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1/98 WWW.“

SIGNALS FOR THE ELECTROWEAK SYMMETRY BREAKING ASSOCIATED
WITH THE TOP QUARK

By

Timothy Maurice Paul Tait

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1999

ABSTRACT

SIGNALS FOR THE ELECTROWEAK SYMMETRY BREAKING ASSOCIATED
WITH THE TOP QUARK

By

Timothy Maurice Paul Tait

The mechanism of the electroweak symmetry-breaking (EWSB) is studied in the
context of the heavy top quark, whose large mass may provide a clue as to the
mechanism which generates the mass of the Wi and Z bosons. As a result, it seems
quite likely that the top quark may be special in the sense that it is involved in
dynamics not experienced by the light fermions. Examples of this include models
such as the top-condensate model in which a bound state of top quarks condenses,
generating both the top mass and the gauge boson masses, and supersymmetric
models in which the large top Yukawa coupling naturally explains the EWSB by
radiatively driving the squared mass of a scalar particle (which is positive at a large
energy scale) negative at low energies. Specific collider signatures of the third family
result from such scenarios, and can be used to test the hypothesis that the top plays
a role in the EWSB. In particular, single top production, as a measure of the top’s
weak interactions, provides an excellent probe of nonstandard top quark properties.
The physics of single top production at hadron colliders is carefully studied, with a
particular eye towards what can be learned from single top, including the signs of

new physics that may show up in the single top rate.

This dissertation is dedicated to those who have contributed indirectly to its
existence by providing much-needed support and encouragement. Among many
others, this includes my grand-parents, Maurice and Jean Jones, and Mimi, who
taught me to see the beauty inherent in the world; and my Parents, Susan and

Peter Tait, and Rex and Maureen Daysh, who taught me how to live in it. Finally,
this work is for Simona; you are the light of my life. Each day you are in my life

makes it brighter, and in the end it is this that makes it worth living.

iii

ACKNOWLEDGEMENTS

It is a pleasure to acknowledge those people who have helped me grow into the
physicist I am today. Foremost is my thesis advisor, C.—P. Yuan, who has shared with
me many of his ideas on which to pursue research, and encouraged me to learn inter-
esting and invaluable ideas, techniques, and skills. Ed Berger at Argonne National
Lab has also provided invaluable guidance in research, encouragement to pursue tech-
nically challenging calculations, and has supported me financially when the means to
do so did not exist at Michigan State University. I have further benefitted from the
exposure to high energy physicists which have given me very important information,
advice, and a context in which to fit my own work. At Michigan State University
this includes Carl Schmidt, Wu-Ki Tung, Wayne Repko, Dan Stump, Jon Pumplin,
Maris Abolins, Raymond Brock, and Joey Huston. At Argonne, it includes Cosmas
Zachos, Alan White, Geoff Bodwin, Don Sinclair, and Eve Kovacs. All have had a
large impact on my research. I am also grateful to my committee members Vladimir

Zelevinsky and William Hartmann for useful guidance.

I am further grateful to Michael Klasen, Stephen Mrenna, Francisco Larios, Lorenzo
Diaz-Cruz, Ehab Malkawi, Hong-Jian He, and Csaba Balazs, my collaborators, for
fun and interesting work, and to my colleagues in physics, Simona Murgia, Phil Tsai,
Andy Lloyd, Hal Widlansky, Doug Carlson, Mike Wiest, Xiaoning Wang, Hung-
Liang Lai, David Bowser-Chao, Kate Frame, Jim Amundson, Glenn Ladinsky, Chris
Glosser, Pavel Nadolsky, Rocio Vilar, Gervasio Gomez, Miguel Mostafa, Juan Valls,
Andrea Petrelli, Lionel Gordon, Jean-Francios Lagae, Carmine Pagliarone, Zack Sul-

livan, Brian Harris, and Gordon Chalmers for many hours of interesting discussions.

iv

Contents

LIST OF TABLES vii
LIST OF FIGURES viii
1 Introduction : The Standard Model 1
1.1 Yang-Mills Gauge Theory ........................ 2

1.2 The Standard Model ........................... 4
1.2.1 Spin 1 Gauge Boson Fields .................... 4

1.2.2 Spin fi- Matter Fields ....................... 5

1.2.3 Masses and the Higgs Mechanism ................ A 8

1.2.4 Fermion Masses and the CKM Matrix ............. 12

1.3 Theoretical Puzzles of the Standard Model .............. 15
1.3.1 General Considerations ...................... 15

1.3.2 The Electroweak Symmetry-Breaking .............. 17

1.3.3 The Fermion Mass Hierarchy .................. 21

1.3.4 The Electroweak Chiral Lagrangian ............... 22

1.3.5 Final Remarks .......................... 25

2 Single Top Production 26
2.1 Top Quark Properties at a Hadron Collider ............... 27
2.2 Single Top Production in the SM .................... 32
2.2.1 W‘ Production .......................... 33

2.2.2 W-gluon Fusion .......................... 38

2.2.3 tW‘ Production ......................... 43

2.3 New Physics in Single Top Production ................. 50
2.3.1 Additional Nonstandard Particles ................ 50

2.3.2 Modified Top Quark Interactions ................ 62

2.4 Top Polarization ............................. 71

2.4.1 The W+ Polarization: The W-t-b Interaction .......... 71

2.4.2 The Top Polarization ....................... 72

2.4.3 New Physics and Top Polarization ................ 76

2.5 Top Quark Properties ........................... 77

3 Higgs with Enhanced Yukawa Coupling to Bottom 81
3.1 Introduction ................................ 81
3.2 Signal and Background .......................... 82
3.3 Implications for Models of Dynamical EWSB .............. 95
3.3.1 The Two Higgs Doublet Extension of the BHL Model ..... 96

3.3.2 Top-color Assisted Technicolor .................. 97

3.4 Implications for Supersymmetric Models ................ 100
3.5 Conclusions ................................ 103

4 Associated Production of Gauginos with Gluinos at NLO 105
4.1 Leading Order Cross Sections ...................... 107
4.2 Next-to—Leading Order Corrections ................... 109
4.2.1 Virtual Loop Corrections ..................... 109

4.2.2 Real Emission Corrections .................... 111

4.3 NLO Inclusive Cross Sections ...................... 112
4.4 Summary ................................. 116

5 Conclusions 118
LIST OF REFERENCES 120

vi

List of Tables

1.1
1.2
1.3

2.1

2.2
2.3
2.4

2.5
2.6
2.7

2.8

3.1

3.2

3.3

Vector Boson Masses ...........................
Lepton and Quark Masses ........................
Quantum Numbers of the Fermions ...................

The NLO rates of qtj' ——> W‘ —+ t5 (in pb) at the Tevatron Run 11. At

the Tevatron, the rate of 5 production is equal to the t production rate. 35

The NLO rates of qq' —> W“ ——> t5 (in pb) at the LHC. ........
The NLO rates of qrj’ —+ W’ —+ bf (in pb) at the LHC. ........
The NLO rates of bq_—> tq’ (in pb) at the Tevatron Run II. At the

Tevatron, the rate of t production is equal to the rate of t production.

The NLO rates of bq —> tq’ (in pb) at the LHC. ............
The NLO rates of Eq —> t q’ (in pb) at the LHC. ............

The LO (with 0(1/ log mf/mfi) corrections) rates of b g —) tW‘ (in
pb) at the Tevatron Run II. The rate of t production is equal to the
rate of t production. ...........................

The L0 (with 0(1/ log(mf/m§) corrections) rates for by -+ tW" (in
pb) at the LHC. The rate of t production is equal to the rate of t
production. ................................

The signal and background events for 2 fb“lof Tevatron data, assuming
m4, = 100 GeV, 2Am¢ = 26 GeV, and K = 40 after imposing the
acceptance cuts, pr cuts, and reconstructed m¢ cuts described in the
text. (A k-factor of 2 is included in both the signal and the background
rates.) ...................................

The signal and background events for 2 fb‘lof Tevatron data, assuming
rm, 2 100 GeV, 2Am¢ = 26 GeV, and K = 40 for two or more, three
or more, or four b-tags, and the resulting significance of the signal.

Event numbers of signal (NS), for one Higgs boson, and background

(NB) for a 2 fb-lof Tevatron data and a 100 fb’lof LHC data, for
various values of m¢, after applying the cuts described in the text,
and requiring 4 b—tags. An enhancement of K = 40 is assumed for
the signal, though the numbers may be simply scaled for any KMW by
multiplying by (Knew/40)2 .........................

vii

36
37

40
41
42

48

49

87

91

92

List of Figures

1.1

1.2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

A Feynman diagram illustrating quantum corrections to the scalar
mass coming from a self-interaction ....................

Feynman diagrams illustrating quantum corrections to the fermion
mass coming from interactions with a scalar or a vector boson .....

Representative Feynman diagrams showing QCD production of t f pairs:

qrj,gg—)tt- .................................

Feynman diagram showing the top decay into W+ b, including the lep-
tonic decay W+ -—> 6+ 11,, ..........................

The SM rate of the three modes of single top production, as a function
of m, (summing the rates of t and { production), at the Tevatron
(upper figure) and LHC (lower figure). The solid curve is the NLO
t-channel rate, taken as the average of the results from CTEQ4M and
MRRS(R1) PDF’s. The dashed curve is the NLO s-channel rate, taken
as the average of the results from CTEQ4M and MRRS(R1) PDF’s.
The dotted curve is the LO tW‘ rate, including large log corrections,
taken as the average of the CTEQ4L and MRRS(R1) results ......

Feynman diagram for the s-channel mode of single top production:
qq’ —-> W‘ -—> tb. .............................

Feynman diagrams for the t-channel mode of single top production:
bq -> tq’. A second process in which the incoming light quark is
switched with a light (7 is also possible. .................

Feynman diagrams for the tW’ mode of single top production: 9 b —>
t W‘ .....................................

Representative Feynman diagrams for corrections to the tW‘ mode of
single top production corresponding to (a) large log corrections asso-
ciated with the_b PDF and (b) LO tt production followed by the LO
decay t- —> W“ b. .............................

Feynman diagram for s-channel production of a single top and a b’:

qi—Hb’. .................................

The NLO rates (in pb) for the process qq’ —+ W‘ —> tb’ for various b’
masses at the Tevatron (solid curve) and LHC (dashed curve), assum-
ing Vt! = 1. At the Tevatron, the rates of qq’ —-> W" -> f b’ is equal to
the tb’ rate. The t b’ rate at the LHC is shown as the dotted curve.

viii

18

19

27

28

31

33

38

43

44

53

55

2.10

2.11

2.12

2.13

2.14

2.15

2.16

2.17

2.18

2.19

2.20

2.21

3.1

3.2

Feynman diagrams illustrating how a W_’ boson can contribute to single
top production through qt? —-> W’ —> tb. ................ 56

The NLO rate of qq’ —> W, W’ —+ tb (in pb) at the Tevatron (lower
curves) and LHC (upper curves), for the top-flavor model with sin2 (b =
0.05 (solid curves) and sin2 ()5 = 0.25 (dashed curves), as a function of
M z: 2 MW. The Tevatron cross sections are multiplied by a factor
of 10. At the Tevatron, the f production rate is equal to the t rate.
At the LHC the 5 rates are shown for sin2 (b = 0.05 (dotted curve) and

sin2 43 = 0.25 (dot-dashed curve). .................... 58
Feynman diagram illustrating how_a charged top-pion can contribute
to single top production through cb —-) 1r+ —> tb ............. 59

The L0 rate of single top production through the reaction cb —+ 7r+ —>
tb as a function of Mfli, assuming a tR-cR mixing of 20%. These rates
include t and 5 production, which are equal for both Tevatron and LHC. 61

Feynman diagrams for associated production of a neutral scalar and
single top quark: qb —> q’ t h ........................ 62

Feynman diagrams showing FCNC top decays through (a) t ——> Z c,
(b)t—>'yc, and (c)t—+gc. ....................... 65

Feynman diagram showing how a FCNC Z-t-c interaction contributes
to the s—channel mode of single top production through qq -) Z ‘ -+ t5. 66

Feynman diagrams showing how a FCNC Z -t-c interaction contributes
to the exotic mode of single top production gc —* t Z .......... 67

Feynman diagram showing how a FCNC Z-t-c interaction contributes
to the t-channel mode of single top production through cq ——> tq. . . . 68

The correlation between the maximum cross section of (161' —> t6, ate,
and the minimum BR(t —> W b) assuming the t-c—g operator is the only
source of nonstandard physics in top decays, bmin ............ 69

A diagram indicating schematically the correlation between the charged
lepton (6*) from a top decay, and the top spin, in the top rest frame.
The arrows on the lines indicate the preferred direction of the momen-
tum in the top rest frame, while the large arrows alongside the lines
indicate the preferred direction of polarization. As shown, the 8+ (6‘)
from a t (f) decay prefers to travel along (against) the direction of the
t (f) polarization. ............................. 73

The location of the Tevatron SM point (the diamond) in the arc;
plane, and the 30 deviation curve. Also shown are the points for the
top-flavor model (with M '2 = 900 GeV and sin2 (1) = 0.05) as the square,
the FCNC Z-t-c vertex (xi = 0.29) as the circle, and a model with a
charged top-pion (mfli = 250 GeV and t R—cR mixing of ~ 20%) as the
cross. All cross sections sum the t and 5 rates. ............. 79

Representative leading order Feynman diagrams for ¢bb production at

a hadron collider. The decay d) —+ bb is not shown ............ 83
Representative Feynman diagrams for leading order Z bb production at
a hadron collider. The decay Z -> bb is not shown. .......... 83

ix

3.3

3.4

3.5

3.6

3.7

4.1

4.2

Representative leading order Feynman diagrams for QCD bbbb produc-
tion at a hadron collider ..........................

Representative leading order Feynman diagrams for QCD bbj j produc-
tion at a hadron collider ..........................

In the upper figure is the model-independent minimum enhancement
factor, Km, excluded at 95% CL. as a function of scalar mass (m¢) for

the Tevatron Run II with 2 fb_1(solid curve), 10 fb—1(dashed curve)

and 30 fb‘1(dotted curve). The lower figure shows the same factor,
Kmin, excluded at 95% CL. (solid curve) and discovered at 50 (dashed

curve) as a function of m, for the LHC with 100 fb‘1 ..........

The reach of the Tevatron and LHC for the models of (a) 2HDE and
(b) TCATC. Regions below the curves can be excluded at 95%C.L.
In (b), the straight lines indicate 31:01 = m.) for typical values of the
top-color breaking scale, A. yb is predicted to be very close to 11,.

The regions above the curves in the tan fl-mA plane can be probed at
the Tevatron and LHC with a 95% CL. The soft breaking parameters
correspond to the LEP Scan A2 set. The region below the solid line
will be covered by LEP II. ........................

Total hadronic cross sections for the associated production of gluinos
and gauginos at Run II of the Tevatron. NLO results are shown as
solid curves, and LO results as dashed curves. We vary the SUGRA
scenario as a function of mm 6 [100; 400] GeV and provide the cross
sections as a function of the physical gluino mass mg. The chargino
cross sections are summed over positive and negative chargino rates. .

Total hadronic cross sections for the associated production of gluinos
and gauginos at the LHC. NLO results are shown as solid curves, and
LO results as dashed curves. We vary the SUGRA scenario as a func-
tion of mm 6 [100; 400] GeV and provide the cross sections as a func-
tion of the physical gluino mass mg. The chargino cross sections are
summed over positive and negative chargino rates ............

83

84

94

98

102

114

115

Chapter 1

Introduction : The Standard
Model

The Standard Model of Particle Physics [1, 2] (SM) contains, in principle, a com-
plete description of all of the phenomena currently observed in high energy physics
experiments, including the strong and electroweak interactions. However, as will be
explained below, the model contains a number of theoretical puzzles which indicate
that it is not the “ultimate” theory, but instead should be replaced by some more
fundamental theory at higher energy scales. In fact, the striking success of the SM
at explaining the currently available data places strong constraints on the nature of
any theory that hopes to extend or supplant it at higher energies. As this work is
an examination of several such models, we will begin with a presentation of the SM,
examining its strengths and short-comings, in order to better understand these theo-
ries which hope to replace it. We shall see that the one of the great mysteries of the
SM is the mechanism for the electroweak symmetry breaking (EWSB) that provides
masses for both the weak bosons and the fermions. Because of their large masses, the
fermions of the third family, and the top quark in particular, provide a natural place
to explore hypotheses concerning the EWSB. In the succeeding chapters we examine

specific ways in which experiments at supercolliders can study the possibility of a

connection between the EWSB and heavy top through single top production, pro-
duction of scalars in association with bottom quarks, and supersymmetric particle

production.

The SM is a quantum field theory of fermionic matter particles interacting with
bosonic vector particles. The interactions are fixed by requiring the theory to be
locally gauge invariant under transformations in the group SU(3)C x SU(2)L x U(1)Y.
It is quite remarkable that this condition is enough to uniquely fix the structure of

the renormalizable interactions between fermions and vector particles.

1.1 Yang-Mills Gauge Theory

We begin with a brief presentation of the construction of a Lagrangian invariant
under local gauge transformations, as these ideas form the basic building blocks of
the SM. In the Yang-Mills gauge theory [3] invariant under Lie group 9 with N group
generators T“ (a = 1..N), we can express the generators as Hermitian matrices with

commutators,
[T“,T”] = i fabc Tc, (1.1)

where the f“"‘ are the structure constants of g. An element of the local gauge

transformation acting on a set of Dirac fermions may be expressed as
‘Il(a:) —) 6‘0”“)? ‘Il(:r), (1.2)

where the real function a°(:r) is the local transformation parameter. Clearly the
usual free field Lagrangian density for a set of Dirac spinors is not invariant under
this transformation, because the transformation of 6,,\Il(z) will generate a term in

which the derivative acts on a“(a:). This is remedied by introducing a covariant

derivative,
D“ = 6,, + i g T“ A:(:r), (1.3)

which insures that Dfll transforms like \II under the gauge group provided that the

real vector field A;(:::) transforms according to,
T“ Aflx) —+ ei°“<$>T“ {Tb A2(x) — $6,} (eia°<$>"‘)*. (1.4)
This allows us to write down the gauge-invariant kinetic terms for the Dirac fermion,
£FK =iV7"Dy\II—m_\Il-\II, (1.5)

where for brevity we no longer explicitly write the fields as functions of space-time.
The presence of the covariant derivative, dictated by the gauge invariance, has thus
forced us to include an interaction between the fermion fields \II and the vector fields
A“. It should be noted that a four-component Dirac spinor may be written in terms of
a left-handed and a right-handed two-component Weyl spinor [4]. One can formulate
a theory of massless fermions in two-component form, which proceeds much as it is

described above, but with no mass term in £12K.

In order for the vector field to be dynamical it must also have kinetic terms in the

Lagrangian. It is easy to verify that,
1:0,, = —%F““” F“,,., (1.6)
with,
Fat = apA: — 6qu — g f“ A2. A: (1.7)

will servel, and itself respects gauge invariance, with g the same coupling that appears

in the covariant derivative for the fermion. In the case in which g is a non-Abelian

 

1A term of the form gF““”F‘:, with 17“?” = 1/26uyapFaafi is also gauge invariant, and could be
included in EGK. For an Abelian youp g this term corresponds to a total derivative, and thus does
not contribute to the dynamics. In the case of a non-Abelian group this term is related to the CP
properties of the theory. For simplicity we will not consider such a term here.

group (and thus the structure constants do not vanish), this term will contain cubic
and quadratic interactions of the gauge field, in addition to the kinetic terms. It is
important to note that Lax does not contain a mass term for the vector field. In
fact, such a term is forbidden by gauge invariance, and thus the vector fields are
necessarily massless as a result of the gauge symmetry. This theory may now be
quantized by employing, i.e., the Faddeev-Popov formalism [5] to quantize only the

physical degrees of freedom.

In addition to fixing the form of the renormalizable interactions, the gauge sym-
metry plays a further role in the construction of a model in that it corresponds
to conserved currents as predicted by Noether’s theorem2 [6], and implies relations
among the Green’s functions known as Ward-Takahashi identities [7]. The Ward iden-
tities, and thus the gauge symmetry itself, play an important role in the proof of the
decoupling of the ghost states from physical amplitudes [8], and in proving the uni-
tarity and renormalizability [7, 8, 9] of the Yang-Mills theory. The gauge symmetry

is therefor seen as an essential ingredient for a theory of vector particles.

1.2 The Standard Model

1.2.1 Spin 1 Gauge Boson Fields

Having briefly reviewed the general Yang-Mills theory of a set of fermions interacting
with gauge bosons as is dictated by local invariance under a symmetry group g, we
now specify to the SM, with symmetry group g = SU(3)C x SU(2)L x U(1)Y. The
SU(3)C gauge symmetry corresponds to the strong interaction of quantum chromody-

namics (QCD). Its eight gauge bosons G2 are known as gluons. The SU(2)L x U(1)y

 

2Noether’s theorem guarantees that a continuous symmetry corresponds to a conserved current
at the classical level. Some symmetries, known as anomalous symmetries, are broken by quantum
effects. The requirement that a desired classical symmetry survives quantization can provide non-
trivial constraints on the theory.

sector contains the combined electromagnetic and weak interactions, generally re
ferred to as the electroweak symmetry. The three SU(2)L bosons are denoted W;
and couple to the weak iso-spin, whereas the U(1)y gauge boson, Bu couples to by-
percharge. Since the gauge bosons of one of the symmetry subgroups in 9 do not
transform under the other gauge symmetries in the product of groups, the gauge

kinetic term may be simply written as a sum of the individual gauge kinetic terms,
L — 13 8"” 1w‘ WW 10° G““"
where,

B)“, = (ha—8,13,, (1.9)
W2", = aflwj—aVWg—gzéikwg wf,

Gan" = (LG: - BVGZ “ 93 fabc G]: G2:

with g,- the gauge couplings, and Eijk and f“"6 the structure constants for SU(2) and

SU(3), respectively.

As will be explained in detail below, the electroweak symmetry is spontaneously
broken, resulting in mixing between the B“ and W3 fields, and non-zero masses for
three of the gauge bosons (Wi and Z“). The photon (A) remains massless, due
to a residual U(1)EM gauge symmetry that remains unbroken. The physical (mass-

eigenstate) gauge bosons and their masses are shown in Table 1.1.

1-2.2 Spin % Matter Fields

The SM contains three families (also called generations) of spin % matter fields, in
the fundamental representation of the gauge groups. Each family is a “copy” of the
0ther families with respect to gauge quantum numbers, but have diverse masses. Each

generation contains a charged and a neutral lepton, which interact electroweakly, and

Table 1.1: Vector Boson Masses

Particle Symbol Mass (GeV)

Photon A 0 Electromagnetic Force
W Boson Wi 80.33 Charged Weak Force
Z Boson Z 0 91.187 Neutral Weak Force
Gluon G“ 0 Strong Force

 

an up-type and a down-type quark, which interact both electroweakly and with the

gluons. A list3 of the fermions, including their masses, is presented in Table 1.2.

In Table 1.3 can be found the transformation properties of the fermions of the
first family under the gauge groups. Since the second and third families are copies
of the first family as far as the quantum number assignment is concerned, only the
first family is presented. The left- and right-chiral fermions have different quantum

numbers, with the left-chiral fields arranged in doublets,

LL = (:8); QL = (3)1: (1'10)

and the right-chiral fermions are in singlets,
83, Hg, (13. (1.11)

The gauge invariance under SU(2)L thus forbids the presence of a mass term for the
fermions. As we will see below, in the SM, fermion masses are generated by the same

spontaneous symmetry-breaking Higgs mechanism that provides mass for the weak

 

3It is worth mentioning that the association of a particular doublet of leptons with a particular
doublet of quarks in order to form a generation is arbitrary in the SM, because there are no local
interactions between quarks and leptons. The SM makes this identification in a natural way by iden-
tifying the doublet containing the heaviest charged lepton with the doublet containing the heaviest
quarks (and so on), but one could in principle associate any quark doublet with any lepton doublet
and call that a family.

Table 1.2: Lepton and Quark Masses

 

Particle Symbol Mass (GeV)

Electron neutrino Va 0

Electron 6 0.00051 First
Up quark 21 0.002 to 0.008 Generation
Down quark d 0.005 to 0.015

Muon neutrino u), 0

Muon p 0.106 Second
Charm quark c 1.0 to 1.6 Generation
Strange quark s 0.1 to 0.3

Tau neutrino u, 0

Tau T 1.78 Third
Top quark t 175 Generation
Bottom quark b 4.1 to 4.5

 

 

bosons. There is no right-handed neutrino in the SM, and thus the neutrino is a mass-
less Dirac field. With respect to the color gauge group of quantum chromodynamics

the quark fields are arranged in triplets,

Qr
q: (‘19): (1.12)
4b

where we have used the common convention of referring to the SU(3)C indices as red

(1'), green (9), and blue (b).
The gauge invariant kinetic Lagrangian for a particular fermion, ‘II, is given in

Equation 1.5 with the covariant derivative given by,

. Y . - - . a a
D],=0p+291§Bp+z9271Wfl+zg3A G“, (1.13)

with Y the hypercharge of the fermion, and Tj and A“ the generators of SU(2) and

Table 1.3: Quantum Numbers of the Fermions

Chirality Q T}; Y C

V“, 0 1/2 -1 0
8L -1 -1/2 -1 0
in, 2/3 1/2 1/3 r,g,b
dL -1/3 -1/2 1/3 r,g,b
CR -1 0 -2 0
113 2/3 0 4/3 r,g,b

d3 -1/3 0 -2/3 r,g,b

 

SU(3) in the representation appropriate for \II. The hypercharge has been normalized

such that the electric charge of the fermion is given by Q = T2 + %.

1.2.3 Masses and the Higgs Mechanism

As we have seen, the gauge symmetries of the SM forbid explicit masses for vector
bosons (as is true for any gauge theory) and fermions (as is true in the case of the SM,
in which left- and right-chiral fermions transform differently). In order to describe the
world seen in particle physics experiments, these objects must acquire masses. This
may be resolved by introducing a spontaneous breaking of the electroweak symme-
try, through the Higgs Mechanism [10]. The spontaneous symmetry-breaking (SSB)
occurs when the Lagrangian is invariant under the gauge transformations, but the
vacuum state does not respect the symmetry. In the SM this is accomplished by

introducing a weak iso-spin doublet of complex scalar fields, the Higgs doublet. This

doublet carries hypercharge +1, and thus can be expressed as,

,, = «Limfm z (Way (035)). (1.14)

where the second form (which displays the space-time dependence of the four fields Bi
and a explicitly for clarity) illustrates an interesting property of the Higgs doublet,
which can be seen by noting that under a SU(2)L x U(1)y gauge transformation, the

doublet transforms as,
<I> —+ a“? e‘°‘<x)7‘ <I>, (1.15)

and thus comparing Equations 1.14 and 1.15 indicates that provided the expectation
value of a is non-zero (which indicates that SSB has occurred), one may choose
a particular gauge in which three of the four real degrees of freedom of the Higgs
doublet vanish. Since on the one hand it is possible to “gauge away” these fields,
while on the other physical quantities are independent of the choice of gauge, this
indicates that these degrees of freedom are unphysical. As we shall see below, under
SSB these unphysical scalars reappear as the longitudinal polarizations of the weak

bosons.

The scalar field can be given gauge invariant terms in the Lagrangian,
c, = (1),,<I>)’r (D"<I>) — ”21>t <I> — A (<I>T <I>)2, (1.16)

with,

1

Dp=6p+i912

3,, + z‘g2rJ'Wg, (1.17)

where the first (kinetic) term in Equation 1.16 is required by gauge invariance, and
the remaining terms correspond to a mass-like term and a self-interaction of the <1)

field. These latter two terms together are generally referred to as the Higgs potential.

10

One can also construct gauge invariant Yukawa couplings between the doublet and

the fermions,

Me:

3
LYukawa =

Z (112m 731*.» <1" L}? + yg‘m‘ I? <I> e33) (1.18)

m-

amFWQZ‘ + yzm'TMA)

:
ll
H

+
A A“
0:

+
Q

:WWQ': + mama),
with,
<i> :17? ch", (1.19)

and the sum over n and m is over the three families of fermions.

Spontaneous symmetry-breaking is exhibited by assuming4 #2 < 0. Under these

conditions the minimum of the Higgs potential shifts (in field space) from <I> = 0 to,

(pig) = ¢¥+¢g+¢§+n2=:§J—=v2. (1.20)

The field thus acquires a non-zero vacuum expectation value (VEV). Expressing this
condition in terms of the real scalars (131"3 and 17, we see that the minimization condi-
tion allows for any of these (or a combination of them) to carry the VEV. Choosing

< 17 >= 1), we expand about 1),

¢+
<1> = (1%”) , (1.21)

with «15“ = ($1 +i (1)2) / J5 and n = v+h. Inserting this into Equation 1.16, we see after
some algebra (which can be simplified by working in the unitary gauge, (1)1"3 = 0)

that tree-level mass terms for the h field and gauge bosons are present,

 

‘It is important to note that A must be positive in Equation 1.16 in order for the theory to possess
a stable vacuum.

11

 

2 2
LMB = — (2);” ) ’12 + (222—2)) w+fl W; (1.22)

+ (”Z-)2 (92W: — 913..) (92W3" — 918") ,
along with many other interaction terms. The fields W: are defined as W: =
(W; IF iW3)/\/§ to be electric charge eigenstates. Physically, the appearance of
mass terms for the gauge bosons after SSB can be explained by the gauge bosons
absorbing (“eating”) the unphysical would-be Goldstone bosons, ¢1"3, which serve
as the longitudinal degrees of freedom that distinguish massive from massless vector
fields. This is referred to as the electroweak symmetry-breaking and can be denoted
SU(2)L x U(1)y —+ U(1)EM, because the U(1) to which the photon corresponds re-
mains unbroken. An interesting heuristic picture for the Higgs mechanism is that the
Higgs potential generates dynamics which “fills” the vacuum with Higgs field. The
resulting masses for the vector bosons and fermions are then seen as a result of these

particles interacting with this “medium” as they move through the vacuum.

As was mentioned previously, the SSB has mixed the B” and W3 gauge bosons.

The mass eigenstates thus consist of the massive Z boson and massless photon,

Zn _ cos 0w — sin ow Wg)
(An) — (sin0w cosaw ) (Bu 1 (1.23)
with the weak mixing angle 9w given by,

tan 0w = 5 (1.24)

92

It is conventional to discuss the SM couplings in terms of the coupling of the photon
to the electron, e, the weak mixing parameter, sin2 0w, and the masses of the Z and
Higgs bosons, Mz and Mh, as opposed to the original couplings, 91, 92, {12, and A in
which the theory was formulated. From the presentation above, it should be clear how
to relate these two sets of parameters at tree—level. At higher orders in perturbation

theory, the relations depend on the renormalization scheme.

12

It is worth noting that the Wi and Z mass terms in Equation 1.22 arose from
the covariant derivative part of Equation 1.16 (the term that was fixed by gauge
invariance). This has two interesting consequences for the masses generated. The first
is that once the gauge couplings 91 and 92 are specified, the Wi and Z masses are
determined by v and the representation of (I) (this property is general for the Higgs
mechanism). The second property is specific to the particular quantum numbers

assigned to the SM Higgs doublet; the quantity,

MW

P = m (”5)

is equal to one at tree level in the SM, and thus provides a test of the SM realization

of 883 compared to other models.

1.2.4 Fermion Masses and the CKM Matrix

We saw in the previous section how the SSB provides masses for the gauge bosons. In
the SM, the same mechanism provides masses for the fermions through the Yukawa
interactions in Equation 1.18. As the values of these couplings are not fixed by the
gauge symmetry, they can be tuned to correspond to the particular fermion masses
observed in nature. This is complicated by the fact that in general the interaction
eigenstates need not be the same as the mass eigenstates because of the off-diagonal (in
family-space) interactions between fermions and the Higgs doublet in Equation 1.18.
In terms of the 3 x 3 interaction eigenstate mass matrices, Mu, Md, and Me, the

fermion mass terms can be expressed,
LMF = fin Mu 111, + an Md dz, + 53 Me eL + H.C., (1.26)

where +H.c. indicates the Hermitean conjugate of the preceeding terms. The bold-

faced fermion fields now indicate a vector containing the fields of a given type for all

13

three families,

UL dL 8L
“L = 81, , dL = 8L , e1, = ”L , (1.27)
tL bL TL

and similar notation for me, d3, and eg. To make the connection between EMF and

Equation 1.18 explicit, we present as an example the the mass matrix for up—type

quarks after SSB,

«2 1’22 ”it: 3’11: x
Mu = 75 yu ya ya ' (1'28)
1131 1132 1133

To express the fermion masses in terms of mass eigenstates, one uses the fact that
it is possible to rotate the left- and right-chiral fields among the three generations. A

general unitary rotation of the fields may be denoted,

UR —) Ru 111;, (IR -) Rd (11;, en -) Re 83, (1.29)

11L —+ Lu 11L, dL —‘> La (11., 9L —* Le 6L,

and the condition for mass eigenstates may be expressed by requiring that these
transformations diagonalize the interaction eigenstate mass matrices. Employing D,-

(with 2' = u, d, e) to indicate the diagonalized matrix, this may be written,
D,, = RI, Mu Lu, D, = R3, Md Ld, D, = R1, M. L... (1.30)

The requirement that the nine free entries in Du, Dd, and De correspond to the
fermion masses observed in nature provides some information about the Yukawa in-
teractions in Equation 1.18. The remaining information must come from studying

the effect of the quark mixing on the quark interactions with other particles.

Having transformed to mass eigenstates, it is still necessary to examine the effect

of these rotations on the interactions of the fermions with the vector and Higgs bosons.

14

From the fact that the mass terms came from the fermion interactions with the Higgs
doublet, it is clear that the interactions of h with the fermions are diagonal in the
mass eigenbasis. For the gauge bosons, it is simple to show that the left- and right-
handed pieces of the A, G, and Z coupling to fermion f are proportional to L]L, = 1
and R}Rf = 1, respectively. Thus, these interactions are the same in mass and
interaction eigenbasis. On the other hand, the W* interactions with the quarks pick
up a factor of LLLd, which allows the W:t interactions to couple up- and down-type
quarks of different generations. The lepton sector has no equivalent effect, because in
the SM the massless neutrinos have no mass diagonalization requirement, and thus

may always be rotated such that the W‘t couplings are diagonal in the generations.

Thus, the only observable matrix related to the generational rotations is V =
LLLd, the Cabibbo—Kobayashi-Maskawa (CKM) matrix [11]. By convention, the ma-
trix L1, is set to the unit matrix, and in that case V = Ld. Since in the SM only
the combination is of physical relevance, this does not result in a loss of generality
(though it should be kept in mind that for a more general model it may be important

to recall that V = LLLd is the true relation). Thus we write,

(I Vud Vus Vub d
S = Vcd Va Va, 3 , (1.31)
b Weak Vw V,, V”, b

Mass
with, in the “standard parameterization” advocated by the Particle Data Group [12],

—:'6
012613 312013 3136 ’3

_ _.6 _.
V — —S12023 — 0128238138 ’ ‘3 012623 — 3123238138 "5‘3 323013 - (1-32)
_.6 _.
312323 - 6126233136 ' ‘3 -Ci2323 - 8120238136 '6“ 023013
In this equation, c,-J- = cos 9.)- and s,-,- = sin 0.3-, with i and j labeling the generations.

613 is a complex phase that can induce C'P violating effects. While a general 3 x 3

unitary matrix has three independent phases, this parameterization has used the

15

fact that we may redefine the quark fields to include a complex phase such that
only 613 remains. In a model containing physics beyond the SM, extended fermion
interactions may allow for effects related to more of these mixing matrices than the
single combination that is the CKM matrix. For example, a model that includes
masses for the neutrinos, it may be appropriate to introduce a CKM-like matrix to

include a Wi coupling to leptons of various mixed generations.

Experimental measurements of the hadrons containing various types of quarks
provide information about all of the CKM elements except th. As we will see in
Chapter 2, Va, can be measured by studying single top production. In fact, in the SM
there are only three generations of fermions and thus the requirement that the CKM
matrix be unitary already provides strong limits that Va, be close to one. Nonetheless,
it is important to directly measure Vw, since a deviation from the SM limit on V»
would be a signal of physics beyond the Standard Model. In the very least, one could
find an indication of a fourth generation of quarks that is strongly mixed with the
third family, but almost unmixed with the first two families by measuring V“, to be

considerably smaller than unity.

1.3 Theoretical Puzzles of the Standard Model

1.3.1 General Considerations

In spite of its enormous success in explaining high energy physics phenomena, the
SM still contains a number of theoretical flaws and puzzles that lead us to believe it
should be replaced by a more fundamental theory at higher energy scales. As there
are a large number of opinions and approaches to this question, the discussion below
will necessarily be somewhat personalized and incomplete. In this section we will

discuss some general puzzles of the SM, followed by more detailed discussion of issues

16

concerning the EWSB and the fermion mass hierarchy that will be explored in the

rest of this work.

In fact it is quite obvious that the SM is “only an effective theory” because it
does not include gravity. A truly fundamental theory should explain all four of the
forces observed in nature. The SM contains a description of the electromagnetic,
weak, and strong nuclear forces, but does not address how to include gravitational
interactions within its framework. In fact, because of the negative mass dimension
of the gravitational coupling constant (1 /M%,a,w,,), simple power counting of loop
diagrams indicate that a field theory of gravity is not expected to be renormalizable.
Thus, there is no way to consistently include quantum corrections to gravitational
phenomena within a field theory of point-like objects such as the SM. Further, the
evolution of the structure of the universe under gravitational interactions is sensitive
to the cosmological constant, which is observed to be very small (or zero). Why
this constant is so small compared to typical particle physics energy scales remains a

mystery.

Even if one were to focus on only a more modest goal of a theory without grav-
ity, the SM still contains many puzzling features. For example, it includes 18 free
parameters, including the three gauge couplings 6, sin2 0W, and 93; the two Higgs
potential couplings that may be expressed as M z and Mh; and 9 fermion masses and
4 CKM mixing parameters that contain the physical information about the Higgs
Yukawa interactions with the fermions. As the Higgs boson has not been observed,
its mass is the only undetermined quantity in the 8M5. This large number of parame-

ters can itself be seen as a drawback of the SM. It would seem more attractive if there

I5Recall that in the SM Va, is fixed by the unitarity of the CKM matrix. It is worth noting that
precision measurements have become sensitive to radiative corrections involving the Higgs. Thus,
indirect constraints on M]. already exist, as well as excluded regions of M). that correspond to
predictions for signals that have not been observed at colliders [13, 14].

 

17

were a deeper symmetry or structure that could explain the origin of these seemingly

arbitrary quantities in terms of a smaller set of parameters.

A seemingly straight-forward means to accomplish this would be to invoke a larger
gauge symmetry, with the quarks and leptons put together in its multiplets. The
SM contains a separate symmetry for the strong interactions, and a mixture of two
symmetries results in the weak and electromagnetic interactions. Each symmetry has
its own coupling constant, and thus the theory still includes three separate forces.
Following a reductionist mentality, it is an attractive idea that these three forces
should be unified into one single force. This “grand unified” theory (GUT) could
then be spontaneously broken (by SSB similar to the EWSB, for example) at a high

energy scale (MGUT), resulting in the three symmetry groups we see at low energies.

However, it is still not clear how this works (if it works at all). A grand unified
theory should have one coupling constant at MGUT, which indicates that the three
couplings we see at low energies should converge at some high energy scale. By
running the SM couplings, one finds that they approximately unify at ~ 1015 GeV, but
do not quite meet at a single energy scale. Of course, in carrying out this computation
one must assume that there is no additional heavy matter between the weak scale
and the GUT scale, and so one must be careful in drawing conclusions. Even if one
were to resolve this issue, however, it would still raise the question why the grand
unified symmetry is apparently broken to SU(3)C x SU(2)L x U(1)y at MGUT ~ 1015
GeV, whereas the electroweak symmetry is broken, SU(2)L x U(1)Y —-> U(1)EM at the

weak scale 1) ~ 246 GeV. Such a large hierarchy seems unnatural.

1.3.2 The Electroweak Symmetry-Breaking

As we saw above, the SM uses a Higgs doublet to generate the EWSB. Thus, there is

a physical Higgs boson 11 remaining after generation of the Wit and Z masses. The

Figure 1.1: A Feynman diagram illustrating quantum corrections to the scalar mass
coming from a self-interaction.

Higgs boson has yet to be discovered experimentally, and thus we still lack direct
evidence that the method of SSB employed by the SM is correct. In fact, from the
discussion above it should be clear that the Higgs sector of the SM contains most of
the assumptions that went into the SM. For example, the interactions of gauge bosons
with each other and the fermions is fixed in terms of the three gauge couplings 91,
92, and 93. On the other hand, all fifteen of the remaining parameters in the SM are

related to the Higgs interactions with the fermions and with itself [15].

The Higgs boson is also a source of fine-tuning in the SM. If one computes the
quantum corrections to its mass coming from the self-interaction in the Higgs poten-
tial (from Feynman diagrams such as that shown in Figure 1.1), one finds that the

corrections have the form,
6M3 ~ ,\ A2, (1.33)

where A is a cut-off introduced to represent the high energy scale at which the SM
ceases to be a good description of nature. We have already seen that such a scale is
expected to occur at the Planck mass (though if there is new physics at lower energy
scales then it could also be lower). This indicates that whatever the mass of the Higgs
is at tree-level, quantum corrections tend to push it to the scale A. In order for the
Higgs to have mass around the weak scale, one must require an amazing degree of
cancellation between the bare Higgs mass and the quantum corrections to occur such
that we subtract two quantities of order A (possibly as high as M plane); ~ 1019 GeV)

and arrive at a difference on the order of the weak scale, 1). It is in this sense that

19

V

‘ ¥ M
r r V

Figure 1.2: Feynman diagrams illustrating quantum corrections to the fermion mass
coming from interactions with a scalar or a vector boson.

the Higgs mass requires fine-tuning. It seems quite unnatural that such a delicate

cancellation should occur between these two a priori unrelated quantities.

It is not very satisfying to simply accept that the Higgs might be an extremely
heavy particle. If that is the case, then one must ask the question why the EWSB
contains these two very different energy scales, Mh, and M z- This itself seems quite
unnatural. Further, if M). > about 1 TeV, the interaction between the longitudinal
Wi and Z bosons becomes non-perturbative [16], and the perturbative expansion
of the SM will no longer suffice to accurately compute scattering amplitudes. In
fact, precision measurements at LEP [13] and the Tevatron [14] indicate that for
consistency with the SM, the data prefers a Higgs boson lighter than a few hundred
GeV.

While on the subject of radiative corrections to particle masses, we note that the
situation is very different for fermion masses, which have quantum corrections (from

Feynman diagrams such as the ones shown in Figure 1.2),

6772f N g2 m, log ("7%) , (1.34)

where g is the coupling of the fermion to the boson in Figure 1.2. In this equation,
we see two features. The first is that the correction to m, is proportional to m f

itself“. The second occurs because the requirement that 6mf be proportional to m f

 

“This can be easily understood from the fact that a theory of fermions with no masses contains
a chiral symmetry that protects the fermion mass from acquiring quantum corrections. Introducing
a fermion mass, m I: breaks this symmetry, and since m f is now the order parameter that indicates
that the symmetry is broken, the quantum corrections are proportional to it.

20

means any function of lambda multiplying m f must be dimensionless (and because
of ultraviolet (UV) singularities in the loop integrals, divergent as A —+ 00). Thus
the correction depends only on logA and it is clear that the corrections to m f are

naturally of the same order as m f itself.

A further weakness of the SM coming from the Higgs sector is the problem of
“triviality”. This problem arises in any theory of a scalar field interacting with itself
via a quartic interaction. From Equation 1.16 and the discussion following it, it is
clear that such a term is vital in the SM to induce SSB. The problem can be studied
by examining the running coupling for the quartic scalar interaction. From next-to-

leading order (NLO) in perturbation theory, the running coupling may be expressed,

lo
M”) — 1- Mlogfi‘fi’

4W2

 

(1.35)

with A0 = A010) the value of the coupling at some reference energy scale. This
expression shows that for a given A010), there is a large energy scale for which the
denominator goes to zero, and thus the coupling blows up. If the SM is to remain
perturbatively valid all the way to the Planck scale, this limits the size of A010) at
the weak scale. As we saw above, A010) is related to Mh, and so this statement
can be reformulated as saying that if the SM is to remain valid to the Planck scale,
M), must be smaller than about 1 TeV. The precise M), for which the break-down
occurs also depends on contributions from the heavy top quark, and is best studied
non-perturbatively (i.e., on a lattice) because as the coupling becomes large, results
based on the perturbative expansion are not expected to be very reliable. This issue
is generally referred to as “triviality” because the only way to guarantee that the SM
is valid to an arbitrarily high energy scale is to take Mpg) —) 0, which results in a

trivial, non-interacting scalar theory.

Because of these apparent problems (triviality and fine-tuning) associated with

21

scalar fields, there are a number of proposed extensions of the SM that hope to
address these weaknesses. There are two widely considered models that fall into this
category. The first class of models, the supersymmetry (SUSY) models [17], invoke
an additional symmetry relating bosons and fermions to stabilize the Higgs sector of
the SM. Under SUSY, each field of the SM is given a partner with identical charges,
but spin differing by %. Loops of fermions appear in the quantum corrections to
the Higgs mass with a negative sign relative to the scalar contributions, and cancel
the quadratic divergences, thus removing the fine-tuning problem. In the minimal
supersymmetric standard model (MSSM) [18], the quartic Higgs interaction is related
to the gauge interactions, with a different behavior under the renormalization group

than the quartic interaction of the SM. This takes care of the triviality problem.

The second class of model can be generically referred to as dynamical symmetry
breaking models. There are many models of this kind, with the common feature that
the Higgs mechanism results not from a fundamental scalar field acquiring a VEV,
but from a composite scalar operator condensing. This composite operator may be
built from heavy fermion fields whose masses, as shown above, are not subject to
large quantum corrections. Thus, the problems with a scalar field are side-stepped by
requiring that the scalar is not fundamental. At high enough energies the low energy
effective theory in terms of a scalar particle breaks down, and one is left with a theory
without scalar particles, whose high energy behavior is thus improved. Examples of
this type of theory include technicolor models [19], top-condensate models [20], and

top-color models [21].

1.3.3 The Fermion Mass Hierarchy

Having discussed some of the issues involved in using the Higgs mechanism to generate

masses for the gauge bosons, we now examine the fermions. A deep puzzle of the SM

22

is the question as to why there exist three generations, interacting identically with
the gauge bosons, but very differently with the Higgs doublet, as can be seen by the
wide range of masses listed in Table 1.2. Outstanding issues include the questions of
why there are three families (and not some other number), why neutrinos are massless
whereas the other fermions are massive, why the top quark is so much heavier than
the other fermions, why the light fermions have masses so much smaller than D (and
masses that are so diverse from one another), and why the CKM matrix is almost

diagonal and has such a small CP violating phase.

A particular puzzle is the top quark. The top is the only quark to have a mass
on the same order as v, and thus a Yukawa interaction close to 1. From that point
of view, it seems that the top is the “natural” quark, while all of the other quarks
are odd in that their Yukawa couplings are very very small. Another point of view
is that the top quark is heavy because it is special in some way, perhaps having
been given a special role in the mechanism of EWSB (as, for example, in the top—
condensation models which provide the large top mass and the EWSB through the
same mechanism). Following this line of thought, it is very natural to study the top
quark very carefully. If the top is special in some way, then studies of top should
reveal in what way the top is special, and what that means for the EWSB. In the
very least, careful study of the top interaction with the Higgs would indicate whether
or not the mechanism that generates the top mass is identical to that which generates

the boson masses.

1.3.4 The Electroweak Chiral Lagrangian

As we have seen, the Higgs sector of the SM represents the single largest source
of our ignorance concerning particle physics : the mechanism of the EWSB. Thus,

it seems reasonable that one could expect new phenomena to appear at energies

23

not much greater than the weak scale, and thus within the range of supercollider
experiments currently underway and planned for the future. In order to search for
signals of new physics effectively, there are generally two sorts of deviations from the
SM that one could consider a sign of new physics. The first is some sort of exotic
particle beyond those predicted by the SM. The supersymmetric partners present
in a supersymmetrized SM are one example of this type of new phenomenon, and
composite scalar bound states of top quarks (or some other heavy fermion) that often
arise in dynamical EWSB models are another. Searches for particles of this type
are necessarily model-dependent, because one must specify how the “new particle”
interacts with the known ones, thus determining how it is produced, what (if anything)
it decays into, and even how it interacts with the material of a particle detector. The
second class of new phenomenon involves modified properties of the known particles
of the SM. This type of modification could be caused, for example, by quantum
effects from particles too heavy to be directly produced at colliders. This kind of
new phenomenon can be tested in a model-independent way by carefully measuring
various masses and interactions of the known particles (and being careful to avoid

“assuming the SM” in interpreting the results).

A powerful tool with which one can examine new phenomena is the electroweak
chiral Lagrangian (EWCL) [22]. The philosophy behind the EWCL is that since
we observe the masses of the W“: and Z bosons, in some sense we have already
seen the would-be Goldstone bosons [23]. Using these ingredients, one can construct
the most general effective Lagrangian that realizes the SU(2)L x U(1)Y symmetry
nonlinearly while preserving SU(3)C x U(1)EM. The result is an effective theory that is
constructed to encapsulate what is known about the presence of the gauge symmetry,
while allowing for more freedom in how the symmetry is spontaneously broken than

the particular realization of the Higgs mechanism employed in the SM. Further, such a

24

construction allows one to search for new phenomena in a model-independent fashion.
The results may then be applied to learn something about what sort of new physics
is consistent with observed data, or to confirm or rule out a given model of physics
beyond the Standard Model. Whenever possible, we will present results in the context

of the EWCL, in order to be as model-independent as possible.

As the EWCL is to be regarded as an effective theory, one generally includes
non-renormalizable interactions. Such interactions have coupling constants with di-
mensions of inverse mass, and are thus attributed to residual low energy effects from
high energy physics (which presumably are renormalizable if one were to know the
full high energy theory). Thus, by observing such effects one hopes to learn some-
thing about the scale at which these effects become important, and the details of the
full theory could be studied. An example of such an operator is a flavor-changing
neutral current (FCNC) operator which connects the top quark, the charm quark,
and the gluon. In order to respect the SU(3)C gauge symmetry, the lowest possible

mass dimension of operator is dimension 5 and may be written [24],

[.gtc = {1qu (K1 50“” A“ c + n2 f0” '75 A“ c + H.c.) , (1.36)
gtc

where 191,2 parameterize the strength of the interaction in terms of ()5 = 93 and Agtc,
which contains the mass dimension of the coupling, may be thought of as the scale
at which the effective theory breaks down. The matrix 0“” is related to the Dirac

matrices by,
V i V i V V
0“ =§[7",7]=-2-('7“7 -7 7")- (1.37)
In constructing this operator, we have followed the usual EWCL procedure of defining
the fermion fields such that they transform under SU(2) L x U(1)y the same way they

7

transform under U(1)EM . As we shall see in Chapter 2, this operator may have

 

7This may be accomplished by including an exponential of the Goldstone bosons in the definition
of the fermion field. In the Unitary gauge, this corresponds with the usual definition.

25

important implications for single top production.

1.3.5 Final Remarks

In presenting the SM, and in exposing its weaknesses, we have obtained some sense of
what a theory that hopes to improve our understanding of the electroweak symmetry-
breaking needs to accomplish. Many theories propose a wide variety of ways to ac—
complish this, and finding ways to prove or disprove these theories is one of the current
challenges for experimental high energy physics. The remainder of this dissertation is
an exploration of several classes of models, with an eye towards the question of how
we could discover whether or not these models represent a viable picture of reality. As
we have argued, the top quark is a likely place to find new phenomena because of its
huge mass. Thus, we begin by studying the process of single top production, which is
expected to provide us with the first real understanding of the top’s weak interactions.
We will employ a mixture of model-independent tools (such as the EWCL) as well as
predictions of specific models to show that Run II of the Fermilab Tevatron and the
CERN Large Hadron Collider (LHC) represent a wealth of information about the top

quark itself, and thus most likely about the EWSB as well.

Chapter 2

Single Top Production

As we saw in Chapter 1, the SM suffers from a number of weaknesses that are in
one way or another related to the mechanism of the EWSB, both in the generation
of the gauge boson masses and the fermion masses. The attractive idea that the top
quark may play a special role in the EWSB was introduced. The definitive test of
this hypothesis must come from studying the properties of the top quark. Careful
measurement will reveal if it is indeed a SM top, or something different. Indeed, if
signals of something beyond the SM exist in top quark observables, careful study of
them will provide a means to determine what properties the more fundamental theory

must possess in order to explain the observed deviation.

The question of how to discover physics beyond the SM related to the top quark
reduces to the question of how the top’s properties may be determined in a model-
independent fashion, and without making strong assumptions that will bias the inter-
pretation of the measurements. Experiments at the Tevatron1 Run II and the LHC2
will observe thousands of top quarks, and thus it is important to examine various
ways to probe top quark properties. In this chapter we present a detailed exposition

of the available means to obtain information about top, weighing the strengths and

 

1 Run II of the Tevatron will involve p 13 collisions with an expected center-of-mass energy of
([8- = 2 TeV.
“We use LHC to denote a pp collider with center-of-mass energy J? = 14 TeV.

26

27

gm~+—t

Figure 2.1: Representative Feynman diagrams showing QCD production of ti pairs:
(16. gg -* t?-

 

H.

II

weaknesses of each. In particular we will see that single top production at a hadron
collider represents a vital means to study the weak interactions of the top quark,
and thus test the possibility of a relationship between top and EWSB. In this entire

chapter, we assume a top mass of m, = 175 GeV, unless explicitly noted otherwise.

2.1 Top Quark Preperties at a Hadron Collider

At hadron colliders, the dominant mechanism for producing top quarks is to pro-
duce pairs of t and f through the strong interaction [25], as is shown in Figure 2.1.
As dictated by the QCD-improved parton-model [26], the cross section for hadrons
scattering into tf pairs is computed by considering the partonic reactions qrj —> tf
through a virtual gluon and through fusion of two gluons, g g —> if. These partonic
cross sections are then convolved with universal parton distribution functions (PDF’s)
[27] which contain non-perturbative information about the likelihood of finding a par-
ticular parton inside a parent hadron carrying a given fraction of the parent hadron’s
momentum. It is well known that the gluon distribution function is much larger at
very low momentum fraction than the corresponding valence quark distributions, but
falls much more rapidly as the momentum fraction increases. For the production of
massive top quarks, this has the consequence that at a collider with relatively low

center of mass energy such as the Tevatron the dominant subprocess will be from qq

28

Figure 2.2: Feynman diagram showing the top decay into W+ b, including the leptonic
decay W+ —) 6+ Ve.

fusion, whereas a collider with a much higher center of mass energy such as the LHC
has a dominant contribution from g 9 fusion. It is clear that the rate of t 5 production
(coming from either subprocess) represents a measure of the top’s coupling to the

gluons.

The large rate of ti production (about 7.55 pb at the Tevatron Run II and 760
pb at the LHC [25, 28]) insures that it is an important means to study the top quark.
As we have seen, it is an important measure of the top’s strong interactions, and
could also be sensitive to some kind of new physics resonance in if production. It
also allows one to measure the top quark mass, mt, by reconstructing the top mass
from the top decay products. From Run I of the Tevatron, a combined CDF and D0
measurement of m, = 174.3 :1: 5.1 GeV based on direct observation of top has been
made, and it is hoped that improved statistics at Run II of the Tevatron will allow a

more precise measurement of :i:2 GeV [14].

Top quarks are identified by their decay products. In the SM, the top decays into
a W+ boson and a down-type quark (predominantly bottom because Va, >> V13. Val)
[29], through Feynman diagrams such as that shown in Figure 2.2. Its width can thus
be computed in terms of the top mass, the gauge couplings, and the CKM elements
V“, V“, and V». The SM prediction is found to be about 1.5 GeV, much larger

than for any other quark. This large decay width indicates that the top decays very

29

quickly, before it has time to hadronize3 [30]. This fact means that even aside from
the strong motivation to study top as a means to understand the EWSB, there is also
interest in top because it is the only quark that we are able to study “bare”. Since
the top decays into a W+ and b with a branching ratio (BR) close to one, top decays
are distinguished by the W+ decay products. The hadronic decays (W+ —+ qq’) are
dominant (with BR ~ 6 / 9), however the leptonic decays (W+ -> 3" 11,) generally

provide a clean signature at a hadron collider.

Clearly studying top decays provides some information about the top’s weak in-
teractions. However, there is an important fact to keep in mind while considering top
decays; a study of decays can measure BR’s but since it does not actually measure
the decay width itself, it is not directly proportional to the coupling, and thus cannot
measure the magnitude of the W-t-b coupling. Thus, if the W-t-b vertex is modified,
but no new decay modes appear, the BR for t —-> W+ b will remain close to 1, despite
the fact that new physics is affecting the structure of the interaction. A further prob-
lem in using top decays to search for new physics is that exotic top decays may be
unobservable or unrecognized as originating from top quarks, and therefore could be
missed. Despite these weaknesses, as we shall see in Section 2.4, top decays are an
excellent opportunity to explore the Dirac structure of the W-t-b interaction, testing

the left-handed nature of the SM weak interactions of the top.

A powerful probe of the top’s weak interactions is provided by single top pro-
duction, in which a top (or anti-top) quark is produced singly through the weak
interaction. There are three important modes of single top production in the SM: the
s-channel W“ mode [31] in which a virtual off-shell W boson is produced which then

decays into tb; the t-channel W-gluon fusion mode [32] in which a W is exchanged

 

3A simple heuristic way to understand this is to notice that the t0p width (1.5 GeV) is very
much larger than AQCD ~ 200 MeV, the scale at which non-perturbative effects in the strong force
become important.

30

between a bottom quark and a light quark, resulting in a top and a jet; and the tW’
mode [33] in which a bottom quark radiates an on-shell W' boson, resulting in a
tW‘ final state. The SM rates of these three processes at the Tevatron and LHC, as

a function of the top mass, are presented in Figure 2.3.

Single top production represents a genuine opportunity to probe the magnitude of
the W—t—b vertex because in this case the size of the cross section is directly propor-
tional to the W-t-b coupling. In the SM, this allows one to measure V». In a model
of new physics involving the top, this could lead to a discovery of the new physics.
In the following sections we will discuss the three modes of single top production in
some detail, first in the context of the SM, and then in regards to their sensitivity to
new physics effects. The issue of the top polarization will also be discussed, and it
will be demonstrated that not only can the polarization of the top be observed, but
it can provide interesting information about the structure of the interactions of the

top.

31

 

YTY'IYTTIIIITTTY'TTI'IYFITYIY

1.00 r ‘.

-—_-
"‘---__-
-—--_-- ‘
----‘

0.60 [-

Cross Section (pb)

0.10 _._._ .........

o..-
.....
-----
.....
o
................
.....
.....
.......
oc-

 

 

)-
0.“ A A 1 A l l A A A L L L A A l A I l I l A A A A l; A A L
170 172 174 170 178 100 182

 

 

 

m. (00V)
T IIIIIIIIIIIIIIIIIIIIIII
l l l I I ,
1
A
a 100,— e,
V t ‘
g , .................................... .
a m )- ............................... a
a 1
= II
0
6 r-
10_— “““----------_-1
)- d
L
-AL1LIAAJAIAJIIIIAAAIIAALIAAAA

 

 

 

170 172 174 176 178 180 182
mi. (GOV)

Figure 2.3: The SM rate of the three modes of single top production, as a function
of 1m (summing the rates of t and 1? production), at the Tevatron (upper figure) and
LHC (lower figure). The solid curve is the NLO t-channel rate, taken as the average
of the results from CTEQ4M and MRRS(R1) PDF’s. The dashed curve is the NLO
s-channel rate, taken as the average of the results from CTEQ4M and MRRS(R1)
PDF’s. The dotted curve is the LO tW‘ rate, including large log corrections, taken
as the average of the CTEQ4L and MRRS(R1) results.

32

2.2 Single Top Production in the SM

The production mechanisms for the three modes of single top production are quite
different, and it is worth spending some time discussing the particular physics aspects
of each mode individually. In this discussion, we avoid detailed consideration of
the particular kinematics and detection strategy for each mode, as this has been
considered elsewhere [31, 32, 34]. Instead we concentrate on the inclusive rates and
the effects of nonstandard physics on each process, as our goal is to understand
how single top production serves as an important probe of new physics effects. We
begin with the SM rates, and discuss the theoretical issues involved in SM single top

production.

In fact, it will be shown that the three modes are separately susceptible to quite
different types of new physics [35], and can potentially be observed independently
from each other [34]. Thus, each mode is an independent source of information about
the top quark. One sometimes finds in the literature [36, 37] analyses that treat
all of the single top modes together as one signal. This practice of combining three
signals with quite distinct kinematic signatures together is not good physics; it wastes
the information contained in each mode separately. As we will see, the three modes
provide complimentary information about the top, and thus are worth examining
independently. Further, because the W-gluon fusion rate is generally much larger
than the other two modes, these “combined analyses” are really optimized to see that
mode, with a small fraction of the other modes that manages to fake the characteristics
of a typical t-channel event included as well. Thus there is little practical difference
between a combined analysis and one focused on the t-channel process. For these
reasons, it is highly preferable to avoid thinking of single top as one process, when it

is really three separate ones.

33

q’ E

Figure 2.4: Feynman diagram for the s-channel mode of single top production: q (7’ —>
W" —-> tb.

2.2.1 W“ Production

The W“ mode of single top production proceeds through an s—channel W boson, as
shown in Figure 2.4. The final state consists of a top quark and a central jet containing
a b. Because the initial partons include both a quark and an anti-quark, this process
is relatively large at a high energy p13 collider such as the Tevatron, where valence
anti-quarks are present in the 13. It has been computed at NLO in QCD corrections
[38], and it has been found that the corrections coming from initial state radiation
of soft and collinear gluons are rather strong and substantially increase the cross
section. The resulting NLO cross section is a, = 0.84 pb at the Tevatron Run II and
a, = 11.0 pb at the LHC. The rather small increase in the cross section in going from
the Tevatron to LHC (compared to the other channels) can be understood from the
fact that the LHC is a pp collider, and thus has no valence anti-quarks. In Tables 2.1,
2.2 and 2.3, we present the NLO cross section, as a function of the top quark mass for
the Tevatron Run II and LHC, for several choices of the scale“ and the CTEQ4M [39]
and MRRS(R1) [40] PDF’s. The mean cross section is defined to be the average of
the CTEQ4M and MRRS(R1) results evaluated at the canonical scale choice. These
rates are for the production process, qi’ -—> tb only, and do not include the branching

ratios for any particular top decay. V“) has been assumed to be one, and Vt, and th

 

‘In principle the factorization scale and the renormalization scale may be chosen independently.
In practice, we follow the usual procedure of choosing them to be equal to each other.

34

have been neglected, as they are so small that their effect on the cross section is much

less than 1% of the s-channel rate.

The theoretical prediction for the cross section shows a rather small dependence
on the renormalization and factorization scales of about i5%, when the scale is
varied from the default value of p3 = \/E by a factor of 2, indicating that the uncom-
puted higher order QCD corrections are probably small. The distribution functions of
quarks and anti-quarks in the proton are relatively well-determined by deeply inelastic
scattering (DIS) data, and thus this important input to the theoretical prediction is
rather well understood. The mass of the top quark is another important quantity that
will affect the predicted cross section. The W’ mode is particularly sensitive to this
quantity, because it not only determines the phase space of the produced particles,

but also controls how far off-shell the virtual W“ boson must be.

35

Table 2.1: The NLO rates of qi’ —) W‘ —> tb (in pb) at the Tevatron Run II. At the
Tevatron, the rate of f production is equal to the t production rate.

CTEQ4M MRRS(R1)
mt (GeV) 11 = 113/2 u = #3 u = 2113 u = 113/2 [1 = [15 p = 2,113 agmean)

 

170 0.53 0.49 0.46 0.495 0.46 0.425 0.475
171 0.52 0.485 0.445 0.485 0.45 0.415 0.465
172 0.51 0.475 0.435 0.475 0.435 0.405 0.455
173 0.495 0.46 0.425 0.46 0.425 0.395 0.44
174 0.48 0.445 0.415 0.45 0.415 0.385 0.43
175 0.465 0.43 0.405 0.44 0.405 0.375 0.42
176 0.45 0.415 0.395 0.425 0.395 0.365 0.41
177 0.445 0.405 0.385 0.415 0.385 0.36 0.405
178 0.435 0.40 0.375 0.405 0.375 0.35 0.39
179 0.43 0.395 0.365 0.395 0.365 0.34 0.38
180 0.42 0.39 0.355 0.385 0.355 0.335 0.37
181 0.41 0.38 0.345 0.375 0.35 0.325 0.36
182 0.395 0.365 0.34 0.37 0.34 0.315 0.355

36

Table 2.2: The NLO rates of qq' —> W’ —-> tb (in pb) at the LHC.

CTEQ4M MRRS(R1)
mt (GeV) u=u3l2 u=u3 u=2u3 u=p3/2 p=p3 ”=2“; 05mm)

 

170 7.2 7.5 7.8 7.0 7.3 7.4 7.4

171 7.1 7.4 7.6 6.8 7.1 7.25 7.25
172 6.9 7.2 7 .5 6.7 6.9 7.1 7.05
173 6.8 7.1 7.3 6.55 6.8 6.95 6.95
174 6.7 6.9 7.1 6.4 6.65 6.8 6.78
175 6.5 6.8 7.0 6.3 6.5 6.65 6.65
176 6.4 6.6 6.9 6.2 6.4 6.5 6.5

177 6.3 6.5 6.7 6.05 6.25 6.4 6.38
178 6.1 6.4 6.6 5.9 6.1 6.25 6.25
179 6.0 6.3 6.4 5.8 6.0 6.1 6.15
180 5.9 6.1 6.3 5.7 5.9 6.0 6.0

181 5.8 6.0 6.2 5.6 5.75 5.9 5.88

182 5.7 5.9 6.1 5.5 5.65 5.8 5.78

 

37

Table 2.3: The NLO rates of ch’ —+ W“ —-+ bf (in pb) at the LHC.

CTEQ4M MRRS(R1)
m: (GeV) fl = {13/2 p = ”0 p = 2);?) p = ”3/2 .11 = #3 ,1 = 2,13 05mm)

170 4.5 4.7 4.8 4.2 4.4 4.5 4.55
171 4.4 4.6 4.7 4.1 4.3 4.4 4.45
172 4.3 4.5 4.6 4.0 4.2 4.3 4.35
173 4.2 4.4 4.5 3.9 4.1 4.2 4.25
174 4.1 4.3 4.4 3.85 4.0 4.1 4.15
175 4.0 4.2 4.3 3.8 3.9 4.0 4.05
176 3.9 4.1 4.2 3.7 3.8 3.9 3.95
177 3.8 4.0 4.1 3.6 3.7 3.8 3.85
178 3.75 3.9 4.0 3.5 3.65 3.75 3.78
179 3.7 3.8 3.9 3.45 3.6 3.7 3.7

180 3.6 3.7 3.85 3.4 3.5 3.6 3.6

181 3.5 3.65 3.8 3.3 3.4 3.5 3.53

182 3.45 3.6 3.7 3.25 3.35 3.4 3.48

 

38

b

Figure 2.5: Feynman diagrams for the t-channel mode of single top production: bq —>
tq’. A second process in which the incoming light quark is switched with a light (1' is
also possible.

2.2.2 W-gluon Fusion

The W-gluon fusion production mode involves a t-channel W exchange, as shown
in Figure 2.5. Thus, its final state consists of a top quark and a jet that tends to
be forward. It relies on the possibility of finding bottom quarks inside the hadrons
involved in a high energy collision in order to produce a single top quark. The name
“W-gluon fusion” can be understood in that the physical picture is that the process
actually involves a virtual gluon splitting into a bb pair, with one of the bottom
quarks participating in the high energy scattering. One could thus compute the in-
clusive cross section starting from a quark-gluon initial state, but the result is not
perturbatively reliable because the kinematic region in which the bb pair from the
gluon splitting is approximately collinear with the initial gluon produces a contribu-
tion that is proportional to as log mf/mg, which for m, ~ 175 GeV, 172,, ~ 4.5 GeV,
and as ~ 0.1 is over-all of order 1. In fact, the nth order correction always contains a
collinear piece which has the behavior (as log rug/mg)“, which spoils the perturbative
description of this process. A convergent perturbative expansion can be restored by
resumming these large logarithms into a bottom quark parton distribution function

[41]. This PDF is different from the light quark PDF’s in that it is actually perturba-

39

tively derived from the gluon distribution function. In fact, this two particle to two
particle (2 —> 2) description of the scattering represents the most important part of
the W-gluon fusion kinematics, because the dominant kinematic configuration is one
in which the incoming bottom is collinear with the gluon [42]. It should be kept in
mind that the b PDF has effectively integrated out the b kinematics, so this formalism
does not accurately describe the kinematic region in which the b has large transverse
momentum (p1). In this region, a description based on the two to three scattering is
more appropriate (and since this is precisely the region in which the b is not collinear
with the incoming gluon, it is well—defined in perturbation theory). The resulting

NLO cross section is at = 2.53 pb at the Tevatron and 241 pb at the LHC.

This strong dependence on the gluon PDF is a large source of uncertainty in the
prediction for the W-gluon fusion cross section. The DIS experiments are much less
sensitive to the gluon density than to the quark density, and thus the gluon density is
much less well determined, particularly in the high momentum fraction region relevant
for single top production. Though it is nota quantitative measure of the uncertainty
from the PDF, this fact is reflected in the larger dependence of the W-gluon fusion

rate on the choice of PDF in the computation.

The NLO QCD corrections to W-gluon fusion are slightly negative at both the
Tevatron and LHC [43]. The NLO cross section varies by about :i:6% at the Tevatron
and 35% at the LHC when the natural scale choice of pf, = W is varied by
a factor of 2, where Q2 is related to the W boson momentum by Q2 = —p%,,. This
again indicates that the NLO inclusive rate is expected to be fairly insensitive to the
uncomputed higher order QCD corrections. In Tables 2.4, 2.5, and 2.6, we show at
for various top masses, PDF choices, and scales at the Tevatron Run II and LHC.

These rates are for the production process bq -—> tq’, with V“, = 1 and V”, th = 0.

40

Table 2.4: The NLO rates of bq —> tq’ (in pb) at the Tevatron Run II. At the
Tevatron, the rate of 5 production is equal to the rate of t production.

CTEQ4M

MRRS(R1)
deeV) n=#6/2 u=u3 [1:214 u=u3/2 u=u3 u=21u3 at

(mean)

 

170
171
172
173
174
175
176
177
178
179
180
181
182

1.255
1.235
1.215
1.195
1.175
1.155
1.135
1.115
1.095
1.075
1.06
1.045
1.03

1.31
1.285
1.26
1.24
1.225
1.205
1.19
1.17
1.115
1.14
1.12
1.10
1.08

1.365
1.355
1.34
1.32
1.30
1.275
1.25
1.225
1.20
1.175
1.155
1.14
1.125

1.18
1.16
1.14
1.12
1.105
1.085
1.07
1.05
1.035
1.02
1.00
0.985
0.97

1.22
1.195
1.175
1.155
1.135

1.12
1.105
1.085

1.07
1.055
1.035
1.015
0.995

1.26
1.235
1.215
1.195
1.175
1.155
1.135

1.12

1.10

1.08
1.065
1.045

1.03

1.265
1.24
1.22
1.20
1.18

1.165
1.15
1.13

1.115
1.10
1.08
1.06
1.04

41

Table 2.5: The NLO rates of bq —+ tq’ (in pb) at the LHC.

CTEQ4M MRRS(R1)
m. (Gav) # = #3/2 u = #3 u = 2113 u = 113/2 u = #3 u = 2,13 aim")

 

170 156 161 165 154 157 164 159

171 154 160 164 153 155 162 157.5
172 152 159 163 152 153 161 156

173 150 157 162 150 152 160 154.5
174 149 156 160 149 151 158 153.5
175 147 155 159 148 150 157 152.5
176 146 154 158 147 149 155 151.5
177 144 153 157 146 148 154 150.5
178 142 152 155 145 147 152 149.5
179 141 151 154 143 146 151 148.5
180 140 150 153 142 145 149 147.5
181 139 148 152 141 144 148 146

182 138 147 151 140 143 147 145

 

42

Table 2.6: The NLO rates of bq —-> f q’ (in pb) at the LHC.

 

CTEQ4M MRRS(R1)

m. (GeV) 11 = 143/2 u = #3 l1 = 2113 u = 113/2 u = #3 u = 2:43 05M")
170 90 93 96 90 98 95 95.5
171 89.5 91 95 89.5 96 95 93.5
172 89 89.5 94 89 94 93 91.8
173 88.5 89 93.5 88.5 92 92 90.5
174 88 88 93 88 90 91 89
175 87.5 87.5 92 87 89 90 88.3
176 87 87 91 86 88 89 87.5
177 86.5 86.5 90 84 87 88 86.8
178 86 86 89 83 86 87 86
179 85 85.5 88 82 85 86 85.3
180 84 85 86 81 84 85 ' 84.5
181 83 84 85 80 83 84 83.5

182 82 83 84 78 82 82 82.5

 

43

g W'

.8 .

Figure 2.6: Feynman diagrams for the tW‘ mode of single top production: gb —>
tW‘.

2.2.3 tW" Production

The tW' mode of single top production proceeds from Feynman diagrams such as
those presented in Figure 2.6. The final state consists of an on-shell W‘ (which can
decay either to quarks or leptons) and a top quark. It should be quite clear that
the signature of this process is very distinct from the other two modes, because of
the extra W’ decay products in the final state. This process also involves finding
a bottom quark inside one of the colliding hadrons, and the same issues related to
this fact that were present in the W-gluon fusion process discussed above are also
important here, in particular a rather strong dependence on the gluon PDF used to

obtain the prediction.

Though the complete NLO QCD corrections are not available, one can improve
the LO estimates by including the 0(1/ log mf/mg) corrections coming from Feyn-
man diagrams such as those in Figure 2.7 There are two subtle points that must be
carefully dealt with in carrying out this procedure. The first is that when the b PDF
was defined, the collinear contributions from these diagrams was already resummed
into what we called the LO contribution. Thus, in order to avoid double-counting

this collinear region one must subtract out the piece already included in the LO

44

 

 

 

 

5
8 MAMA/V W 8 : t
1) A W'
8 : t 6 mm 5
(a) (b)

Figure 2.7: Representative Feynman diagrams for corrections to the tW‘ mode of

single top production corresponding to (a) large log corrections asso_ciated with the b
PDF and (b) LO tf production followed by the LO decay f —} W‘ b.

contribution. The full cross section for A B ——> tW‘ may thus be expressed as,
am = 0°(AB —> tW') + 01(AB —> tw- 5) — 05(AB —> tW‘ 5), (2.1)
with the individual terms given by,

o°<AB —+ tW-> = / dz. 014209.954) mam. #)0(bg-> tW‘) (2.2)
+ fb/A($la u) fg/a($2. It) 0(5)!) —+ t W‘)}
01(AB—> tW' 5) = [(16:1de fg/A(a:1,p)fg/B(x2,u)0(99->tW‘ 6)
05MB 9 WV?) = / 441442015962) 79/862, u)0(b9-> W")

+ 79/191, M) Jib/392, u) a(gb -+ tW-)}.

The “modified b PDF”, fb/H, contains the collinear logarithm and splitting function

Pb._g convoluted with the gluon PDF,

ib/H($,H) = 9;%108 (E5) I: 'di [22 + (i _ 2?] fg/H (an) . (2.3)

mb Z

 

Having included this subtraction piece, the problem of double-counting the collinear

region is resolved.

45

The second subtle point in evaluating the large log contributions is that they
contain contributions such as those found in Figure 2.7b that correspond to LO 9 g —->
tf production followed by the LO decay t- —+ W" b. This expresses the fact that as one
considers higher orders in perturbation theory, the distinction between tf production
and various types of single top production is blurred. However, when considering
quantities that are properly defined, these corrections are small, and there is no
problem distinguishing these processes. As a matter of book keeping, the corrections
to t W‘ production involving an on-shell top are more intuitively considered a part
of the LO tf rate, and thus it is important to subtract them out to avoid double
counting in this kinematic region. This may be done by noting that in the region
where the invariant mass of the bW’ system, Mm, is close to the top mass, the

behavior of the partonic cross section 0(gg ——> tW‘b) may be expressed,

do
(1M Wb

mt FLO({ —) W_0)
(Mfr/b - "1%)2 + m? Ff]

m.1‘.BR(t‘ —> w-E)
7r [(Mfiw. - m?)2 + m? I‘?]

—> 0“0(gg —> t0 BRE‘7 W-B)6(M12Vb ’ m?)

 

 

(99—>tW'5) = 0L0(99->tt—)1r[ (2.4)

= 0"°(99 —> t1f)

 

where 0L0(g g -+ ti) and I‘L0(t- —> W‘b) are the LO cross section and partial width,
and I", is the inclusive top decay width. The last distribution identity holds in the limit
1‘; << ing. Having identified this LO on-shell piece, it may now be simply subtracted
from 0(g g —-) tW‘ b). The advantage to this formulation of the subtraction is that
by taking the narrow width limit, one removes all of the on-shell f contribution. The
interference terms between one of the on-shell f amplitudes and an amplitude without
an on-shell f involve a Breit-Wigner propagator of the form, (Maw, — m? + i mtI‘t)‘1,
which in the limit of small I}, may be expressed as a principle valued integral in MWb-

Following this prescription, and choosing a canonical scale choice of #0 = J3, leads

46

to a large log correction to the tW" rate of —9.5% at the LHC, which is consistent

with previous experience from the W-gluon fusion mode [34].

This problem of the on-shell top was dealt with another way in [37], where a cut
was applied on MWb, to exclude the region of IMWb — mtl S 3I‘t. Following this
prescription, one finds a much larger correction of about +50% to the tW‘ rate
at the LHC. However, this is misleading because the large corrections are mostly
coming from the region where the f is close to on-shell (though still at least 3 top
widths away). In other words, the large positive correction comes from the tails of the
Breit-Wigner distribution for on-shell t- production. This can be simply understood
by taking the prescription in [37] and varying the cut by increasing the interval about
the on-shell t- region that is excluded. One finds that the correction computed in this
way varies quite strongly with the cut, and reproduces the subtraction method we
have employed for the cut [MWb — rm] 3 12 I}. A further theoretical advantage of the
subtraction method is that when one determines the ti and tW‘ rates, one would
like to actually fit the data to the sum of the two rates, and thus the subtraction
method allows one to simply separate this sum into the two contributions without

introducing an arbitrary cut-off into the definition of the separation.

Even if one were to use a cut-off to effect the separation, there is a further problem
in employing the cut [MWb — mt| S 3 F, to remove on-shell tf production. This is that
from a purely practical point of view 3 Ft ~ 4.5 GeV, which is much smaller than the
expected jet resolution at the Tevatron or LHC. Thus, it is not experimentally possible
to impose this definition of the separation between tW' and ti. A more realistic
resolution is about 15 GeV [44], which corresponds to a subtraction of [MWb — ml 3
10 I“. As we have seen above, this choice of the MW), cut agrees rather well with our

subtraction method result.

47

The t W‘ process has been studied much less intensively than the other two modes,
mostly owing to the fact that it has a small rate at the Tevatron Run II (otw = 0.094
pb) that is probably unobservable. On the other hand, the rate is fairly considerable
at the LHC (atw = 55.7 pb) and it may be observable there. However, detailed
simulations studying means by which the signal may be extracted from the large tf
background are still underway. For completeness, we include in Tables 2.7 and 2.8 the
LO rate (including the large log corrections described above) of t W‘ production at
the Tevatron and LHC, for various choices of 1m, PDF, and scale, with the canonical
scale choice set to yo = J3. As usual, the t and f rates have been summed, V“, = 1,
V“, V“ = 0 and no decay BR’s are included. From these results, we see that varying
the scale by a factor of two produces a variation in the resulting cross section of about
:l:25% at the Tevatron and i15% at the LHC. This large scale dependence signals
the utility in having a full NLO in as computation of this process in order to have a

more theoretically reliable estimate for the cross section.

48

Table 2.7: The LO (with 0(1/ log mf/mg) corrections) rates of b g —-> tW' (in pb)
at the Tevatron Run II. The rate of f production is equal to the rate of t production.

CTEQ4L

MRRS(R1)
mt (GeV) # = Ito/2 H = #0 (”8““)

u=2uo u=uo/2 u=uo u=2no atW

 

170 0.0645 0.0505 0.0405 0.076 0.058 0.046 0.0545
171 0.063 0.049 0.0395 0.074 0.0565 0.0445 0.053
172 0.061 0.048 0.0385 0.072 0.055 0.0435 0.0515
173 0.0595 0.0465 0.0375 0.07 0.053 0.042 0.05
174 0.0575 0.045 0.0365 0.068 0.0515 0.041 0.0485
175 0.056 0.044 0.0355 0.066 0.05 0.0395 0.047
176 0.0545 0.0425 0.0345 0.064 0.049 0.0385 0.046
177 0.053 0.0415 0.0335 0.062 0.0475 0.0375 0.0445
178 0.0515 0.0405 0.0325 0.06 0.046 0.0365 0.0435
179 0.05 0.039 0.0315 0.0585 0.0445 0.0355 0.042
180 0.0485 0.038 0.0305 0.057 0.0435 0.0345 0.041
181 0.0475 0.037 0.03 0.0555 0.0425 0.0335 0.040
182 0.046 0.036 0.029 0.054 0.041 0.0325 0.0385

49

Table 2.8: The LO (with 0(1/ log(m,2/m§) corrections) rates for by -—> tW‘ (in pb)
at the LHC. The rate of 5 production is equal to the rate of t production.

CTEQ4L MRRS(R1)
m¢(GBV) “=H0/2 p=po #:2110 ”=I‘0/2 [1:110 #:2I10 03380")

 

170 33.0 28.2 24.5 39.0 33.0 28.4 30.6
171 32.2 27.5 24.0 38.3 32.5 27.9 30.0
172 31.6 27.1 23.6 37.6 31.8 27.4 29.4
173 31.1 26.6 23.1 38.0 31.3 26.9 28.9
174 30.5 26.1 22.7 36.2 30.7 26.4 28.4
175 29.9 25.6 22.2 35.4 30.1 26.0 27.9
176 29.4 25.2 21.8 34.8 29.6 25.5 27.4
177 28.9 24.7 21.5 34.2 28.9 25.0 26.8
178 23.3 24.2 21.1 33.6 28.4 24.6 26.3
179 27.8 23.7 20.7 33.0 27.9 24.1 25.8
180 27.2 23.3 20.3 32.4 27.4 23.7 25.4
181 26.8 22.9 20.0 31.8 26.9 23.2 24.9

182 26.3 22.5 19.6 31.2 26.4 22.9 24.5

50

2.3 New Physics in Single Top Production

As we have argued above, single top production is an important place to search for
physics beyond the SM. This is reflected in the growing body of literature in which the
effect of loops of new particles on the single top rate is examined [45]. In this section,
we analyze several possible signals for new physics that could manifest themselves
in single top production. These signals can be classified as to whether they involve
the effects of a new particle (either fundamental or composite) that couple to the
top quark, or the effect of a modification of the SM coupling between the top and
other known particles. In fact these two classifications can be seen to overlap in the
limit in which the additional particles are heavy and decouple from the low energy
description. In this case the extra particles are best seen through their effects on the

couplings of the known particles.

2.3.1 Additional Nonstandard Particles

Many theories of physics beyond the SM predict the existence of particles beyond
those required by the SM itself. Examples include both the fundamental super-
partners in a theory with SUSY, and the composite top-pions found in top-condensation
and top—color models. In order for some kind of additional particle to contribute to
single top production at a hadron collider, the new particle must somehow couple the
top to one of the lighter SM particles. Thus, the new particle may be either a boson
(such as a W’ vector boson that couples to top and bottom) or a fermion (such as a

b’ quark that couples to the W boson and top).

Additional fermionic particles can couple the top and either one of the gauge
bosons or the Higgs boson. In order to respect the color symmetry, this requires that

the extra fermion occurs in a color triplet, and thus it is sensible to think of it as

51

some type of quark. In order to be invariant under the electromagnetic symmetry,
this new “quark” should have either electric charge (Q) +2/3 or —1/3 in order for
one to be able to construct couplings between the extra quark, the top quark, and
the known bosons. Generally, we can refer to a Q = +2/ 3 extra quark as a t’ and
a Q = —1/3 extra quark as a b’, though this does not necessarily imply that the
extra quarks are in the same representation under SU(2)L x U(1)Y as the SM top
and bottom. Additional fermions are not generally expected to be a large source of
new contributions to single top production, because of strong constraints from other
observables. On the other hand we will see that there are models with additional

fermions to which single top production is a sensitive probe.

“Extra” bosons can contribute to single top production either by coupling top to
the down-type quarks, in which case the boson must have electric charge Q = :l:1
in order to maintain the electromagnetic symmetry, or by coupling top to the charm
or up quarks, in which case the boson should be electrically neutral. One could also
imagine a boson carrying an odd combination of color and electric charge that would
allow it to couple to both top and a lepton field. Such bosons carrying both baryon-
and lepton- number (leptoquarks) could arise, for example, from the generators of
the part of the gauge group of a GUT that connects the electroweak and strong
sectors of the GUT. In that case one would naturally suppose that these bosons
have mass of the order of MGUT, and thus may not play an important role in single
top production at the weak scale. This GUT picture has the leptoquark as part
of the gauge interactions, so the question as to whether or not top observables are
an interesting means to study leptoquarks becomes a question as to whether or not
the leptoquark has some reason to prefer to couple to the top quark. One could
imagine that the grand unified interaction contains a sector corresponding to a family

symmetry that could somehow cause this to be the case. Another interesting picture

52

of leptoquarks is one in which the SM quarks and leptons are bound states of some
more fundamental set of particles (which we will refer to as preons). In that case the
question as to whether or not the top quark is a good place to look for evidence of
the preons depends on how the model arranges the various types of preons to build
quarks and leptons. However, at a hadron collider the possible light parton initial
states available are not suitable for production of a single leptoquark, and thus are
not particularly interesting in the context of single top production“. For this reason,

we will not focus on leptoquarks in the discussion below.
Extra Quarks

A simple extension of the SM is to allow for an extra set of quarks. Such objects
exist in a wide variety of extensions to the SM. Examples of such theories include the
top seesaw versions of the top-color [46] and top-flavor [47] models, which rely on
additional fermions to participate in a see-saw mechanism to generate the top mass;
SUSY theories with gauge mediated SUSY breaking that must be communicated from
a hidden sector in which SUSY is broken to the visible sector through the interactions
of a set of fields with SM gauge quantum numbers [48]; and even models with a fourth

generation of fermions.

Direct search limits on extra quarks require that they be quite massive (mg: 2
46 - 128 GeV at the 95% C. L., depending on the decay mode [12]), and thus they
cannot appear as partons in the incident hadrons at either Tevatron or LHC. This
prevents them from significantly affecting the W-gluon fusion and tW‘ rates. Thus,
they are best observed either through their mixings with the third family (and thus

their effect on the top couplings), or through direct production.

 

5It is interesting to note that a leptoquark with Q = +2/ 3 could play an important role in top
decays through a process such as t -) u L —+ ub£+. This leads to a final state that is identical to a
SM top decay, but with a very distinct kinematic structure

53

9’ 5'
Figure 2.8: Feynman diagram for s-channel production of a single top and a b’:
qQ’ —-> tb’.

As a particular example, a fourth generation of quarks could mix with the third
generation through a generalized CKM matrix, and this could allow V“ to deviate
considerably from unity. In this case, all three modes of single top production would
be expected to have considerably lower cross sections than the SM predicts. This
already shows how the separate modes of single top production can be used to learn
about physics beyond the SM. Other types of new physics could scale the three rates
independently. Thus, if all three modes are measured to have cross sections that are
the same fraction of the SM rates, it is an indication that the new physics modifies
the top’s coupling to the bottom and W (and not another pair of light particles),
and further that the modification is the same regardless of the momentum flowing

through the vertex (as is the case with the W-t-b interaction in the SM).

In addition to mixing effects, one could also hope to observe direct production of
one of the fourth generation quarks, through reactions such as qrj’ —> tb’, shown in
Figure 2.8. If the b’ is somewhat heavier than the top, and V»: is large, this process
could be more important than QCD production of b’ 5’ because of the greater phase
space available to the lighter top. The production rates will depend on the magnitude
of the W-t-b’ coupling (ll/“,42 in the model with a fourth family) and the mass of the
b’. In Figure 2.9 we present the NLO rate for tb’ production (as well as fb’ production)

without any decay BR’s. Since the IV»: |2 dependence may be factored out, these rates

54

assume Vw = 1. The collider signatures resulting from such a process depend on the
decay modes available to the b’. If my > mt + mw, it is likely to decay into a top
quark and a W‘, and the events will have a ti pair with an additional W“: boson.
If this decay mode is not open, loop induced decays such as b’ —> by may become
important, resulting in a signature tb plus a hard photon whose invariant mass with

the b quark will reconstruct the mass of the b’.

Extra Gauge Bosons

Another simple extension of the SM is to postulate the existence of a larger gauge
group which somehow reduces to the SM gauge group at low energies. Such theories
naturally have additional gauge bosons, some of which may prefer to couple to the
top (or even the entire third family). Examples of such theories include the top-color
[21] and top-flavor [47, 49] models, which give special dynamics to the third family
in order to explain the large top mass. As a specific example, we will consider the
top-flavor model with an extra SU(2)h gauge symmetry that generates a top mass

through a see-saw effect [47].

This model has an over-all gauge symmetry of SU(2)),X SU(2);x U(1)y, and thus
there are three additional weak bosons (W’i and Z’). The first and second generation
fermions and third family leptons transform under SU(2)), while the third generation
quarks transform under SU(2),,. As was alluded to before, in order to cancel the
anomaly and provide a see-saw mechanism to generate the top mass, an additional
doublet of heavy quarks whose left-handed components transform under SU(2), and

right-handed components transform under SU(2)h is also present.

A set of scalar fields transforming under both SU(2); and SU(2))I acquire a VEV,
u, and break the symmetry to SU(2);+hx U(1)y. From here the usual electro-weak

symmetry breaking can be accomplished by introducing a scalar doublet which ac-

 

55

 

.1 IIIIIII
l llllll

I
I
l

100 \4

1 I Will
I

1 111m:

     
   
   
 

 

 

 

:3 ..
3 I— d
‘ -1
5’ 1° :5 "1
10‘“ E‘ 1
t l L 1 +1 I 1 PL 14 l l l l l L l l l l I l 1 d
50 100 150 200 250 300

mg. (GeV)

Figure 2.9: The NLO rates (in pb) for the process qti' -+ W' —) tb’ for various b’
masses at the Tevatron (solid curve) and LHC (dashed curve), assuming V»: = 1. At
the Tevatron, the rates of qrj’ —> W" —) f b’ is equal to the tb’ rate. The fb’ rate at
the LHC is shown as the dotted curve.

 

56

q’ 5
Figure 2.10: Feynman diagrams illustrating how a W’ boson can contribute to single
top production through qi’ —> W’ —> tb.
quires a VEV 22, further breaking the gauge symmetry to U(1)EM. We write the

covariant derivatives for the fermions as,

Y
D” = 8" + 7:91 [If War +19}; T: W“): + 7:91 3 B”, (2.5)

where {(30 are the generators for SU(2)¢(h), Y is the hyper-charge generator, and
W“f‘(h) and B" are the gauge bosons for the SU(2)“) and U(1)y symmetries. The

gauge couplings may be written,

_ e e e
g; _ sin 0w cos (b ’ “h “1

 

(2.6)

sin 9w sin (b ’ cos 0W ’

where (b is a new parameter in the theory. Thus this theory is determined by two addi-
tional quantities a; = 12/22, the ratio of the two VEV’s, and sin“ (b, which characterizes

the mixing between the heavy and light SU(2) gauge couplings.

At leading order, the heavy bosons are degenerate in mass,

2: sin“ (b)
7

 

(2.7)

M2 I I = M2 . +
Z’W 0 (sm“¢c082¢ coszqi

 

where Mo“ = 43in“ its, 0W. We can thus parameterize the model by the heavy boson
mass, Mzr, and the mixing parameter“, sin2 (1). Low energy data requires that the

mass of these heavy bosons, M zz, be greater than about 900 GeV [50].

 

“As shown in [49], for sin“ 45 g 0.04, the third family fermion coupling to the heavy gauge bosons
can become non-perturbative. Thus we restrict ourselves to considering 0.95 2 sin“ ¢ _>_ 0.05.

57

The additional W’ boson can contribute to the s-channel mode of single top pro-
duction through virtual exchange of a W’ as shown in Figure 2.10 [51]. In particular,
if enough energy is available, the W’ may be produced close to on-shell, and a res-
onant enhancement of the signal may result. Since the additional diagrams involve
a virtual W’, they will interfere with the SM W-exchange diagrams, and thus the
net rate of single top production can be increased or decreased as a result, though
the particular model under study always results in an increased s-channel single top
rate. In Figure 2.11 the resulting NLO s-channel rate for qrj’ —+ W,W’ —) tb at
Tevatron and LHC is shown, as a function of the W’ mass, for a few values of sin“ 45.
The rate for 1? production through the same process is shown as well. While the final
state particles for this case are the same as the SM s—channel mode, the distribution
of the invariant mass of the tb system could show a Breit-Wigner resonance effect
around MW], which serves to identify this type of new physics. However, if the mass
of the W’ is large compared to the collider energy, and its width broad, the resonance
shape can be washed out even at the parton level. Jet energy smearing from detector

resolution effects will further make such a resonance difficult to identify.

A t-channel exchange of the W’ is also possible, but in that case a negligible
effect is expected because the boson must have a space—like momentum, and thus the
additional contributions are suppressed by 1 /Mw:, and are not likely to be visible.
This argument applies quite generally to any heavy particle’s effect on, single top
production. The s-channel rate is quite sensitive to a heavy particle because of the
possibility of resonant production, whereas the t-channel rate is insensitive because

the space-like exchange is suppressed by the heavy particle mass.

Clearly, the existence of a W’ will not influence the rate of t W" production, but
it could allow for exotic production modes such as b g —> tW’. If the W’ has a

strong coupling with the third family, then one would expect that its dominant decay

58

 

25

‘

IIITIIIIFYIIIIUIIIIIIIUIII

TVIT
/

I

20

15

IITIIIII
/
I

0. (pb)

 

 

 

 

Tevatron (x 10) ‘

 

 

ll[L11llllllLllllllllllllll

1000 1200 1400 1600 1800 2000
mz. (GeV)

 

Figure 2.11: The NLO rate of qq’ —> W,W’ —> tb (in pb) at the Tevatron (lower
curves) and LHC (upper curves), for the top-flavor model with sin“ (b = 0.05 (solid
curves) and sin“ 43 = 0.25 (dashed curves), as a function of M z: 2 MW. The Tevatron
cross sections are multiplied by a factor of 10. At the Tevatron, the 5 production rate
is equal to the t rate. At the LHC the t- rates are shown for sin“ (b = 0.05 (dotted
curve) and sin“ 4) = 0.25 (dot-dashed curve).

59

Cl
6

Figure 2.12: Feynman diagram illustrating how a charged top-pion can contribute to
single top production through cb —> 7r+ —> t b.

should be into bf, and thus a final state of if b would result with the tb invariant
mass reconstructing the W’ mass. Current limits on the W’ mass in the top—flavor
model make this mode nonviable at the Tevatron and unpromising at the LHC, with
a cross section of 1.14 pb for MW: = 900 GeV and sin“ ti) = 0.05 including the large
log contributions described in Section 2.2.3. However, an observation of this signal

would be a clear indication of the nature of the new physics.

Extra Scalar Bosons

Scalar particles appear in many theories, usually associated with the spontaneous
breaking of a symmetry. In the SM and the minimal supersymmetric extension,
fundamental scalar fields of both neutral and charged character are present in the
theory, and are expected to have a strong coupling with the top because of the
role they play in generating fermion masses. In dynamical models such as the top-
condensate and top-color assisted technicolor models, scalar particles exist as bound
states of top and bottom quarks (as was seen in Chapter 1 this is how these models
deal with the fine-tuning and naturalness problems of the SM). These composite
scalars also have a strong coupling to the top because of their role in the generation
of the top mass. This illustrates the fact that the large top mass naturally makes it

a likely place to look for physics associated with the EWSB.

60

An illustrative example is provided by the charged composite top-pions (7?) of
the top-color model, which can be produced in the s-channel through cb fusion [52],
cb —) 7r+ —> tb. The leading order Feynman diagram is shown in Figure 2.12. In this
case the strong 7r+-c-b coupling comes from mixing between the t and c quarks. In
order to avoid constraints from the CKM matrix, this requires the mixing to occur
between right-handed t and c quarks, and thus this interaction has a right-handed

nature that will prove interesting when we study top polarization below.

Like the W’, the 11'“: contributes to the s—channel topology of single top production
and can allow large resonant contributions. However, unlike the W’, the 1r+ does not
have a significant interference with the SM amplitudes, because the SM contribution
is mostly from light quarks (u and cf). In Figure 2.13, we present the NLO single top
rate from the top-pion process [53], for a variety of 1ri masses with the tR-cR mixing
set equal to 20%. The two other modes of single top production are once again
relatively insensitive to the 1E". The t-channel process has additional contributions
suppressed by l/Mfi and the fact that the xi does not couple to light quarks. The
tW' mode is insensitive because presumably the 1f“ is generally distinguishable from

a Wi boson, and so gb —+ 7r‘ t —-) fbt will not be mistaken for t W' production.

Different types of scalar particles that couple top and bottom can be analyzed
in a similar fashion. The s-channel mode allows for resonant production, which
can show a large effect, where-as the t-channel mode is suppressed by the space-like
momentum (and large mass) of the exchanged massive particle. The tW‘ mode is
insensitive because in that case the W is actually observed in the final state. The
technipions in a technicolor model can contribute to single top production in this way
[54]. Another example is provided by SUSY models with broken R—parity, in which
the scalar partners of the leptons (sleptons) can couple with the top and bottom

quarks, and will contribute to single top production [55].

 

61

 

7

10 Y ‘I' 1 I 17" 1 I I l I Y I I
3
- :l: J
10“ 99/ 99 -M X —.
1
10“ .3
1
1 4
104 LHC (14 TGV) 1
A i:
.o :
'0- ..

V
.(
b 103 _a
Tevatron :
.(
102 _E
1 2.0 TeV E
.[
10 —.
1.8 ToV E
‘ l 1 l L L l I I l L l I l I 1 .l
200 400 soo 800 1000

 

 

Mass of 1r“: (GeV)

Figure 2.13: The L0 rate of single top production through the reaction cb —> 77+ —> t b
as a function of Mfli, assuming a tR-cR mixing of 20%. These rates include t and 5
production, which are equal for both Tevatron and LHC.

62

 

q q q q
h h
b t b t

 

   

Figure 2.14: Feynman diagrams for associated production of a neutral scalar and
single top quark: q b ——) q’ t h.

As a final note, there is the interesting process in which a neutral scalar (like the
Higgs of the SM) is produced in association with a single top quark [56]. Feynman
diagrams are shown in Figure 2.14. This process is of interest because while the
magnitude of the h-W-W and h-t—t couplings can be measured independently by
studying qq’ -—) W’ —> Wh and qq (g g) -> htf, the relative phase between the
couplings can be found from the process qb —> q’ th, as that phase information is
contained in the interference between the two diagrams shown in Figure 2.14. This
process is extremely small compared to the other two mentioned (with a SM cross
section of 6 x 10’5 pb at the Tevatron and 0.1 pb at the LHC for m), = 100 GeV
and including both t and 5 production), and thus it is not promising a discovery
mode. The small SM rate results from the fact that the interference term provides a
strong cancellation of the rate, reducing it by a factor of about 5. This indicates that
this process is very strongly sensitive to any physics that modifies the relative phase
between the h-t-t and h—W-W couplings from the SM relation. Thus, it contains

important information not available in the other two processes.

2.3.2 Modified Top Quark Interactions

Another interesting set of properties of the top that can be studied in single top

production are the top couplings to light particles. As was shown in Section 1.3.4,

63

the electroweak chiral Lagrangian provides a powerful way to study such effects model-
independently. Following the EWCL approach, we write an effective Lagrangian to

describe low energy physics as,
.68” = LSM + £4 + L5, (2.8)

where [ISM refers to the usual SM Lagrangian described in Chapter 1, and £4 and
£5 are the Lagrangians containing deviations from the SM in terms of operators of

mass dimension 4 and 5, respectively.

Terms which have the potential to modify single top production include mass

dimension 4 operators [57],

_ e + Li¢"“n an-n
£4 -— fisin OWW ”(nwwe Wtbb'y PLt+nW,,,e Wtbb'y Pat) (2.9)

 

2 sin 9w cos 9w “ ( “w?” Si“ 9Z5; 57" PLt +

cw?“ cos 02,667“ PRt) + H.c.,

which can be classified as two operators which modify the SM top weak interactions,
as well as two flavor-changing neutral current operators involving the Z boson, t, and
c quarks. Additional dimension 4 FCNC operators with the c quark replaced by the
u quark are also possible. We have included the CP violating phases (1)522“) in the
interactions for generality, though they are not always considered in the literature. In
addition there are dimension 5 operators that involve interactions between new sets
of particles and the top7 and can contribute to single top production. These include

the FCNC operators,

 

7There are also dimension five operators involving the sets of particles that already appear in
Equation 2.9 [58]. Since naive dimensional analysis [59] suggests that at low energies these operators
are less significant than their dimension four counterparts, we limit .65 to the dimension 5 operators
which involve only new sets of fields.

64

2 G“ .
£5 = 91%;”: (emf... sin 05,0 5 T“ 0"" PL t (2.10)
9‘0
+ ’85:. 3.95:, ET“ .w p. .)
2 2 F . .
+—‘:{—;\e—’w ( emf“ sin 05’“ E 0"” PL t + 6%?“ cos 0.5““. E 0"” PR t) + H.c.,
“rte

which couple the charm quark to the top and gluon or photon fields. Once again,
we have included CP violating phases 03:83“) which are not generally considered in
the literature. Additional operators with the charm replaced by the up quark are
also possible. As dimension 5 operators, these terms have couplings with dimension
of inverse mass that have been written in the form of 1/A,,tc and 1/A71c If the
underlying theory is strongly coupled, these mass scales may be thought of as the
energy scale in which the SM breaks down and must be replaced with the underlying
theory. However, it should be kept in mind that if the underlying theory is weakly
coupled, this interpretation is somewhat obscured by the fact that the energy scales
A will also include small factors of the fundamental interaction strength and loop
suppression factors. Even in this case, an experimental constraint on A is very useful

because it will provide constraints on the parameters of an underlying model that

makes a prediction for it.

The dimension 4 terms which modify the W-t-b vertex will clearly have a large im-
pact on single t0p production [34]. However, “ii/:6 is already very strongly constrained

by low energy b —) 8') data [60], which requires [61],
—.0035 2 ( n5“, cos 41%,, + 20 mg,“ ) 3 0.0039, (2.11)

provided that new is somewhat smaller than 1. Given this strong constraint, it is
unlikely that further information about 16%,, can be gleaned from single top produc-

tion, so we will assume Isa/a, = 0 in the discussion below. On the other hand, all three

65

0\
0‘
Z 7 033g

(2!) (b) (C)

Figure 2.15: Feynman diagrams showing FCNC top decays through (a) t —+ Z c,
(b) t —> 70, and (c) t-> gc.
modes are sensitive to Isa/w, and will be proportional to (1 + 16%,,“ + 2 “6m cos 651mb)

much the same way that they will all be sensitive to V”, in the SM“.

The flavor-changing neutral current terms in £4 and .65 will also contribute to
single top production, and since they involve particles lighter than the top mass,
will also contribute to top decays through Feynman diagrams such as those shown in
Figure 2.15, which illustrate FCNC t decays to c. The FCNC interactions between t
and u will allow for exotic decays of the same type, but with the c quark exchanged
with a u quark. One could hope to learn about these anomalous FCNC couplings
both by studying single top production and top decays. However, this brings us back
to the problem with using top decays to determine the magnitude of a coupling —
the decay can provide information about the relative branching fraction of the exotic
decay compared to the SM top decay t —> W+ b, but since it does not allow one
to measure the top decay width, it cannot provide a limit on the size of the exotic
operator without first making an assumption concerning the nature of the W-t-b
interaction. In fact, one might think that single top would suffer from the same

difficulty in distinguishing the magnitude of new physics in the W-t-b interaction

 

8This is because the dimension 4 term that is proportional to 51W“ in L4 does not depend on
the momenta of the interacting particles, as is the case for the SM W-t—b interaction. For higher
dimension W-t-b operators, which may depend on the momenta, each single t0p mode will respond
differently to the new interaction, and thus could be used to distinguish one Operator from another.

66

'CI 5

Figure 2.16: Feynman diagram showing how a FCNC Z-t-c interaction contributes
to the s-channel mode of single top production through qq —> Z ’ —> t6.

from new physics in a FCNC interaction. However, as we shall see, one can use the
three modes of single top production separately to disentangle the FCNC new physics

from the possibility of W—t-b new physics.

The three FCNC operators have a similar structure of a light c (or u) quark
interacting with a top and a neutral vector boson. Thus, we can discuss their impact
on the three single top processes rather generally by considering the specific example
of the Z-t-c operator. In examining the FCNC operators in Equations 2.9 and 2.10,
we note that they can have left-handed and right-handed interactions with different
interaction coefficients (and even different phases). For now we will restrict our
discussion to the case where all of the phases are zero, and discuss only the magnitude
of the interactions, set by A9“, A7“, and reg... We will return to the subject of

exploring their chiral structure when we consider top polarization in Section 2.4.

The Z-t-c operator will allow for additional contribution to the s-channel mode
of single top production through reactions such as qq —+ Z’ —) t5, shown in Fig-
ure 2.16. This reaction has different initial and final state from the SM s—channel
mode, and thus there is no opportunity for interference between SM and new physics
contributions. The fact that the new physics process has a '0’ instead of a b in the final
state has a drastic practical consequence that the new physics production mechanism

probably cannot be experimentally extracted at all, because in order to separate the

67

g I

Figure 2.17: Feynman diagrams showing how a FCNC Z-t-c interaction contributes
to the exotic mode of single t0p production gc —+ tZ.

s-channel mode from the large if and W-gluon fusion backgrounds, it is necessary to
tag the 5 produced in association with the top in the s-channel mode, in addition to
the b from the top decay. Thus, while a FCNC operator could contribute to s-channel

production of a single top, it will not be counted as such“.

The tW‘ mode cannot receive a contribution from a FCNC, though a FCNC will
generally allow for new exotic production mechanisms such as 90 —-+ tZ shown in
Figure 2.17. From this consideration, along with the analysis of the tW‘ mode in
Section 2.3.1, we see that the tW" mode has a special quality because both the top
and the W are in the final state (and thus identifiable). Thus, it is sensitive to new
physics which modifies the W-t-b interaction”, but it is not sensitive to nonstandard
physics involving new particles or FCNC’s. Thus, the tW‘ mode represents a chance

to study the W-t-b vertex without contamination from other types of new physics.

The W-gluon fusion mode of single top production is quite sensitive to a FCNC
involving the top and one of the light partons, through processes such as cq —> tq,

from Feynman diagrams such as those shown in Figure 2.18. The FCNC operators

 

9It could be possible to search for s-channel production via a FCNC with a Specialized stratey
differing from the usual one employed to extract the W" process, but such a search will suffer from
large backgrounds from t f and W-gluon fusion single top processes.

10Of course it is also sensitive to the W-t-s and W-t-d interactions, but these have been measured
to be small [12].

68

C t

Figure 2.18: Feynman diagram showing how a FCNC Z-t-c interaction contributes
to the t—channel mode of single top production through cq —) tq.

involve a diflerent set of spectator quarks in the reaction, and thus they do not in-
terfere with the SM t-channel process. In fact, because the W-gluon fusion mode
requires finding a b inside a hadron, which has less probability than finding a lighter
parton, the FCNC’s involving u or c quarks already receive an enhancement relative
to the SM t-channel rate purely from the parton densities. This can somewhat com-
pensate for a (presumably) smaller FCNC coupling. This shows the sense in which
the t-channel single top mode is sensitive to the top quark’s decay properties. The
same type of new physics which opens up new top decay modes (and thus modifies
the top’s total width) will also modify the t—channel rate of single top production,
because the same light partons into which the top may decay are also responsible for
producing single tops in the t-channel process. Thus, one can think of the t-channel

process as a kind of measure of the inclusive top width.

Because of the strong motivation to use single top production to study FCNC op-
erators involving the top quark, detailed simulations of the effect of the g-t—c operator
on single top production were performed [24], and found that this operator could be
constrained by the process qq —+ t6 to A9,, 2 4.5 TeV at Run II of the Tevatron if

no new physics signal were to be found. Further refinements on this idea [62] showed

that it could be improved by including other reactions such as gc —) t, gc —-) gt,

 

 

69

 

 

 

 

 

Figure 2.19: The correlation between the maximum cross section of mi -+ t5, 0...,
and the minimum BR(t —> Wb) assuming the t-c-g operator is the only source of
nonstandard physics in top decays, bun-n.

70

qc —-> tq, and gg -> it? to Am 2 10.9 TeV at the Tevatron Run II. In [63] it was
pointed out that since the same g-t—c operator contributes to both single top produc-
tion and the top decay t —+ cg, one can look for the correlation in the BR(t —+ cg) and
the anomalous rate of single top production to verify that this particular operator is
responsible for a given observation of a new physics efiect. In Figure 2.19 we present
the correlation between the BR for the FCNC decay (assuming no other new physics
is present) and the process qq —+ t5 expected from the g—t—c operator. Observation
of this correlation (or one relating a different single top production process with the
t —> gc decay) could be the smoking gun in identifying [the g-t-c operator as being

responsible for a deviation in single top production.

Detailed simulations of the Z-t—c and 'y-t-c operators have so far been confined to
studies of top decays [64, 65]. The quantity [16.2.c sin 92ml is constrained by low energy
data on flavor-mixing processes to be less than the order of magnitude of 0.05 [64].
These studies indicate that from Run II at the Tevatron top decays should provide
constraints of Ag“ 2 7.9 TeV, 162,, S 0.29, and will not improve the bounds on Am
from the current b —> s 7 limit of about 5 TeV. Of course, as we have argued before, it
was necessary to assume a SM W-t-b interaction in order to use decays to say anything
at all about these operators. The effect of the Z -t-c operator to the inclusive t-channel
production rate is to contribute an additional 0.13 pb at the Tevatron Run II and
12.6 pb at the LHC, assuming test, = 0.29, and including the NLO QCD corrections
for both t and f production. Low energy constraints indicate that nzm = 0.29 requires
[sin 92th S 0.17, and the inclusive cross sections are insensitive when 02.. is varied in
this range. The 7—t-c operator can be studied at a hadron collider through the reaction

'7 c —> t (where the photon is treated as a parton inside the proton) [66], though this

exotic production mechanism suffers from potentially large SM backgrounds.

 

 

 

beca
Sure
may 1

0”? he

71

2.4 Tap Polarization

The polarization of top quarks represents another way to probe the properties of
top interactions. In the SM, the W-t-b vertex is entirely left-handed, which means
that the top polarization information is passed on to the W boson and b quark into
which the top decays. Since the W interaction with the light fermions into which it
decays is also left-handed, the W polarization information is thus also reflected in the
kinematics of its decay products. The same weak interaction is also responsible for
single top production, which has the consequence that single tops also show a large
degree of polarization. The discussion below is based on the SM amplitudes for top

production and decay presented in [34].

2.4.1 The W+ Polarization: The W-t-b Interaction

In order to probe the chiral structure of the W-t-b interaction, it is enough to consider
the W polarization of top decays. As was shown in [34], the left-handed nature of
the SM interaction demands that the produced W bosons be either left-handed or

longitudinally polarized, and predicts the specific ratio of

2

No _ mt ~
N_ .. 2%,, +1”, _ 70%. (2.12)

The degree of W polarization from top decays can be reconstructed by studying the
angle between the W momentum and the charged lepton momentum, in the W rest

frame.

It is desirable to employ top decays in order to probe the W-t-b interaction,
because in the case of a top decay, the W and b are observed, and thus one can be
sure that it is this interaction that is responsible for the effect one is seeing, which
may not be the case if there is new physics in single top production. Further, once

one has probed the chiral structure of the W-t-b interaction, one can then employ

72

this information to unfold the top decay and reconstruct the polarization of the top

itself, as will be explained below.

2.4.2 The Top Polarization

Once the chiral structure of the W-t-b interaction has been probed through top decays,
and the SM left-handed structure verified, the top decay products can be used in order
to study the polarization of the produced top quarks themselves. As we will see,
this can be very useful in determining what sort of new physics is responsible for an
observed deviation in single top production. Currently, there are two important bases
for describing the top polarization. The usual helicity basis measures the component
of top spin along its axis of motion (in the center of mass frame - because the top
mass is large its helicity is not a Lorentz invariant quantity). The so—called “optimized
basis” [67] relies on the SM dynamics responsible for single top production in order
to find a direction (either along the direction of one of the incoming hadrons or
produced jets) which results in a larger degree of polarization for the top quark. In
the discussion below, we will describe the modes of single top production in both

bases, and analyze the particular strengths and weaknesses of each.

Before looking at a particular process or basis, it is worth describing how one
can determine the top polarization from its decay products [34]. A simple heuristic
argument based on the left-handed nature of the W interactions and the conservation
of angular momentum can be made, and is displayed diagrammatically in Figure 2.20.
The analysis is carried out in the rest frame of the top quark, and is slightly different
for a left-handed or a longitudinally polarized W boson participating in the decay. In
the left-handed W case, the fact that the b quark must be left-handed forces it to move
along the direction of the top polarization. The W thus moves against this direction.

When the W decays, the charged lepton (8*) must be right-handed, so it prefers to

 

73

 

 

¢ 3+ & 3+
:5 :5 fl = <=
1 ‘ ¢ :
b t W” v V. W+ => t b
§ c
(a) (b)

 

.. Va
Ve
P fl 2 => 6: = W :5 4,
e- 4: W 't' 5 5 f
e-
(C) ((1)

Figure 2.20: A diagram indicating schematically the correlation between the charged
lepton (8*) from a top decay, and the top spin, in the top rest frame. The arrows on
the lines indicate the preferred direction of the momentum in the top rest frame, while
the large arrows alongside the lines indicate the preferred direction of polarization.
As shown, the 6+ (6’) from at (f) decay prefers to travel along (against) the direction
of the t (t) polarization.

move against the W direction, in the same direction as the top polarization. When
the W boson is longitudinally polarized, it prefers to move in the same direction
as the top spin. Its decay products prefer to align along the W polarization, and
since the W is boosted in the direction of the top polarization, the charged lepton
again prefers to move along the top spin axis. As shown in Figure 2.20, a similar
argument can be made for the 5 spin, but in this case the charged lepton prefers to
move against the 5 spin axis. From this point onward, we restrict our discussion to
top quarks, but it should be clear how they apply to f as well. The simple angular

momentum argument is reflected in a more detailed computation of the distribution,

1 df‘
I‘dcosO

 

(t—->W*b—+£+u¢b) = %(1+cosfl), (2.13)

where 0 is the angle between the top polarization and the direction of the charged

lepton, in the top rest frame, and I‘ is the partial width for a semi-leptonic top decay

 

74

in the SM. In principle, one has only to decide on a scheme for relating the top

polarization to some axis, and one can fit the distribution,
F(cos 0) = 13(1 + cos 0) + L;_4 (1 — cos 0) , (2.14)

to determine the degree of polarization (A) along this axis. In practice, there are
complications arising from the fact that the endpoints of the distribution tend to be
distorted by the cuts required to isolate the signal from the background, and the fact
that in reconstructing the top rest frame, the component of the unobserved neutrino
momentum along the beam axis (p13) is unknown. One may determine this quantity
up to a two-fold ambiguity by requiring the top decay products to have an invariant
mass that is close to 1m. However, the ambiguity in this procedure will also have
some effect on the distribution, and so careful study is required. One can also use the
asymmetry between events with cos0 > 0 and c080 < 0 to characterize the degree
of polarization of the top, which may be helpful if the data set is limited by poor

statistics.

W’ Production

The degree of top polarization in the W‘ process is straight-forward to compute in
the helicity basis [34]. Using the CTEQ4M PDF’s, we find that about 75% of the top
quarks produced through the s—channel process at the Tevatron are left-handed, and

76% of them are left-handed at the LHC.

The optimized basis improves the helicity basis result at the Tevatron by noting
that in the SM, the W‘ process produces top quarks whose polarization is always along
the direction of the initial anti-quark involved in the scattering. At the Tevatron, the
vast majority (~ 97%) of these anti-quarks come from the 17 (which has valence anti-

quarks). Thus, one expects that by choosing to measure the top polarization along

 

75

the if direction in the top rest frame, one can raise the degree of polarization from
75% to 97%. This represents a large improvement for Tevatron polarization studies of
the W‘ process. However, at the LHC there are no valence anti-quarks, and thus no
optimized basis to analyze the W‘ top polarization (though as we have seen, at the

LHC the helicity basis results in a fair degree of left-handed top production anyway).

W-gluon Fusion

The discussion of polarization in the W-gluon fusion process is somewhat tricky,
mostly owing to the fact that as we have seen above, the detailed kinematics of
this process are sensitive to higher orders of perturbation theory. It is clear that the
kinematic region described by the process q b —> q’ t is the dominant one, but a precise
calculation of the interplay between the 2 —> 2 scattering contribution and the 2 —> 3
scattering contribution is still lacking. Thus, one must be careful in claiming what

degree of polarization results from a particular basis.

In the helicity basis, the 2 —+ 2 description has the top quarks 100% left-handed
when produced from the ab —> dt sub—process. In fact, at both Tevatron and LHC
the of b —+ a t sub-process is quite small, and thus the over-all degree of polarization is
about 97%. On the other hand, the 2 -> 3 description shows a degree of polarization
that is much lower, and depends on the choice of the bottom mass used in the com-
putation. This is an indication that this method of computation is not perturbatively
stable. Thus, it is fair to say that the degree of polarization in the helicity basis is

high, but at the moment no reliable determination is available.

The optimized basis once again makes use of the fact that the top polarization
is 100% along the direction of the spectator anti-quark in the reaction. At both
Tevatron and LHC, this is dominantly the spectator jet in the final state. This basis

thus results in a top which is about 96% polarized along the direction of the spectator

76

jet. In [67], it was shown that this basis is also not sensitive to the value of the bottom

mass, and thus is perturbatively reliable.

2.4.3 New Physics and Top Polarization

As we have seen, new physics may alter the structure of single top production. It may
be that the new physics effects will reveal themselves, and tell us something about
their nature by causing a large deviation in one or more of the single top production
cross sections. In that case one can study the distribution of the top polarization
in order to learn something further about the nature of the nonstandard production

mechanism.

In Section 2.3.1, it was demonstrated that either a charged scalar top-pion or W’
gauge boson can have a substantial effect on single top production in the s-channel
mode. Assuming for the moment that such a deviation has been observed, one can
then use the top polarization in order to narrow down the class of models responsible
for such an effect. The W’ boson couples to the left-handed top and bottom quarks,
and thus an analysis of the resulting top polarization will be the same as the SM
prediction. Namely, the helicity basis will show 75% of the tops to be left-handed
(76% at the LHC) and the optimized basis will show 97% at the Tevatron. However,
the 17* has a right-handed interaction, completely at odds to the SM. In fact, there
is another difference between the W’ and the 17* that is also very important. Like
the SM W boson, the W’ is a vector particle, and thus carries angular momentum
information between the initial state and final state in the s-channel process. However,
the 17*, as a scalar particle, does not carry such information. Thus, the optimized
basis, which relies on the correlation between top spin and the initial (7 momentum
fails to apply to a scalar production mechanism, and if one were to use it to analyze

the polarization of the top coming from this type of new physics effect, one would

 

77

come to the wrong conclusion that the produced tops were unpolarized. On the other
hand, in the helicity basis the top quarks produced from the 77* show very close to
100% right-handed polarization. This demonstrates the utility of using both bases.
If there is new physics in single top production, not only is it unclear at the outset
which basis will show a larger degree of polarization, but we can use them together
to distinguish a vector from a scalar exchange, thus learning about the nature of the

new particle without directly observing it.

Study of polarization can also be useful in disentangling the operators in the
effective Lagrangian in Equations 2.9 and 2.10. As we saw, those operators have
left-handed and right-handed versions, and thus the distribution of top polarizations
will depend on the relative strength of the two. Thus, by studying top polarization,
one could begin to disentangle the chiral structure of the operator responsible for a
deviation in single top production, giving further insight into the nature of the full

theory that accurately describes higher energies.

2.5 TOp Quark Properties

Having gone over in detail the physics one can probe with single top production,
it is worth summarizing what we have learned and examining how one can use the
difl'erent top quark observables to extract information about the top that maximizes
the available information. In the preceeding sections we have seen that single top
production allows one to measure the magnitude of the top’s weak interactions (unlike
top decays). The three modes of single top production are sensitive to different types
of new physics. All three modes are sensitive to modification of the W-t—b interaction,
with the tW‘ mode distinguished by the fact that it is rather insensitive to any other

types of new physics. The s-channel mode is sensitive to certain types of additional

78

particles. And the t-channel mode is sensitive to physics which modifies the top decay
properties, in particular to FCNC interactions. In this light, it is rather unfortunate
that the tW‘ mode is so small at the Tevatron that it is not likely to be useful there,
as it can allow one to measure the strength of the W-t-b vertex, which would be a

good first step in disentangling the information from the s— and t-channel modes.

Without the tW‘ mode, one will most likely have to study the correlation of the
s- and t— channel rates in the plane of a, — at in order to attempt to understand
if a new physics effect is present, and how one should interpret it if it is observed.
In Figure 2.21 we show this plane, including the SM point (with the contour of 30
deviation around it) and the points from the the top-flavor model (with M z: = 900
GeV and sin“¢ = 0.05), the top-color model with a charged top-pion (with mass
m,* = 250 GeV and tR-cR mixing of 20%) and a FCNC Z—t-C operator (with NZ“; =
0.29, sin Oz“, = 0.2, and (11%“ = (3%“. = 0). This illustrates how to use the knowledge
we have about the sensitivity of the W" and W-gluon fusion modes to find a likely
explanation for a new physics effect. A deviation in a, that is not also reflected in at
is most likely due to the effect of nonstandard particles. A deviation in at that is not
also seen in a, is likely from a FCNC. A deviation that is comparable in both rates
is most likely from a modification of the W-t—b interaction. In the very least, if the
SM is a sufficient description of single top production, the fact that the two rates are
consistent will allow one to use them to extract V» with confidence that new physics

is not distorting the measurement.

Additional information is provided by polarization information. By studying the
W polarization from top decays, one learns about the nature of the W-t-b interac-
tion. By studying the top polarization, in both the helicity and optimized bases, one
can learn more about the chiral structure of nonstandard top interactions, either by

probing the chiral structure of the interactions, or even the scalar/ vector nature of a

 

79

 

4,0 rlrrlrrrrlrlrrlrrrr

3.5

3.0

a t (Pb)

2.5

2.0

llllllllLlllllllllJlllLJ

 

 

1 5 l l I 1 l L l l 1 l l J 1 l l l I l l
O

 

0.. (pb)

Figure 2.21: The location of the Tevatron SM point (the diamond) in the 03-0; plane,
and the 30 deviation curve. Also shown are the points for the top-flavor model (with
M’z = 900 GeV and sin“ <13 = 0.05) as the square, the FCNC Z-t-c vertex (16,“c = 0.29)
as the circle, and a model with a charged top-pion (771,4 = 250 GeV and tR-cR mixing
of ~ 20%) as the cross. All cross sections sum the t and 1? rates.

80

virtual particle participating in single top production.

Chapter 3

Higgs with Enhanced Yukawa
Coupling to Bottom

3.1 Introduction

As we have seen, the mystery of the EWSB is one of the primary challenges for
modern particle physics. The large top mass, of the same order as the EWSB scale,
suggests that top may play a special role in the generation of mass. This occurs in
models with dynamical top-condensate or top-color scenarios [20, 21] as well as in
SUSY theories [18]. Since the bottom quark is the iso-spin partner of the top quark,
its Yukawa coupling with a Higgs boson can be closely related to that of the top
quark. In [68], we demonstrated that because of the small mass of bottom (711;, ~ 4.5
GeV) relative to top (111, ~ 175 GeV), studying the b Yukawa coupling can effectively

probe new physics beyond the SM.

In this Chapter, we study the detection of a Higgs boson (111) at hadron colliders
in the context of models where the bottom has an enhanced Yukawa coupling (go) to
the scalar Higgs. We begin with a model—independent analysis for Higgs production
associated with bb jets, through the reactions p p —) ¢bb -—> bbbb, and pp —> ¢bb —->
bbbb at the Tevatron Run II and LHC, to determine their ability to probe models of

dynamical EWSB and SUSY theories through this process.

81

 

82

3.2 Signal and Background

We are interested in studying production of (1155 —> bbbb at the Run II of the Tevatron
and the LHC. The signal events result from QCD production of a primary bb pair,
with a Higgs boson (¢) radiated from one of the bottom quark lines as shown in
Figure 3.1. The Higgs boson then decays into a secondary bb pair to form a bbbb final
state. Because our detection strategy relies upon observing the primary b quarks in
the final state (and thus demands that they have large transverse momentum), our
calculation of the ¢bb signal rate from diagrams such as those shown in Figure 3.1 is
expected to be reliable. This is in contrast to the inclusive rate of (5 production at a
hadron collider, in which one does not require a final state topology with four distinct
jets. In this case a calculation based upon Feynman diagrams such as those shown
in Figure 3.1 may not be reliable. It would be better to consider the Higgs boson
production via bottom quark fusion, such as bb —> 65 and gb —-> ¢b, with cares to avoid
double counting its production rate [69]. (This calculation would resum some large
logarithms which are included in the definition of the bottom parton distribution
function within the proton, much as was true for single top production in Chapter 2.)
We have chosen to search in the four jet final topology because the QCD background
for 3 jets is much larger than that for 4 jets, and thus it would be more difficult
to extract a 3 jet signal. Since the signal consists of four b (including 5) jets, the
dominant backgrounds at a hadron collider come from production of Z bb —> bbbb, seen
in Figure 3.2, purely QCD production of bbbb, seen in Figure 3.3, and bbj j , where j
indicates a light quark or a gluon, shown in Figure 3.4 which can occasionally fake a

b—jet signature in the detector.

In order to derive model-independent bounds on the couplings of the scalar par-

ticles with the bottom quark, we consider K, the square-root of the enhancement

83

  

.DI
c‘l
U‘I

g

Figure 3.1: Representative leading order Feynman diagrams for ¢bb production at a
hadron collider. The decay 43 —+ bb is not shown.

q b b
g

Z Z

a E '6
g

Figure 3.2: Representative Feynman diagrams for leading order Z bb production at a
hadron collider. The decay Z —) bb is not shown.

 

b

g b
, '5

g ‘ 5

 

Figure 3.3: Representative leading order Feynman diagrams for QCD bbbb production
at a hadron collider.

 

84

 

 

 

 

b
_ b
9 b s q
V A
< i
g B
b
q 5'

 

’ q
s WLHfir/oe 8
Figure 3.4: Representative leading order Feynman diagrams for QCD bbj j production
at a hadron collider.

 

85

factor for the production of qub —> bbbb over the SM prediction. By definition,

_ yb
K — (yb)SM , (3.1)

 

in which (115)3M = fimb/v is the SM bottom Yukawa coupling and yb is the bot-
tom Yukawa coupling in the new physics model under the consideration. The decay
branching ratio of 63 to bb is model-dependent, and is not included in the calculations
of this section. (Namely, BR(q‘) -> bb) is set to one). When analyzing the specific

models in the following sections, we include the appropriate BR for that model.

We compute the signal and the backgrounds at the parton level, using leading
order (LO) results from the MADGRAPH package [70] for the signal and the back-
grounds, including the sub—processes initiated by qq and gg (and in the case of bbj j ,
qg and fig). While the complete next-to—leading order (NLO) calculations are not
currently available for the signal or background cross sections, we draw upon existing
results for high pr bb production at hadron colliders [71] and‘thus estimate the NLO
effects by including a k-factor of 2 for all of the signal and background rates. We
will estimate the theoretical uncertainty in the signal and background cross sections
below. We use the CTEQ4L [39] parton distribution functions (PDF’s) and set the
factorization scale, so, to the average of the transverse masses of the primary 5 quarks,
and the boson (43 or Z) transverse mass1 for the ¢bb and Z bb processes, and use a
factorization scale of [10 = J9, where 9 is the square of the partonic center of mass
energy, for the bbbb and bbj j background processes. It is expected that a large part
of the total QCD bbbb and bbj j rates at the Tevatron or LHC energies will come from
fragmentation effects, which we have neglected in our LO matrix element calculation.
However as we shall see below, due to the strong pT and isolation cuts which are

necessary to improve the signal-to-background ratio, we expect that these effects will

 

. . 2
1The transverse mass of particle i is given by mg?) 5 \/ m,“ + p35) .

86

be suppressed, and thus will only have a small effect on our results. Similarly, we
expect that after imposing the necessary kinematic cuts, the signal and the back-
ground rates are less sensitive to the choice of the factorization scale. In this section,
unless otherwise noted, we will restrict our discussion of numerical results to a signal
rate corresponding to a scalar mass of m¢ = 100 GeV, and an enhancement factor of
K = int/ms z 40. We will consider the experimental limits which may be placed on

K as a function m¢ below.

In order to simulate the detector acceptance, we require the p1- of all four of
the final state jets to be pr 2 15 GeV, and that they lie in the central region of
the detector, with rapidity I17] 3 2. We also demand that the jets are resolvable as
separate objects, requiring a cone separation of AR 2 0.4, where AR E W.
(Acp is the separation in the azimuthal angles.) In the second column of Table 3.1 we
present the number of events in the signal and background processes at the Tevatron
Run II which satisfy these acceptance cuts, assuming 2 fb‘lof integrated luminosity.
As can be seen, the large background makes it difficult to observe a signal in the
absence of a carefully tuned search strategy to enhance the signal-to-background
ratio. In presenting these numbers, we have assumed that it will be possible to trigger
on events containing high pr jets (and thus retain all of the signal and background

events). This capability is essential for our analysis.

The typical topology of the bottom quarks in the signal events is a “lop-sided”
structure in which one of the bottom quarks from the Higgs decay has a rather high
pr of about m4, / 2, whereas the other three are typically much softer. Thus, the signal
events typically have one bottom quark which is much more energetic than the other
three. On the other hand, the QCD bbbb (or bbj j ) background is typically much more

symmetrical, with pairs of bottom quarks (or fake b’s) with comparable p7». In order

87

Table 3.1: The signal and background events for 2 fb'lof Tevatron data, assuming
m4, = 100 GeV, 2Am¢ = 26 GeV, and K = 40 after imposing the acceptance cuts, pT
cuts, and reconstructed m¢ cuts described in the text. (A k-factor of 2 is included in
both the signal and the background rates.)

Process Acceptance Cuts pr Cuts AR Cut AM Cut

366 4923 1936 1389 1389
be 1432 580 357 357
55513 5.1 x 104 3760 1368 1284
bbjj 1.2 x 107 1.5 x 106 6.3 x 105 5.9 x 105

 

to exploit this, we order the b quarks by their transverse momentum,
1 2 3 4
111’ Z 101’ 2 111’ 2 191’. (3-2)

and require that the bottom quark with highest transverse momentum have p(T1 ’ 2 50
GeV, and that pg?) 2 30 GeV and p53“) 2 20 GeV. In the third column of Table 3.1
we show the effect of these cuts on the signal and backgrounds. As can be seen,
these cuts reduce the signal by about 60%, while drastically reducing the QCD bbbb
background by about 90%.

Since the pT spectrum of the leading jets is determined by the mass of the scalar
boson produced, the leading pr cuts can be optimized to search for a particular ms.
From the discussion above, the optimal cut on p513) can be seen to be close to m¢/ 2
whereas the optimal cut on pg“) is somewhat lower (generally closer to m¢/3). We
adopt these optimized p7 cuts for each mass considered, when estimating the search

reach of the Tevatron or LHC.

Another effective method for reducing the QCD background is to tighten the
isolation cut on the bottom quarks. In the QCD bbbb background, one of the bb pairs

is preferentially produced from gluon splitting. Because of the collinear enhancement,

 

88

the invariant mass of this bb pair tends to be small, and the AR separation of these
two b’s prefers to be as small as possible. On the contrary, in the signal events,
the invariant mass of the bb pair from the tb-decay is on the order of m¢, and the
AR separation is large because the angular distribution of b in the rest frame of the
scalar d) is flat. Thus, by increasing the cut on AR to AR 2 0.9 we can improve
the significance of the signal. As shown in column four of Table 3.1, this cut further
decreases the signal by about 30%, and the QCD bbbb background by about 65%. In

the end, their event rates are about the same.

One can further improve the significance of the signalby attempting to reconstruct
the mass of the scalar resonance. This can be difficult in principle, because one does
not know a priori what this mass is, or which bottom quarks resulted from the 03
decay in a given event. It may be possible to locate the peak in the invariant mass
distribution of the secondary b quarks resulting from the (b decay, though with limited
statistics and a poor mass resolution this may prove impractical. However, one can
also scan through a set of masses, and provide 95% CL. limits on the presence of
a Higgs boson (with a given enhancement to the cross section, K) in the bbbb data
sample for each value of m4, in the set. In order to do this, we assume a Higgs mass,
and find the pair of b quarks with invariant mass which best reconstructs this assumed
mass. We reject the event if this “best reconstructed” mass is more than 2Am¢ away
from our assumed mass, where 2Am¢ is the maximum of either twice the natural
width of the scalar under study (F4) or the twice experimental mass resolution. We

estimate the experimental mass resolution for an object of mass 111,, to be,

 

Under this assumption, the natural width of the bosons in the specific models of new

physics considered below are usually smaller than this experimental mass resolution.

89

As shown in the fifth column of Table 3.1, this cut has virtually no eflect on the signal
or be background (for a 100 GeV Higgs) while removing about another 10% of the
bbbb background.

As will be discussed below, the natural width of the Higgs bosons in both the
MSSM and the models of strong EWSB that we wish to probe in this paper are
generally much smaller than our estimated experimental mass resolution, and thus
one might think that an improved experimental mass resolution could considerably
improve the limits one may place on a scalar particle with a strong b interaction.
However, the models in which we are interested generally have one or more nearly
mass-degenerate bosons with similarly enhanced bottom Yukawa couplings. If the
extra scalars are much closer in mass than the experimental mass resolution (and
the natural width of the bosons), the signal can thus include separate signals from
more than one of them. Thus there is potentially a trade-off in the AM cut between
reduction of the background and acceptance of the signal from more than one scalar
resonance. In order to estimate the potential improvement for discovering a single
Higgs boson, we have examined the effect on the significance one obtains if the cut
on the invariant mass which best reconstructs m4, is reduced to Am.) as opposed to
2Am¢ as was considered above. We find that this improved mass resolution further
reduces the QCD bbbb background by about another 40%. Assuming four b tags (as
discussed below), this improved mass resolution increases the significance of the signal
from about 12.2 to 14.6, which will improve the model-independent lower bound on
K by about 10%. Thus, an improved mass resolution would most likely be helpful in

this analysis.

Another method to further suppress background rate is to observe that in the
background events, the b quarks whose invariant mass best reconstructs m¢ come from

the same gluon. This is because, after imposing all the kinematical cuts discussed

90

above, the matrix elements are dominated by Feynman diagrams in which one very far
ofl-shell gluon decays into a bb pair, as opposed to interference of many production
diagrams, which dominates the lower invariant mass region. Thus, for m¢ greater
than about 100 GeV, the background event produces b quarks with the characteristic
angular distribution of a vector decaying into fermions, 1 + cos“ 0, in the rest frame
of the bb system. This is distinct from the signal distribution, which comes from a
scalar decay, and is flat in cos 0. Thus, for masses above 100 GeV, we further require
|cos0| S 0.7 after boosting back to the rest frame of the bb pair which we have

identified as coming from the scalar boson d).

In order to deal with the large QCD bbj j background, it is important to be able
to distinguish jets initiated by b quarks from those resulting from light quarks or
gluons. We estimate the probability to successfully identify a b quark passing the
acceptance cuts outlined above to be 60%, with a probability of 0.5% to misidentify
a jet coming from a light quark or gluon as a b jet [72]. In Table 3.2 we show the
resulting number of signal and background events passing our optimized cuts at the
Tevatron, assuming 2 fb‘lof integrated luminosity, after demanding that two or more,
three or more, or four b-tags be present in the events, and the resulting significance
of the signal (computed as the number of signal events divided by the square root of
the number of background events). We find that requiring 3 or more b—tags results
in about the same significance of 12.20 as requiring 4 b—tags. However, we see that
for the chosen parameters (m), = 100 GeV and K = mt/mb z 40), even with only
2 or more b-tags, one arrives at a significance of about 30, and thus has some ability
to probe a limited region of parameters. From the large significance, we see that the
Tevatron may be used to place strong constraints on Higgs particles with enhanced
bottom quark Yukawa couplings, and that the ability to tag 3 or more of the bottom

quarks present in the signal can probe a larger class of models (or parameter space of

91

Table 3.2: The signal and background events for 2 fb"lof Tevatron data, assuming
m4. = 100 GeV, 2Am¢ = 26 GeV, and K = 40 for two or more, three or more, or four
b—tags, and the resulting significance of the signal.

 

Process 2 or more b-tags 3 or more b-tags 4 b-tags
¢bb_ 1139 660 180
be 293 170 46
bbbb 1054 610 166
bbjj 1.2 x 105 2141 4

Significance 3.3 12.21 12.25

 

the models) as compared to what is possible if only 2 or more of the bottom quarks
are tagged. In the analysis below, to allow for the possibility that the bbjj background
may be somewhat larger than our estimates, we require 4 b-tags, though as we have
demonstrated above, we do not expect a large change in the results if 3 or 4 b-tags

were required instead.

This analysis can be repeated for any value of m¢, using the corresponding pr
for that particular mass described above. It is interesting to note that the signal
composition in terms of the gg or qq initial state depends on the collider type and the
mass of the produced boson, which controls the type of PDF and the typical region
of :2: ~ m: / S at which it is evaluated. At the Tevatron, for m, = 100 GeV, the signal
is 99% gg initial state before cuts, and 87% after cuts, while for m,» = 200 GeV, it
is 99% 99 initial state before cuts, and 85% after cuts. Thus, at the Tevatron, one
ignores about 15% of the signal if one relies on a calculation employing only the gg
initial state. At the LHC, for m, = 100, the signal is very close to 100% gg initial

state before cuts and 99% after cuts, and for rm» 2 500 GeV, it is 99% 99 initial state

92

Table 3.3: Event numbers of signal (Ns), for one Higgs boson, and background (N B)
for a 2 fb’lof Tevatron data and a 100 fb’lof LHC data, for various values of m4),
after applying the cuts described in the text, and requiring 4 b-tags. An enhancement
of K = 40 is assumed for the signal, though the numbers may be simply scaled for
any Knew by multiplying by (Knew/40)“.

 

Tevatron LHC
m¢ (GBV) N3 N3 N5 N3

75 583 640 3.4 x10“ 4.8 x10“
100 180 216 2.0 x106 3.0 x10“
150 58 92 9.2 x105 1.2 x106
200 17 31 4.2 ><105 5.6 x105
250 4.8 8.8 1.9 x105 2.0 x105
300 1.3 2.1 83000 70000
500 12000 5700
800 1500 406
1000 407 70

 

before cuts, and 99% after cuts. This indicates that at the LHC, very accurate results
are possible from a calculation considering only the gg initial state. The resulting
numbers of signal and (total) background events after cuts for various boson masses

are shown in Table 3.3.

From these results, one may derive the minimum value of K, Kmin, for a scalar
boson with mass 711,) to be discovered at the Tevatron or the LHC via the production
mode bb¢(—+ bb). Similarly, if signal is not found, one can exclude models which pre-
dict the enhancement factor K to be larger than Kmin. To give a model-independent
result, we assume that the width of the (15 is much less than the estimated experi-
mental mass resolution defined above, which is the case for the models studied in
this paper. We determine Kmin by noting that in the presence of a Higgs boson with

enhanced bottom Yukawa couplings, the number of expected signal events passing

93

NéSM’, where N§SM) is the number of

our selection criterion is given by Ns = K “
signal events expected for a scalar of mass m, with SM coupling to the b quark (as-
suming Br(¢ —+ bb) = 1), whereas the number of background events expected to pass
our cuts, N3, is independent of K. Thus, requiring that no 95% CL. deviation is

observed in the bbbb data sample (and assuming Gaussian statistics) determines

l 1.96 \/NB
Kmin : N(SM) , (3.4)
.S'

where 1.960 is the 95% CL. in Gaussian statistics. In Figure 3.5, we show the result-
ing 95% CL. limits one may impose on Kmin as a function of m¢ from the Tevatron
with 2, 10, and 30 fb‘land from the LHC with 100 fb'l, as well as the discovery reach
of the LHC at the 50 level. Our conclusions concerning the LHC’s ability to probe
a Higgs boson with an enhanced b Yukawa coupling are very similar to those drawn
in [73], but are considerably more optimistic than those in [74], where the conclusion
was that the bbj j background is considerably larger than our estimate (though there
are elements of the search strategy which differ between those of [74] and ours as
well, and their simulation of the ATLAS detector is certainly more sophisticated).
In [74] the backgrounds were simulated using PYTHIA [75] to generate two to two
hard scatterings and then generating the additional jets from a parton showering al-
gorithm. As noted above, in the light of the strong (ordering of) pr and isolation cuts
applied to select the signal events, we feel that a genuine four body matrix element

calculation such as was used in our analysis provides a more reliable estimate of this

background.

We have examined the scale and PDF dependence of our calculation for the signal
and background rates at the Tevatron, and find that in varying the scale between one
half and twice its default choice (defined above), [.7 = 110/2 and u = 2110, the ¢bb signal

and be background rates both vary from the result at u = #0 by about 30%, while

94

 

120 IT I 7 I r 7 r r I r 71w

     
 

100

IIVIUTITWTIF

Minimum Enhancement
8

-
e
e
O.-
.
.-

e
5".
.e

  

T11TTTIUUWTI'

e
e...
e

l l

 

 

 

 

25 TII'IYW—IIIIIII'IUITTYY

YTII
j

20

15

Irrlfrr1l
\
1

Minimum Enhancement

 

‘

 

 

I+AAIIIILIIILLILLLL

200 400 600 800 1000
Hia- l-u (00V)

 

Figure 3.5: In the upper figure is the model-independent minimum enhancement
factor, Kmin, excluded at 95% CL. as a function of scalar mass (ms) for the Tevatron
Run II with 2 fb—1(solid curve), 10 fb'1(dashed curve) and 30 fb’1(dotted curve).
The lower figure shows the same factor, Kmin, excluded at 95% CL. (solid curve) and
discovered at 50 (dashed curve) as a function of 1nd, for the LHC with 100 fb‘“.

95

the bbbb and bbj j backgrounds vary by about 45%. This strong scale dependence
is indicative of the possibility of large higher order corrections to the leading order
rate. Thus, in order to better understand the true signal and background rates, it
would be useful to pursue these calculations to NLO. We have also compared the
difference in the results from the MRRS(R1) PDF [40] and the CTEQ4L PDF, and
find a variation of about 10% in the resulting signal and background rates. Since these
separate sources of uncertainty (from PDF and scale dependence) are non-Gaussianly
distributed, there is no way to rigorously combine them. Thus, we conservatively
choose to add them linearly, finding a total uncertainty of about 40% in the signal
rate (NéSM’), and 50% in the background rate (NB). From the derivation of Kmin
above, we see that these uncertainties in signal and background rate (which we assume

to be uncorrelated) combine to give a fractional uncertainty in Kmin,

_ (SM) 2 N 2
6Km1n = (“NS + 6 B , (35)
I{min 2 NévSM) 4 NB

where 6N§SM’ and 6N3 are the absolute uncertainties in N SM) and N s, respectively.

 

From this result, we see that in terms of a more precise theoretical determination of
Kmin, one gains much more from a better understanding of the signal rate than a bet-
ter determination of the backgrounds. Applying our estimate of the uncertainty from
PDF and scale dependence to Eq. (3.5), we find an over-all theoretical uncertainty in

Kmin of about 25%.

3.3 Implications for Models of Dynamical EWSB

Examples of the strongly interacting EWSB sector with composite Higgs bosons are
top-condensate and top-color models [20, 21], in which new strong dynamics asso-

ciated with the top quark play a crucial role for top and W, Z mass generation. A

96

generic feature of these models is a naturally large Yukawa coupling of the bottom
quark, of the same order as that of top (y. ~ 1), due to the infrared quasi-fixed-point
structure [76] and particular boundary conditions for (yb,yt) at the compositeness

scale.

3.3.1 The Two Higgs Doublet Extension of the BHL Model

The effective theory of the top—condensate model is the SM without its elementary
Higgs boson, but with 4-Fermi interaction terms induced from (unspecified) strong
dynamics at a high scale A instead. The minimal Bardeen-Hill-Lindner (BHL) top—
condensate model with three families [20], contains only one type of 4-Fermi vertex
for < it > condensation which generates the masses for the top, and the W, and Z
bosons. However, the top mass required to obtain the correct boson masses is too
large to reconcile with experiment. Thus, we consider the two Higgs doublet extension
(2HDE) [78] as an example (which, with some improvements [20, 46], is expected to
produce an acceptable 1m), and examine its prediction for the ¢bb rate. The 4-Fermi
interactions of the 2HDE model produce condensates in both the ti and bb channels,
which generate the EWSB and induce two composite Higgs doublets <I>t and (Pb. The

Yukawa interactions take the form,
yt (‘31, (Pt ta + H.c.) + yb (‘31, T1, 03 + HUG) . (36)

In the above equation, III, is the left-handed third family quark doublet and t}; is
the right-handed top, and so forth. This model predicts EMA) = yb(A) >> 1 at
the scale A [20, 78]. In fact, one finds that y¢(u) z yb(p) for any [1 < A, because
the renormalization group equations governing the running of 3;. and yb are identical
except for the small difierence in the the t and b hypercharges [20, 76]. Due to the

dynamical < it > and < bb > condensation, the two composite Higgs doublets develop

97

< (D. > = (v.,0)T /\/2 (3.7)

<¢b> = (0,Ub)T/\/2.

The bottom mass is given by m, = yb vb/\/2, and must match the experimental value
at scale 11 = 111,, ~ 4.5 GeV. Assuming the Yukawa couplings yt ~ yb ~ 1, this requires

the two VEV’s to have ratio,

1’1 2: 39 = tan 5. (3.8)

vb

We thus see that in this model, the bottom quark has a Yukawa coupling of the same

order as the top quark, which implies that tan 6 = 1),/11,, is naturally large.

The 2HDE has three neutral scalars, the lightest with enhanced bottom coupling

being the pseudoscalar,
P = fish. 3 Im <53 + cos 5 Im <52), (3.9)

whose mass (Mp) is less than about 233 GeV for A = 1015 GeV [78]. Given yo and
Mp, one can calculate the production rate of P bb(-—> bbbb) at hadron colliders, and
thus for a given Mp one can determine the minimal 31), value needed for the Tevatron
and LHC to observe the signal. As shown in Figure 3.6a, the Tevatron Run II data
with 2 fb'1 will exclude such a model with Mp ~ 200 GeV at 95%C.L.

3.3.2 Top-color Assisted Technicolor

The top-color-assisted technicolor models (TCATC) [21] postulate the gauge structure
9 = SU(3)1 x SU(3)2 x U(1)1 x U(1)2 x SU(2)L at the scale above A to explain
the dynamic origin of the 4-Fermi couplings described above. At A ~ 1 TeV, g

spontaneously breaks down to SU(3)C x U(1)Y x SU(2)“ and additional massive

98

 

 

 

     

 

 

 

I I I I l I I I I I I I I I 1 I r I: I I j I I I I I I Ifi I I I I I I
C (0) 5 (b) _
, Theor. Upper Limit [
(A=1C[ “GeV) 5 [
‘02 7 Ii [ 1o2
[- é ‘
t tonfi=m,/m. E I
.. : [
1' . 1
Q. : [ a;
c ,. : I \
.8 L : 1 >1
Tevatron E [
,0 I Tevatron :10
:3015" 5 i 3
.- “..e+' [
r- .ofl'”......”.LHC i [ I OOfbd "
.--' :
7..--’"10075" : ’ ‘
1 a LHC _
E I
1 _ l - 1
_111L1L_1_Ll LIL—11111111 _1__l 1 liIJil 1 11141 I.
100 200 200 400
M,(GeV) M..(GeV)

Figure 3.6: The reach of the Tevatron and LHC for the models of (a) 2HDE and (b)
TCATC. Regions below the curves can be excluded at 95%C.L. In (b), the straight
lines indicate y¢(p = 7m) for typical values of the top-color breaking scale, A. yb is
predicted to be very close to 3],.

Q.‘

99

gauge bosons are produced in color octet (8“) and singlet (2’) states. Below the

scale A, the effective 4-Fermi interactions are generated in the form,

 

4 2 - - — -
£4F = A—72r [(K + Q—ICVLC) ‘1”, t3 tn ‘1’], + (K - 9,2,6) ‘1’], b3 b3 \I’L] , (3.10)

where K and m originate from the strong SU(3)1 and U (1)1 dynamics, respectively.
In the low energy effective theory at the EWSB scale, two composite Higgs doublets

are induced with the Yukawa couplings

yt = \/47r (K. + 2931/.) g (3.11)

y" _ I/‘m ('9 9N.) ’

It is clear that, unless n1 is unnaturally larger than 5, yo is expected to be only slightly

 

 

 

 

below y,. The U (1)1 force is attractive in the < t-t > channel but repulsive in the
< 5b channel, thus t, but not b, acquires a dynamical mass, provided yb(A) < ycm =
M < y¢(A). (In this model, b acquires a mass mainly from a top-color instanton
effect [21].) Furthermore, the composite Higgs doublet in, but not in, develops a
VEV, i.e., vt aé O and Up, = 0.

In TCATC, the top-color interaction generates mt, but is not responsible for the
entire EWSB. Thus, A can be as low as 0(1 — 10) TeV (which avoids the severe fine-
tuning needed in the minimal models [20, 78]), and correspondingly, v; = 64- 88 GeV
for A = 1 — 5 TeV by the Pagels—Stokar formula. The smaller value of 12, predicted in
the TCATC model, compared to v = 246 GeV makes the top coupling to <I>t stronger,
i.e., y, = 2.8 — 3.9 at p = 1m, than in the SM (yt ~ 1). As explained above, this
results in a large bottom Yukawa interaction, yb, as well. Thus, the neutral scalars
hb and Ab in the doublet in, which are about degenerate in mass, have an enhanced

coupling to the b—quark.

100

In Figure 3.6b, we show the minimal value of yb/ (315)“, needed to observe the
TCATC model signal as a function of Mhb. As shown, if MM is less than about
400 GeV, the Tevatron Run II data can effectively probe the scale of the top-color
breaking dynamics, assuming the TCATC model signal is observed. If the signal is
not found, the LHC can further explore this model up to large Mh.- For example,
for M1,, = 800 GeV, the required minimal value of yb/(yb)3M is about 9.0 at 95%C.L.
Similar conclusions can be drawn for a recent left—right symmetric extension [79] of

the top-condensate scenario, which also predicts a large b-quark Yukawa coupling.

3.4 Implications for Supersymmetric Models

The EWSB sector of the MSSM model includes two Higgs doublets with a mass
spectrum including two neutral CP-even scalars h” and H0, one CP-odd pseudoscalar
A0 and a charged pair H i. The Higgs sector is completely determined at tree level by
fixing two parameters, conventionally chosen to be the ratio of the VEV’s, tan fl, and
the pseudoscalar mass, m A [80]. At loop level, large radiative corrections to the Higgs
boson mass spectrum are dominated by the contributions of top quarks and squarks
in loops [81]. In this study we employ the full one loop results [82] to generate the
Higgs mass spectrum assuming the sfermion masses, u, scalar tri-linear parameters,
and SU(2)L gaugino masses at the electro-weak scale are those chosen in the LEP
Scan A2 set. There is some sensitivity to this choice of parameters, coming from the

Higgs mass spectrum and coupling to bottom quarks [68, 83].

The parameter tan fl is free in the MSSM, and the Higgs mass is constrained by
mh, mA > 75 GeV for tan ,6 > 1 [13]. Since the couplings of h°-b—I}, Ho-b-5 and Ao-b-E
are proportional to sin a/ cos 5, cos a/ cos fl and tan ,8, respectively, they can receive

large enhancing factor when tan ,6 is large. This can lead to detectable bob?) signal

101

events at the LHC, as was previously studied in [73]. We calculate the enhancement
factor K predicted by the MSSM for given values of tan fl and mA. In Figure 3.7 we
present the discovery reach of the Tevatron and the LHC, assuming the LEP Scan
A2 soft-breaking parameters, and that all the superparticles are so heavy that Higgs
bosons will not decay into them at tree level. For comparison, the region that will be

covered by LEP II is also shown.

The BR for 45 —+ b5 is close to one for most of the parameter space above the
discovery curves. Moreover, for tan 5 >> 1, the ho is nearly mass-degenerate with
the A0 (if m4 is less than ~120 GeV) and otherwise with H0. We thus include
both scalars in the signal rate provided their masses differ by less than Amh. The
MSSM can also produce additional b13125 events through production of h°Z —) bbe;
and h°A° —-) bI-JbI-J, however these rates are expected to be relatively small when the
Higgs-bottom coupling is enhanced, and the resulting kinematics are different from
the $115 signal. Thus we conservatively do not include these processes in our signal

rate.

From Figure 3.7 we deduce that if a signal is not found, the MSSM with tan ,8 >
45 (30, 20) can be excluded for mA up to 200 GeV at the 95% CL. by Tevatron data
with a luminosity of 2 (10, 30) fb‘l; while the LHC can exclude a much larger mA
(for m A = 800 GeV, the minimal value of tan fl is about 5). These Tevatron bounds
thus improve a recent result obtained by studying the 5511 channel [84]. We note
that studying the ¢b5 mode can probe an important region of the tan fl-mA plane
which is not easily covered by other production modes at hadron colliders, such as
pp —+ tt-+ ¢(—+ 77) + X and pp -—> d>(—) ZZ“) + X [85]. Also, in this region of

parameter space the SUSY Higgs boson ho is clearly distinguishable from a SM one.

The above results provide a general test for many SUSY models, for which the

102

 

 

l l

 

OllllIlllllllllIllJll_1_L|*

100 200 300
M,(GeV)

Figure 3.7: The regions above the curves in the tan fi-mA plane can be probed at the
Tevatron and LHC with a 95% CL. The soft breaking parameters correspond to the
LEP Scan A2 set. The region below the solid line will be covered by LEP II.

 

103

MSSM is the low energy effective theory. In the MSSM, the effect of SUSY breaking is
parametrized by a large set of soft—breaking (SB) terms (~O(100)), which in principle
should be derived from an underlying model. We discuss, as examples, the Supergrav-
ity and Gauge-mediated (GM) models with large tan ,6. In the supergravity-inspired
model [86] the SUSY breaking occurs in a hidden sector at a very large scale, of
0(101°“11) GeV, and is communicated to the MSSM through gravitational interac—
tions. In the simplest model of this kind, all the SB parameters are expressed in
terms of 5 universal inputs. The case of large tan 5, of 0(10), has been examined
within this context [87], and it was found that in such a case mA ~ 100 GeV. Hence,

these models can be cleanly confirmed or excluded by measuring the bhbf) mode at

the Tevatron and LHC.

The GM models assume that the SUSY-breaking scale is much lower, of 0(104‘5)
GeV, and the SUSY breaking is communicated to the MSSM superpartners by ordi-
nary gauge interaction [48]. This scenario can predict large tan ,6 (~ 30). However,
some models favor m A Z, 400 GeV [88], which would be difficult to test at the Teva—
tron, though quite easy at the LHC. Nevertheless, in some other models, a lighter
pseudoscalar is possible (for instance, tan fl = 45 and m A = 100) [89], and the bbbh

mode at hadron colliders can easily explore such a SUSY model.

3.5 Conclusions

In conclusion, the large QCD production rate at a hadron collider warrants the de-
tection of a light scalar with large ¢-b—5 coupling. This process can provide useful
information concerning dynamical models of EWSB and on the MSSM, either through

discovery or by limiting the viable region of parameters in the model.

At LEP—II and future e+e‘ linear colliders, because of the large phase space sup-

104

pression factor for producing a direct 3-body final state as compared to first producing
a 2-body resonant state, the Iii-2A” and bI-2h0 rates predicted by the MSSM are domi-
nated by the production of A0 h0 and h° Z pairs via electroweak interactions. Hence,
the 6+8_ collider is less able to directly probe the ¢-b—5 coupling. This has the effect
that our process is complimentary to the the LEP studies, in that it is sensitive to a

different region of SUSY parameter space.

Chapter 4

Associated Production of Gauginos
with Gluinos at NLO

As we saw in Chapter 1, one of the attractive solutions to the problems with the
Higgs sector of the SM is to introduce weak scale supersymmetry, which removes the
instability of the Higgs mass under quantum corrections, and deals with the triviality
problem. Thus, the discovery of SUSY would constitute a major development in
understanding the EWSB. In this chapter we demonstrate how the NLO SUSY-QCD
corrections to the production of a gaugino()2) in association with a gluino (g) are
moderately sizable, and significantly improve the theoretical stability of the cross
section [90], which is important in interpreting experimental data in terms of a SUSY

discovery or exclusion.

Supersymmetry predicts the existence of supersymmetric partners for each of the
particles of the standard model. The search for these sparticles is a principal moti-
vation of the forthcoming Run II of the Fermilab Tevatron collider and of the CERN
Large Hadron Collider (LHC) program. A potentially important, but heretofore
largely overlooked, discovery channel is the associated production of a spin-1/2 gaug-
ino with a spin-1/2 gluino or with a spin-0 squark (d). Color-neutral gauginos couple

with electroweak strength, whereas the colored squarks and gluinos couple strongly.

105

106

Associated production is therefore a semi-weak process in that it involves one some-
what smaller coupling constant than the pair production of colored sparticles. How-
ever, in popular models of SUSY breaking [48, 86], the mass spectrum favors much
lighter masses for the low-lying neutralinos and charginos than for the squarks and
gluinos. This mass hierarchy means that the phase space for production of neutrali-
nos and charginos, the corresponding partonic luminosities, and the production cross
sections will be greater than those for gluinos and squarks. These advantages are po-
tentially decisive at a collider with limited energy, such as the Tevatron. Furthermore,
associated production has a clean experimental signature. For example, the lowest
lying neutralino is the (stable) lightest supersymmetric particle (LSP) in supergravity
(SUGRA) models [86], manifest as missing energy in the events, and it is the second
lightest in gauge-mediated models [48]. In models with a very light gluino [91], there
could be large rates for g); production, with simple signatures, whereas fig production

suffers from large hadronic jet backgrounds.

Experimental investigations are facilitated by firm theoretical understanding of
the expected sizes of the cross sections for production of the superparticles. In the
case of hadron-hadron colliders, the large strong coupling strength (as) results in po-
tentially large contributions to cross sections from terms beyond leading order (L0)
in a perturbative quantum chromodynamics (QCD) evaluation of the cross section.
For accurate theoretical estimates, it is necessary to extend the calculations to next-
to—leading order (NLO) or beyond. NLO contributions generally reduce and stabilize
dependence on undetermined parameters such as the renormalization and factoriza-
tion scales. To date, associated production has been calculated only in L0 [92],

but NLO results exist for hadroproduction of gluinos and “light” squarks1 [93], top

 

lBy light squarks, we refer to the squarks which are the superpartners of light quarks (u, d, s, c,
and b). In most models of SUSY breaking these scalars have masses on the order of a few hundred
GeV.

107

squarks [94], sleptons [95, 96], and gauginos [96]. Studies have begun to incorporate

these NLO results into Monte Carlo simulations [97, 98].

In this Chapter we present the first NLO (in SUSY-QCD) calculation of hadropro-
duction of a g in association with a )2, including contributions from virtual loops of
colored sparticles and particles and three-particle final states involving the emission of
light real particles. We extract the ultraviolet, infrared, and collinear divergences by
use of dimensional regularization and employ standard MS renormalization and mass
factorization procedures. In the course of computing the virtual contributions, we en-
countered new divergent four-point functions. The contributions from real emission
of light particles are treated with a phase space slicing method. We provide predic-
tions for inclusive cross sections at Tevatron and LHC energies. We focus on the 5755
final state, rather than on the associated production of (3')}, because at the energy of
the Tevatron the LO cross sections for g); are 3 to 6 times greater than those for (i)?
when mg = m4 = 300 GeV, and 6 to 15 times greater when mg = m, = 600 GeV. In

obtaining the if)? cross sections, we sum over five flavors of squarks and anti-squarks.

4.1 Leading Order Cross Sections

In LO of SUSY-QCD, the associated production of a gluino and a gaugino proceeds
through the subprocess qq —-> 552 with a t-channel or a u-channel squark exchange. We
assume that there is no mixing between squarks of different generations and that the
squark mass eigenstates are aligned with the squark chirality states, equivalent to the
assumption that the two squarks of a given flavor are degenerate in mass. We ignore
the n; = 5 light quark masses in all of the kinematics and couplings. Under these
assumptions, the massless incoming quarks and antiquarks have a particular helicity,

and thus the Feynman diagrams in which a right-handed squark is exchanged cannot

108

interfere with those mediated by a left-handed squark. In evaluating the Feynman
diagrams involving Majorana and explicitly charge-conjugated fermions, we follow
the approach of [99]. In the case of charged gauginos, only the left-handed squarks
participate, whereas neutral gauginos receive contributions from both left- and right-

handed squarks.

The L0 matrix element summed (averaged) over the colors and helicities of the

outgoing (incoming) particles has the analytic form [92]

8mg Xttth 2Xtusmng Xu uguX

[MB] = 9 W—(t—m %)(u- mg“) m]. (4.1)

 

 

Here, mg," is the mass of the squark exchanged in the t- and u—channels, and (is =
jig/411' is the coupling between quarks, squarks, and gluinos (at leading order it is
equal to the gauge coupling constant as). JIM”, stand for the weak couplings of
quarks, squarks, and gauginos which will be explained below, and the quantities s, t,

and u are the usual Mandelstam invariants at the partonic level with t“ = t — mg”?
2

and ng— = u— mm.

For production of a neutralino of type 22, the X are given by [100]

A A A e 2

Xt=Xu=Xtu=2 eeq ;1+sin0W COSOW (Tq—eq Sin20W) ;2

 

(4.2)

 

In the expressions above, e is the electric charge, 0w the weak mixing angle, Tq the
third component of the weak isospin for the squark, and 6,, is the charge of the quark
in units of e. For up-type quarks eq = 2/ 3 and for down-type quarks eq = —1/3. The
matrix N’ is the transformation from the interaction to mass eigenbasis defined in

[100]. The expressions for production of positive chargino of type )2?” are

62

Xi = Sin 20W —_]V:71|2’ (43)

62

Xtu = sin 20w ——_M( Ujl)’

109

x 62 2
Xu = lUjll a

Sill2 9w

 

and for the negative chargino )2; they have the form,

 

 

X = U- 2, 4.4
A e2 a:
Xtu = sin2 0w 118(le Uj 1),
2
‘ .. _e__ . 2
Xu — sin20w|V31| a

where U and V are the chargino transformation matrices from interaction to mass
eigenstates defined in [100]. As was mentioned above, in the case of chargino produc-

tion, the exchanged squark is always left-handed.

4.2 N ext-to-Leading Order Corrections

At NLO in SUSY-QCD the cross section receives contributions from virtual loop
diagrams and from real parton emission diagrams. The virtual contributions arise
from the interference of the Born amplitudes with the related one-loop amplitudes
containing self-energy corrections, vertex corrections, and box diagrams. We include
the full supersymmetric spectrum of strongly interacting particles in the virtual loops,

i.e. squarks and gluinos as well as quarks and gluons.

4.2.1 Virtual Loop Corrections

Since the virtual loop contributions are ultraviolet and infrared divergent, we regular-
ize the cross section by computing the phase space and matrix elements in n = 4 - 26
dimensions. We calculate the traces of Dirac matrices using the “naive” '75 scheme
in which 75 anticommutes with all other 7,, matrices. This choice is justified for

anomaly-free one-loop amplitudes. The '75 matrix enters the calculation through

110

both the quark-squark-gluino and quark-squark-gaugino couplings. We simplify the
integration over the internal loop momenta by reducing all tensorial integration ker-
nels to expressions that are only scalar in the loop momentum [101]. The resulting
one, two-, three, and some of the four-point functions were computed in the context
of other physical processes [93]. However, we compute two previously unknown diver-
gent four—point functions; these new functions arise because the final state gluino and
gaugino generally have different masses. We evaluate the scalar four-point functions
by calculating the absorptive parts with Cutkosky cutting rules and the real parts

with dispersion techniques.

The ultraviolet (UV) divergences are manifest in the one- and two-point functions
as poles in 1/6. We remove them by renormalizing the coupling constants in the
m scheme at the renormalization scale Q and the masses of the heavy particles
(squarks and gluinos) in the on-shell scheme. The self-energies for external particles
are multiplied by a factor of 1/ 2 for proper wave function renormalization. A difficulty
arises from the fact that spin-1 gluons have n -— 2 possible polarizations, whereas spin-
1/2 gluinos have 2, leading to broken supersymmetry in the W3 scheme. The simplest
procedure to restore supersymmetry is with finite shifts in the quark-squark-gluino

and quark-squark—gaugino couplings [102].

In addition to the ultraviolet singularities, the virtual corrections have collinear
and infrared singularities that show up as 1/6 or l/os2 poles in the derivatives of
the two-point function and in the three- and four-point functions. These infrared
singularities appear as factors times parts of the Born matrix elements. They can be
separated into CF and NC color classes, depending on the color flow and the Abelian
or non-Abelian nature of the correction vertices. They are cancelled eventually by
corresponding soft and collinear singularities from the real three particle final state

corrections.

111

4.2.2 Real Emission Corrections

The real corrections to the production of gluinos and gauginos arise from three par-
ticle final-state subprocesses in which additional gluons and massless quarks and

antiquarks are emitted: qrj —+ gig, qg —-+ gig, and (19 —> @2611

The n-dimensional phase space for 2 —> 3 scattering may be factored into the
phase space for 2 -) 2 scattering and the phase space for the subsequent decay of one
of the two final state particles with squared invariant mass 84 = (pl + p3)2 -— m? into
two particles with momenta p1 and p3, parametrized in the rest frame of particles 1
and 3 [103]. We follow the procedure of [103] and reduce all of the angular integrals.

to the form

1,257,!) = [0 " sin1‘2‘(01)d01 j: sin—2‘(02)d62 (4.5)

x (a + bcos 01)”‘(A + B cos 01 + Csin 01 cos 02)".

Analytic expressions for the integrals 15,“) for different k,l may be found in [103]. The
angular integrations involving negative powers of t’ = (1),, — p3)2 and u’ = (pa — p3)2,
where pa and 1),, are the four-momenta of the incoming partons, produce poles in 1/5
which correspond to the collinear singularities in which particle 3 is collinear with
particle a or b. Because these singularities have a universal structure, they may be
removed from the cross section and absorbed into the parton distribution functions

according to the usual mass factorization procedure [104].

In addition to the collinear singularities described above, the corrections involving
real gluon emission also have infrared (IR) singularities arising when the energy of the
emitted gluon approaches zero. These singularities appear as poles in .94 in the cross
section and must also be extracted so that they can be combined with corresponding

terms in the virtual corrections and shown to cancel. In order to make this cancellation

112

conveniently, we slice the gluon emission phase space into hard and soft pieces,

 

d265- [A (130.5 jam" (130.11
0

dtidui = 34 dti dug (£34 + A d84 dti dag (184,

(4.6)
where A is an arbitrary cut-off between soft and hard gluon radiation. When the
cut-off is much smaller than the other invariants, the 3.. integration for the soft term
becomes simple and can be evaluated analytically, leading to explicit logarithms of
the form log A /m§, log2 A/mg. The hard term is free from infrared and, after mass
factorization, also collinear singularities and can be evaluated numerically in four
dimensions. This procedure leads to an implicit logarithmic dependence of the hard

term on the cut-off A which cancels the explicit logarithmic dependence in the soft

term.

4.3 NLO Inclusive Cross Sections

To obtain numerical results for the cross sections, we work within a particular SUGRA
scheme, though the cross sections depend principally on the masses of the )2 and 57 and
are otherwise fairly independent of the details of the SUSY breaking. The physical
gluino and gaugino masses as well as the gaugino mixing matrices are calculated from
the minimal SUGRA scenario. We choose the common scalar and gaugino masses at
the GUT scale to be mo = 100 GeV, m1 /2 = 150 GeV, trilinear coupling A0 = 300
GeV, and the ratio-of the Higgs vacuum expectation values tan 5 = 4. The absolute
value of the Higgs mass parameter p is fixed by electroweak symmetry breaking, and

we choose [1 > 0. For this set of parameters, we find the neutralino masses m to

-o
X1—4

be 55, 104, 283, and 309 GeV with mfg < 0 inside a polarization sum. The chargino
masses mm are 102 and 308 GeV and therefore almost degenerate with the masses
of the mfg and mfg, respectively. This is a fairly general feature of the mSUGRA

spectrum.

113

The total hadronic cross section is obtained from the partonic cross section through

the convolution

iJ=9.q,q

0h1h2(5,Q2) = z jldxl /: dxg (4.7)

ihl ($1, Q2)f3h2 (1'2, Q2)6ij ($133251 Q2),

where 'r = 5333-, with m = (mg + m,-() / 2, and S is the square of the hadronic center-
of-mass energy For the NLO predictions, we employ the CTEQ4M parametrization
[39] for the parton densities f(a:,Q2) in the proton or antiproton and a two-loop
approximation for the strong coupling constant as with N5) = 202 MeV. To compute
LO quantities we use the CTEQ4L LO PDF’s and the one-loop approximation for as
with M5) =181 MeV.

In Figures 4.1 and 4.2 we present predictions for total hadronic cross sections at
the Tevatron and the LHC, as a function of the physical gluino mass. To obtain
these results, we use the average produced mass as the hard scale Q in Equation 4.8,
Q = m/ 2. We vary the SUGRA parameter mm» between 100 and 400 GeV and keep
the other SUGRA parameters fixed at the values listed above. As the gluino mass
increases over the range shown in the figure, the corresponding gaugino mass ranges
are 31 to 163 GeV for 52?, 62 to 317 GeV for g and if, 211 to 666 GeV for 2g,
and 240 to 679 GeV for )"(2 and 562*. The chargino cross sections are summed over

production of positive and negative charges.

We observe that the cross sections for 523 and 21* and those for 522 and 525; are
very similar in magnitude at the Tevatron, as are their respective masses. One might
expect the largest cross section for the lightest gaugino 53?. However, its coupling is
dominantly Bino-like and smaller than the W3ino-like coupling2 of 553 which therefore

has a larger cross section at small mg despite its larger mass. The heavier gauginos

 

2The Bino and W3ino are the superpartners of the B and W3 gauge bosons discussed in Chapter 1.

114

pp-egiatVS=2.0TeV

  
 

 

 

-——— NLO I
------ L01
1
'3
:1
3 2
o .
10 E— ~ E
I __01
i‘ 811 ‘
104—
: ~~o
E 312 .
.- -l
l-
l— ”“1
10 -5 . 821
300 350 400 450 500 550 600 650 700 750

In; [GeV]

Figure 4.1: Total hadronic cross sections for the associated production of gluinos
and gauginos at Run II of the Tevatron. NLO results are shown as solid curves,
and LO results as dashed curves. We vary the SUGRA scenario as a function of
m1/2 E [100; 400] GeV and provide the cross sections as a function of the physical
gluino mass mg. The chargino cross sections are summed over positive and negative
chargino rates.

115

pp-egiat‘JS: l4TeV

0 [pbl

 

.4
10 Ԥ
llllllllllllllllllllllLLLLlJllJlLl‘

3004005006007008009001000
m;[GeVl

§

Figure 4.2: Total hadronic cross sections for the associated production of gluinos and
gauginos at the LHC. NLO results are shown as solid curves, and LO results as dashed
curves. We vary the SUGRA scenario as a function of "ll/2 6 [100; 400] GeV and
provide the cross sections as a function of the physical gluino mass mg. The chargino
cross sections are summed over positive and negative chargino rates.

116

23,4 and )2; are dominantly higgsino-like and their cross sections are suppressed by

more than an order of magnitude with respect to those of the lighter gauginos.

At the Tevatron, the NLO contributions increase the cross sections by 5 to 15% at
the hard scattering scale Q = (m;2 + m) / 2, depending on the channel considered and
the values of the masses. At the LHC, the increases are in the range of 15 to 35%.
The purely NLO qg incident channel contributes significantly at the LHC, in addition
to the qq channel, particularly for the lighter gauginos, whereas the qg channel plays
an insignificant role at the Tevatron. In the event sparticles are not observed, the

predicted increases translate into more restrictive experimental mass limits.

The enhancements of the cross sections are modest and, as such, underscore the
validity of perturbative predictions for the processes considered. A further important
benefit of the NLO computation is the considerable reduction in theoretical uncer-
tainty associated with variation of the renormalization and factorization scale Q. For
the processes studied here, this dependence is typically :l:10% at the Tevatron when
Q is varied over the interval Q /m from 0.5 to 2, compared to 21:25% in leading order.
At the LHC, the dependences are i9% at NLO and 21:12% at LO.

We limit ourselves to total cross sections. Differential distributions in the trans-
verse momentum pT and the rapidity 17 of the produced sparticles will be published
elsewhere [105], along with figures of scaling functions, renormalization / factorization
scale dependence, K-factors, and several appendices containing a detailed exposition

of the calculation.

4.4 Summary

In summary, we provide NLO predictions of the cross sections for the associated

production of gauginos and gluinos at hadron colliders. If supersymmetry exists at

117

the electroweak scale, the cross section for this process is expected to be large and
observable at the Fermilab Tevatron and/or the CERN LHC. It is enhanced by the
large color charge of the gluino and the (in many SUSY models) small mass of the light
gauginos. The cross sections for gig and éfif production are comparable, and the
largest, because of their Wino-like couplings. As we have seen, the NLO predictions

are modestly larger than the LO values but considerably more stable.

Chapter 5

Conclusions

In this work, we have seen that the Standard Model ‘of particle physics, while fabu-
lously successful at describing high energy physics experiments, suffers from a number
of puzzles that indicate that it is not a fundamental theory, but should be replaced
by something else to describe the physics at very short distances. The primary puzzle
confronting particle physicists today is the understanding of the electroweak symme-
try breaking, responsible for the large masses of the weak bosons and the top quark.
The fact that the top is so much heavier than the other fermions seems to indicate
that it may play some special role. Its large mass further indicates that it is a natural

laboratory to test hypotheses concerning the nature of the symmetry breaking.

If the top does play a special role in nature, one must discover this fact through
careful study of its properties. In particular, the electroweak interactions are likely
to feel the effect of the true mechanism for the weak symmetry breaking, and are
perhaps the most interesting properties to examine. Single top production is a vital
means to study these weak interactions at a hadron collider, and thus we have spent
considerable time describing the physics of single top production, to see how one can
hope to use it as a tool to study the top’s electroweak interactions. We have seen

that the three modes of single top production, along with studies of top decays and

118

119

top polarization, represent a wealth of information about the top quark.

We have further studied the bottom quark, whose special partnership with the
top may allow it to inherit some of the top’s nonstandard properties. In particular,
we have seen that this can result in an enhancement of the bottom coupling to scalar
particles. It has been demonstrated that processes involving associated production of
Higgs bosons with bottom quarks can provide interesting information about a wide
class of models of the weak symmetry breaking, from supersymmetric theories to

theories with dynamical EWSB.

We have also seen that the superpartners of the electroweak gauge bosons, the
charginos and neutralinos can be produced in association with the gluino, superpart-
ner to the gluon. This process provides an interesting means to search for evidence of
supersymmetry at hadron colliders such as the Tevatron and LHC. In order to obtain
a reliable theoretical prediction for the cross section, one must include higher orders
in SUSY QCD. We have shown that the NLO corrections to this process are fairly
large, and dramatically increase the stability of the theoretical prediction, indicating

the necessity to include them.

With the advent of the Tevatron Run II, to be followed in a few years time by the
LHC, we stand on the threshold of a wealth of information concerning the mechanism
of the EWSB. The processes described above represent vital means to interpret this
information. Regardless of whether one favors one particular description or another,
it is an exciting time, as our conceptions of the symmetry breaking are confronted

with reality.

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