.e.,

Dµν (k) = −i

2
gµν − kµ kν /MW
2
k 2 − MW

−−−−→ O(1)
k→∞

(A.3)

rather falling with ∼ 12 , in the way the proton propagator does in QED. This can be seen
k
as a result of the presence of the k µ k ν term. In addition, with only two charged vector
bosons, the charged current is not conserved. This can be illustrated with one set of l , νl
by inspecting that the time evolution of Q− , Q− ∼

†

d3 xνl (1 − γ 5 )l is not zero and is

proportional to a charge corresponding to a neutral current; i.e.,

Q˙ − ∼ [Q− , H] =

†

d3 x l† (1 − γ 5 )l + νl (1 − γ 5 )νl W − .

(A.4)

Neutral current weak interaction, which involves fermions with both left- and righthanded chiralities, can also be described by a Hamiltonian similar to (A.1) with an electricallyneutral massive vector boson Z, with MZ ∼ 90 GeV, at high-energy limit. This kind of inter-

191

action has a distinctive experimental feature that processes involving changes of a fermion’s
flavor is highly suppressed. Similarly to the immediate charged vector boson, the theory
including neutral vector boson also has issues with high-energy limit. The prescription of
the theory for weak interaction which does not have issue at high energy is hinted by the
symmetry structure of the fermion contents.
The inclusion of neutral current, together with various experimental evidence, indicate
that under charged weak interactions, left-handed fermions are treated equally; i.e., as a
doublet of l − νl while the right-handed charged lepton appears as a singlet (as it does not
participate in the interaction). Observed phenomena so far provided no strong evidence of
the right-handed (Dirac) neutrinos, hence they have been excluded from the current picture.
This structure under weak interaction is written as








 νµ 
 
  ,
 
µ−

 νe 
 
  ,
 
e−

L

L





 ντ 
 
  ,
 
τ−

e−
R,

µ−
R,

−.
τR

(A.5)

L

Quarks also share similar structure except that all flavors have their right-handed partners
involved in weak interaction processes:
 

u
 
  ,
 
d
L

 

c
 
  ,
 
s
L

 

t
 
  ,
 
b

uR ,

dR ,

cR ,

sR ,

tR ,

bR ,

(A.6)

L

where the prime indicate there is a difference between mass and “gauge” eigenstates, which
can be safely omitted for the current discussion. It turns out that the prescription that
could explain the interaction of this nature while avoiding issues aforementioned is a gauge
theory which treats the W ± and Z as gauge fields. In this picture, the “neutral charge”
192

appearing in (A.4) is a consequence of the closure property of the SU (2) gauge group; namely,
[T + , T − ] = T 3 , where T a = τ a /2 and T ± = T 1 ± iT 2 with τ a being the Pauli matrices,
the generators of the Lie algebra of SU (2). However, a gauge group SU (2) alone does not
provide a complete picture for weak interaction.
The picture of a gauge field explaining weak interaction having W ± and Z as mediators
inevitably has to include photons as the W ± are charged under electromagnetic interaction.
In other words, the explanation of weak interaction as a gauge theory automatically demands
a unification of electromagnetic and weak interactions. The minimum gauge group having
4 gauge bosons which correctly explains the observed phenomena turns out to be a product
group SU (2) × U (1) [13, 14, 15]. The U (1), hence its gauge field, in SU (2) × U (1) does not
correspond directly to electromagnetism. This is because its charge cannot be the electromagnetic charge Q — Q has a preferred direction in the weak isospin space, distinguishing
between the two members of the left-handed doublets. In other words, Q is not a constant
of motion as it does not commute with the SU (2) generators. The quanta of gauge field W 3
coupled to T 3 cannot be treated as a photon for similar reason1 - photon does not couple
to T 3 . The quantum number corresponding to an operator commuting with the SU (2) generators is called, historically, a weak hypercharge Y ; hence the U (1) group being designated
U (1)Y . As the SU (2) part also affects only fermions with left-handed chirality, it is common
to denote the gauge group for electroweak interaction as SU (2)L × U (1)Y . The connection
between the gauge boson of U (1)Y , called the B µ , and photon Aµ , hence the form of the
U (1)Y operator, can be identified using the known form of electromagnetic interaction.
The form of U (1)Y charge is deduced from the form of electric charge for the left-handed
particles. The charge corresponding to the right-handed fermions automatically commutes
1 This is also the reason one cannot describe weak interaction using just an SU (2) gauge group.

193

with the T a of SU (2)L . The argument applies on both lepton and quark families where the
former is chosen for illustration. Denoting the lepton doublet as




 νl 
 
Ll =   ,
 
l−

(A.7)

the corresponding electric charge,

QLeft = T3 −

1
2

†

d3 xLl Ll

(A.8)

has its second part commuting with T a ’s of the SU (2)L . This leads to the form of the Y ;
namely, Y ∝ Q − T3 is an invariant, reflecting the feature of generators of Abelian symmetry
which has arbitrary normalization. We use a convention

Y = Q − T3 .

(A.9)

With this convention quantum numbers of particles content of the SU (2)L ×U (1)Y are listed
in table A.1. This leads to the expression of electromagnetic current

1
− T3 Ll + ¯lR γ µ QlR
2

(l)µ
¯lγµ
Jem = ¯lγ µ Ql = QL

(A.10)

for leptons as well as for other fermions to be written using currents corresponding to
SU (2)L × U (1)Y ,
µ

µ

µ

Jem = J3 + JY ,

194

(A.11)

where
µ
¯ f γ µ T a Lf ,
Ja = L

(A.12)

µ
¯ f γ µ Yf Lf + f¯R γ µ Yf fR
JY = L

(A.13)

and

Knowing the form of the U (1)Y charge, interactions between currents and the fields to be
regarded as gauge fields allows us to write down the form of gauge theory for electroweak
interaction.
Table A.1: Hypercharge assignments for fermions.
fermions

T

T3

νe , νµ , ντ
eL , µL , τL
eR , µR , τR
uL , cL , tL
dL , sL , bL
uR , cR , tR
dR , sR , bR

1/2
1/2
0
1/2
1/2
0
0

Q

Y

1/2
0
-1/2
-1/2 -1 -1/2
0
-1
-1
1/2 2/3 1/6
-1/2 -1/3 1/6
0
2/3 2/3
0
-1/3 -1/3

The connection between the gauge fields for SU (2)L × U (1)Y , Wµa and Bµ and the
physical gauge fields Wµ± , Zµ , and Aµ can be obtained by demanding that the electrically
neutral part of the gauge interaction term eventually be expressed using the physical fields
Zµ and Aµ
µ

µ

µ

µ

gJa Wµa + g JY Bµ ⊃ gZ JZ Zµ + eJem Aµ ,

(A.14)
µ

where g, g , gZ , e denote appropriate couplings to different fields and JZ is to be determined.

195

This means Z and A are linear combinations of W 3 and B; hence the “rotation”

Zµ = −Bµ sin θW + Wµ3 cos θW

(A.15)

Aµ = Bµ cos θW + Wµ3 sin θW ,

(A.16)

where θ is known as weak mixing angle or Weinberg angle2 . Observe that only Z can interact
with T 3 . This leads, by inspecting the left-handed side of (A.14) which contains the term
T 3 (gW 3 − g B), to
sin θW =

g
g2 + g 2

,

cos θW =

g
g2 + g 2

,

(A.17)

once a proper normalization has been included. This agrees with the requirement that
photon couples to electromagnetic current Jem with strength e. This in turn demands that
both g and g are not smaller than e; i.e.,

e = g sin θW = g cos θW .

(A.18)

At this stage, we have the interaction term for the SU (2)L × U (1)Y interaction expressed in
terms of the physical fields at low-energy,

g
µ
µ
gJa Wµa + g JY Bµ = √ (J +µ Wµ+ + J −µ Wµ− )
2
g
µ
µ
+
(J 3µ − sin2 θW Jem )Zµ + eJem Aµ
cos θW

(A.19)

√
where Wµ± = (Wµ1 ∓ iWµ2 )/ 2, which determines gZ = cosgθ hence JµZ introduced in
W
2 The mixing was introduced by Glashow [13].

196

(A.14).
To formally introduce a gauge theory where its gauge fields eventually have interactions
presented in (A.19), one introduces a covariant derivative

Dµ = ∂µ − igT a Wµa − ig Y Bµ ,

(A.20)

as well as the field strength

a = ∂ W a − ∂ W a + gε W b W c ,
Fµν
µ ν
ν µ
abc µ ν

Bµν = ∂µ Bν − ∂ν Bµ .

(A.21)

Then the gauge theory for electroweak interaction has the form

L =

¯ i γ µ Dµ Li +
iL
l
l

i γ µ D li
i¯lR
µR

l

l

+

¯ iq γ µ Dµ Liq +
iL
i

id¯iR γ µ Dµ diR

i¯
uiR γ µ Dµ uiR +
i

i

1 a µνa 1
F
− Bµν B µν ,
− Fµν
4
4

(A.22)

where the superscript i = 1, 2, 3 denotes the fermion generations.
The gauge theory for electroweak interaction having interactions shown in (A.22) alone
does not provide a consistent theory explaining observed phenomena for an obvious fact: it
cannot describe massive gauge fields, which are required as weak interaction is short-ranged.
Observe that left-handed and right-handed fermions do not appear in a symmetric manner
in the Lagrangian. This means that they have to be massless; otherwise the gauge symmetry
is violated. We will come back to the issue of fermion masses later. For now, we consider
the requirements for a theory with a massive gauge field, which are well-illustrated in the

197

case of an Abelian theory, QED.
The masslessness of photon in QED is a result of the location of a pole in its full propaµν

gator. Starting from the free propagator D0 = g µν k 2 − k µ k ν /k 2 , one constructs the full
propagator by collecting a set of 1-particle-irreducible diagrams (whose cut on one internal
line will not result in two diagrams where each has two external photon lines) to all orders,

µν

µα

iDµν = iD0 + iD0

βν

µα

iΠαβ iD0 + iD0

βγ

iΠαβ iD0

iΠγδ iD0δν + . . .

µν

D0
= i
1 + Π(q 2 )

(A.23)

Here Πµν is a vacuum polarization tenser defined from electromagnetic current J(x)

iΠµν (k 2 ) = e2

d4 xeikx 0|T J µ (x)J ν (0)|0 ≡ i(g µν k 2 − k µ k ν )Π(k 2 ) ,

(A.24)

which is conserved kµ Πµν (k) = 0 (i.e., it has only transverse polarizations), reflecting that
photon interacts with a conserved current. The exact propagator (A.23) implies that photon
have mass µ2 if Π(k 2 ) had a pole at k 2 = 0 with Π(k 2 ) → µ2 /k 2 . This pole could arise if
there were a massless particle in an intermediate state of the diagram, which is not possible
in QED as all the contributions to Π(k 2 ) are from one-photon-irreducible diagrams (hence
no one-photon intermediate state by construction). Gauge invariance does not prohibit such
a pole, though. In order to obtain that, one expects an addition of a massless particle, say
π, having coupling to the gauge field (conserved current) that allows a transition A − π.
This transition naturally has a coupling ieF k µ , where k µ is the photon’s momentum and
F is a constant. The process A − π − A is allowed to be included in the one-photonirreducible diagrams and eventually contributes to the polarization tensor via the k µ k ν term

198

as −ie2 F 2 k µ k ν /k 2 (where 1/k 2 comes from the propagator of the massless π). Thus, noting
that gauge invariance forces the presence of the g µν term, we arrive at the singular part of
the vacuum polarization tensor

iΠµν (k 2 ) −−−→ i(g µν k 2 − k µ k ν )e2 F 2 /k 2
k→0

(A.25)

So the full propagator is
µν

iDµν (k)

D0 (k)
−−−→ i
,
k→0 1 + e2 F 2 /k 2

(A.26)

which could lead to a gauge boson that still couples to a conserved current while being
massive. Notice that the propagator retains safe high-energy behavior; i.e., Dµν (k) →
O

1
k2

as k → ∞.

It was the mechanism of spontaneous breaking of local symmetry, developed by Brout,
Englert, Higgs, Guralnik, Hagen, Kibble, and Anderson [27, 28, 29, 30, 31], that provides
a way to incorporate mass of the gauge fields while preserving gauge symmetry in the Lagrangian; hence maintaining safe behavior at high energy. There, the additional massless
particle π is the Nambu-Goldstone boson of a hidden global symmetry, which eventually
shows up as an extra unphysical degree of freedom that can be “gauged away”. We provide
a review of how the mechanism works in the next section.

199

A.2

Hidden Symmetry and the “Brout–Englert–Higgs–
Guralnik–Hagen–Kibble–Anderson Mechanism”

Electroweak symmetry does not manifest at low energy as we know that masses of the gauge
fields corresponding to the group SU (2) × U (1) are not degenerate: photon is massless while
the rest have masses of O(100 GeV). The symmetry describing the interactions has to be
hidden by some means, leaving only the U (1) for electromagnetism apparent. The mechanism
of spontaneous breaking, or hiding, of a local symmetry provides a way to incorporate
massive gauge bosons into a theory without spoiling the much needed ingredient, a gauge
symmetry. It is a result of works pioneered by various physicists including Nambu, Goldstone,
Brout, Englert, Guralnik, Hagen, Higgs, Kibble, and Anderson. For brevity, we refer to this
mechanism as the BEH mechanism. The mechanism illustrates the interplay between a
spontaneous breaking of global symmetry and the gauge theory for electroweak interaction;
i.e., how gauge bosons become massive due to interactions with the so-called would-be
Nambu-Goldstone bosons of the hidden global symmetry. In this section, we review its
foundations and its applications to electroweak symmetry.

A.2.1

Spontaneous hidden of a global symmetry

One aspect of the foundations of the BEH mechanism is the spontaneous hidden, or broken, of a global continuous symmetry. This happens when the symmetry respected by the
Lagrangian does not manifest in the ground state of the system, the vacuum. In other
words, generators T a (a = 1, . . . , dim(G)) corresponding to the symmetry group G of the
system can be partitioned to two parts T a = (X r , Y h ) where X r are called broken generr r
h h
h h
ators, eiε X |0 = |0 , and eiε Y |0 = |0 . If eiε Y forms a group H, we say that the

200

symmetry is broken down from G to H, where the orientation of H in G is arbitrary. In
this case, h = 1, . . . , dim(H) and i = 1, . . . , dim(G) − dim(H). Now since X i |0 = 0, there
are dim(G) − dim(H) states |π r (p) that are connected to the vacuum via conserved currents J µr (x) constructed from broken generators. Their non-zero matrix elements can be
parametrized as

0|J µr (x)|π s (p) = ipµ F rs e−ip·x ,

r, s = 1, . . . , dim(G) − dim(H)

(A.27)

where F rs is the element of a constant matrix which can be diagonalized F rs = δrs F . Since
the current J µr is conserved; i.e., ∂µ J µr = 0,

0 = 0|∂µ J µr (x)|π s (p) = −p2 F δrs e−ip·x .

(A.28)

2
This implies ms2
π = p = 0; namely there are dim(G) − dim(H) massless states, one for each

broken generator X r that does not annihilate the vacuum. These massless states are referred
to as Nambu-Goldstone bosons. Apparently, they have not shown up in observed physical
spectrum either. It is the BEH mechanism that explains the absence of these NambuGoldstone bosons, at the benefits of having massive gauge fields, when the symmetry that
is hidden is a local gauge symmetry.

A.2.2

The BEH mechanism

The idea of BEH mechanism is as follows. The Lagrangian describing the theory of interest
can be partitioned into two parts: one preserving a gauge symmetry (the corresponding gauge
group is called here W ) and the other having a global symmetry group G large enough to

201

contain that gauge group, while being spontaneously broken down to a subgroup H. Then,
conserved currents and the gauge fields are allowed to interact in a gauge invariant manner
(i.e., via the “gJ µ Aµ ” term where g is the gauge coupling). This results in what is usually
µa

referred to as “gauging the global symmetry”. From the conserved currents JW constructed
from generators belonging to the gauge group W , there will be linear combinations of those
that do not annihilate the vacuum; i.e., those corresponding to broken generators. These
gauge currents will allow transitions between a vacuum and the states |π r (p) with the
following amplitude
µa

0|JW (x)|π s (p) = ipµ F as e−ip·x .

(A.29)

This leads to an interaction between the gauge fields, called here Aµa and the states |π r
with coupling iga pµ F ar where ga represents gauge couplings (no summation over a). These
particular states |π r will contribute to the transition Aa −π r −Ab in the vacuum polarization
µ ν
tensor Πab
µν (k) of the gauge fields via the “k k ” terms as

ga gb F ar F br k µ k ν /k 2

(A.30)

r

(summation over r, the various |π r states, only). Since the tensor Πab
µν is gauge invariant
by construction, it has to have the following asymptotic k → 0 form
2
µ ν
iΠab
µν (k) −−−→ i(gµν k − k k )
k→0

202

ga gb F ar F br /k 2 .
r

(A.31)

In a manner similar to what we saw in the previous section on the propagator of a photon,
this vacuum polarization leads a full propagator whose residue is interpret as mass:

m2ab =

ga gb F ar F br .

(A.32)

r

In other words, due to gauge symmetry, the states that would have been the NambuGoldstone bosons in the physical states were “eaten” so that the gauge fields become massive. The gauge fields have the longitudinal degree of freedom at the expense of the would-be
Nambu-Goldstone bosons. Next, we consider a specific application of the mechanism in the
electroweak sector of the standard model.

A.2.3

BEH mechanism and electroweak interaction

The gauge group SU (2)L × U (1)Y describing electroweak interaction is relevant at high
energy while only the U (1)em of QED manifests at low energy compared to the weak scale
of O(100 GeV). So the group G of the global symmetry breaking part of the Lagrangian
has to be at least as large as SU (2) × U (1). The choice of G will actually determine the
structure of the symmetry breaking. We will consider the minimal case of G = SU (2)×U (1).
Then, all the gauge bosons of SU (2)L × U (1)Y couples to 4 generators T a of G, while only
3 of those do not annihilate a vacuum. This means there will be 4 conserved currents
µa

µ

µ

µ

µ

JW = (J1 , J2 , J3 , JY ) connecting between the 3 Nambu-Goldstone bosons and a vacuum
with amplitudes

µa

0|JW (x)|π s (p) = ipµ F as e−ip·x

203

a = 1, 2, 3, Y, s = 1, 2, 3

(A.33)

and a constraint, whose form of the current determined by (A.11), indicating that electric
µ

charge is conserved; i.e., J3ν + JY |0 = 0. This means
µ

µ

0|JY (x)|π 3 (p) = − 0|J3 (x)|π 3 (p) .

(A.34)

Thus, singular parts (as p2 → 0) of the vacuum polarization tensors become
Πab (p2 ) =

g 2 F 2 δ ab
,
p2

ΠY Y (p2 ) =

g 2F 2
,
p2

ΠaY (p2 ) = −

gg F 2 δ a3
,
p2

(A.35)

which, after being incorporated to full propagators, lead to a mass matrix, with rows and
µ

µ

µ

columns constructed from W1 , W2 , W3 , B µ :




g 2




g2

2
2
M =F 







g2
−gg







.


−gg 



g2

(A.36)

The 2 × 2 lower-right block can be diagonalized to diag[(g 2 + g 2 )F 2 , 0] using exactly the
same states Zµ and Aµ defined in (A.17). So masses of the gauge fields are

2 = g2F 2 ,
MW

MZ2 = (g 2 + g 2 )F 2 ,

MA = 0 ,

(A.37)

with a constraint
2
MW

MZ2 cos2 θW

204

= 1.

(A.38)

This also indicates that F ∼ O(MW , MZ ). In short, masses of the gauge fields are the result
of their interactions with the Nambu-Goldstone bosons of the hidden global symmetry.
It is important to emphasize that in the BEH mechanism, there is no specific requirement
the structure of an object that lead to spontaneous hidden of a symmetry by developing its
vacuum expectation value under a certain circumstance. Apart from the requirement that
it has to be a Lorentz scalar, the object could be either an elementary scalar such as the
complex scalar “Higgs” doublet in the standard model, or a composite scalar showing up
¯
as result of a fermion condensate 0|ΨΨ|0
. The Standard Model utilizes the elementary
scalar as the agent of electroweak symmetry breaking. Strong dynamics models implement
the condensate. In this appendix, we focus on the BEH mechanism in the Standard Model.

A.2.4

The “standard” BEH mechanism in the standard model

In the standard model, the field whose non-zero vacuum expectation value breaks electroweak
symmetry is a fundamental scalar. The choice of a representation of the scalar field under
the electroweak gauge group SU (2)L × U (1)Y is arbitrary. Only experiments can tell which
one is preferred. The minimal model introduced by Weinberg [14] uses a fundamental scalar
doublet of SU (2) and lead to predictions in good agreement with experiments. A complex
doublet H that causes a symmetry breaking has self-interaction structure determined by the
potential terms of
†

†

†

LH = ∂µ H ∂ µ H − µ2 H H − λ(H H)2 .

205

(A.39)

The assumption that the symmetry breaks; i.e., the condition µ2 < 0, implies that the
vacuum corresponds to one of equivalent configurations of non-zero 0|H|0 that satisfies

†

v 2 ≡ 0|H H|0 = −

µ2
.
λ

(A.40)

The particular vacuum that leads to the pattern SU (2)L × U (1)Y → U (1)em is
Q 0|H|0 = T 3 + Y

0|H|0 = 0 .

(A.41)

In the standard convention usually employed, the doublet satisfying this condition is




h+ 
 
H= ,
 
h0

with





 0 
 
0|H|0 =  
 

(A.42)

√v
2

where its form of vacuum expectation value is reachable from any other vacua by an SU (2)
transformation. When electroweak gauge interaction is introduced (which, in this case,
is equivalent to gauging the aforementioned global symmetry) the kinetic term of (A.39)
contains
LH ⊃ Dµ H

2

=

∂µ − igT a Wµa − ig Y Bµ H

2

.

(A.43)

Couplings between the gauge fields and the H doublet explains the masses of the gauge fields,
when H develops a vacuum expectation value, at the expense of the unphysical degrees of
freedom of H. The disappearance of the unphysical degrees of freedom is apparent when

206

they are parameterized as







a a  0 
H = eiT π /v 
,



(A.44)

v+h
√
2

which can be removed by a transformation allowed by the gauge freedom. With the definition
of the gauge fields in (A.17), the kinetic term (A.43) contains
1 2 2
h 2 1 1 2
h 2
µ+
2
2
µ+
LH ⊃ g v W Wµ− 1 +
g +g
v Z Zµ 1 +
+
,
4
v
2 4
v

(A.45)

explaining the massiveness of the gauge fields as well as their interactions with the scalar
field h. The vacuum expectation value v can be identified with the constant F in (A.37) as
v = 2F . Quantities in the brackets are interpret as masses of the gauge fields. Notice that
there is no gauge choice that removes all the four degrees of freedom of H. The field h thus
remains in the physical spectrum.
An elementary scalar particle is a feature of the BEH mechanism when the gauge symmetry is broken by a scalar field. This is the Higgs boson. As indicated in (A.45), it has cubic
and quartic interactions with the gauge fields (as hhV and hhV V ) with coupling strengths
proportional to their masses. Its mass and self-interaction is described by the potential term
in (A.39), under the same gauge choice that leads to (A.45), as
gm2h 3 g 2 m2h 4
1
h −
h ,
LH ⊃ − m2h h2 −
2
2
2MW
8MW

(A.46)

2 ). In other words, Higgs self-coupling, hence the
where we have replaced λ = g 2 m2h /(2MW

potential driving the electroweak symmetry breaking, is predicted as a function of the Higgs
mass.

207

A.2.5

Fermion masses in the standard model

Fermions have masses. However, the Standard Model is a chiral theory. This means that
masses of the fermions cannot be explained via a typical Dirac mass form of mf (f¯L fR +f¯R fL )
as only the left-handed (chirality) fermions carry the weak isospin charge. Unlike the gauge
bosons, fermions do not automatically become massive under the BEH mechanism when the
gauge group and the representation of the Higgs have been chosen. Their masses have to be
explained from phenomenological requirements via appropriate interactions with the Higgs
fields.
Fermion masses in the Standard Model are a result of interactions with the Higgs field
or its complex conjugate
H = iσ2 H ∗ ,

(A.47)

where σ2 is the Pauli matrix. Denoting different generations with a superscript i, j, the
interactions are written in the (electroweak) gauge eigenstate as

ij

j

j

ij

ij

j

− yu QiL HuR − yd QiL HdR − ye LiL HeR + h.c. .

(A.48)

ij

The coupling matrix fu/d/e , the Yukawa couplings, are not necessarily diagonal and have to
be determined from experiments. The mixing between generations in the gauge eigenstates
can be removed here via a transformation to the mass eigenstate by unitary matrix U

(u/d)L/R

leaving the Lagrangian in the flavor-diagonal form. This in turns means that the charged
currents have flavor-changing nature, quantified conventionally by VCKM ≡ U uL (U dL )†
where VCKM is known as the Cabibbo-Kobayashi-Maskawa matrix. Once the Higgs develops
a non-zero vacuum expectation value ∼ v, the fermions will have mass mf ∼ yv.

208

Appendix B
Strong Top Dynamics

B.1

TC2 in the NJL Approximation

In this appendix, we calculate the top-Higgs and top-pion spectrum in topcolor assisted
technicolor (TC2) models [100], using the Nambu–Jona-Lasinio (NJL) [76] approximation
for the topcolor dynamics. On phenomenological grounds [104], we expect the “cutoff” Λ
of the NJL topcolor theory (which is of order the mass of the gauge-bosons of the topcolor
model, i.e. the top-gluon and Z ) to be much higher than the technicolor scale ΛT C , which
is of order 1 TeV. We can therefore construct the low-energy theory which we use to compute
the scalar spectrum in two stages.
First, as described in the next section, we integrate out the strong topcolor-induced fourfermion operators using the Nambu–Jona-Lasinio approximation, and construct an effective
theory involving a composite top-Higgs field coupled to the third-generation quarks and the
technifermions. This effective theory will be valid at energies below the topcolor cutoff and
above the scale at which the technicolor interactions become strong. Next, as described in
the third section, we match to an effective technicolor chiral Lagrangian valid at low energies.
In the fourth section we use this effective Lagrangian to compute the scalar spectrum of the
theory. Custodial isospin violation is necessarily present in the theory so as to explain the
top-bottom mass difference. In the fifth section we consider what constraints the limits on
the custodial isospin violating parameter ∆T place on the parameters of the model, and
209

what these restrictions imply for the scalar mass spectrum. In the last section, we consider
the mass splitting between the charged- and neutral-top-pions.

B.1.1

TC2 Dynamics

In the NJL approximation,1 the interactions of interest in this model include
ηgt2 ¯
gt2 ¯
¯
(
ψ
t
)(
t
ψ
)
+
(ψL0 tR2 )(U¯R QL ) + h.c.
Λ2 L0 R2 R2 L0
Λ2

(B.1)

where the first four-fermion operator is the traditional topcolor interaction responsible for
top-quark condensation and the second, arising from ETC interactions [82, 81], couples the
top-quark to the weak-doublet and singlet technifermions [79, 80] QL and UR . Here gt and
Λ represent the top-color coupling and cutoff, respectively. We expect the second operator
to arise from ETC interactions at a scale larger than Λ, and for convenience we characterize
the strength of these interactions (relative to topcolor) through the small dimensionless
parameter η. All weak, color, and technicolor indices implicit in Eq. (B.1) are summed.
For strong gt , we expect that the topcolor interactions will give rise to a bound electroweak
scalar state with the quantum numbers of the standard model Higgs boson. In the NJL
approximation, this may be seen directly. The interactions Eq. (B.1) may be recast as
gt2 ¯
¯ L UR
ψ t + ηQ
Λ2 L0 R2

η2g2 ¯
¯
t¯R2 ψL0 + η U¯R QL − 2t (Q
L UR )(UR QL ) ,
Λ

(B.2)

which, following [96], may be rewritten in terms of an auxiliary electroweak doublet scalar
1 The NJL approximation [76, 96, 294] involves two parts. First we approximate the effects of exchange
of heavy top-color gauge bosons by four-fermion contact interactions and include only those parts of the
interaction responsible for coupling left- and right-handed fermion currents. Second, as discussed below, we
analyze the effect of these interactions in the “fermion bubble” approximation. Here, and in the following,
we also neglect additional TC2 interactions involving the right-handed bottom quark.

210

field Φ (with SU (2) × U (1) quantum numbers 2−1/2 )
2 2
¯ L UR )Φ + h.c. − η gt (Q
¯ L UR )(U¯R QL ) .
− Λ2 Φ† Φ − gt (ψ¯L0 tR2 + η Q
Λ2

Φ
Φ

ψL , QL

Φ†

tR , UR

ψL , QL
Φ†

tR , UR

(B.3)

Φ†
ψL , QL

tR , UR

Φ

Figure B.1: Diagrammatic representation of “fermion bubble” approximation yielding the
kinetic energy and mass (left) and self-couplings (right) of the composite Φ field. The twopoint function is resumed to generate the kinetic energy term for the composite scalar field.
In the “fermion bubble” approximation [76, 96, 294] illustrated in Fig. B.1, and close
to the critical point for chiral symmetry breaking, the auxiliary field Φ becomes a light
propagating composite state. To leading order in the number of fermions (colors for quarks
or technicolors for technifermions), the effects of the strong topcolor interactions at a scale
µ

Λ may be summarized by the effective Lagrangian
˜
λ
Ltc = Dµ ΦDµ Φ − m
˜ 2Φ Φ† Φ − g˜t (ψ¯L0 tR2 Φ + h.c.) − (Φ† Φ)2
2
2
2
¯ L UR Φ + h.c.) − η gt (Q
¯ L UR )(U¯R QL )
−η˜
gt (Q
Λ2

(B.4)

with the couplings
(4π)2
,
(NC + η 2 NT C ) ln(Λ2 /µ2 )
(4π)2
˜
λ(µ)
= 2
.
(NC + η 2 NT C ) ln(Λ2 /µ2 )

g˜t2 (µ) =

211

(B.5)
(B.6)

Here, in order to have a conventional kinetic energy term, we have rescaled the field Φ by

1/2
ZΦ

=

gt2
Λ2
2N
(N
+
η
)
ln
TC
(4π)2 C
µ2

1/2

.

(B.7)

The mass parameter for the composite field Φ is given by,
2
−1 Λ2 − 2gt (N + η 2 N
2
2
m
˜ 2Φ = ZΦ
T C )(Λ − µ )
(4π)2 C

The composite Higgs is light, and the effective theory valid, when µ

.

(B.8)

Λ and gt is close to

the critical coupling gt∗ for topcolor chiral symmetry given by
2gt∗ 2
(N + η 2 NT C ) = 1 .
(4π)2 C

(B.9)

For convenience, we conclude this section with a brief discussion of the η → 0 limit. As
we will see, η will be rather small and many of the parametric estimates that follow will
derive from this limit. If we define f as the expectation value of Φ through




 √f 
 2
Φ =  ,
 
0

(B.10)

we see from Eqs. (B.4) and (B.5) that
g˜2 (mt )f 2
(4π)2 f 2
m2t = t
=
,
2
2NC ln(Λ2 /m2t )

(B.11)

where we choose µ = mt as appropriate in evaluating the top-quark mass. This expression

212

is usually rewritten as
f2 =

2NC m2t
log
(4π)2

Λ2
m2t

,

(B.12)

and reproduces the Pagels-Stokar relation [295] appropriate in this limit [76, 96, 294]. Note
that, in the effective Lagrangian of Eq. (B.4), the top quark receives mass only through
its coupling to the composite Higgs. Therefore, to the extent that η is small, this relation
continues to be true even after including the effects of technicolor.

B.1.2

Technicolor

Next, we consider matching2 the Lagrangian in Eq. (B.4) to the chiral Lagrangian valid at
scales below the scale of technicolor chiral symmetry breaking, ΛT C . In what follows, we
will use the Naive Dimensional Analysis (NDA) [296, 297, 180] estimate ΛT C

4πF , where

F is the technicolor pion decay constant (the analog of fπ ≈ 93 MeV in QCD). To keep track
of the chiral symmetry properties of the technifermion – scalar coupling in Ltc we introduce
the 2 × 2 matrix
M = η˜
gt (Φ 0) ,

(B.13)

which serves as a spurion “transforming” as M → LMR† under the SU (2)L × SU (2)R
chiral symmetries of the technifermions. The coupling of the technifermions to the field Φ,
then, is similar to the mass term in QCD, and hence we expect the effective Lagrangian

(2)

LT C =

F2
c
tr[Dµ Σ† Dµ Σ] + 4πF 3 1 tr[M† Σ + Σ† M] ,
4
2

(B.14)

2 In principle, if Λ/Λ
1, we should also include the scaling of the operators in Eq. (B.4) due to the
TC
technicolor interactions. In practice, all of the relevant corrections can be absorbed into a redefinition of η
– and hence will be neglected in what follows.

213

where c1 is an unknown chiral coefficient related to the magnitude of the technifermion
condensate which, in QCD, is approximately 2.3 Here Σ is the SU (2)L × SU (2)R /SU (2)V
nonlinear sigma model field associated with electroweak symmetry breaking, and is to be
associated with Σ01 Σ12 in the triangle Moose model described in Sec. 3.2 above.
The second term in Eq. (B.14) arises from the ETC coupling of the top quark and is
of particular interest since it couples the top-color and technicolor chiral symmetries – and
hence will give rise to the top-pion masses. To analyze this term, it is convenient to rewrite
Σ in terms of a two-component complex unimodular vector ξ

Σ=

ξ −iσ2 ξ ∗

=

ξ ξ˜

,

(B.16)

where
ξξ † + ξ˜ξ˜† = I2×2 ,

ξ † ξ = ξ˜† ξ˜ = 1,

ξ † ξ˜ = ξ˜† ξ = 0 .

(B.17)

By the usual convention, ξ has the following vacuum expectation value
 

1
 
ξ =  ,
 
0

(B.18)

in unitary gauge. With this convention the combined Φ − ξ potential is a special case of a
two-Higgs potential (with F ξ playing the role of a second “Higgs”), and the second term in
3 More properly, the corresponding term in the QCD chiral Lagrangian gives
2
m2
π = 4πfπ c1 (mu + md ) ≈ (135 MeV) ·
.

214

mu + md
8 MeV

·

c1
2

.

(B.15)

Eq. (B.14) becomes

F m2M ix [Φ† ξ + h.c.] ,

(B.19)

with mass-squared

m2M ix (ΛT C ) = 4πF 2

c1
η˜
gt (ΛT C ) ,
2

(B.20)

renormalized at scale ΛT C = 4πF .
˜
˜ 2Φ continue to renormalize
At scales µ < ΛT C , the parameters λ(µ),
m2M ix (µ), and m
through the top-quark loop diagrams illustrated in Fig. (B.1), i.e. the formulas in Eqs.
(B.5), (B.6), and (B.8) continue to apply with η → 0. The complete effective Lagrangian at
scale µ is

(2)

LT C2 (µ) =

F2
tr[Dµ Σ† Dµ Σ] + Dµ ΦDµ Φ − m
˜ 2Φ (µ)Φ† Φ + F m2M ix (µ)[Φ† ξ + h.c.]
4
˜
λ(µ)
−˜
gt (µ)(ψ¯L0 tR2 Φ + h.c.) −
(Φ† Φ)2
(B.21)
2

In what follows we will need the values of these parameters evaluated at low energies, µ
We will find that η

mt .

1; hence, in the derivations below we will apply Eqs. (B.5), (B.6),

and (B.8) in the η → 0 limit.

B.1.3

Minimizing the Potential and the Scalar Spectrum

We are interested in identifying the region of parameter space where topcolor and technicolor
jointly yield electroweak symmetry breaking, i.e. Φ has the vacuum expectation value shown

215

in Eqs. (B.10) and (B.18), with4

f = v sin ω ,

F = v cos ω ,

(B.22)

and where v ≈ 246 GeV is the usual weak scale. We will assume that all of the low-energy
mass parameters (the masses of all the scalars in the spectrum and the top-quark) have the
same order of magnitude, and we adopt µ

mt implicitly in what follows.

The Φ − ξ potential may be written
˜
λ
(Φ† Φ)2 + m
˜ 2Φ Φ† Φ − F m2M ix [Φ† ξ + h.c.]
2
2 2
˜
ftc
λ
†
=
Φ Φ−
− F m2M ix [Φ† ξ + h.c.] + const.
2
2

V (Φ, ξ) =

(B.23)

˜ Requiring the minimum of the potential to occur at (B.22), we see
where, ftc = −2m
˜ 2Φ /λ.
that
˜
√
∂V
λ
2 ) − 2m2
= 0 ⇒ f (f 2 − ftc
M ix F = 0
∂f Φ , ξ
2

(B.24)

2 in favor of f 2 and m2
Using Eq. (B.24) to eliminate ftc
M ix , the potential can be rewritten

as
˜
λ
V (Φ, ξ) =
2
=

˜
λ
2

√

2

2m2M ix F
− F m2M ix [Φ† ξ + h.c.] + const.
˜
λf
√
2
2m2M ix F
f2
f 2
†
√
Φ Φ−
+
Φ−
ξ + const. ,
(B.25)
2
f
2
Φ† Φ −

f2
+
2

4 Note that the value of F here differs from that in the Top Triangle Moose model, Eq. (3.4) since there
electroweak symmetry breaking occurs collectively through the symmetry breaking encoded through both
Σ01 and Σ12 .

216

which is precisely the form found in [125].
From this we find
2 =
MΠ

√

v2
,
Ff

(B.26)

cos ω
c1
·η·
.
2
sin2 ω

(B.27)

2m2M ix

and, using Eqs. (B.11) and (B.20),

2 = 8πvm ·
MΠ
t

Note that this leading contribution to the top-pion masses yields degenerate charged- and
neutral-top-pions. The same potential also yields the top-higgs mass MHt ,
2
MH
t

=

˜ 2
λf

+

√

2m2M ix F
2 cos2 ω ,
= 4m2t + MΠ
f

(B.28)

where the form of the relation between MHt , MΠ , and cos ω is fixed from the form of the
˜ and mt is fixed in the NJL approximation [96].
potential [125], and the relation between λ
Note that the TC2 theory in the NJL limit is specified primarily by four parameters: gt ,
Λ, η, and F . Physical quantities will only depend on these four parameters, up to coefficients
in the chiral Lagrangian (such as c1 ) of order 1. Using Eqs. (B.12), (B.22), and (B.27), we
will trade the parameters gt , Λ, η, and F for mt , v, sin ω, and MΠ .

B.1.4

Constraints from ∆T

The physics giving rise to the top-quark mass violates custodial isospin, causing deviations
in the low-energy parameter ∆ρ = α∆T . Consider the Lagrangian shown in Eq. (B.4). The
Yukawa interaction between the composite Higgs Φ and the top-quark gives rise to the usual
top-quark mass dependent contribution – just as in the standard model. The last two terms
217

Σ

Φ†

UR

Σ

QL
Φ

†

QL
UR

UR

Σ

QL

Σ

QL
Σ†

UR

Σ†

Figure B.2: Diagrams corresponding to the two leading contributions to α∆T in the TC2
model. The diagram on the left gives rise to the operator shown in Eq. (B.29). The
diagram on the right arises from the four-technifermion operator shown in Eq. (B.32). The
small black circles in these diagrams represent the dynamical technifermion mass arising
from technicolor chiral symmetry breaking, as parameterized by the field Σ in the chiral
Lagrangian of Eq. (B.14).
in this Lagrangian, the Yukawa interaction between the technifermions and the composite
Higgs and the four-technifermion operator, give rise to new contributions which we consider
below.
Consider first the technifermion Yukawa coupling. This operator violates custodial isospin
by one unit, ∆I = 1, and therefore the leading contribution to α∆T arises through two
insertions of this operator as shown in left-hand panel of Fig. B.2. This diagram yields an
operator of the form

cT
tr[M† (Dµ Σ)M† (Dµ Σ)] ,
(4π)2

(B.29)

where, consistent with NDA [296, 297, 180], the constant cT is expected to be of order 1.
Computing the effect of this operator on the W and Z masses, we find

α|∆T | =

2|cT |η 2 m2t
,
(4πv)2

218

(B.30)

2 , we find
or, alternatively, rewriting the dependence on η in terms of MΠ

α|∆T | =
If we require |∆T |

|cT |
·
2

2 2
1
·
|c1 |
cos2 ω

MΠ sin ω 4
.
4πv

0.5, we find from Eq. (B.30) that η

(B.31)

0.6. The equivalent constraint,

in terms of MΠ , from Eq. (B.31) is shown as the red solid line in Fig. B.3. This is a rather
weak upper bound, phenomenologically speaking. Theoretically, it is still an interesting
bound because it derives directly from the Yukawa coupling operator in the low-energy chiral
expansion that also gives rise to MΠ withough any dependence on the details of technicolor
dynamics at high energies.
On the other hand, since the ETC interaction between the top quark and technifermions
in Eq. (B.1) couples to both the left-handed current ψ¯L0 γ µ QL and right-handed current
t¯R2 γ µ UR , it is natural to expect that there are ETC gauge bosons that couple to UR with
the same strength. The exchange of such an ETC boson will give rise to the ∆I = 2 operator,
ηgt2 ¯ µ
(U γ UR )(U¯R γµ UR ) ,
Λ2 R

(B.32)

which can contribute directly to ∆T . In particular, the diagram on the right of Fig. B.2
yields the operator

cT ·







2

 
1 0
ηgt2

4 Tr Σ† D Σ 
·
F
 
 ,
µ 
2




Λ
0 0

219

(B.33)

where cT is an unknown chiral coefficient of order 1.5 The correction to |∆T | is
α|∆T | =

4|cT | ηgt2 F 4
·
.
v2
Λ2

(B.34)

To evaluate this expression, we use Eq. (B.27) to rephrase η in terms of MΠ , apply Eq.
(B.22 ) to eliminate F , and approximate gt2 by gt∗ 2 as in Eq. (B.9) [neglecting the term of
order η 2 ]:
α|∆T | =

4π
· |cT | ·
NC

2
|c1 |

· sin2 ω cos3 ω ·

2
vMΠ
.
mt Λ2

(B.35)

This constraint is represented by the blue long-dashed line in Fig. B.3.
Figure B.3 summarizes the approximate constraints on the sin ω − MΠ plane that arise
from limits on α|∆T | as discussed above. The pink-shaded regions are excluded; the area
above the solid red line is excluded due to the impact of the technifermion Yukawa coupling
and the area to the left of the blue long-dashed line is excluded by the effects of ETC gauge
boson exchange. In the left-hand pane, a few dotted curves for different values of η are
shown to indicate how that dimensionless parameter varies with sin ω and MΠ ; in the righthand pane, a few nearly-horizontal purple contours corresponding to several values of the
top-Higgs mass are shown.

B.1.5

Top-Pion Mass Splitting

Finally, we consider the mass splitting between the charged and neutral top-pions. The
leading contribution comes from the same diagram that produces the operator in Eq. (B.29).
In particular, in addition to the derivative operator discussed above, these diagrams give rise
5 In fact, it is exactly equal to 1 in the vacuum insertion approximation.

220

5

50

500

M

GeV

2000
1500
1000
700
500
300

100
0.30

Yukawa
4FRR
Η 0.005
Η 0.05

0.35

0.40

5

0.45 0.50
sinΩ

0.55

50

2000
1500
GeV

T

Allowed

200
150

M

TeV

2500

Excluded by

500

0.60

0.65

TeV

2500

Excluded by

T

1000
700
500
Allowed

300
200
150
100
0.30

MHt
MHt
MHt

0.35

0.40

0.45 0.50
sinΩ

0.55

400 GeV
600 GeV
1000 GeV

0.60

0.65

Figure B.3: Top: Approximate constraints on MΠ and sin ω in the TC2 model in the NJL
approximation coming from bounds on α|∆T |. The constraints shown arise from taking
|∆T | < 0.5 and assuming that c1 /2 = cT = cT = 1; the shaded pink region is excluded.
The red solid line shows the bound arising from the operator in Eq. (B.31) (red line); the blue
long-dashed line shows the bound from Eq. (B.34) (blue dashed line). The dotted purple
curves on the top pane depict contours of constant η from Eq.(B.4); the dashed purple curves
in the bottom pane are contours of constant top-Higgs mass from Eq. (B.28).
to the operators
˜ 4 F 2 Φ† ξξ † Φ + λ
˜ F 2 Φ† ξΦ† ξ + ξ † Φξ † Φ
λ
5

(B.36)

where, using NDA, the parameters λi are

˜ i = ci (η˜
λ
gt )2 ,

221

(B.37)

and the ci are parameters of order 1. Comparing the operators in Eq. (B.36) with those in
the two-Higgs doublet model (ξ transforms precisely as a Higgs, but with fixed magnitude)
we see that these terms each give rise to mass splittings of order

2 = M2 − M2 ∝ λ
˜iv2 .
∆MΠ
0
Π+
Π

(B.38)

From the relations derived previously, we find

2
∆MΠ

∝ ci

4
MΠ
2 2
sin2 ω
,
·
·
c1
32π 2 v 2 cos2 ω

(B.39)

and therefore, ignoring factors of order one

∆MΠ
∝
MΠ

2
sin2 ω
MΠ
·
.
6.2 TeV
cos2 ω

(B.40)

From this we see that, for the allowed range of MΠ , the mass-splitting between the charged
top-pion and the neutral top-pion is typically very small, and always less than O(10%). For
MΠt of order 200 GeV, the mass splitting is of order 100 MeV.
Based on this analysis, it is clear that the classic TC2 dynamics do not lead to large
splittings between the top and neutral top-pions. A model with a large splitting must contain
additional isospin violation, beyond the minimum required to generate the top quark’s mass.

222

B.2

Alternative Fermion Couplings and Constraints
from b → sγ

The couplings of the top-pion and top-Higgs to fermions are model dependent. In this
appendix we discuss the relation between the assumptions about the flavor structure that
are used in chapter 3 and the simpler form of the fermion couplings used in [119].
The form for the light fermion masses given in [119] is




 uR



L = MD  L ψ¯L0 Σ01 ψR1 + ψ¯R1 ψL1 + ψ¯L1 Σ12 


0



0   uR2



dR2
dR





 . (B.41)


We have denoted the Dirac mass that sets the scale of the heavy fermion masses as MD .
Here, L is a parameter that describes the degree of delocalization of the left handed fermions
and is assumed to be universal for the light quark generations and the leptons. All the flavor
violation for the light fermions is then encoded in the last term; the delocalization parameters
for the right handed fermions, f R , which can be adjusted to realize the masses and mixings
of the up and down type fermions. The mass of the top quark arises from similar terms with
a unique left-handed delocalization parameter tL and also from a unique Lagrangian term
reflecting the coupling of the top-Higgs to the top quark:

Ltop = −λt ψ¯L0 Φ tR + h.c.

(B.42)

If this simple picture for the fermion masses is correct, then top-color provides mass only
to the top-quark while the three-site/technicolor sector provides mass to both the top-quark
and all lighter-quarks. In this case, insofar as the third-generation quarks are concerned, the
223

b
0
G

sL

s
L

L

+
t

t

t

+
t

+
t

t
−
t

t

t
b

b
R

L

b
R

(b) b

g
Zbb

Figure B.4: Loop corrections to δgZbb and b → sγ arising from exchange of charged top-pions.
pattern of top-pion couplings is the same as the pattern of charged-Higgs couplings in “typeII” two-Higgs-doublet models [158] – with the top-Higgs playing the role of the Higgs-doublet
coupling to top-quark and the technicolor-sector playing the role of the Higgs-doublet giving
mass to the bottom.
√
Lyukawa = (2 2GF )1/2

u¯i (cot ω mui Vij PL + tan ω Vij mdj PR )dj Π+ + h.c.
i,j

√
⊃ (2 2GF )1/2 [mt Vtb cot ω t¯R bL + mt Vts cot ω t¯R sL + mb Vtb tan ω t¯L bR ] Π+
+h.c.

(B.43)

These couplings imply significant corrections from charged top-pion exchange to the processes
Z → ¯bb and b → sγ, as illustrated in Fig. B.4. The correction to the process Z → ¯bb comes
predominantly from the first term in Eq. (B.43) – and is characteristic of top color theories
[203]. As explained in [125], the top-color corrections to Z → ¯bb can be compensated for by
an adjustment of the top-quark delocalization parameter tL .
The potential corrections to b → sγ, however, are more problematic. These arise from
vertices involving both the second interaction in Eq. (B.43) [which is necessary since the
process involves the strange-quark] and either the first or third one. These contributions are

224

Table B.1: Lower bound on MΠ+ from b → sγ assuming the fermion couplings in Eq. (B.43).
t

sin ω
MΠ+ (GeV )

0.16
754

t

0.19
685

0.23
617

sin ω
0.53
MΠ+ (GeV ) 332
t

0.26
551

0.60
311

0.30
500

0.70
289

0.34
440

0.83
270

0.40
396

0.46
363

0.96
254

particularly severe6 in the case of small sin ω. Translating the bounds in two-Higgs models
to the case at hand [298], we find that the couplings of Eq.(B.43) imply the stringent lower
bounds on the charged top-pion shown in Table B.1. Charged top-pion masses of this order,
and hence neutral pion and top-Higgs masses which are expected to be of the same order,
would be very difficult to observe at the LHC. As discussed in the text, this constraint does
not apply if left-handed mixing is purely in the up-quark sector.

B.3

Formulas for the top-Higgs decay widths

The couplings of the top-Higgs, along with its decay widths to the relevant channels W W ,
+ −
∓
0
0 0
ZZ, tt¯, Π±
t W , Πt Z, Πt Πt , and Πt Πt , were given in Ref. [126]. For completeness, we

reproduce the key formulas below.
For the limit-setting in Sec. 3.5, we compute the top-Higgs production cross section with
the aid of the 7 TeV LHC SM Higgs cross sections in the gluon fusion and VBF modes from
6 The role of β in type-II two-Higgs-doublet models is played here by ω. In two-Higgs models one often
considers tan β mt /mb
1 – while here, we are mostly interested in tan ω = f /F 1.

225

Ref. [204, 299]. To the extent that the narrow-width approximation is valid, we can write
σ(pp → Ht → W W )
=
σ(pp → HSM → W W )

σgg (pp → Ht ) + σVBF (pp → Ht ) BR(Ht → W W )
σgg (pp → HSM ) + σVBF (pp → HSM ) BR(HSM → W W )

1 σ (pp → H
2
gg
SM ) + sin ω σVBF (pp → HSM )
sin2 ω
σgg (pp → HSM ) + σVBF (pp → HSM )

×

BR(Ht → W W )
,
BR(HSM → W W )

(B.44)

and analogously for the ZZ final state [note that BR(Ht → ZZ)/BR(HSM → ZZ) =
BR(Ht → W W )/BR(HSM → W W )]. The approximation in the second line is exact insofar
as (i) the QCD corrections to Higgs production are the same for the top-Higgs and the SM
Higgs and (ii) the efficiencies of the inclusive LHC Higgs searches are the same for events
arising from gluon fusion and VBF.
For decays to a top-pion and a gauge boson,
cos2 ω 3 3
M β ,
8πv 2 Ht W
cos2 ω 3 3
Γ(Ht → Π0t Z) =
M β ,
16πv 2 Ht Z

∓
Γ(Ht → Π±
t W ) =

(B.45)

where
βV2

≡ 1−

(MΠt + MV )2
2
MH

t

1−

(MΠt − MV )2
2
MH

.

(B.46)

t

For decays to two top-pions,

λ2 + −
+
−
HΠ Π
Γ(Ht → Πt Πt ) =
16πMHt
λ2 0 0
0
0
HΠ Π
Γ(Ht → Πt Πt ) =
32πMHt
226

4M 2 +
1−

Πt
2
MH
t

,

4M 2 0
1−

Πt
2
MH
t

,

(B.47)

where

1
2 cos2 ω − M 2 + 2M 2 sin2 ω ,
MH
t
Π+
Π+
v sin ω
t
t
1
2 cos2 ω − M 2 + 2M 2 sin2 ω .
λHΠ0 Π0 =
MH
t
Π0t
Π+
v sin ω
t

λHΠ+ Π− =

(B.48)

For decays to top-quark pairs,
3m2t
Γ(Ht → tt¯) =
MHt
8πv 2 sin2 ω

4m2
1 − 2t
MH

3/2

.

(B.49)

t

By comparison, the width to gauge-bosons is suppressed by sin2 ω:

Γ(Ht →

W +W −)

=

Γ(Ht → ZZ) =

3 sin2 ω
MH
t

3
1 − xW 1 − xW + x2W ,
4

t

3
1 − xZ 1 − xZ + x2Z ,
4

16πv 2
3 sin2 ω
MH
32πv 2

2 .
where xV = 4MV2 /MH
t

227

(B.50)

Appendix C
Uncertainty of Intrinsic Width
Measurement
The uncertainty in the intrinsic width of the resonance plays a key role in the estimation of the
uncertainties in Dcol , the variable we propose to distinguish a color-octet and a color-singlet
state. In this section we extract the uncertainty of the intrinsic width from a measurement of
the total width and a knowledge of the systematic uncertainties in measuring that width. The
systematic uncertainties relevant to width measurement are the di-jet mass resolution and
the uncertainty in the di-jet mass resolution; we model them to have a Gaussian distribution
and ignore correlations between them.
The standard deviation of the observed invariant mass distribution (σT ) is related to the
standard deviation of the intrinsic width (σΓ

Γ/2.35 assuming a Gaussian distribution)

and that of the detector mass resolution (Mres ) by

σΓ =

2 .
σT2 − Mres

(C.1)

This implies that


(∆σΓ )2 = 

σT ∆σT

2



2

 +  Mres ∆Mres  ,
2
2
σT2 − Mres
σT2 − Mres
228

(C.2)

or
∆σΓ 2
=
σΓ

M2
1 + res
σΓ2

2

∆σT 2
+
σT

2
Mres
σΓ2

2

∆Mres 2
,
Mres

(C.3)

where Eq. (C.1) was used to obtain Eq. (C.3).
For N observed signal events, where N is sufficiently large, the uncertainty of the observed
width ∆σT is given by σT / 2(N − 1) [300]. So Eq. (C.3) leads to
∆σΓ
=
σΓ

1+

2
Mres 2
1
+
σΓ
2(N − 1)

Mres 4
σΓ

∆Mres 2
,
Mres

(C.4)

where ∆σΓ /σΓ = ∆Γ/Γ. Note that for large N , the above expression simplifies to
∆Γ
=
Γ

Mres 2
σΓ

∆Mres
Mres

=

Mres 2
Γ/2.35

∆Mres
Mres

.

(C.5)

This expression shows that the uncertainty in the intrinsic width is inversely proportional to
Γ/Mres which implies that the uncertainty in the intrinsic width is small (large) when the
intrinsic width is large (small).

229

Appendix D
Curves with Constant Dcol
In this section we present the relationship between the values of color discriminant variable
and the input parameters of the models; namely,

2
σtt¯
gu
2 +g 2 , σjj ,
gu
d

σ¯

and σ bb , without taking
jj

uncertainties into account. This provides a set of “benchmarks” illustrating how a set of
curves for fixed Dcol shifts with changing up ratio and mass, and how well each relevant
observable must be measured to distinguish colorons from Z’ bosons.
In each subplot of Fig. D.1 and D.2, we show models of colorons and leptophobic Z ’s
leading to the same (here, exact) value of Dcol . Curves corresponding to three illustrative
g2

values of the experimentally inaccessible up-ratio 2 u 2 are displayed; we used the minimum
gu +g
d
(0) and maximum (1) values of the ratio, along with an intermediate value (0.5). The sets of
plots for resonances of masses 3.0 TeV and 3.5 TeV are shown in Fig. D.1 (top and bottom
panels, respectively), while similar plots for 4.0 TeV and 4.5 TeV are shown in Fig. D.2.

230

10

Dcol = 0.0001

σb¯b /σjj

8

6

4

4

2

2

10

0

2

4

6

8 10

Dcol = 0.001

8

0
10

C

0

2

6

8 10

Dcol = 0.005

8

10

4

4

4

2

2

2

2

4

6

8 10

0

0

2

10

10
6

4

4

2

2

10

0

2

4

6

8 10

Dcol = 0.001

8

8 10

0

0
10

0

2

4

6

8 10

Dcol = 0.005

8

10

4

2

2

2

σtt¯/σjj

8 10

0

0

2

4

6

8 10

σtt¯/σjj

Dcol = 0.01

8

4

6

8 10

0
0.5
1
0
0.5
1

Z

4

4

6

C

6

2

4

M = 3.5 TeV
gu2 /(gu2 + gd2 ) =

6

0

2

σtt¯/σjj

6

0

0

Dcol = 0.0005

8

6

0

6

σtt¯/σjj

Dcol = 0.0001

8

4

Dcol = 0.01

8
6

σtt¯/σjj

σb¯b /σjj

4

6

0

0
0.5
1
0
0.5
1

Z

6

0

σb¯b /σjj

M = 3.0 TeV
gu2 /(gu2 + gd2 ) =

Dcol = 0.0005

8

6

0

σb¯b /σjj

10

0

0

2

4

6

8 10

σtt¯/σjj

Figure D.1: The region of parameter space for the colorons (in blue) and Z bosons (in green)
that is consistent with measurements
of the ratios σtt¯ and σb¯b , the mass, cross section, and
√
Dcol at the LHC with s = 14 TeV. The value of Dcol in a given plot is listed at upper
right. The three blue (green) lines in a given plot correspond to colorons (Z bosons) with
three different values of the experimentally inaccessible coupling ratio (the gu2 /(gu2 + gd2 )).
These plots illustrate the measurement precision in σtt¯ and σb¯b that is required to distinguish
between the coloron and the leptophobic Z . The set of plots for resonances with a mass of
3.0 TeV (3.5 TeV) is in the top (bottom) panel.
231

10

Dcol = 0.0001

σb¯b /σjj

8

6

4

4

2

2

10

0

2

4

6

8 10

Dcol = 0.001

8

0
10

C

0

2

6

8 10

Dcol = 0.005

8

10

4

4

4

2

2

2

2

4

6

8 10

0

0

2

10

10
6

4

4

2

2

10

0

2

4

6

8 10

Dcol = 0.001

8

8 10

0

0
10

0

2

4

6

8 10

Dcol = 0.005

8

10

4

2

2

2

σtt¯/σjj

8 10

0

0

2

4

6

8 10

σtt¯/σjj

Dcol = 0.01

8

4

6

8 10

0
0.5
1
0
0.5
1

Z

4

4

6

C

6

2

4

M = 4.5 TeV
gu2 /(gu2 + gd2 ) =

6

0

2

σtt¯/σjj

6

0

0

Dcol = 0.0005

8

6

0

6

σtt¯/σjj

Dcol = 0.0001

8

4

Dcol = 0.01

8
6

σtt¯/σjj

σb¯b /σjj

4

6

0

0
0.5
1
0
0.5
1

Z

6

0

σb¯b /σjj

M = 4.0 TeV
gu2 /(gu2 + gd2 ) =

Dcol = 0.0005

8

6

0

σb¯b /σjj

10

0

0

2

4

6

8 10

σtt¯/σjj

Figure D.2: Same as Fig. D.1, but for 4.0 TeV (top panel) and 4.5 TeV (bottom panel)
resonances.

232

BIBLIOGRAPHY

233

BIBLIOGRAPHY
[1] R. S. Chivukula, P. Ittisamai, E. H. Simmons, and J. Ren, Technipion Limits from
LHC Higgs Searches, Phys.Rev. D84 (2011) 115025, arXiv:1110.3688 [hep-ph].
Erratum: ibid. D85 (2012) 119903.
[2] R. S. Chivukula, P. Ittisamai, J. Ren, and E. H. Simmons, Technicolor in the LHC
Era, arXiv:1202.1505 [hep-ph].
[3] R. S. Chivukula, P. Ittisamai, E. H. Simmons, B. Coleppa, H. E. Logan, et al.,
Discovering Strong Top Dynamics at the LHC , Phys.Rev. D86 (2012) 095017,
arXiv:1207.0450 [hep-ph].
[4] A. Atre, R. S. Chivukula, P. Ittisamai, E. H. Simmons, and J.-H. Yu, Probing Color
Octet Couplings at the Large Hadron Collider , Phys.Rev. D86 (2012) 054003,
arXiv:1206.1661 [hep-ph].
[5] E. H. Simmons, A. Atre, R. S. Chivukula, P. Ittisamai, N. Vignaroli, et al., Coloron
Models and LHC Phenomenology, arXiv:1304.0255 [hep-ph].
[6] A. Atre, R. S. Chivukula, P. Ittisamai, and E. H. Simmons, Distinguishing
Color-Octet and Color-Singlet Resonances at the Large Hadron Collider , Phys.Rev.
D88 (2013) 055021, arXiv:1306.4715 [hep-ph].
[7] R. S. Chivukula, P. Ittisamai, and E. H. Simmons, Distinguishing Flavor
Non-universal Colorons from Z’ Bosons at the LHC , arXiv:1406.2003 [hep-ph].
[8] R. Feynman, A Relativistic cutoff for classical electrodynamics, Phys.Rev. 74 (1948)
939–946.
[9] J. S. Schwinger, On Quantum electrodynamics and the magnetic moment of the
electron, Phys.Rev. 73 (1948) 416–417.
[10] J. Schwinger, On Radiative Corrections to Electron Scattering, Phys.Rev. 75 (1949)
898–899.
[11] S.-I. Tomonaga and J. Oppenheimer, On Infinite Field Reactions in Quantum Field
Theory, Phys.Rev. 74 (1948) 224–225.

234

[12] F. Dyson, The Radiation theories of Tomonaga, Schwinger, and Feynman, Phys.Rev.
75 (1949) 486–502.
[13] S. Glashow, Partial Symmetries of Weak Interactions, Nucl.Phys. 22 (1961) 579–588.
[14] S. Weinberg, A Model of Leptons, Phys.Rev.Lett. 19 (1967) 1264–1266.
[15] A. Salam, Weak and Electromagnetic Interactions, Conf.Proc. C680519 (1968)
367–377.
[16] G. ’t Hooft and M. Veltman, Regularization and Renormalization of Gauge Fields,
Nucl.Phys. B44 (1972) 189–213.
[17] Particle Data Group Collaboration, K. Olive et al., Review of Particle Physics,
Chin.Phys. C38 (2014) 090001.
[18] C.-N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge
Invariance, Phys.Rev. 96 (1954) 191–195.
[19] H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Advantages of the Color Octet Gluon
Picture, Phys.Lett. B47 (1973) 365–368.
[20] O. Greenberg, Spin and Unitary Spin Independence in a Paraquark Model of Baryons
and Mesons, Phys.Rev.Lett. 13 (1964) 598–602.
[21] D. J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories,
Phys.Rev.Lett. 30 (1973) 1343–1346.
[22] H. D. Politzer, Reliable Perturbative Results for Strong Interactions? , Phys.Rev.Lett.
30 (1973) 1346–1349.
[23] S. R. Coleman and E. J. Weinberg, Radiative Corrections as the Origin of
Spontaneous Symmetry Breaking, Phys.Rev. D7 (1973) 1888–1910.
[24] Y. Nambu, Axial vector current conservation in weak interactions, Phys.Rev.Lett. 4
(1960) 380–382.
[25] J. Goldstone, Field Theories with Superconductor Solutions, Nuovo Cim. 19 (1961)
154–164.

235

[26] J. Goldstone, A. Salam, and S. Weinberg, Broken Symmetries, Phys.Rev. 127 (1962)
965–970.
[27] F. Englert and R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons,
Phys.Rev.Lett. 13 (1964) 321–323.
[28] P. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys.Rev.Lett.
13 (1964) 508–509.
[29] P. W. Higgs, Broken symmetries, massless particles and gauge fields, Phys.Lett. 12
(1964) 132–133.
[30] G. Guralnik, C. Hagen, and T. Kibble, Global Conservation Laws and Massless
Particles, Phys.Rev.Lett. 13 (1964) 585–587.
[31] P. W. Anderson, Plasmons, Gauge Invariance, and Mass, Phys.Rev. 130 (1963)
439–442.
[32] D. Ross and M. Veltman, Neutral Currents and the Higgs Mechanism, Nucl.Phys.
B95 (1975) 135.
[33] ATLAS Collaboration, ATLAS: Detector and physics performance technical design
report. Volume 1 , Tech. Rep. CERN-LHCC-99-14, CERN, 1999.
http://alice.cern.ch/format/showfull?sysnb=0317330.
[34] ATLAS Collaboration, ATLAS: Detector and physics performance technical design
report. Volume 2 , Tech. Rep. CERN-LHCC-99-15, CERN, 1999.
[35] CMS Collaboration, G. Bayatian et al., CMS physics: Technical design report, Tech.
Rep. CERN-LHCC-2006-001, CERN, 2006.
http://cdsweb.cern.ch/record/922757.
[36] CMS Collaboration, G. Bayatian et al., CMS technical design report, volume II:
Physics performance, J.Phys. G34 (2007) 995–1579.
[37] UA2 Collaboration, M. Banner et al., Observation of Single Isolated Electrons of
High Transverse Momentum in Events with Missing Transverse Energy at the CERN
anti-p p Collider , Phys.Lett. B122 (1983) 476–485.
[38] UA2 Collaboration, P. Bagnaia et al., Evidence for Z0 —&gt; e+ e- at the CERN
anti-p p Collider , Phys.Lett. B129 (1983) 130–140.
236

[39] UA1 Collaboration, G. Arnison et al., Further Evidence for Charged Intermediate
Vector Bosons at the SPS Collider , Phys.Lett. B129 (1983) 273–282.
[40] CDF Collaboration, F. Abe et al., Observation of top quark production in p¯p
collisions, Phys.Rev.Lett. 74 (1995) 2626–2631, arXiv:hep-ex/9503002 [hep-ex].
[41] D0 Collaboration, S. Abachi et al., Observation of the top quark , Phys.Rev.Lett. 74
(1995) 2632–2637, arXiv:hep-ex/9503003 [hep-ex].
[42] DONUT Collaboration, K. Kodama et al., Observation of tau neutrino interactions,
Phys.Lett. B504 (2001) 218–224, arXiv:hep-ex/0012035 [hep-ex].
[43] ALEPH Collaboration, DELPHI Collaboration, L3 Collaboration, OPAL
Collaboration, SLD Collaboration, LEP Electroweak Working Group, SLD
Electroweak Group, SLD Heavy Flavour Group Collaboration, S. Schael et al.,
Precision electroweak measurements on the Z resonance, Phys.Rept. 427 (2006)
257–454, arXiv:hep-ex/0509008 [hep-ex].
[44] ATLAS Collaboration, G. Aad et al., Observation of a new particle in the search for
the Standard Model Higgs boson with the ATLAS detector at the LHC , Phys.Lett.
B716 (2012) 1–29, arXiv:1207.7214 [hep-ex].
[45] CMS Collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of
125 GeV with the CMS experiment at the LHC , Phys.Lett. B716 (2012) 30–61,
arXiv:1207.7235 [hep-ex].
[46] ATLAS Collaboration, G. Aad et al., Measurements of Higgs boson production and
couplings in diboson final states with the ATLAS detector at the LHC , Phys.Lett.
B726 (2013) 88–119, arXiv:1307.1427 [hep-ex].
[47] ATLAS Collaboration, G. Aad et al., Measurement of Higgs boson production in the
diphoton decay channel in pp collisions at center-of-mass energies of 7 and 8 TeV
with the ATLAS detector , arXiv:1408.7084 [hep-ex].
[48] ATLAS Collaboration, G. Aad et al., Measurements of fiducial and differential
cross
√
sections for Higgs boson production in the diphoton decay channel at s = 8 TeV
with ATLAS , JHEP 1409 (2014) 112, arXiv:1407.4222 [hep-ex].
[49] CMS Collaboration, V. Khachatryan et al., Observation of the diphoton decay of the
Higgs boson and measurement of its properties, Eur.Phys.J. C74 (2014) no. 10, 3076,
arXiv:1407.0558 [hep-ex].
237

[50] CMS Collaboration, S. Chatrchyan et al., Evidence for the direct decay of the 125
GeV Higgs boson to fermions, Nature Phys. 10 (2014) , arXiv:1401.6527 [hep-ex].
[51] CMS Collaboration, S. Chatrchyan et al., Evidence for the 125 GeV Higgs boson
decaying to a pair of τ leptons, JHEP 1405 (2014) 104, arXiv:1401.5041 [hep-ex].
[52] CMS Collaboration, S. Chatrchyan et al., Measurement of the properties of a Higgs
boson in the four-lepton final state, Phys.Rev. D89 (2014) 092007, arXiv:1312.5353
[hep-ex].
[53] CMS Collaboration, S. Chatrchyan et al., Measurement of Higgs boson production
and properties in the WW decay channel with leptonic final states, JHEP 1401
(2014) 096, arXiv:1312.1129 [hep-ex].
[54] B. Pontecorvo, Mesonium and anti-mesonium, Sov.Phys.JETP 6 (1957) 429.
[55] B. Pontecorvo, Inverse beta processes and nonconservation of lepton charge,
Sov.Phys.JETP 7 (1958) 172–173.
[56] S. Bilenky, The History of neutrino oscillations, Phys.Scripta T121 (2005) 17–22,
arXiv:hep-ph/0410090 [hep-ph].
[57] WMAP Collaboration, D. Spergel et al., First year Wilkinson Microwave Anisotropy
Probe (WMAP) observations: Determination of cosmological parameters,
Astrophys.J.Suppl. 148 (2003) 175–194, arXiv:astro-ph/0302209 [astro-ph].
[58] G. Bertone, D. Hooper, and J. Silk, Particle dark matter: Evidence, candidates and
constraints, Phys.Rept. 405 (2005) 279–390, arXiv:hep-ph/0404175 [hep-ph].
[59] WMAP Collaboration, D. Spergel et al., Wilkinson Microwave Anisotropy Probe
(WMAP) three year results: implications for cosmology, Astrophys.J.Suppl. 170
(2007) 377, arXiv:astro-ph/0603449 [astro-ph].
[60] WMAP Collaboration, E. Komatsu et al., Seven-Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Cosmological Interpretation,
Astrophys.J.Suppl. 192 (2011) 18, arXiv:1001.4538 [astro-ph.CO].
[61] Planck Collaboration, P. Ade et al., Planck 2013 results. XVI. Cosmological
parameters, Astron.Astrophys. (2014) , arXiv:1303.5076 [astro-ph.CO].

238

[62] G. F. Giudice, Naturally Speaking: The Naturalness Criterion and Physics at the
LHC , arXiv:0801.2562 [hep-ph].
[63] W. A. Bardeen, On naturalness in the standard model , tech. rep., Fermilab, 1995.
http://cds.cern.ch/record/295811.
[64] K. G. Wilson, Renormalization group and critical phenomena. 2. Phase space cell
analysis of critical behavior , Phys.Rev. B4 (1971) 3184–3205.
[65] G. Altarelli, The Higgs: so simple yet so unnatural , EPJ Web Conf. 71 (2014) 00005.
[66] B. W. Lee, C. Quigg, and H. Thacker, Weak Interactions at Very High-Energies: The
Role of the Higgs Boson Mass, Phys.Rev. D16 (1977) 1519.
[67] H. Georgi and S. Glashow, Unity of All Elementary Particle Forces, Phys.Rev.Lett.
32 (1974) 438–441.
[68] P. Langacker and M.-x. Luo, Implications of precision electroweak experiments for
Mt , ρ0 , sin2 θW and grand unification, Phys.Rev. D44 (1991) 817–822.
[69] U. Amaldi, W. de Boer, and H. Furstenau, Comparison of grand unified theories with
electroweak and strong coupling constants measured at LEP , Phys.Lett. B260 (1991)
447–455.
[70] H. E. Haber and G. L. Kane, The Search for Supersymmetry: Probing Physics
Beyond the Standard Model , Phys.Rept. 117 (1985) 75–263.
[71] H. P. Nilles, Supersymmetry, Supergravity and Particle Physics, Phys.Rept. 110
(1984) 1–162.
[72] S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from
noncritical string theory, Phys.Lett. B428 (1998) 105–114, arXiv:hep-th/9802109
[hep-th].
[73] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity,
Int.J.Theor.Phys. 38 (1999) 1113–1133, arXiv:hep-th/9711200 [hep-th].
[74] J. Bardeen, L. Cooper, and J. Schrieffer, Theory of superconductivity, Phys.Rev. 108
(1957) 1175–1204.

239

[75] J. Bardeen, L. Cooper, and J. Schrieffer, Microscopic theory of superconductivity,
Phys.Rev. 106 (1957) 162.
[76] Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on
an Analogy with Superconductivity. I., Phys.Rev. 122 (1961) 345–358.
[77] Y. Nambu and G. Jona-Lasinio, Dynamical Model Of Elementary Particles Based On
An Analogy With Superconductivity. II , Phys.Rev. 124 (1961) 246–254.
[78] S. Weinberg, Implications of Dynamical Symmetry Breaking, Phys.Rev. D13 (1976)
974–996.
[79] S. Weinberg, Implications of Dynamical Symmetry Breaking: An Addendum,
Phys.Rev. D19 (1979) 1277–1280.
[80] L. Susskind, Dynamics of Spontaneous Symmetry Breaking in the Weinberg-Salam
Theory, Phys.Rev. D20 (1979) 2619–2625.
[81] S. Dimopoulos and L. Susskind, Mass Without Scalars, Nucl.Phys. B155 (1979)
237–252.
[82] E. Eichten and K. D. Lane, Dynamical Breaking of Weak Interaction Symmetries,
Phys.Lett. B90 (1980) 125–130.
[83] P. Sikivie, L. Susskind, M. B. Voloshin, and V. I. Zakharov, Isospin Breaking in
Technicolor Models, Nucl.Phys. B173 (1980) 189.
[84] B. Holdom, Raising the Sideways Scale, Phys.Rev. D24 (1981) 1441.
[85] B. Holdom, Techniodor , Phys.Lett. B150 (1985) 301.
[86] K. Yamawaki, M. Bando, and K.-i. Matumoto, Scale Invariant Technicolor Model
and a Technidilaton, Phys.Rev.Lett. 56 (1986) 1335.
[87] T. W. Appelquist, D. Karabali, and L. Wijewardhana, Chiral Hierarchies and the
Flavor Changing Neutral Current Problem in Technicolor , Phys.Rev.Lett. 57 (1986)
957.
[88] T. Appelquist and L. Wijewardhana, Chiral Hierarchies and Chiral Perturbations in
Technicolor , Phys.Rev. D35 (1987) 774.
240

[89] T. Appelquist and L. Wijewardhana, Chiral Hierarchies from Slowly Running
Couplings in Technicolor Theories, Phys.Rev. D36 (1987) 568.
[90] R. S. Chivukula, An Introduction to dynamical electroweak symmetry breaking,
arXiv:hep-ph/9701322 [hep-ph].
[91] R. S. Chivukula, Weak Isospin Violation in ’Walking’ Technicolor , Phys.Rev.Lett.
61 (1988) 2657.
[92] V. Miransky, M. Tanabashi, and K. Yamawaki, Dynamical Electroweak Symmetry
Breaking with Large Anomalous Dimension and t Quark Condensate, Phys.Lett.
B221 (1989) 177.
[93] V. Miransky, M. Tanabashi, and K. Yamawaki, Is the t Quark Responsible for the
Mass of W and Z Bosons? , Mod.Phys.Lett. A4 (1989) 1043.
[94] Y. Nambu, bootstrap Symmetry Breaking In Electroweak Unification, Tech. Rep.
EFI-89-08, Chicago U., EFI, 1989.
[95] W. Marciano, heavy Top Quark Mass Predictions, Phys.Rev.Lett. 62 (1989)
2793–2796.
[96] W. A. Bardeen, C. T. Hill, and M. Lindner, Minimal Dynamical Symmetry Breaking
of the Standard Model , Phys.Rev. D41 (1990) 1647.
[97] G. Cvetic, Top quark condensation, Rev.Mod.Phys. 71 (1999) 513–574,
arXiv:hep-ph/9702381 [hep-ph].
[98] C. T. Hill, Topcolor: Top quark condensation in a gauge extension of the standard
model , Phys.Lett. B266 (1991) 419–424.
[99] C. T. Hill and E. H. Simmons, Strong dynamics and electroweak symmetry breaking,
Phys.Rept. 381 (2003) 235–402, arXiv:hep-ph/0203079 [hep-ph].
[100] C. T. Hill, Topcolor assisted technicolor , Phys.Lett. B345 (1995) 483–489,
arXiv:hep-ph/9411426 [hep-ph].
[101] K. D. Lane and E. Eichten, Natural topcolor assisted technicolor , Phys.Lett. B352
(1995) 382–387, arXiv:hep-ph/9503433 [hep-ph].

241

[102] K. D. Lane, Symmetry breaking and generational mixing in topcolor assisted
technicolor , Phys.Rev. D54 (1996) 2204–2212, arXiv:hep-ph/9602221 [hep-ph].
[103] M. B. Popovic and E. H. Simmons, A Heavy top quark from flavor universal
colorons, Phys.Rev. D58 (1998) 095007, arXiv:hep-ph/9806287 [hep-ph].
[104] F. Braam, M. Flossdorf, R. S. Chivukula, S. Di Chiara, and E. H. Simmons,
Hypercharge-Universal Topcolor , Phys.Rev. D77 (2008) 055005, arXiv:0711.1127
[hep-ph].
[105] B. A. Dobrescu and C. T. Hill, Electroweak symmetry breaking via top condensation
seesaw , Phys.Rev.Lett. 81 (1998) 2634–2637, arXiv:hep-ph/9712319 [hep-ph].
[106] R. S. Chivukula, B. A. Dobrescu, H. Georgi, and C. T. Hill, Top quark seesaw theory
of electroweak symmetry breaking, Phys.Rev. D59 (1999) 075003,
arXiv:hep-ph/9809470 [hep-ph].
[107] H.-J. He, C. T. Hill, and T. M. Tait, Top quark seesaw, vacuum structure and
electroweak precision constraints, Phys.Rev. D65 (2002) 055006,
arXiv:hep-ph/0108041 [hep-ph].
[108] R. S. Chivukula, D. A. Dicus, and H.-J. He, Unitarity of compactified
five-dimensional Yang-Mills theory, Phys.Lett. B525 (2002) 175–182,
arXiv:hep-ph/0111016 [hep-ph].
[109] R. S. Chivukula and H.-J. He, Unitarity of deconstructed five-dimensional Yang-Mills
theory, Phys.Lett. B532 (2002) 121–128, arXiv:hep-ph/0201164 [hep-ph].
[110] R. S. Chivukula, D. A. Dicus, H.-J. He, and S. Nandi, Unitarity of the higher
dimensional standard model , Phys.Lett. B562 (2003) 109–117,
arXiv:hep-ph/0302263 [hep-ph].
[111] C. Csaki, C. Grojean, H. Murayama, L. Pilo, and J. Terning, Gauge theories on an
interval: Unitarity without a Higgs, Phys.Rev. D69 (2004) 055006,
arXiv:hep-ph/0305237 [hep-ph].
[112] C. Csaki and D. Curtin, A Flavor Protection for Warped Higgsless Models, Phys.Rev.
D80 (2009) 015027, arXiv:0904.2137 [hep-ph].
[113] G. Cacciapaglia, C. Csaki, G. Marandella, and J. Terning, A New custodian for a
realistic Higgsless model , Phys.Rev. D75 (2007) 015003, arXiv:hep-ph/0607146
[hep-ph].
242

[114] G. Cacciapaglia, C. Csaki, C. Grojean, and J. Terning, Curing the Ills of Higgsless
models: The S parameter and unitarity, Phys.Rev. D71 (2005) 035015,
arXiv:hep-ph/0409126 [hep-ph].
[115] G. Cacciapaglia, C. Csaki, C. Grojean, and J. Terning, Oblique corrections from
Higgsless models in warped space, Phys.Rev. D70 (2004) 075014,
arXiv:hep-ph/0401160 [hep-ph].
[116] C. Csaki, C. Grojean, L. Pilo, and J. Terning, Towards a realistic model of Higgsless
electroweak symmetry breaking, Phys.Rev.Lett. 92 (2004) 101802,
arXiv:hep-ph/0308038 [hep-ph].
[117] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Large N field
theories, string theory and gravity, Phys.Rept. 323 (2000) 183–386,
arXiv:hep-th/9905111 [hep-th].
[118] R. S. Chivukula, B. Coleppa, S. Di Chiara, E. H. Simmons, H.-J. He, et al., A Three
Site Higgsless Model , Phys.Rev. D74 (2006) 075011, arXiv:hep-ph/0607124
[hep-ph].
[119] R. Sekhar Chivukula, N. D. Christensen, B. Coleppa, and E. H. Simmons, The Top
Triangle Moose: Combining Higgsless and Topcolor Mechanisms for Mass
Generation, Phys.Rev. D80 (2009) 035011, arXiv:0906.5567 [hep-ph].
[120] ATLAS Collaboration, G. Aad et al., Measurement of the Higgs boson mass from the
H → γγ and H → ZZ ∗ → 4 channels with the ATLAS detector using 25 fb−1 of pp
collision data, Phys.Rev. D90 (2014) 052004, arXiv:1406.3827 [hep-ex].
[121] E. Eichten, I. Hinchliffe, K. D. Lane, and C. Quigg, Super Collider Physics,
Rev.Mod.Phys. 56 (1984) 579–707.
[122] E. Eichten, I. Hinchliffe, K. Lane, and C. Quigg, Signatures for Technicolor ,
Phys.Rev. D34 (1986) 1547.
[123] R. S. Chivukula, R. Rosenfeld, E. H. Simmons, and J. Terning, Strongly coupled
electroweak symmetry breaking: Implication of models, arXiv:hep-ph/9503202
[hep-ph].
[124] A. Belyaev, A. Blum, R. S. Chivukula, and E. H. Simmons, The Meaning of Higgs:
tau+ tau- and gamma gamma at the Tevatron and the LHC , Phys.Rev. D72 (2005)
055022, arXiv:hep-ph/0506086 [hep-ph].
243

[125] R. S. Chivukula, E. H. Simmons, B. Coleppa, H. E. Logan, and A. Martin, Top-Higgs
and Top-pion phenomenology in the Top Triangle Moose model , Phys.Rev. D83
(2011) 055013, arXiv:1101.6023 [hep-ph].
[126] R. S. Chivukula, E. H. Simmons, B. Coleppa, H. E. Logan, and A. Martin, LHC
Limits on the Top-Higgs in Models with Strong Top-Quark Dynamics, Phys.Rev.
D84 (2011) 095022, arXiv:1108.4000 [hep-ph].
[127] CDF Collaboration, T. Aaltonen et al., Search for charged Higgs bosons in decays of
top quarks in p anti-p collisions at s**(1/2) = 1.96 TeV , Phys.Rev.Lett. 103 (2009)
101803, arXiv:0907.1269 [hep-ex].
[128] D0 Collaboration, V. Abazov et al., Search for charged Higgs bosons in top quark
decays, Phys.Lett. B682 (2009) 278–286, arXiv:0908.1811 [hep-ex].
[129] P. Langacker, The Physics of Heavy Z Gauge Bosons, Rev.Mod.Phys. 81 (2009)
1199–1228, arXiv:0801.1345 [hep-ph].
[130] P. Langacker, R. W. Robinett, and J. L. Rosner, New Heavy Gauge Bosons in p p
and p anti-p Collisions, Phys.Rev. D30 (1984) 1470.
[131] R. M. Harris and K. Kousouris, Searches for Dijet Resonances at Hadron Colliders,
Int.J.Mod.Phys. A26 (2011) 5005–5055, arXiv:1110.5302 [hep-ex].
[132] LHC/LC Study Group Collaboration, G. Weiglein et al., Physics interplay of the
LHC and the ILC , Phys.Rept. 426 (2006) 47–358, arXiv:hep-ph/0410364
[hep-ph].
[133] F. Yu, Di-jet resonances at future hadron colliders: A Snowmass whitepaper ,
arXiv:1308.1077 [hep-ph].
[134] A. Leike, The Phenomenology of extra neutral gauge bosons, Phys.Rept. 317 (1999)
143–250, arXiv:hep-ph/9805494 [hep-ph].
[135] J. D. Anderson, M. H. Austern, and R. N. Cahn, Measurement of Z-prime couplings
at future hadron colliders through decays to tau leptons, Phys.Rev. D46 (1992)
290–302.
[136] F. del Aguila, M. Cvetic, and P. Langacker, Determination of Z-prime gauge
couplings to quarks and leptons at future hadron colliders, Phys.Rev. D48 (1993)
969–973, arXiv:hep-ph/9303299 [hep-ph].
244

[137] M. Cvetic and S. Godfrey, Discovery and identification of extra gauge bosons,
arXiv:hep-ph/9504216 [hep-ph].
[138] M. Dittmar, A.-S. Nicollerat, and A. Djouadi, Z-prime studies at the LHC: An
Update, Phys.Lett. B583 (2004) 111–120, arXiv:hep-ph/0307020 [hep-ph].
[139] F. Petriello and S. Quackenbush, Measuring Z couplings at the CERN LHC ,
Phys.Rev. D77 (2008) 115004, arXiv:0801.4389 [hep-ph].
[140] M. Cvetic and P. Langacker, V-prime Z and V-prime W production as tests of heavy
gauge boson couplings at future hadron colliders, Phys.Rev. D46 (1992) 4943,
arXiv:hep-ph/9207216 [hep-ph].
[141] ATLAS Collaboration, G. Aad et al., Search for the Standard Model Higgs boson in
the two photon decay channel with the ATLAS detector at the LHC , Phys.Lett. B705
(2011) 452–470, arXiv:1108.5895 [hep-ex].
[142] CMS Collaboration, CMS Collaboration, Search for a Higgs boson decaying into two
photons in the CMS detector , Tech. Rep. CMS-PAS-HIG-11-021, CERN, 2011.
http://cds.cern.ch/record/1376642.
[143] ATLAS Collaboration, ATLAS Collaboration, Update of the Combination of Higgs
Boson Searches in pp Collisions at sqrt(s) = 7 TeV with the ATLAS Experiment at
the LHC , Tech. Rep. ATLAS-CONF-2011-135, CERN, 2011.
http://cds.cern.ch/record/1383838.
[144] ATLAS Collaboration, ATLAS Collaboration, Search for neutral MSSM Higgs bosons
decaying to tau+tau- pairs in proton-proton collisions at sqrt(s) = 7 TeV with the
ATLAS detector , Tech. Rep. ATLAS-CONF-2011-132, CERN, 2011.
http://cds.cern.ch/record/1383835.
[145] CMS Collaboration, CMS Collaboration, Search for Neutral Higgs Bosons Decaying
to Tau Pairs in pp Collisions at sqrts=7 TeV , Tech. Rep. CMS-PAS-HIG-11-020,
CERN, 2011. http://cds.cern.ch/record/1378096.
[146] A. Manohar and L. Randall, Searching for neutral pseudoGoldstone bosons in Z0
decays, Phys.Lett. B246 (1990) 537–540.
[147] L. Randall and E. H. Simmons, Signatures of neutral pseudoGoldstone bosons from
technicolor , Nucl.Phys. B380 (1992) 3–21.

245

[148] G. Rupak and E. H. Simmons, Limits on pseudoscalar bosons from rare Z decays at
LEP , Phys.Lett. B362 (1995) 155–163, arXiv:hep-ph/9507438 [hep-ph].
[149] V. Lubicz and P. Santorelli, Production of neutral pseudoGoldstone bosons at LEP-2
and NLC in multiscale walking technicolor models, Nucl.Phys. B460 (1996) 3–36,
arXiv:hep-ph/9505336 [hep-ph].
[150] R. Casalbuoni, A. Deandrea, S. De Curtis, D. Dominici, R. Gatto, et al., Detecting
and studying the lightest pseudoGoldstone boson at future pp, e+ e− and µ+ µ−
colliders, Nucl.Phys. B555 (1999) 3–52, arXiv:hep-ph/9809523 [hep-ph].
[151] K. R. Lynch and E. H. Simmons, Composite scalars at CERN LEP: Constraining
technicolor theories, Phys.Rev. D64 (2001) 035008, arXiv:hep-ph/0012256
[hep-ph].
[152] E. Farhi and L. Susskind, Technicolor , Phys.Rept. 74 (1981) 277.
[153] K. D. Lane and M. Ramana, Walking technicolor signatures at hadron colliders,
Phys.Rev. D44 (1991) 2678–2700.
[154] K. D. Lane, Technihadron production and decay in low scale technicolor , Phys.Rev.
D60 (1999) 075007, arXiv:hep-ph/9903369 [hep-ph].
[155] S. Dimopoulos, S. Raby, and G. L. Kane, Experimental Predictions from Technicolor
Theories, Nucl.Phys. B182 (1981) 77–103.
[156] J. R. Ellis, M. K. Gaillard, D. V. Nanopoulos, and P. Sikivie, Can One Tell
Technicolor from a Hole in the Ground? , Nucl.Phys. B182 (1981) 529–545.
[157] B. Holdom, A Phenomenological Lagrangian for Hypercolor , Phys.Rev. D24 (1981)
157.
[158] J. F. Gunion, H. E. Haber, G. L. Kane, and S. Dawson, The Higgs Hunter’s Guide,
Front.Phys. 80 (2000) 1–448.
[159] LHC Higgs Cross Section Working Group Collaboration, S. Dittmaier et al.,
Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables, arXiv:1101.0593
[hep-ph].

246

[160] A. Djouadi, J. Kalinowski, and M. Spira, HDECAY: A Program for Higgs boson
decays in the standard model and its supersymmetric extension,
Comput.Phys.Commun. 108 (1998) 56–74, arXiv:hep-ph/9704448 [hep-ph].
[161] ATLAS Collaboration, ATLAS Collaboration, An update to the combined search for
the Standard Model Higgs boson with the ATLAS detector at the LHC using up to 4.9
fb1 of pp collision data at s = 7 TeV , Tech. Rep. ATLAS-CONF-2012-019, CERN,
2012. http://cds.cern.ch/record/1430033.
[162] ATLAS Collaboration, ATLAS Collaboration, Search for the Standard Model Higgs
boson in the decay mode H -&gt; tau+ tau- -&gt; ll + 4 neutrinos in Association with
jets in Proton-Proton Collisions at sqrt(s) =7 TeV with the ATLAS detector , Tech.
Rep. ATLAS-CONF-2011-133, CERN, 2011. http://cds.cern.ch/record/1383836.
[163] ATLAS Collaboration, ATLAS Collaboration, Search for the Standard Model Higgs
boson in the H → τ + τ − decay mode with 4.7 fb−1 of ATLAS data at 7 TeV , Tech.
Rep. ATLAS-CONF-2012-014, CERN, 2012. http://cds.cern.ch/record/1429662.
[164] A. Azatov, R. Contino, and J. Galloway, Model-Independent Bounds on a Light
Higgs, JHEP 1204 (2012) 127, arXiv:1202.3415 [hep-ph].
[165] F. Sannino and K. Tuominen, Orientifold theory dynamics and symmetry breaking,
Phys.Rev. D71 (2005) 051901, arXiv:hep-ph/0405209 [hep-ph].
[166] R. Foadi, M. T. Frandsen, T. A. Ryttov, and F. Sannino, Minimal Walking
Technicolor: Set Up for Collider Physics, Phys.Rev. D76 (2007) 055005,
arXiv:0706.1696 [hep-ph].
[167] A. Belyaev, R. Foadi, M. T. Frandsen, M. Jarvinen, F. Sannino, et al., Technicolor
Walks at the LHC , Phys.Rev. D79 (2009) 035006, arXiv:0809.0793 [hep-ph].
[168] CMS Collaboration, S. Chatrchyan et al., Search
√ for the standard model Higgs boson
decaying into two photons in pp collisions at s = 7 TeV , Phys.Lett. B710 (2012)
403–425, arXiv:1202.1487 [hep-ex].
[169] W. Bernreuther, P. Gonzalez, and M. Wiebusch, Pseudoscalar Higgs Bosons at the
LHC: Production and Decays into Electroweak Gauge Bosons Revisited , Eur.Phys.J.
C69 (2010) 31–43, arXiv:1003.5585 [hep-ph].
[170] G. Burdman, C. E. Haluch, and R. D. Matheus, Is the LHC Observing the
Pseudo-scalar State of a Two-Higgs Doublet Model ? , Phys.Rev. D85 (2012) 095016,
arXiv:1112.3961 [hep-ph].
247

[171] B. Holdom, A pseudoscalar at the LHC: technicolor vs a fourth family, Phys.Lett.
B709 (2012) 381–384, arXiv:1201.0185 [hep-ph].
[172] E. Cervero and J.-M. Gerard, Minimal violation of flavour and custodial symmetries
in a vectophobic Two-Higgs-Doublet-Model , Phys.Lett. B712 (2012) 255–260,
arXiv:1202.1973 [hep-ph].
[173] M. T. Frandsen and F. Sannino, Discovering a Light Scalar or Pseudoscalar at The
Large Hadron Collider , arXiv:1203.3988 [hep-ph].
[174] J. Moffat, Has A 125 GeV Pseudoscalar Resonance Been Observed at the LHC? ,
arXiv:1204.4702 [hep-ph].
[175] A. Barroso, P. Ferreira, R. Santos, and J. P. Silva, Probing the scalar-pseudoscalar
mixing in the 125 GeV Higgs particle with current data, Phys.Rev. D86 (2012)
015022, arXiv:1205.4247 [hep-ph].
[176] I. Low, J. Lykken, and G. Shaughnessy, Have We Observed the Higgs (Imposter)? ,
Phys.Rev. D86 (2012) 093012, arXiv:1207.1093 [hep-ph].
[177] B. Coleppa, K. Kumar, and H. E. Logan, Can the 126 GeV boson be a pseudoscalar? ,
Phys.Rev. D86 (2012) 075022, arXiv:1208.2692 [hep-ph].
[178] C. T. Hill, S. Pokorski, and J. Wang, Gauge invariant effective Lagrangian for
Kaluza-Klein modes, Phys.Rev. D64 (2001) 105005, arXiv:hep-th/0104035
[hep-th].
[179] N. Arkani-Hamed, A. G. Cohen, and H. Georgi, (De)constructing dimensions,
Phys.Rev.Lett. 86 (2001) 4757–4761, arXiv:hep-th/0104005 [hep-th].
[180] H. Georgi, A Tool Kit for Builders of Composite Models, Nucl.Phys. B266 (1986)
274.
[181] R. Casalbuoni, S. De Curtis, D. Dominici, and R. Gatto, Effective Weak Interaction
Theory with Possible New Vector Resonance from a Strong Higgs Sector , Phys.Lett.
B155 (1985) 95.
[182] R. Casalbuoni, A. Deandrea, S. De Curtis, D. Dominici, R. Gatto, et al., Degenerate
BESS model: The Possibility of a low-energy strong electroweak sector , Phys.Rev.
D53 (1996) 5201–5221, arXiv:hep-ph/9510431 [hep-ph].

248

[183] R. S. Chivukula, E. H. Simmons, H.-J. He, M. Kurachi, and M. Tanabashi,
Multi-gauge-boson vertices and chiral Lagrangian parameters in Higgsless models with
ideal fermion delocalization, Phys.Rev. D72 (2005) 075012, arXiv:hep-ph/0508147
[hep-ph].
[184] R. S. Chivukula, E. H. Simmons, H.-J. He, M. Kurachi, and M. Tanabashi, Ideal
fermion delocalization in Higgsless models, Phys.Rev. D72 (2005) 015008,
arXiv:hep-ph/0504114 [hep-ph].
[185] R. S. Chivukula, E. H. Simmons, H.-J. He, M. Kurachi, and M. Tanabashi,
Deconstructed Higgsless models with one-site delocalization, Phys.Rev. D71 (2005)
115001, arXiv:hep-ph/0502162 [hep-ph].
[186] M. Kurachi, R. S. Chivukula, E. H. Simmons, H.-J. He, and M. Tanabashi, Oblique
corrections in deconstructed Higgsless models, arXiv:hep-ph/0409134 [hep-ph].
[187] R. S. Chivukula, E. H. Simmons, H.-J. He, M. Kurachi, and M. Tanabashi, Universal
non-oblique corrections in Higgsless models and beyond , Phys.Lett. B603 (2004)
210–218, arXiv:hep-ph/0408262 [hep-ph].
[188] R. S. Chivukula, E. H. Simmons, H.-J. He, M. Kurachi, and M. Tanabashi, The
Structure of corrections to electroweak interactions in Higgsless models, Phys.Rev.
D70 (2004) 075008, arXiv:hep-ph/0406077 [hep-ph].
[189] K. Lane and A. Martin, An Effective Lagrangian for Low-Scale Technicolor ,
Phys.Rev. D80 (2009) 115001, arXiv:0907.3737 [hep-ph].
[190] R. Foadi, S. Gopalakrishna, and C. Schmidt, Higgsless electroweak symmetry breaking
from theory space, JHEP 0403 (2004) 042, arXiv:hep-ph/0312324 [hep-ph].
[191] R. Foadi, S. Gopalakrishna, and C. Schmidt, Effects of fermion localization in
Higgsless theories and electroweak constraints, Phys.Lett. B606 (2005) 157–163,
arXiv:hep-ph/0409266 [hep-ph].
[192] R. Foadi and C. Schmidt, An Effective Higgsless theory: Satisfying electroweak
constraints and a heavy top quark , Phys.Rev. D73 (2006) 075011,
arXiv:hep-ph/0509071 [hep-ph].
[193] M. Bando, T. Kugo, and K. Yamawaki, Nonlinear Realization and Hidden Local
Symmetries, Phys.Rept. 164 (1988) 217–314.

249

[194] M. Bando, T. Kugo, and K. Yamawaki, On the Vector Mesons as Dynamical Gauge
Bosons of Hidden Local Symmetries, Nucl.Phys. B259 (1985) 493.
[195] M. Bando, T. Kugo, and K. Yamawaki, Composite Gauge Bosons and ’Low-energy
Theorems’ of Hidden Local Symmetries, Prog.Theor.Phys. 73 (1985) 1541.
[196] M. Bando, T. Kugo, S. Uehara, K. Yamawaki, and T. Yanagida, Is rho Meson a
Dynamical Gauge Boson of Hidden Local Symmetry? , Phys.Rev.Lett. 54 (1985) 1215.
[197] M. Bando, T. Fujiwara, and K. Yamawaki, Generalized Hidden Local Symmetry and
the A1 Meson, Prog.Theor.Phys. 79 (1988) 1140.
[198] M. E. Peskin and T. Takeuchi, Estimation of oblique electroweak corrections,
Phys.Rev. D46 (1992) 381–409.
[199] D. Kominis, Flavor changing neutral current constraints in topcolor assisted
technicolor , Phys.Lett. B358 (1995) 312–317, arXiv:hep-ph/9506305 [hep-ph].
[200] G. Buchalla, G. Burdman, C. Hill, and D. Kominis, GIM violation and new dynamics
of the third generation, Phys.Rev. D53 (1996) 5185–5200, arXiv:hep-ph/9510376
[hep-ph].
[201] H.-J. He and C. Yuan, New Method for Detecting Charged Scalars at Colliders,
Phys.Rev.Lett. 83 (1999) 28–31, arXiv:hep-ph/9810367 [hep-ph].
[202] G. Burdman, Scalars from top condensation models at hadron colliders,
Phys.Rev.Lett. 83 (1999) 2888–2891, arXiv:hep-ph/9905347 [hep-ph].
[203] G. Burdman and D. Kominis, Model independent constraints on topcolor from R(b),
Phys.Lett. B403 (1997) 101–107, arXiv:hep-ph/9702265 [hep-ph].
[204] LHC Higgs √
Cross Section Working Group (summer 2011), SM Higgs production cross
sections at s = 7 TeV (update in CERN Report 3), tech. rep., CERN, 2011. https:
//twiki.cern.ch/twiki/bin/view/LHCPhysics/CERNYellowReportPageAt7TeV.
[205] M. Spira, A. Djouadi, D. Graudenz, and P. Zerwas, Higgs boson production at the
LHC , Nucl.Phys. B453 (1995) 17–82, arXiv:hep-ph/9504378 [hep-ph].
[206] ATLAS Collaboration, G. Aad et al., Search for the Standard
√ Model Higgs boson in
−1
the diphoton decay channel with 4.9 fb of pp collisions at s = 7 TeV with
ATLAS , Phys.Rev.Lett. 108 (2012) 111803, arXiv:1202.1414 [hep-ex].
250

[207] CMS Collaboration, S. Chatrchyan et al., Search for anomalous
production of highly
√
boosted Z bosons decaying to dimuons in pp collisions at s = 7 TeV , Phys.Lett.
B722 (2013) 28–47, arXiv:1210.0867 [hep-ex].
[208] ATLAS Collaboration, G. Aad et al., Search for charged Higgs
√ bosons decaying via
+
H → τ ν in top quark pair events using pp collision data at s = 7 TeV with the
ATLAS detector , JHEP 1206 (2012) 039, arXiv:1204.2760 [hep-ex].
[209] CMS Collaboration, S. Chatrchyan et al., Combined
results of searches for the
√
standard model Higgs boson in pp collisions at s = 7 TeV , Phys.Lett. B710 (2012)
26–48, arXiv:1202.1488 [hep-ex].
[210] P. H. Frampton and S. L. Glashow, Chiral Color: An Alternative to the Standard
Model , Phys.Lett. B190 (1987) 157.
[211] P. H. Frampton and S. L. Glashow, Unifiable Chiral Color With Natural Gim
Mechanism, Phys.Rev.Lett. 58 (1987) 2168.
[212] R. Chivukula, A. G. Cohen, and E. H. Simmons, New strong interactions at the
Tevatron? , Phys.Lett. B380 (1996) 92–98, arXiv:hep-ph/9603311 [hep-ph].
[213] E. H. Simmons, Coloron phenomenology, Phys.Rev. D55 (1997) 1678–1683,
arXiv:hep-ph/9608269 [hep-ph].
[214] O. Antunano, J. H. Kuhn, and G. Rodrigo, Top quarks, axigluons and charge
asymmetries at hadron colliders, Phys.Rev. D77 (2008) 014003, arXiv:0709.1652
[hep-ph].
[215] P. Ferrario and G. Rodrigo, Constraining heavy colored resonances from top-antitop
quark events, Phys.Rev. D80 (2009) 051701, arXiv:0906.5541 [hep-ph].
[216] P. H. Frampton, J. Shu, and K. Wang, Axigluon as Possible Explanation for p anti-p
—&gt; t anti-t Forward-Backward Asymmetry, Phys.Lett. B683 (2010) 294–297,
arXiv:0911.2955 [hep-ph].
[217] G. Rodrigo and P. Ferrario, Charge asymmetry: A Theory appraisal , Nuovo Cim.
C033 (2010) no. 4, 221–228, arXiv:1007.4328 [hep-ph].
[218] R. S. Chivukula, E. H. Simmons, and C.-P. Yuan, Axigluons cannot explain the
observed top quark forward-backward asymmetry, Phys.Rev. D82 (2010) 094009,
arXiv:1007.0260 [hep-ph].
251

[219] G. Marques Tavares and M. Schmaltz, Explaining the t-tbar asymmetry with a light
axigluon, Phys.Rev. D84 (2011) 054008, arXiv:1107.0978 [hep-ph].
[220] D. Dicus, C. McMullen, and S. Nandi, Collider implications of Kaluza-Klein
excitations of the gluons, Phys.Rev. D65 (2002) 076007, arXiv:hep-ph/0012259
[hep-ph].
[221] K. Lane and S. Mrenna, The Collider phenomenology of technihadrons in the
technicolor straw man model , Phys.Rev. D67 (2003) 115011, arXiv:hep-ph/0210299
[hep-ph].
[222] I. Antoniadis, A Possible new dimension at a few TeV , Phys.Lett. B246 (1990)
377–384.
[223] G. Senjanovic and R. N. Mohapatra, Exact Left-Right Symmetry and Spontaneous
Violation of Parity, Phys.Rev. D12 (1975) 1502.
[224] H. Georgi, E. E. Jenkins, and E. H. Simmons, Ununifying the Standard Model ,
Phys.Rev.Lett. 62 (1989) 2789.
[225] H. Georgi, E. E. Jenkins, and E. H. Simmons, The Ununified Standard Model ,
Nucl.Phys. B331 (1990) 541.
[226] D. J. Muller and S. Nandi, Top flavor: A Separate SU(2) for the third family,
Phys.Lett. B383 (1996) 345–350, arXiv:hep-ph/9602390 [hep-ph].
[227] E. Malkawi, T. M. Tait, and C. Yuan, A Model of strong flavor dynamics for the top
quark , Phys.Lett. B385 (1996) 304–310, arXiv:hep-ph/9603349 [hep-ph].
[228] R. Chivukula, E. H. Simmons, and J. Terning, A Heavy top quark and the Z b anti-b
vertex in noncommuting extended technicolor , Phys.Lett. B331 (1994) 383–389,
arXiv:hep-ph/9404209 [hep-ph].
[229] J. L. Hewett and T. G. Rizzo, Low-Energy Phenomenology of Superstring Inspired
E(6) Models, Phys.Rept. 183 (1989) 193.
[230] L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension,
Phys.Rev.Lett. 83 (1999) 3370–3373, arXiv:hep-ph/9905221 [hep-ph].
[231] L. Randall and R. Sundrum, An Alternative to compactification, Phys.Rev.Lett. 83
(1999) 4690–4693, arXiv:hep-th/9906064 [hep-th].
252

[232] H. Davoudiasl, J. Hewett, and T. Rizzo, Phenomenology of the Randall-Sundrum
Gauge Hierarchy Model , Phys.Rev.Lett. 84 (2000) 2080, arXiv:hep-ph/9909255
[hep-ph].
[233] J. Bijnens, P. Eerola, M. Maul, A. Mansson, and T. Sjostrand, QCD signatures of
narrow graviton resonances in hadron colliders, Phys.Lett. B503 (2001) 341–348,
arXiv:hep-ph/0101316 [hep-ph].
[234] UA1 Collaboration, G. Arnison et al., Measurement of the Inclusive Jet
Cross-Section at the CERN p anti-p Collider , Phys.Lett. B172 (1986) 461.
[235] UA2 Collaboration, J. Alitti et al., A Search for new intermediate vector mesons and
excited quarks decaying to two jets at the CERN p¯p collider , Nucl.Phys. B400 (1993)
3–24.
[236] CDF
Collaboration, F. Abe et al., The Two jet invariant mass distribution at
√
s = 1.8 TeV , Phys.Rev. D41 (1990) 1722–1725.
[237] CDF Collaboration,
F. Abe et al., Search for new particles decaying to dijets in p¯
p
√
collisions at s = 1.8 TeV , Phys.Rev.Lett. 74 (1995) 3538–3543,
arXiv:hep-ex/9501001 [hep-ex].
[238] CDF Collaboration, F. Abe et al., Search for new particles decaying to dijets at
CDF , Phys.Rev. D55 (1997) 5263–5268, arXiv:hep-ex/9702004 [hep-ex].
[239] D0 Collaboration, V. Abazov et al., Search for new particles in the two jet decay
channel with the D0 detector , Phys.Rev. D69 (2004) 111101,
arXiv:hep-ex/0308033 [hep-ex].
[240] CDF Collaboration, T. Aaltonen et al., Search for new particles decaying into dijets
in proton-antiproton collisions at s**(1/2) = 1.96-TeV , Phys.Rev. D79 (2009)
112002, arXiv:0812.4036 [hep-ex].
[241] ATLAS Collaboration, G. Aad et al., Search for New Particles in Two-Jet Final
States in 7 TeV Proton-Proton Collisions with the ATLAS Detector at the LHC ,
Phys.Rev.Lett. 105 (2010) 161801, arXiv:1008.2461 [hep-ex].
[242] CMS Collaboration, V. Khachatryan et al., Search for Dijet Resonances in 7 TeV pp
Collisions at CMS , Phys.Rev.Lett. 105 (2010) 211801, arXiv:1010.0203 [hep-ex].

253

[243] CMS Collaboration, S. Chatrchyan et al., Search for Resonances in the Dijet Mass
Spectrum from 7 TeV pp Collisions at CMS , Phys.Lett. B704 (2011) 123–142,
arXiv:1107.4771 [hep-ex].
[244] ATLAS Collaboration, G. Aad et al., Search for New
√ Physics in the Dijet Mass
−1
Distribution using 1 fb of pp Collision Data at s = 7 TeV collected by the
ATLAS Detector , Phys.Lett. B708 (2012) 37–54, arXiv:1108.6311 [hep-ex].
[245] ATLAS Collaboration, ATLAS Collaboration, Search for New Physics in Dijet Mass
and Angular Distributions using 4.8/fb of pp Collisions at sqrt(s) = 7 TeV collected
by the ATLAS Detector , Tech. Rep. ATLAS-CONF-2012-038, CERN, 2012.
http://cds.cern.ch/record/1432206.
[246] ATLAS Collaboration, G. Aad et al., ATLAS search for
√ new phenomena in dijet
mass and angular distributions using pp collisions at s = 7 TeV , JHEP 1301
(2013) 029, arXiv:1210.1718 [hep-ex].
[247] ATLAS Collaboration, ATLAS Collaboration, Search for New Phenomena in the
Dijet Mass Distribution updated using 13.0 f b−1 of pp Collisions at sqrts = 8 TeV
collected by the ATLAS Detector , Tech. Rep. ATLAS-CONF-2012-148, CERN, 2012.
http://cds.cern.ch/record/1493487.
[248] CMS Collaboration, S. Chatrchyan et al.,
√ Search for narrow resonances using the
dijet mass spectrum in pp collisions at s=8TeV , Phys.Rev. D87 (2013) no. 11,
114015, arXiv:1302.4794 [hep-ex].
[249] CMS Collaboration, CMS Collaboration, Search for Narrow Resonances using the
Dijet Mass Spectrum with 19.6fb-1 of pp Collisions at sqrts=8 TeV , Tech. Rep.
CMS-PAS-EXO-12-059, CERN, 2013. http://cds.cern.ch/record/1519066.
[250] B. A. Dobrescu, K. Kong, and R. Mahbubani, Massive color-octet bosons and pairs of
resonances at hadron colliders, Phys.Lett. B670 (2008) 119–123, arXiv:0709.2378
[hep-ph].
[251] R. S. Chivukula, E. H. Simmons, and N. Vignaroli, Same-Sign Dileptons from
Colored Scalars in the Flavorful Top-Coloron Model , Phys.Rev. D88 (2013) 034006,
arXiv:1306.2248 [hep-ph].
[252] D0 Collaboration, V. Abazov et al., First measurement of the forward-backward
charge asymmetry in top quark pair production, Phys.Rev.Lett. 100 (2008) 142002,
arXiv:0712.0851 [hep-ex].

254

[253] CDF Collaboration, T. Aaltonen et al., Forward-Backward Asymmetry in Top Quark
Production in p¯
p Collisions at sqrts = 1.96 TeV , Phys.Rev.Lett. 101 (2008) 202001,
arXiv:0806.2472 [hep-ex].
[254] D0 Collaboration, V. M. Abazov et al., Forward-backward asymmetry in top
quark-antiquark production, Phys.Rev. D84 (2011) 112005, arXiv:1107.4995
[hep-ex].
[255] CDF Collaboration, T. Aaltonen et al., Evidence for a Mass Dependent
Forward-Backward Asymmetry in Top Quark Pair Production, Phys.Rev. D83
(2011) 112003, arXiv:1101.0034 [hep-ex].
[256] CDF Collaboration, CDF Collaboration, Study of the Top Quark Production
Asymmetry and Its Mass and Rapidity Dependence in the Full Run II Tevatron
Dataset, Tech. Rep. CDF-Note-10807, Fermilab, 2012. http://www-cdf.fnal.gov/
physics/new/top/2012/LepJet_AFB_Winter2012/CDF10807.pdf.
[257] B. A. Dobrescu and F. Yu, Coupling-mass mapping of dijet peak searches, Phys.Rev.
D88 (2013) no. 3, 035021, arXiv:1306.2629 [hep-ph].
[258] Y. Bai, J. L. Hewett, J. Kaplan, and T. G. Rizzo, LHC Predictions from a Tevatron
Anomaly in the Top Quark Forward-Backward Asymmetry, JHEP 1103 (2011) 003,
arXiv:1101.5203 [hep-ph].
[259] U. Haisch and S. Westhoff, Massive Color-Octet Bosons: Bounds on Effects in
Top-Quark Pair Production, JHEP 1108 (2011) 088, arXiv:1106.0529 [hep-ph].
[260] K. Gumus, N. Akchurin, S. Esen, and R. M. Harris, CMS Sensitivity to Dijet
Resonances, Tech. Rep. CMS-NOTE-2006-070, CERN, 2006.
http://cds.cern.ch/record/962025.
[261] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, and T. Stelzer, MadGraph 5 :
Going Beyond , JHEP 1106 (2011) 128, arXiv:1106.0522 [hep-ph].
[262] J. Pumplin, D. Stump, J. Huston, H. Lai, P. M. Nadolsky, et al., New generation of
parton distributions with uncertainties from global QCD analysis, JHEP 0207 (2002)
012, arXiv:hep-ph/0201195 [hep-ph].
[263] ATLAS Collaboration, ATLAS Collaboration, Reconstruction of High Mass tt
Resonances in the Lepton+Jets Channel , Tech. Rep. ATL-PHYS-PUB-2009-081,
CERN, 2009. http://cds.cern.ch/record/1177410.
255

[264] CMS Collaboration, CMS Collaboration, Search for narrow resonances in top-pair
production close to threshold in the semileptonic muon channel at sqrt(s) = 10 TeV ,
Tech. Rep. CMS-PAS-TOP-09-009, CERN, 2009.
http://cds.cern.ch/record/1194523.
[265] CMS Collaboration, CMS Collaboration, Search for heavy narrow t-tbar resonances
in muon-plus-jets final states with the CMS detector , Tech. Rep.
CMS-PAS-EXO-09-008, CERN, 2009. http://cds.cern.ch/record/1194498.
[266] ATLAS Collaboration, ATLAS Collaboration, New ATLAS event generator tunes to
2010 data, Tech. Rep. ATL-PHYS-PUB-2011-008, CERN, 2011.
http://cds.cern.ch/record/1345343.
[267] ATLAS Collaboration, Studies of Sensitivity to New Dilepton and Ditop Resonances
with an Upgraded ATLAS Detector at a High-Luminosity LHC , Tech. Rep.
ATL-PHYS-PUB-2013-003, CERN, 2013.
http://cdsweb.cern.ch/record/1516108.
[268] ATLAS Collaboration, ATLAS Collaboration, Physics at a High-Luminosity LHC
with ATLAS , arXiv:1307.7292 [hep-ex].
[269] CMS Collaboration, CMS Collaboration, Projected Performance of an Upgraded
CMS Detector at the LHC and HL-LHC: Contribution to the Snowmass Process,
arXiv:1307.7135.
[270] Top Quark Working Group Collaboration, K. Agashe et al., Working Group Report:
Top Quark , arXiv:1311.2028 [hep-ph].
[271] I. Iashvili, S. Jain, A. Kharchilava, and H. B. Prosper, Discovery potential for heavy
tt resonances in dilepton+jets final states in pp collisions at ps= 14 TeV ,
arXiv:1309.7684 [hep-ex].
[272] UTfit Collaboration, M. Bona et al., Model-independent constraints on ∆F = 2
operators and the scale of new physics, JHEP 0803 (2008) 049, arXiv:0707.0636
[hep-ph].
[273] H.-L. Lai, J. Huston, S. Mrenna, P. Nadolsky, D. Stump, et al., Parton Distributions
for Event Generators, JHEP 1004 (2010) 035, arXiv:0910.4183 [hep-ph].
[274] ATLAS Collaboration, A search for tt¯ resonances
in the lepton plus jets final state
√
1
with ATLAS using 14 fb of pp collisions at s = 8 TeV , Tech. Rep.
ATLAS-CONF-2013-052, CERN, 2013. http://cds.cern.ch/record/1547568.
256

[275] CMS Collaboration, S. Chatrchyan et al., Search
for Z ’ resonances decaying to tt¯ in
√
dilepton+jets final states in pp collisions at s = 7 TeV , Phys.Rev. D87 (2013)
072002, arXiv:1211.3338 [hep-ex].
[276] K. Agashe, O. Antipin, M. Backovic, A. Effron, A. Emerman, et al., Warped Extra
Dimensional Benchmarks for Snowmass 2013 , arXiv:1309.7847 [hep-ph].
[277] CMS Collaboration, CMS Collaboration, Search for Heavy Resonances Decaying into
bb and bg Final States in pp Collisions at sqrt(s) = 8 TeV , Tech. Rep.
CMS-PAS-EXO-12-023, CERN, 2012. http://cds.cern.ch/record/1542405.
[278] M. Czakon, P. Fiedler, and A. Mitov, Resolving the Tevatron top quark
forward-backward asymmetry puzzle, arXiv:1411.3007 [hep-ph].
[279] D0 Collaboration, V. M. Abazov et al., Measurement of the forward-backward
asymmetry in top quark-antiquark production in ppbar collisions using the lepton+jets
channel , Phys.Rev. D90 (2014) no. 7, 072011, arXiv:1405.0421 [hep-ex].
[280] V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak, and L. L. Yang, The top-pair
forward-backward asymmetry beyond NLO, Phys.Rev. D84 (2011) 074004,
arXiv:1106.6051 [hep-ph].
[281] W. Hollik and D. Pagani, The electroweak contribution to the top quark
forward-backward asymmetry at the Tevatron, Phys.Rev. D84 (2011) 093003,
arXiv:1107.2606 [hep-ph].
[282] CMS Collaboration, S. Chatrchyan et al., Measurement √
of the charge asymmetry in
top-quark pair production in proton-proton collisions at s = 7 TeV , Phys.Lett.
B709 (2012) 28–49, arXiv:1112.5100 [hep-ex].
[283] ATLAS Collaboration, G. Aad et al., Measurement
of the charge asymmetry in top
√
quark pair production in pp collisions at s = 7 TeV using the ATLAS detector ,
Eur.Phys.J. C72 (2012) 2039, arXiv:1203.4211 [hep-ex].
[284] P. Ferrario and G. Rodrigo, Massive color-octet bosons and the charge asymmetries of
top quarks at hadron colliders, Phys.Rev. D78 (2008) 094018, arXiv:0809.3354
[hep-ph].
[285] J. Alwall, P. Demin, S. de Visscher, R. Frederix, M. Herquet, et al.,
MadGraph/MadEvent v4: The New Web Generation, JHEP 0709 (2007) 028,
arXiv:0706.2334 [hep-ph].
257

[286] T. Sjostrand, S. Mrenna, and P. Z. Skands, PYTHIA 6.4 Physics and Manual , JHEP
0605 (2006) 026, arXiv:hep-ph/0603175 [hep-ph].
[287] J. Conway et al., PGS 4 Pretty Good Simulation of high energy collisions,
http://www.physics.ucdavis.edu/ conway/research/software/pgs/pgs4-general.htm,
June, 2012.
[288] CDF Collaboration, Measurement of the Inclusive Forward-Backward Asymmetry
and its Rapidity Dependence Af b (|∆y|) of tt Production in 5.3/fb of Tevatron Data,
Tech. Rep. Conf. Note 10224, Fermilab, 2010.
http://www-cdf.fnal.gov/physics/new/top/2010/tprop/Afb/cdfnote_10224_
public_v02_for_10185_ttbarAfbDeltay.pdf.
[289] D0 Collaboration, Measurement of the forward-backward production asymmetry of t
and tbar quarks in ppbar to ttbar events, Tech. Rep. D0-Note 6062-CONF, Fermilab,
2010.
http://www-d0.fnal.gov/Run2Physics/WWW/results/prelim/TOP/T90/T90.pdf.
[290] A. Atre, G. Azuelos, M. Carena, T. Han, E. Ozcan, et al., Model-Independent
Searches for New Quarks at the LHC , JHEP 1108 (2011) 080, arXiv:1102.1987
[hep-ph].
[291] E. Fermi, An attempt of a theory of beta radiation. 1., Z.Phys. 88 (1934) 161–177.
[292] R. Feynman and M. Gell-Mann, Theory of Fermi interaction, Phys.Rev. 109 (1958)
193–198.
[293] E. Sudarshan and R. Marshak, Chirality invariance and the universal Fermi
interaction, Phys.Rev. 109 (1958) 1860–1860.
[294] R. S. Chivukula and H. Georgi, Large N and vacuum alignment in topcolor models,
Phys.Rev. D58 (1998) 075004, arXiv:hep-ph/9805478 [hep-ph].
[295] H. Pagels and S. Stokar, The Pion Decay Constant, Electromagnetic Form-Factor
and Quark Electromagnetic Selfenergy in QCD, Phys.Rev. D20 (1979) 2947.
[296] S. Weinberg, Phenomenological Lagrangians, Physica A96 (1979) 327.
[297] A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model ,
Nucl.Phys. B234 (1984) 189.

258

[298] O. Deschamps, S. Descotes-Genon, S. Monteil, V. Niess, S. T’Jampens, et al., The
Two Higgs Doublet of Type II facing flavour physics data, Phys.Rev. D82 (2010)
073012, arXiv:0907.5135 [hep-ph].
[299] LHC Higgs Cross Section Working Group Collaboration, S. Heinemeyer et al.,
Handbook of LHC Higgs Cross Sections: 3. Higgs Properties, arXiv:1307.1347
[hep-ph].
[300] Particle Data Group Collaboration, J. Beringer et al., Review of Particle Physics
(RPP), Phys.Rev. D86 (2012) 010001.

259