FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

SO(10) SUPERSYMMETRIC GRAND UNIFIED THEORIES: FROM

COSMOLOGY TO COLLIDERS

By

HEAYA ANN SUMMY

A Dissertation submitted to theDepartment of Physics

in partial fulfillment of therequirements for the degree of

Doctor of Philosophy

Degree Awarded:Fall Semester, 2008

The members of the Committee approve the Dissertation of Heaya Ann Summy defended

on September 12, 2008.

Howard BaerProfessor Directing Dissertation

Mark SussmanOutside Committee Member

Laura ReinaCommittee Member

Horst WahlCommittee Member

Efstratios ManousakisCommittee Member

The Office of Graduate Studies has verified and approved the above named committee members.

ii

To my parents: Jeong-Chi Kang and Yeon-Hwa Kim; my siblings: Huni and Meaya; andmy Ponce.

iii

ACKNOWLEDGEMENTS

There are so many people in this department and at FSU that I would like to give thanks

to, and I hope I have included everyone.

I am most grateful to Howie Baer, a terrific and fun to work with advisor. I would also like

to give a special thanks to Laura Reina and Jeff Owens for their conscientious instruction.

Thanks also to Pedro Schlottmann for his academic wisdom and great communication of

physics. Thanks to Azar Mustafayev, Alexander Belyaev, Xerxes Tata, and Horst Wahl, for

their excellent and illuminating discussions.

Thank you to my NSF fellowship advisor, Ellen Granger, an exceptionally talented, wise,

and good-hearted mentor. In addition to those mentioned above, I would like to thank Don

Robson and Jorge Piekarewicz for their insightfulness. Thanks to Robert Fulton, Lev Gelb,

and Bruno Linder for their kindness and patience when I was new to science. Thanks to

Stratos Manousakis, Marc Sussman, and Nancy Greenbaum for serving on my committee

and doing a fantastic job above-and-beyond.

Many thanks to some of my closest and dearest friends that I met while here and

other long-time friends, who by definition are among the best people I know: Hanoh Lee,

Jon Rinkenberger, Justin Bartee, Mathis Wiedeking, Thomas Rutishauser, Guler Arsal,

and Alexei Bazavov. Thanks to the following friends and coworkers for their support:

Chenggang Tao, Quoc Doan, Yi Cheng, Andrew Hornig, Jason Pratti, Kim Feinzilberg,

Sonnie Nguyen, Terri Volsh, Tianqing Liao, William Leparulo, William Gilmore, Carole

Koski, Ginger Martin, Brian Roeder, Eun-Kyung Park, Sherry Tointigh, Rob Westerling,

John Whetsel, Kathy Mork, Sara Stanley, and Karimah Wright. Thanks to Jack Tyndall

for being always so prompt and helpful during the tough times of defense preparation.

—

iv

TABLE OF CONTENTS

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

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Minimal Supersymmetric Standard Model . . . . . . . . . . . . . . 91.2.2 Supergravity and the Minimal Supergravity Model . . . . . . . . . 17

1.3 SO(10) SUSY Grand Unification . . . . . . . . . . . . . . . . . . . . . . 18

2. Yukawa-unified SO(10) SUSY GUTs . . . . . . . . . . . . . . . . . . . . . . . 202.1 HS v. DT Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Random Scan in HS model . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Random scan results . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Three proposals to reconcile Yukawa-unified models with dark mat-

ter relic density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Dark matter solution via neutralino decay to axino . . . . . . . . . . 30Dark matter solution via non-universal gaugino masses . . . . . . . . 31Dark matter solution via generational non-universality . . . . . . . . 32

2.3 Discussion of Markov Chain Monte Carlo analysis . . . . . . . . . . . . . 332.3.1 HS model: neutralino annihilation via h resonance . . . . . . . . . 352.3.2 Solutions using weak scale Higgs boundary conditions . . . . . . . 41

2.4 Yukawa-unified benchmark scenarios and LHC signatures . . . . . . . . . 46

3. SUSY Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 SO(10) SUSY GUTs and Yukawa unification . . . . . . . . . . . . . . . 52

3.2.1 The gravitino problem . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.2 Non-thermal leptogenesis . . . . . . . . . . . . . . . . . . . . . . . 563.2.3 Axino dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.4 A consistent cosmology for axino DM from SO(10) SUSY GUTs . 58

4. Collider Searches for New Physics . . . . . . . . . . . . . . . . . . . . . . . . . 60

v

4.1 Early SUSY Discovery using Multi-leptons . . . . . . . . . . . . . . . . . 614.2 Yukawa-unified SO(10) at the Cern LHC . . . . . . . . . . . . . . . . . . 71

4.2.1 Cross sections and branching fractions for sparticles in Yukawa-unified models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2.2 Gluino pair production signals at the LHC . . . . . . . . . . . . . . 754.2.3 Sparticle masses from gluino pair production . . . . . . . . . . . . 834.2.4 Trilepton signal from W1χ

02 production . . . . . . . . . . . . . . . . 90

5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

vi

LIST OF TABLES

1.1 One generation of the MSSM. . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Masses and parameters in GeV units for five benchmark Yukawa unified pointsusing Isajet 7.75 and mt = 171.0 GeV. The upper entry for the Ωχ0

1h2 etc.

come from IsaReD/Isatools, while the lower entry comes from micrOMEGAs;σ(χ0

1p) is computed with Isatools. . . . . . . . . . . . . . . . . . . . . . . . 48

4.1 Events generated and cross sections for various SM background processes plusthe SPS1a′ case study. The C1′ cuts are specified in Eqns. (1− 3). . . . . . 64

4.2 Masses and parameters in GeV units for two cases studies points A and D ofRef. [79] using Isajet 7.75 with mt = 171.0 GeV. We also list the total treelevel sparticle production cross section in fb at the LHC, plus the percent forseveral two-body final states. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Events generated and cross sections (in fb) for various SM background andsignal processes before and after cuts. The C1′ and Emiss

T cuts are specified inthe text. The W + jets and Z + jets background has been computed withinthe restriction pT (W,Z) > 100 GeV. . . . . . . . . . . . . . . . . . . . . . . . 76

4.4 Clean trilepton signal after cuts listed in the text. . . . . . . . . . . . . . . . 92

vii

LIST OF FIGURES

1.1 Evolution of the SU(3)C×SU(2)L×U(1)Y gauge coupling constants from theweak scale to the GUT scale for the case of (a) the SM, (b) the MSSM withtwo Higgs doublets, and (c) the MSSM with four Higgs doublets [63]. . . . . 15

2.1 Plot of R versus various input parameters for a wide (dark blue) and narrow(light blue) random scan over the parameter ranges listed in Eq. (2.6). Wetake µ > 0 and mt = 171 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Plot of R versus various sparticle masses for a random scan over the parameterrange listed in Eq. (2.6). We take µ > 0 and mt = 171 GeV. . . . . . . . . . 28

2.3 Plot of R versus various sparticle masses for a random scan over the parameterrange listed in Eq. (2.6). We take µ > 0 and mt = 171 GeV. . . . . . . . . . 29

2.4 Plot of Ωχ01h2 vs. R for a random scan over the parameter range listed in

Eq. (2.6). We take µ > 0 and mt = 171 GeV. . . . . . . . . . . . . . . . . . 29

2.5 Plot of variation in Ωχ01h2 versus non-universal GUT scale gaugino mass M1

for benchmark point A in Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . 32

2.6 Plot of variation in Ωχ01h2 versus non-universal GUT scale first/second gener-

ation scalar mass m16(1, 2) for benchmark point C in Table 2.1. . . . . . . . 34

2.7 Plot of MCMC results in the m16 vs. m10 plane; the light-blue (dark-blue)points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05)and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.8 Plot of MCMC results in the m16 vs. A0/m16 plane; the light-blue (dark-blue)points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05)and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.9 Plot of MCMC results in the m16 vs. m1/2 plane; the light-blue (dark-blue)points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05)and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

viii

2.10 Plot of MCMC results in the m16 vs. mHd,uplane; the light-blue (dark-blue)

points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05)and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.11 Plot of MCMC results in the mh − 2mχ01

vs. mA − 2mχ01

plane; the light-

blue (dark-blue) points have R < 1.1 (1.05), while for the orange (red) pointsR < 1.1 (1.05) and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . . . . . . . . . . . . 39

2.12 Plot of MCMC results in the R vs. Ωχ01h2 plane; the light-blue (dark-blue)

points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05)and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.13 Plot of MCMC results in the mχ02− mχ0

1vs. mh plane; the light-blue

(dark-blue) points have R < 1.1 (1.05), while for the orange (red) pointsR < 1.1 (1.05) and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . . . . . . . . . . . . 40

2.14 Plot of MCMC results using WSH boundary conditions in the m16 vs. A0/m16

plane; the light-blue (dark-blue) points have R < 1.1 (1.05), while for theorange (red) points R < 1.1 (1.05) and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . 42

2.15 Plot of MCMC results using WSH boundary conditions in the mA vs. µplane; the light-blue (dark-blue) points have R < 1.1 (1.05), while for theorange (red) points R < 1.1 (1.05) and Ωχ0

1h2 < 0.136. . . . . . . . . . . . . . 43

2.16 Plot of MCMC results using WSH boundary conditions in the mh− 2mχ01

vs.

mA−2mχ01

plane; the light-blue (dark-blue) points have R < 1.1 (1.05), while

for the orange (red) points R < 1.1 (1.05) and Ωχ01h2 < 0.136. . . . . . . . . 44

2.17 Plot of MCMC results using WSH boundary conditions in the mh vs.BF (Bs → µ+µ−) plane; the light-blue (dark-blue) points have R < 1.1 (1.05),while for the orange (red) points R < 1.1 (1.05) and Ωχ0

1h2 < 0.136. . . . . . 45

2.18 Contours of R and DM-allowed regions in the m1/2 vs. µ parameter space form16 = 3 TeV, m10/m16 = 1.3, A0/m16 = −1.85, ∆mH = 0.14, tan β = 50.9,mA = 500 GeV and mt = 173.9 GeV, as in Dermisek et al., but using Isajet7.75 for mass spectra generation. . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1 The gravitino problem in generic SUGRA models: an overproduction of grav-itinos followed by late gravitino decay can destroy successful BBN predictions⇒ upper bound on reheating temperature. . . . . . . . . . . . . . . . . . . 55

3.2 Plot of Yukawa unified solutions with R < 1.05 and 5 TeV < m16 < 20 TeV in thema vs.TR plane. The upper band of solutions has ΩNTP

a h2 = 0.01, ΩTPa h2 = 0.10

and fa/N = 1012 GeV, while the lower band of solutions has ΩNTPa h2 = 0.03,

ΩTPa h2 = 0.08 and fa/N = 5× 1011 GeV. . . . . . . . . . . . . . . . . . . . . . 59

ix

4.1 Plot of jet multiplicity from SUSY collider events from SPS1a′ after cuts C1′.We also plot the histograms of various SM backgrounds, plus the total SMbackground (gray histogram). . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Plot of augmented effective mass A′T (without Emiss

T ) from SUSY collider eventsfrom SPS1a′ after cuts C1′. We also plot the histograms of various SMbackgrounds, plus the total SM background (gray histogram). . . . . . . . . 65

4.3 Plot of b-jet multiplicity nb from LHC SUSY events from SPS1a′ after cutsC1′. We also plot the histograms of various SM backgrounds, plus the totalSM background (gray histogram). . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Plot of isolated lepton multiplicity n` from LHC SUSY events from SPS1a′

after cuts C1′. We also plot the histograms of various SM backgrounds, plusthe total SM background (gray histogram). . . . . . . . . . . . . . . . . . . 67

4.5 Plot of signal cross section from mSUGRA model versus mg after cuts C1′

and n` ≥ 3, for m0 = 200 and 1000 GeV. We also take A0 = 0, tan β = 10,µ > 0 and mt = 171 GeV. We also plot the 5σ background level for 0.1 and 1fb−1 of integrated luminosity. . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.6 Distribution in variable A′T from SUSY events from SPS1a′ after cuts C1′ plus

≥ 3` plus ≥ 1 b-jet. We also plot the remaining SM backgrounds (grayhistogram). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.7 Plot of OS/SF dilepton invariant mass from SUSY events from SPS1a′ aftercuts C1′ plus ≥ 3`. We also plot the remaining SM backgrounds (grayhistogram). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.8 Plot of OS/SF dilepton invariant mass from SUSY events from benchmarkSPS1a′ after cuts C1′ plus a OS/SF pair of leptons. We also plot the remainingSM backgrounds (gray histogram). . . . . . . . . . . . . . . . . . . . . . . . 70

4.9 Plot of σ(pp→ ggX) in pb at√s = 14 TeV versus mg. We use Prospino with

scale choice Q = mg, and show LO (solid) and NLO (dashes) predictions inthe vicinity of point A (red) and point D (blue) from Table 4.2. . . . . . . . 74

4.10 Plot of various -ino pair production processes in fb at√s = 14 TeV versus

mχ±1, for mq = 3 TeV and µ = mg, with tan β = 49 and µ > 0. . . . . . . . 74

4.11 Plot of various sparticle branching fractions taken from Isajet for points Aand D from Table 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.12 Plot of distribution in transverse sphericity ST for events with cuts C1′

from benchmark point A and the summed SM background; point D leadsto practically the same distribution. . . . . . . . . . . . . . . . . . . . . . . 77

x

4.13 Plot of jet ET distributions for events with ≥ 4 jets after requiring justST > 0.2, from benchmark point A; distributions for point D are the same. . 78

4.14 Plot of missing ET for events with ≥ 4 jets after cuts C1′, from benchmarkpoints A (full red line) and D (dashed blue line). . . . . . . . . . . . . . . . 79

4.15 Plot of jet multiplicity from benchmark points A (full red line) and D (dashedblue line) after cuts C1′ along with SM backgrounds. . . . . . . . . . . . . . 81

4.16 Plot of b-jet multiplicity from benchmark points A (full red line) and D(dashed blue line) after cuts C1′ along with SM backgrounds. . . . . . . . . 81

4.17 Plot of isolated lepton multiplicity from benchmark points A (full red line)and D (dashed blue line) after cuts C1′ along with SM backgrounds. . . . . 82

4.18 Plot of jet multiplicity in events with isolated SS dileptons from benchmarkpoints A (full red line) and D (dashed blue line) after cut ST > 0.2 along withSM backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.19 SF/OS dilepton invariant mass distribution after cuts C1′ from benchmarkpoints A (full red line) and D (dashed blue line) along with SM backgrounds. 84

4.20 Same as Fig. 4.19 but for same-flavor minus different-flavor subtractedinvariant-mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.21 Plot of m(X1,2) from benchmark points A and D along with SM backgroundsin events with cuts C1′ plus ≥ 4 b-jets and minimizing ∆m(X1−X2); see textfor details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.22 Same as Fig. 4.21 but requiring in addition a pair of SF/OS leptons. . . . . 87

4.23 Plot of m(bb`+`−)min from points A and D along with SM backgrounds. . . 89

4.24 Plot of m(X1,2`+`−)min from points A and D, minimizing ∆m(X1 − X2) as

explained in the text, along with SM backgrounds. . . . . . . . . . . . . . . 89

4.25 Plot of m(`+`−) in the clean trilepton channel from points A and D alongwith SM backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

xi

ABSTRACT

Simple SUSY GUT models based on the gauge group SO(10) require t − b − τ Yukawa

coupling unification, in addition to gauge coupling and matter unification. The Yukawa

coupling unification places a severe constraint on the expected spectrum of superpartners,

with scalar masses ∼ 10 TeV while gaugino masses are quite light. For Yukawa-unified

models with µ > 0, the spectrum is characterized by three mass scales: i). first and second

generation scalars in the multi-TeV range, ii). third generation scalars, µ and mA in the

few-TeV range and iii). gluinos in the ∼ 350− 500 GeV range with chargino masses around

100− 160 GeV. In such a scenario, gluino pair production should occur at large rates at the

CERN LHC, followed by gluino three-body decays into neutralinos or charginos. Discovery

of Yukawa-unified SUSY at the LHC should hence be possible with only 1 fb−1 of integrated

luminosity, by tagging multi-jet events with 2–3 isolated leptons, without relying on missing

ET . A characteristic dilepton mass edge should easily be apparent above Standard Model

background. Combining dileptons with b-jets, along with the gluino pair production cross

section information, should allow for gluino and neutralino mass reconstruction. A secondary

corroborative signal should be visible at higher integrated luminosity in the χ±1 χ02 → 3`

channel, and should exhibit the same dilepton mass edge as in the gluino cascade decay

signal.

A problem generic to all supergravity models comes from overproduction of gravitinos

in the early universe: if gravitinos are unstable, then their late decays may destroy the

predictions of Big Bang nucleosynthesis. We also present a Yukawa-unified SO(10) SUSY

GUT scenario which avoids the gravitino problem, gives rise to the correct matter-antimatter

asymmetry via non-thermal leptogenesis, and is consistent with the WMAP-measured

xii

abundance of cold dark matter due to the presence of an axino LSP. To maintain a consistent

cosmology for Yukawa-unified SUSY models, we require a re-heat temperature TR ∼ 106−107

GeV, an axino mass around ∼ 0.1 − 10 MeV, and a Peccei-Quinn breaking scale fa ∼ 1012

GeV.

xiii

CHAPTER 1

INTRODUCTION

In order to understand the feasibility of SO(10) Supersymmetric Grand Unified Theories

(SUSY GUTs), a foundation must first be laid. For our purposes, this foundation is built

upon the Standard Model (SM), the Minimal Supersymmetric Standard Model (MSSM), and

finally the gauge group SO(10). The theoretical basis must also be reconciled with known

cosmological and experimental constraints, such as Dark Matter (DM) relic abundance and

LEP2 findings. This chapter covers the rudiments needed to justify the models covered in

the remainder of this dissertation.

1.1 Standard Model

Formulated in the 1970s, the Standard Model of particle physics has been undeniably

successful in describing and predicting the properties of matter. The gauge group of the SM

is SU(3)C × SU(2)L × U(1)Y , where the component symmetry groups represent strong and

electroweak interactions. The SU(2)L × U(1)Y group unifies the weak and electromagnetic

interactions; upon spontaneous symmetry breaking via the Higgs mechanism, it breaks down

to just U(1)em. Thus, the SM can be completely characterized by quantum chromodynamics

(QCD) and electroweak theory (EW). It is comprised of 19 free parameters: three lepton

masses, six quark masses, three Cabibbo-Kobayashi-Maskawa (CKM) mixing angles plus

one CP-violating phase δ, three gauge couplings, the QCD vacuum angle θQCD, the Higgs

quadratic coupling µ, and the Higgs self-coupling strength λ.

The matter constituents of the SM are three generations of leptons and quarks with

1

left-handed SU(2) doublets and right-handed singlets:

l1,L =

(νe

e

)L

, eR ; q1,L =

(ud

)L

, uR , dR ;

l2,L =

(νµ

µ

)L

, µR ; q2,L =

(cs

)L

, cR , sR ;

l3,L =

(ντ

τ

)L

, τR ; q3,L =

(tb

)L

, tR , bR .

The Higgs doublet field

ϕ =

(ϕ+

ϕ0

)acquires a non-zero vacuum expectation value (vev) thereby giving mass to the fermions1 and

gauge bosons. The force carriers (gauge bosons) complete the particle content of the SM:

gluons g (in 8 colors, massless, spin-0, mediate the strong force), W± and Z0 (massive, spin-

1, mediate the weak force), and the photon γ (massless, spin-1, mediates the electromagnetic

force).

The SM Langrangian has the form

LSM = LQCD + LEW

= LQCD + Lgauge + Lscalar + Lfermion + LYukawa. (1.1)

Starting with the QCD term, the components of this Lagrangian are as follows:

LQCD = −1

4F i

µνFiµν +

∑r

qrαi 6Dαβq

βr , (1.2)

where

F iµν = ∂µG

iν − ∂νG

iµ − gsf

ijkGjµG

kν (1.3)

is the field strength tensor for the gluon fields Giµ (i = 1, ..., 8), gs is the QCD gauge coupling

constant, and the structure constants fijk (i, j, k = 1, ..., 8) are defined by the commutators

of the Gell-Mann matrices as [λi

2,λj

2

]= if ijkλ

k

2. (1.4)

The gauge covariant derivative in the second term of the QCD Lagrangian is

Daµb = (Dµ)ab = ∂µδ

ab + igsG

iµT

iab , (1.5)

1 However, the Higgs is only partially responsible for masses of neutrinos which are thought to get theirmasses from the see-saw mechanism. Here, neutrinos are classified as Majorana particles with the lightleft-handed neutrinos having heavy right-handed counterparts.

2

where the quarks transform according to the 3 × 3 representation matrices T i = λi/2, qr

indicates the rth quark flavor, and α, β = 1, 2, 3 are color indices.

Secondly,

Lgauge = −1

4F i

µνFiµν − 1

4BµνB

µν , (1.6)

with field strength tensors

Bµν = ∂µBν − ∂νBµ and

Fµν = ∂µWiν − ∂νW

iµ − gεijkW j

µWkν , (1.7)

where W iµ (i = 1, 2, 3) and Bµ are the respective SU(2) and U(1) gauge fields, g (g ′) is the

SU(2) (U(1)) gauge coupling, and εijk is the totally antisymmetric Levi-Civita symbol.

The third term is for the Higgs

Lscalar ≡ Lϕ = (Dµϕ)†Dµϕ− V (ϕ) , (1.8)

where ϕ is the complex Higgs scalar already shown earlier, the gauge covariant derviative is

Dµϕ =

(∂µ + ig

τ i

2W i

µ +ig ′

2Bµ

)ϕ, (1.9)

the τ i are the Pauli matrices, the Higgs potential V (ϕ) has the form

V (ϕ) = +µ2ϕ†ϕ+ λ(ϕ†ϕ)2. (1.10)

It is for µ2 < 0 that spontaneous symmetry breaking occurs giving rise to the SM particle

masses.

Next is the fermionic term

Lfermion ≡ Lf =

3 generations∑m=1

(qmLi 6DqmL+ lmLi 6DlmL+umRi 6DumR+dmRi 6DdmR+emRi 6DemR),

(1.11)

where m is generation or family index, and the L and R refer to the left and right chiral

projections of the fields ψL ≡ (1− γ5)ψ/2 and ψR ≡ (1 + γ5)ψ/2.

Finally, we have the Yukawa term

LYukawa = −3 generations∑

m,n=1

(qmLΓumnϕunR + qmLΓd

mnϕdnR + lmLΓemnϕenR) + h.c., (1.12)

3

where the matrices Γmn describe the Yukawa couplings between the Higgs doublet ϕ and the

various flavors m and n of quarks and leptons.

The SM is comprised of 19 free parameters: three lepton masses, six quark masses, three

Cabibbo-Kobayashi-Maskawa (CKM) mixing angles plus one CP-violating phase δ, three

gauge couplings, the QCD vacuum angle θQCD, the Higgs quadratic coupling µ, and the

Higgs self-coupling strength λ.

Although a host of electroweak precision measurement tests such as the predicted

existence and form of the weak neutral current and the existence and masses of the W and

Z bosons have soundly reinforced the SM, other cosmological and theoretical observations

have not been resolved within the SM. Some remaining issues are how particles attain mass,

the presence of only three generations of quarks and leptons and their mass heirarchy, the

imbalance of matter-antimatter in the universe (a.k.a., the baryogenesis problem), neutrino

masses and mixing, cold dark matter (CDM), dark energy, and gravity (the postulated force

carrier, the graviton, a massless, spin-2 particle has not found a place in the SM). Also, scalar

masses are quadratically divergent in the SM giving rise to what is known as the fine-tuning

problem: at scales above the weak scale, the Higgs mass parameter may require a great deal

of fine tuning to provide the needed cancellation that will maintain a physical Higgs below

its unitarity limit.

Generally, the SM is accepted as an effective field theory applicable to energies up to the

weak scale. Higher than this scale, there are currently a few mainstream theories, e.g., Little

Higgs Models, Universal Extra Dimensions (LED), Technicolor, and Supersymmetry. This

dissertation focuses on the last of these beyond-the-standard-model (BSM) theories.

1.2 Supersymmetry

Supersymmetry (SUSY) was conceived in the late 1960s and early 70s; it is the unique

symmetry that relates the properties of bosons to those of fermions. Supersymmetry requires

that for every boson, a fermion partner should exist, and vice versa. These supersymmetric

partners (or sparticles) serve as the new perturbatively coupled degrees of freedom that act

to cancel the quadratic divergences of the SM. There is a plethora of motivations for studying

SUSY, among them are

• aesthetics of building a super-Poincare extension of the Poincare group,

4

• stability of the scalar potential under radiative corrections (ultra-violet completeness)

making SUSY GUTs natural and extrapolation even to the Planck scale possible,

• gravity is contained within the theory,

• SUSY is an essential ingredient of superstring theories,

• unification of gauge couplings,

• suitable candidates for cold dark matter (CDM) are contained within SUSY,

• can explain radiative breakdown of electroweak symmetry,

• and provides better constraints for the light or SM Higgs boson mass.

Some of these motivations will be discussed in the following sections.

For a brief introduction to SUSY, we follow the Wess-Zumino (WZ) toy model formulated

in 1974. In this model, the Lagrangian takes the form

L = Lkinetic + Lmass,

=1

2(∂µA)2 +

1

2(∂µB)2 +

i

2ψ 6∂ψ +

1

2

(F 2 +G2

)−m

[1

2ψψ −GA− FB

], (1.13)

where A and B are real scalar fields with dimensionality [A] = [B] = 1, ψ is a Majorana

spinor with ψ = ψc = CψT and [ψ] = 32, and F and G are auxiliary (non-propagating) fields

with dimension [F ] = [G] = 2. They can be eliminated by the Euler-Lagrange equations

F = −mB and G = −mA.

The spinorial field expansions are:

ψD(x) =

∫d3k

(2π)3

1

2Ek

∑s

[cksukse

−ikx + d†ksvkseikx]

ψcD(x) =

∫d3k

(2π)3

1

2Ek

∑s

[c†ksvkse

ikx + dksukse−ikx

]ψM(x) =

∫d3k

(2π)3

1

2Ek

∑s

[cksukse

−ikx + c†ksvkseikx].

5

The infinitesimal SUSY field transformations in the WZ model go as

δA = iαγ5ψ,

δB = −αψ,

δψ = −Fα+ iGγ5α+ 6∂γ5Aα− i 6∂Bα,

δF = iα 6∂ψ,

δF = iαγ5 6∂ψ,

with A → A + δA, etc. and where α is a spacetime-independent anticommuting Majorana

spinor parameter with dimension [α] = −1/2. Using Majorana bilinear re-arrangements

(e.g., ψχ = −χψ) along with other algebraic manipulations, we can show that L → L+ δLwith

δLkinetic = ∂µ

(−1

2αγµ 6∂Bψ +

i

2αγ5γµ 6∂Aψ +

i

2Fαγµψ +

1

2Gαγ5γµψ

)δLmass = ∂µ (mAαγ5γµψ + imBαγµψ) .

Since the Langrangian changes by a total derivative, the action is invariant, and thus a WZ

transformation is a symmetry of the action. It can also be shown that the action remains

invariant with the addition of interaction terms and that quadratic divergences all cancel

within this model.

The most general supersymmetry algebra includes anticommutators, and so is referred

to as a graded Lie algebra. Only theories with a single Majorana spinorial generator Qa,

known as N = 1 supersymmetry theories, allow chiral representations. Models with more

than one SUSY charge in the low energy theory do not lead to chiral fermions and so are

excluded for phenomenological reasons. Using the fact that the super-charges Qa are spin-12

objects, a supersymmetric extension of the Poincare algebra or super-Poincare algebra can

6

be written as

[Pµ, Pν ] = 0, (1.14)

[Mµν , Pλ] = i(gνλPµ − gµλPν), (1.15)

[Mµν ,Mρσ] = −i (gµρMνσ − gµσMνρ − gνρMµσ + gνσMµρ) , (1.16)

[Pµ, Qa] = 0, (1.17)

[Mµν , Qa] = −(

1

2σµν

)ab

Qb, (1.18)Qa, Qb

= 2 (γµ)ab Pµ, (1.19)

Qa, Qb = −2 (γµC)ab Pµ, (1.20)Qa, Qb

= 2

(C−1γµ

)abPµ. (1.21)

Since Q is a Majorana spinor charge, the last two anticommutators can be found from the

first anticommutation relation Eq. (1.19).

In order to combine scalar and spinor fields into a single object, we move to superspace

xµ → (xµ, θa), where θa (a = 1 − 4) are four anticommuting dimensions arranged as a

Majorana spinor. Superfields provide us with a convenient procedure of formulating theories

that are guaranteed to be supersymmetric and will help us in our ultimate goal of writing

down the simplest supersymmetric extension of the Standard Model. A superfield results

from combining all three members of an irreducible supermultiplet into a single entity. We

denote the three components S, ψL, and F and can be written in terms of complex fields

S =1√2

(A+ iB) ,

ψL =1− γ5

2ψ, (1.22)

F =1√2

(F + iG) .

Since only one chiral component of the Majorana spinor ψ enters the transformations,

such superfields are referred to as (left) chiral superfields. This is fully a left-chiral scalar

superfield, because the lowest spin component of the multiplet has spin zero.

A general superfield has the form

Φ (x, θ) = S − i√

2θγ5ψ −i

2

(θγ5θ

)M+

1

2

(θθ)N +

1

2

(θγ5γµθ

)V µ

+i(θγ5θ

) [θ

(λ+

i√26∂ψ)]

− 1

4

(θγ5θ

)2 [D − 1

2S]. (1.23)

7

Left and right chiral scalar superfields appear as

SL(x, θ) = S(x) + i√

2θψL(x) + iθθLF(x), (1.24)

SR(x, θ) = S(x†) +−√

2θψR(x†)− iθθRF(x†), (1.25)

where xµ = xµ + i2θγ5γµθ. The product of two left (right) chiral scalar superfields is another

left (right) chiral scalar superfield, and the product of a left and right is a general superfield.

Both the D-term (general superfield) and the F -term (chiral scalar superfield) transform as

a total derivative, thus, they are both candidates for SUSY Lagrangians.

Augmenting the superfields with gauge superfields,

ΦA =1

2

(θγ5γµθ

)V µ

A + iθγ5θ · θA −1

4

(θγ5θ

)2DA (in WZ gauge) or (1.26)

WA(x, θ) = λLA(x) +1

2γµγνFµνA(x)θL − iθθL (6DλR)A − iDA(x)θL, (1.27)

allows one to write a master formula for supersymmetric gauge theories as shown below in

Eq. (1.28).

L =∑

i

(DµSi)† (DµSi) +

i

2

∑i

ψi 6Dψi +∑α,A

[i

2λα,A (6Dλ)α,A −

1

4FµναAF

µναA

]−√

2∑i,α,A

(S†i gαtαAλαA

1− γ5

2ψi + h.c.

)(1.28)

−1

2

∑α,A

[∑i

S†i gαtαASi + ξαA

]2

−∑

i

∣∣∣∣∣ ∂f∂Si

∣∣∣∣∣2

S=S

−1

2

∑i,j

ψi

( ∂2f

∂Si∂Sj

)S=S

1− γ5

2+

(∂2f

∂Si∂Sj

)†

S=S

1 + γ5

2

ψj,

where the covariant derivatives are given by,

DµS = ∂µS + i∑α,A

gαtαAVµαAS, (1.29)

Dµψ = ∂µψ + i∑α,A

gα (tαAVµαA)ψL − i∑α,A

gα (t∗αAVµαA)ψR, (1.30)

(6Dλ)αA = 6∂λα,A + igα

(t†αB 6VαB

)AC

λαC , (1.31)

FµναA = ∂µVναA − ∂νVµαA − gαfαABCVµαBVναC . (1.32)

8

Global SUSY may be spontaneously broken

〈0|Fi|0〉 6= 0 (F−type breaking), or

〈0|DA|0〉 6= 0 (D−type breaking),

as well as explicitly broken by adding soft SUSY breaking (SSB) terms to the Langranian.

In the absence of knowledge about SUSY breaking dynamics, the best that we can do is to

parameterize the effects of SUSY breaking by adding to the Lagrangian all possible SUSY

breaking terms, consistent with all desired (unbroken) symmetries at the SUSY breaking

scale that do not lead to the re-appearance of quadratic divergences, i.e., softly break the

symmetries. Girardello and Grisaru have classified the forms of the soft breaking operators

in a general theory [1]. They have shown that to all orders in perturbation theory, the

following break supersymmetry softly:

• linear terms in the scalar field Si (relevant only for singlets of all symmetries),

• scalar masses S†im2ijSj,

• and bilinear or trilinear operators of the form SiSj or SiSjSk (where SiSj and SiSjSk

occur in the superpotential),

• and finally, in gauge theories, gaugino masses, one for each factor of the gauge group.

1.2.1 Minimal Supersymmetric Standard Model

We are now ready to build a supersymmetric model. It is desirable to build upon that which

we already believe to describe nature at the weak scale, so a supersymmetric version of the

Standard Model would best serve our purposes. The simplest such model is known as the

Minimal Supersymmetric Standard Model (MSSM). It is a direct supersymmetrization of

the SM (except for the fact that one needs to introduce two Higgs doublet fields) and is

minimal in the sense that it contains the smallest number of new particle states and new

interactions consistent with phenomenology. To construct the MSSM, we follow this recipe

1. Choose the gauge symmetry (adopting appropriate gauge superfields for each gauge

symmetry).

2. Select matter and Higgs representations included as left-chiral scalar superfields.

9

3. Choose the superpotential f as a gauge invariant analytic function of left-chiral scalar

superfields; the degree is ≤ 3 for a renormalizable theory.

4. Adopt all allowed gauge invariant soft SUSY breaking terms; these are generally chosen

to parameterize our ignorance of the SUSY-breaking mechanism.

5. Compute the supersymmetric Lagrangian using the master formula Eq. (1.28), aug-

mented by all possible soft SUSY breaking terms.

In the first step, we choose the gauge symmetry of the Standard Model: SU(3)C ×SU(2)L × U(1)Y. The gauge bosons of the SM are promoted to gauge superfields, so in the

Wess-Zumino gauge,

Bµ → B 3 (λ0, Bµ,DB),

WAµ → WA 3 (λA,WAµ,DWA), A = 1, 2, 3, and

gAµ → gA 3 (gA, GAµ,DgA), A = 1, ..., 8.

Secondly, we stipulate the matter content to have three generations of quarks and leptons.

The fermion fields of the SM are promoted to chiral scalar superfields, with one superfield for

each chirality of every SM fermion. We use the left-handed charge conjugates of the right-

handed fermions, since the superpotential must be a function of just left-chiral superfields.

Then the matter superfields consist of(νiL

eiL

)→ Li ≡

(νi

ei

),

(eR)c → Eci ,(

uiL

diL

)→ Qi ≡

(ui

di

),

(uR)c → U ci ,

(dR)c → Dci ,

where, e.g.,

e = eL(x) + i√

2θψeL(x) + iθθLFe(x), (1.33)

while

Ec = e†R(x) + i√

2θψEcL(x) + iθθLFEc(x). (1.34)

10

SM Dirac fermions are constructed out of Majorana fermions via

e = PLψe + PRψEc , (1.35)

where in the chiral representation for γ matrices

ψe =

e1e2−e∗2e∗1

and ψEc =

e∗4−e∗2e3e4

.

Next, we introduce the Higgs multiplets of the theory so that the SM Higgs doublet is

promoted to a doublet of left-chiral superfields:

φ =

(φ+

φ0

)→ Hu =

(h+

u

h0u

)(1.36)

These spin-12

higgsinos with Y = 1 can circulate in triangle anomalies, thus it is necessary

to introduce a second left-chiral scalar doublet superfield with Y = −1,

Hd =

(h−dh0

d

)(1.37)

The MSSM matter and Higgs superfield content along with their gauge transformation

properties and weak hypercharge assignments for a single generation is listed in Table 1.1

We now choose the MSSM superpotential to describe the interactions between the various

chiral superfields,

f = µHauHda +

∑i,j=1,3

[(fu)ijεabQ

ai H

buU

cj + (fd)ijQ

ai HdaD

cj + (fe)ijL

ai HdaE

cj

]. (1.38)

We assume R-parity conservation, R = (−1)3(B−L)+2s, for all work done here, so baryon

and lepton number violating terms in the superpotential are excluded even though they are

gauge invariant and renomalizable.

Finally, we add into the Lagrangian all gauge invariant soft SUSY breaking terms,

Lsoft = −[Q†

im2QijQj + d†Rim

2Dij dRj + u†Rim

2UijuRj

+ L†im2LijLj + e†Rim

2Eij eRj +m2

Hu|Hu|2 +m2

Hd|Hd|2

]− 1

2

[M1λ0λ0 +M2λAλA +M3

¯gB gB

]− i

2

[M ′

1λ0γ5λ0 +M ′2λAγ5λA +M ′

3¯gBγ5gB

]+

[(au)ijεabQ

aiH

buu

†Rj + (ad)ijQ

aiHdad

†Rj + (ae)ijL

aiHdae

†Rj + h.c.

]+ [bHa

uHda + h.c.] ,

11

Table 1.1: One generation of the MSSM.

Field SU(3)C SU(2)L U(1)Y

L =(

νeL

eL

)1 2 −1

Ec 1 1 2

Q =(

uL

dL

)3 2 1

3

U c 3∗ 1 −43

Dc 3∗ 1 23

Hu =(

h+u

h0u

)1 2 1

Hd =(

h−dh0

d

)1 2∗ -1

If we count the number of parameters in the MSSM, the number is daunting.

• g1, g2, g3, θQCD

• gaugino masses M1, M′1, M2, M

′2, M3 (M ′

3 absorbed into g)

• m2Hu

, m2Hd

, µ, b (phase of b absorbed)

• 5× (6 + 3) = 45 in sfermion mass matrices

• 3× (3× 3× 2) = 54 in Yukawa matrices

• 3× (3× 3× 2) = 54 in a-term matrices

• a global U(3)5 transformation in matter allows 45−2 = 43 phases absorbed into matter

sfermions

• total parameters = 9 + 5 + 45 + 54 + 54− 43 = 124

However, we can simplify this number significantly by neglecting all SUSY sources that are

CP -violating or lead to flavor-changing neutral currents (FCNCs).

12

For a robust theory, the observed electroweak symmetry breaking that gives masses to

the W and Z bosons and fermions must be present. To investigate electroweak symmetry

breaking, we must examine the minima of the scalar potential in the MSSM. We can construct

this at tree-level with three parts

VMSSM = VF + VD + Vsoft, (1.39)

with minimization conditions∂V

∂h0u

=∂V

∂h0d

= 0,

and non-trivial, real solutions

〈h0u〉 ≡ vu and 〈h0

d〉 ≡ vd =⇒ tan β ≡ vu

vd

.

Thus, W±, Z0 and SM fermions (e.g., me = fevd) all become massive as in the Standard

Model.

Since states with the same electric charge, color, and spin can mix, SUSY predicts many

new particle states. These predicted new matter states are:

• spin 12

massive color octet: gluino g

• spin 12

bino, wino, neutral higgsinos ⇒ neutralinos χ01, χ

02, χ

03, χ

04

• spin 12

charged wino, higgsinos ⇒ charginos W±1 , W

±2

• spin-0 squarks: uL, uR, dL, dR, sL, sR, cL, cR, b1, b2, t1, t2

• spin-0 sleptons: eL, eR, νe, µL, µR, νµ, τ1, τ2, ντ

• spin-0 higgs bosons: h, H, A, H± (h usually SM-like)

Since we are examining unification of couplings at the GUT scale, we would be remiss

to not give a discussion about renomalization group equations (RGEs). If the MSSM is to

be valid between vastly different mass scales, then we must be able to relate parameters

between these scales. RGEs govern the evolution of gauge couplings, Yukawa couplings, the

µ term, and soft breaking parameters. For gauge couplings, RGEs have the form

dgi

dt= β(gi) with t = logQ, (1.40)

13

where Q is the renormalization scale. In the Standard Model,

β(g) = − g3

16π2

[11

3C(G)− 2

3nFS(RF )− 1

3nHS(RH)

], (1.41)

where C(G) is the quadratic Casimir operator for the adjoint representation of the associated

Lie algebra, S(RF ) is the Dynkin index for representation RF of the fermion fields, S(RH)

is the Dynkin index for representation RH of the scalar fields, nF is the number of fermion

species, and nH is the number of complex scalars. In the MSSM, the gauginos, matter and

Higgs scalars also contribute:

β(g) = − g3

16π2[3C(G)− S(R)] . (1.42)

The precision values of g1, g2 and g3 measured at Q = MZ at LEP2 can be used as boundary

conditions and extrapolated to higher energies. Gauge coupling evolution from the weak to

GUT scales is displayed in Fig. 1.1

The one-loop RGEs for third generation Yukawa couplings of the MSSM are given by

dft

dt=

ft

16π2

(−∑

i=1−3

cig2i + 6f 2

t + f 2b

), (1.43)

dfb

dt=

fb

16π2

(−∑

i=1−3

c′ig2i + f 2

t + 6f 2b + f 2

τ

), (1.44)

dft

dt=

ft

16π2

(−∑

i=1−3

c′′ig2i + 3f 2

b + 4f 2τ

), (1.45)

where ci = (13/15, 3, 16/3), c′i = (7/15, 3, 16/3), c′′i = (9/5, 3, 0), and t = log(Q).

Like the gauge and Yukawa couplings, the various soft SUSY breaking parameters as

well as the superpotential Higgs mass µ, evolve with energy scale. The one-loop RGEs for

the soft SUSY breaking parameters, µ, and for the third generation sfermion masses and

A-parameters are as follows (the first two generations are easily obtained by making the

14

Figure 1.1: Evolution of the SU(3)C × SU(2)L × U(1)Y gauge coupling constants from theweak scale to the GUT scale for the case of (a) the SM, (b) the MSSM with two Higgsdoublets, and (c) the MSSM with four Higgs doublets [63].

15

requisite replacements in the appropriate formulas):

dMi

dt=

2

16π2big

2iMi, (1.46)

dAt

dt=

2

16π2

(−∑

i

cig2iMi + 6f 2

t At + f 2bAb

), (1.47)

dAb

dt=

2

16π2

(−∑

i

c′ig2iMi + 6f 2

bAb + f 2t At + f 2

τAτ

), (1.48)

dAτ

dt=

2

16π2

(−∑

i

c′′i g2iMi + 3f 2

bAb + 4f 2τAτ

), (1.49)

dB

dt=

2

16π2

(−3

5g21M1 − 3g2

2M2 + 3f 2bAb + 3f 2

t At + f 2τAτ

), (1.50)

dµ

dt=

µ

16π2

(−3

5g21 − 3g2

2 + 3f 2t + 3f 2

b + f 2τ

), (1.51)

dm2Q3

dt=

2

16π2

(− 1

15g21M

21 − 3g2

2M22 −

16

3g23M

23 +

1

10g21S + f 2

t Xt + f 2bXb

),

dm2tR

dt=

2

16π2

(−16

15g21M

21 −

16

3g23M

23 −

2

5g21S + 2f 2

t Xt

), (1.52)

dm2bR

dt=

2

16π2

(− 4

15g21M

21 −

16

3g23M

23 +

1

5g21S + 2f 2

bXb

), (1.53)

dm2L3

dt=

2

16π2

(−3

5g21M

21 − 3g2

2M22 −

3

10g21S + f 2

τXτ

), (1.54)

dm2τR

dt=

2

16π2

(−12

5g21M

21 +

3

5g21S + 2f 2

τXτ

), (1.55)

dm2Hd

dt=

2

16π2

(−3

5g21M

21 − 3g2

2M22 −

3

10g21S + 3f 2

bXb + f 2τXτ

), (1.56)

dm2Hu

dt=

2

16π2

(−3

5g21M

21 − 3g2

2M22 +

3

10g21S + 3f 2

t Xt

), (1.57)

where mQ3 and mL3 denote the mass term for the third generation SU(2) squark and slepton

doublet respectively, and

Xt = m2Q3

+m2tR

+m2Hu

+ A2t , (1.58)

Xb = m2Q3

+m2bR

+m2Hd

+ A2b , (1.59)

Xτ = m2L3

+m2τR

+m2Hd

+ A2τ , and (1.60)

S = m2Hu−m2

Hd+ Tr

[m2

Q −m2L − 2m2

U + m2D + m2

E

]. (1.61)

16

1.2.2 Supergravity and the Minimal Supergravity Model

If we allow the parameters in SUSY transformations to become spacetime dependent, i.e.,

α→ α(x) in e−iαQ,

then SUSY becomes a local symmetry. Local SUSY transformations are called supergravity

transformations for reasons that will soon become clear. Just as for gauge theories, we need

to introduce a gauge field to maintain covariance: ψµ(x), a spin 32

vector-spinor (Rarita-

Schwinger) field. In order to maintain local SUSY, we must also introduce a bosonic partner,

a spin 2 field gµν(x) with the properties

• gµν is massless, and in the classical limit obeys Einstein’s GR equations of motion ⇒it couples to the energy-momentum tensor for matter: it is the graviton field,

• usually, gµν(x) is traded for the equivalent vierbein field eaµ(x), where gµν = ea

µebνηab

and ηab is the Minkowski metric.

Supergravity (SUGRA) is inherently non-renormalizable, since gravity itself is non-

renomalizable. Although the Lagrangian for a general non-renormalizable SUSY theory

depends on three independent functions, the Kahler potential K(S†, S), the superpotential

f(S), and the gauge kinetic function fAB(S), SUGRA depends only on the gauge kinetic

function and just one combination of the Kahler potential and superpotential called the

Kahler function,

G(S†, S) = K(S†, S) + log |f(S)|2. (1.62)

Other notable features of SUGRA are

• it can be spontaneously broken just as we saw for SUSY,

• being a local SUSY theory, a super-Higgs mechanism exists wherein the gravitino field

ψµ gains a mass m3/2 while graviton remains massless,

• the MSSM can be embedded into a SUGRA theory along with gauge singlet field(s)

hm and superpotential such that SUGRA is spontaneously broken (hidden sector),

• SUGRA breaking communicated from the hidden sector to the visible sector via gravity

induces soft SUSY breaking terms of order ∼ m3/2.

17

The above features are used to build the Minimal Supergravity model (mSUGRA)

by assuming the MSSM is embedded in a SUGRA theory and SUSY is broken in the

hidden sector with m3/2 ∼ Mweak ∼ 1 TeV. In the simplest models (e.g. using a Polonyi

superpotential in the hidden sector) universal scalar masses m0, gaugino masses m1/2 and

trilinear terms A) are induced as soft SUSY breaking terms. Since this model is inspired by

gauge coupling unification, these universal values are usually taken at Q = MGUT ' 2×1016

GeV. The couplings and soft parameters are evolved from MGUT to Mweak causing m2Hu

to

become negative and thereby breaking EW symmetry. All sparticle masses and mixings are

calculated at Q = Mweak in terms of a small parameter set

m0, m1/2, A0, tan β, sign(µ) (1.63)

Although the mSUGRA model may be too simplistic to be a complete theory for beyond

weak scale physics, it has thus far been the paradigm SUSY model for phenomenological

analysis and is convenient with only five parameters.

1.3 SO(10) SUSY Grand Unification

Grand unified theories (GUTs) based upon the gauge group SO(10) and augmented by

supersymmetry (SUSY) is currently one of the most promising concepts in particle physics.

In addition to gauge group unification, matter unification of each generation occurs within

the SO(10) 16-dimensional spinorial representation ψ(16). Furthermore, even the simplest

SO(10) GUTs allow for Yukawa coupling unification, especially for the third generation.

Triangle anomaly cancellation is automatic in SO(10) theories, thus explaining the ad-hoc

triangle anomaly cancellation in SU(5) GUTs or in the SM. The combination with softly

broken N=1 SUSY allows for stabilization of the weak scale to GUT scale gauge hierarchy

and is experimentally supported by the fact that the measured weak scale gauge couplings

meet at MGUT under MSSM renormalization group evolution. SUSY SO(10) also elegantly

addresses the neutrino mass problem, since one only has matter unification found within the

superfield ψ(16) provided one adds to the set of supermultiplets a SM gauge singlet superfield

N ci containing a right-handed neutrino state. Upon breaking of SO(10), a superpotential

term f 3 12MNi

N ci N

ci leading to a Majorana neutrino mass MNi

is induced in the Lagrangian.

This term is required for implementing the see-saw mechanism for neutrino masses.

18

In the past, GUTs (including SUSY GUTs) formulated in 4-d spacetime have been

plagued with a variety of problems mainly associated with GUT gauge symmetry breaking via

the Higgs mechanism. These include the doublet-triplet splitting problem, lack of observation

of proton decay, and the frequently awkward implementation of GUT symmetry breaking

via at least one large and unwieldy Higgs representation. With the onset of model building

utilizing extra dimensions, it has been shown to be possible to formulate SUSY GUTs in

five or more spacetime dimensions. Then, the GUT gauge symmetry can be broken via

compactification of the extra dimensions on a suitable sub-space, such as an orbifold. In these

5-d and 6-d SUSY GUT models, the large GUT-scale Higgs representations can be dispensed

with, the doublet-triplet problem can be solved, and the proton can be made longer-lived

than current limits or even absolutely stable [75]. The extra-dimensional SUSY GUT models

act as a sort of “proof of principle” of what might be possible in more complicated set-ups

where the SUSY GUT model might arise from compactification of superstring models.

The work presented in this dissertation is chiefly concerned with Yukawa unification

within SO(10) SUSY GUTs. This implies additional restrictions on the general model,

especially splitting of the GUT scale Higgs masses. Further characterization of the specific

models used here and results obtained within these models will be saved for the upcoming

chapters.

19

CHAPTER 2

Yukawa-unified SO(10) SUSY GUTs

To avoid dealing with the unknown physics above the GUT scale, we will assume that nature

is described by an SO(10) SUSY GUT theory at energy scales Q > MGUT ∼ 2 × 1016 GeV

and that the model breaks (either via the Higgs mechanism or via compactification of extra

dimensions) to the MSSM (or MSSM plus right-handed neutrino states) at Q = MGUT.

Thus, below MGUT the MSSM is the correct effective field theory which describes nature.

We will further assume that the superpotential above MGUT is of the form

f 3 fψ16ψ16φ10 + · · · (2.1)

so that the third generation Yukawa couplings ft, fb and fτ are unified at MGUT. It is

simple in this context to include as well the effect of a third generation neutrino Yukawa

coupling fν ; this effect has been shown to be small, although it can help improve Yukawa

coupling unification by a few percent if the neutrino Majorana mass scale is within a few

orders of magnitude of MGUT. Within this ansatz, the GUT scale soft SUSY breaking (SSB)

terms are constrained by the SO(10) gauge symmetry so that matter scalar SSB terms have a

common mass m16, Higgs scalar SSB terms have a common mass m10, and there is a common

trilinear soft breaking parameter A0. As usual, the bilinear soft term B can be traded for

tan β, the ratio of Higgs field vevs, while the magnitude of the superpotential Higgs mass µ is

determined in terms of M2Z via the electroweak symmetry breaking minimization conditions.

Here, electroweak symmetry is broken radiatively (REWSB) due to the large top quark mass.

In order to accomodate REWSB, it is well-known that in Yukawa-unified models, the

GUT scale Higgs soft masses must be split such that m2Hu

< m2Hd

in order to fulfill the

EWSB minimization conditions; this effectively gives m2Hu

a head start over m2Hd

in running

towards negative values at or around the weak scale. We parametrize the Higgs splitting

20

as m2Hu,d

= m210 ∓ 2M2

D. The Higgs mass splitting might originate via a large near-GUT-

scale threshold correction arising from the neutrino Yukawa coupling: see the Appendix

to Ref. [46] for discussion. Thus, the Yukawa unified SUSY model is determined by the

parameter space

m16, m10, M2D, m1/2, A0, tan β, sign(µ) (2.2)

along with the top quark mass. We will take mt = 171 GeV, in accord with recent

measurements from CDF and D0 [3].

In particle phenomenology, we continually strive for agreement with established and/or

forthcoming experimental data in our cutting-edge physics models. With this in mind, one

primary concern is to determine how SO(10) SUSY GUT theories could manifest themselves

in the environment of the LHC detectors while enforcing Yukawa coupling unification and

dark matter (DM) abundance of the universe constraints. This chapter is dedicated to

addressing this concern by means of analyzing complementary Random Scan (RS) and

Markov Chain Monte Carlo (MCMC) scan methods.

2.1 HS v. DT Models

The necessity of splitting the Higgs soft masses at the GUT scale for Yukawa coupling

unification has already been discussed. However, there are two known ways in which to do

this: i.) through D-term contributions to all scalar masses[7] (the DT model), or ii.) via

splitting of only the Higgs soft terms[46] (the HS model). Ensuing is an explanation of why

we choose the HS model method of splitting the Higgs terms followed by our results within

this model.

In Ref. [8], it was found using the Isajet sparticle mass spectrum generator [83] Isasugra

that Yukawa coupling unification to 5% could be achieved in the MSSM using D-term

splitting, but only for µ < 0; for µ > 0, the Yukawa coupling unification was much worse, of

order 30–50%. These parameter space scans allowed m16 values of up to only 1.5 TeV and

used a GUT scale Yukawa unification quantity

R =max(ft, fb, fτ )

min(ft, fb, fτ ), (2.3)

so that, e.g., R = 1.1 would correspond to 10% Yukawa unification. The µ < 0 Yukawa

unification solutions were examined in more detail in Ref. [9], where dark matter allowed

21

solutions were found, and the neutralino A-annihilation funnel was displayed for the first

time.

With the announcement from BNL experiment E-821 that there was a 3σ deviation from

SM predictions on the muon anomalous magnetic moment aµ ≡ (g−2)µ/2, attention shifted

back to µ > 0 solutions. Ref. [45], using the DT model with parameter space scans of m16

up to 2 TeV, found Yukawa-unified solutions with R ∼ 1.3 but only for special choices of

GUT scale boundary conditions:

A0 ∼ −2m16, m10 ∼ 1.2m16, (2.4)

with m1/2 m16 and tan β ∼ 50. In fact, these boundary conditions had been found

earlier by Bagger et al. [82] in the context of models with a radiatively driven inverted scalar

mass hierarchy (RIMH), wherein RG running of multi-TeV GUT scale scalar masses caused

third generation masses to be driven to weak scale values, while first/second generation soft

terms remained in the multi-TeV regime. These models, which required Yukawa coupling

unification, were designed to maintain low fine-tuning by having light third generation

scalars, while solving the SUSY flavor and CP problems via multi-TeV first and second

generation scalars. A realistic implementation of these models in Ref. [10] using 2-loop RGEs

and requiring REWSB found that an inverted hierarchy could be generated, but only to a

lesser extent than that envisioned in Ref. [82], which didn’t implement EWSB or calculate

an actual physical mass spectrum.

Simultaneously with Ref. [45], Blazek, Dermisek and Raby (BDR) published results

showing Yukawa-unified solutions using the HS model solution [46]. Their results also

found valid solutions using the Bagger et al. boundary conditions. BDR used a top-down

method beginning with actual Yukawa unification at MGUT and implemented 2-loop gauge

and Yukawa running but 1-loop soft term running. They extracted physical soft terms at

scale Q = MZ , and minimized a two-Higgs doublet scalar potential to achieve REWSB,

also at scale MZ . Each run generated a numerical value for third generation t, b, and τ

masses and other electroweak and QCD observables. A χ2 fit was performed to select those

solutions which best matched the measured weak scale fermion masses and other parameters.

BDR scanned m16 values up to 2 TeV, and found best fit results with mA ∼ 100 GeV and

µ ∼ 100–200 GeV, in contrast to Ref. [45], where solutions with valid EWSB could only be

22

found if µ ∼ mA ∼ mt1 ∼ 1 TeV.1

In a long follow-up study using Isajet, Auto et al. [76] found that Yukawa-unified solutions

good to less than a few percent could be found in the µ > 0 case using the HS model of

BDR, but only for very large values of m16>∼ 5–10 TeV and low values of m1/2

<∼ 100 GeV,

again using Bagger et al. boundary conditions. Yukawa unification in the DT model was

at best good to 10% (for this reason, in this chapter, we focus only on the HS model). The

spectra were characterized by three mass scales:

1. ∼ 5–15 TeV first and second generation scalars,

2. ∼ 1 TeV third generations scalars, µ term and mA and

3. chargino masses mfW1∼ 100–200 GeV and gluino masses mg ∼ 350–450 GeV.

These Yukawa-unified solutions – owing to very large values of scalar masses, mA and µ –

predicted dark matter relic density values Ωχ01h2 far beyond the WMAP-measured result [49]

of

ΩDMh2 = 0.111+0.011

−0.015 (2σ). (2.5)

Meanwhile, the spectra generated using the BDR program could easily generate Ωχ01h2 values

close to 0.1 since their allowed µ and mA values were far lower, so that mixed higgsino dark

matter or A-funnel annihilation solutions could easily be found. In follow-up papers to the

BDR program [13, 14], the neutralino relic density and branching fraction Bs → µ+µ− were

evaluated. To avoid constraints on BF (Bs → µ+µ−) from the CDF collaboration, the best fit

values of m16 and mA have been steadily increasing, so that the latest papers have mA ∼ 500

GeV and m16 ∼ 3 TeV, while µ can still be of order 100 GeV [14]. In Ref. [15], attempts were

made to reconcile the Isajet Yukawa-unified solutions with the dark matter relic density. The

two solutions advocated were i.) lowering GUT scale first/second generation scalars relative

to the third, to gain neutralino-squark or neutralino-slepton co-annihilation solutions [16],

or ii.) increasing the GUT-scale gaugino mass M1, so the relic density could be lowered by

bino-wino co-annihilation [17].

1 A paper by Tobe and Wells[11] (TW) appeared after Ref. [46]. While TW calculate no actual spectraor address EWSB, they do adopt a semi-model independent approach which favors t−b−τ Yukawa couplingunification if scalar masses are in the multi-TeV regime while gauginos are as light as possible.

23

2.2 Random Scan in HS model

In this section, we explore the parameter space Eq. (2.2) for Yukawa-unified solutions by

means of a random scan. We wish to first check and update results presented in Ref. [76],

using the latest Isajet version and a top quark mass of mt = 171 GeV in accord with recent

measurements from the Fermilab Tevatron [3]. (Note that since the publication of these

results, the top mass has been further updated to mt = 172.6 GeV [78].) The degree of

Yukawa unification, R, is defined in Eq. (2.3), so that, e.g., a value of R = 1.1 corresponds

to 10% Yukawa unification.

For our calculations, we adopt the Isajet 7.75 [83, 84] SUSY spectrum generator Isasugra.

Isasugra begins the calculation of the sparticle mass spectrum with inputDR gauge couplings

and fb, fτ Yukawa couplings at the scale Q = MZ (ft running begins at Q = mt) and evolves

the 6 couplings up in energy to scale Q = MGUT (defined as the value Q where g1 = g2)

using two-loop RGEs.2 At Q = MGUT , the SSB boundary conditions are input, and the set

of 26 coupled two-loop MSSM RGEs [85] are used to evolve couplings and SSB terms back

down in scale to Q = MZ . Full two-loop MSSM RGEs are used for soft term evolution,

while the gauge and Yukawa coupling evolution includes threshold effects in the one-loop

beta-functions, so the gauge and Yukawa couplings transition smooothly from the MSSM to

SM effective theories as different mass thresholds are passed. In Isajet 7.75, the SSB terms of

sparticles which mix are frozen out at the scale Q ≡MSUSY =√mtL

mtR, while non-mixing

SSB terms are frozen out at their own mass scale [84]. The scalar potential is minimized

using the RG-improved one-loop MSSM effective potential evaluated at an optimized scale

Q = MSUSY which accounts for leading two-loop effects [19]. Once the tree-level sparticle

mass spectrum is computed, full one-loop radiative corrections are calculated for all sparticle

and Higgs boson masses, including complete one-loop weak-scale threshold corrections for

the top, bottom, and tau masses at scale Q = MSUSY . These fermion self-energy terms are

critical to evaluating whether or not Yukawa couplings do indeed unify. Since the GUT-scale

Yukawa couplings are modified by the threshold corrections, the Isajet RGE solution must

be imposed iteratively with successive up–down running until a convergent solution for the

spectrum is found. For most of parameter space, there is very good agreement between Isajet

2 As inputs, we take the top quark pole mass mt = 171 GeV. We also take mDRb (MZ) = 2.83 GeV [18]

and mDRτ (MZ) = 1.7463 GeV. The paper Ref. [76] addresses consequences of varying the values of mt and

mb.

24

and the other public spectrum codes SoftSusy, SuSpect, and SPheno, although at the edges

of parameter space agreement between the four codes typically diminishes [20].

We first adopt a wide parameter range scan, and then once the best Yukawa-unified

regions are found, we adopt a narrow scan to try to hone in on the best unified solutions.

The parameter range we adopt for the wide (narrow) scan is

m16 : 0 – 20 TeV (1 – 20 TeV),m10/m16 : 0 – 1.5 (0.8 – 1.4),m1/2 : 0 – 5 TeV (0 – 1 TeV),A0/m16 −3 – 3 (−2.5 – 1.9),MD/m16 : 0 – 0.8 (0.25 – 0.8),tan β : 40 – 60 (46 – 53).

(2.6)

For the random scan, we evaluate Ωχ01h2, BF (b → sγ), ∆aµ and BF (BS → µ+µ−) using

Isatools (a sub-package of Isajet). We plot only solutions for which mfW1> 103.5 GeV, in

accord with LEP2 searches, and for the moment implement no other constraints, such as

relic density, Higgs mass, etc..

2.2.1 Random scan results

Our first results are shown in Fig. 2.1, where we show points from the wide scan (dark blue)

and points from the narrow scan (light blue) in the parameter versus R plane. From frame

a), we see that Yukawa unification to better than 30% (R < 1.3) cannot be achieved for

m16 < 1 TeV, while Yukawa coupling unification becomes much more likely at multi-TeV

values of m16. Frame b) shows that Yukawa-unified models prefer m10 ∼ 1− 1.3m16, while

frame c) shows that a positive value of MD ∼ (0.25 − 0.5)m16 – which yields m2Hu

< m2Hd

– is preferred. In frame d), we see that the best Yukawa-unified solutions are found for

the lowest possible values of m1/2. We note here that – using 1-loop RGEs along with the

LEP2 constraint mfW1> 103.5 GeV – one would expect from models with gaugino mass

unification that since mfW1∼ M2(weak) ∼ 0.8m1/2 that we would have m1/2

>∼ 125 GeV

always. However, the very large values of m16 we probe alter the simple 1-loop gaugino mass

unification condition (that M1

α1= M2

α2= M3

α3) via 2-loop RGE effects. Thus, values of m1/2

much lower than ∼ 125 GeV are possible if m16 is large.

In frame e), we see a sharp dependence that Yukawa-unified solutions can only be obtained

for A0 ∼ −2m16, while frame f) shows that tan β must indeed be large: in the range ∼ 47−53.

Bagger et al. had shown in Ref. [82] that a radiatively-driven inverted scalar mass hierarchy

25

with m(third generation) m(first/second generation) could be derived provided one

starts with unified Yukawa couplings, the boundary conditions

4m216 = 2m2

10 = A20, (2.7)

and one neglects the effect of gaugino masses. Our results in Fig. 2.1 show the inverse

effect: that Yukawa coupling unification can only be achieved if one imposes the boundary

conditions (2.7) along with m16 m1/2. This result holds only in our numerical calculations

for µ > 0 and A0 < 0 and of course m2Hu

< m2Hd

. The results shown in Fig. 2.1 also verify that

the results obtained in Ref. [76] still hold, even with updated spectra code and a lower value

of mt = 171 GeV. In Fig. 2.2, we show various -ino masses3 versus R as generated from our

random scan. In frame a), we see that – owing to the preference of Yukawa-unified solutions

to have m1/2 as small as possible – the chargino mass mfW1is preferred to be quite light, as

close to the LEP2 limit as possible, with mfW1∼ 100–200 GeV. Likewise, in frame b), the

gluino mass should be relatively light, with mg ∼ 350–500 GeV. The lightest neutralino χ01

mass is shown in frame c), and is preferred in the range mχ01∼ 50–100 GeV. Meanwhile, the

mass difference mχ02−mχ0

1is shown in frame d), and is also in the range ∼ 50–100 GeV. This

latter quantity is important because if mχ02−mχ0

1< MZ , two body spoiler decay modes such

as χ02 → χ0

1Z will be kinematically closed, and the three body decays χ02 → χ0

1`¯ (` = e or µ)

should occur at a sufficiently large rate at the LHC that an edge should be visible in them(`¯)

invariant mass distribution at mχ02−mχ0

1[21]. This measureable mass edge can serve as the

starting point for sparticle mass reconstruction in SUSY particle cascade decay events at the

LHC [22]. Thus, in Yukawa-unified models, this mass edge is highly likely to be visible. In

Fig. 2.3, we show the expected masses of a) uL-squark, b) the t1-squark, c), the pseudoscalar

Higgs boson A and d) the superpotential Higgs parameter µ. Frame a) shows that Yukawa-

unified solutions prefer first/second generation squarks and sleptons with masses in the 5–20

TeV range – far higher than values typically examined in phenomenological SUSY studies!

The top squark mass and the A, H and H± Higgs bosons tends to be somewhat lighter: in

the 2–8 TeV range. Finally, frame d) shows that the µ parameter – which is derived from

the EWSB minimization conditions – tends also to be in the 5–15 TeV range. Thus, using

a top-down approach to search for Yukawa-unified solutions in the HS model, we find that

µM1, M2, so that the lighter charginos and neutralinos should be gaugino-like and quite

3 We collectively refer to the set of all gluinos, charginos and neutralinos as -inos.

26

Figure 2.1: Plot of R versus various input parameters for a wide (dark blue) and narrow(light blue) random scan over the parameter ranges listed in Eq. (2.6). We take µ > 0 andmt = 171 GeV.

27

Figure 2.2: Plot of R versus various sparticle masses for a random scan over the parameterrange listed in Eq. (2.6). We take µ > 0 and mt = 171 GeV.

light, while the heavier charginos and neutralino will be in the multi-TeV range and nearly

pure higgsino-like states. In particular, the lightest SUSY particle (LSP )– the neutralino

χ01 – is nearly pure bino-like. In Fig. 2.4, we plot R vs. Ωχ0

1h2 for LEP2 allowed points from

our random scan. It is clear that R ∼ 1 points predict an extremely large value of Ωχ01h2 of

30–30,000. On the other hand, if we require consistency with the WMAP-measured value of

Ωχ01h2 ' 0.1, then we generate Yukawa-unified solutions to 40% unification with the random

scan. This plot underscores the difficulty of finding sparticle mass spectra solutions which

are compatible with both the measured dark matter abundance and t–b–τ Yukawa coupling

unification.

28

Figure 2.3: Plot of R versus various sparticle masses for a random scan over the parameterrange listed in Eq. (2.6). We take µ > 0 and mt = 171 GeV.

Figure 2.4: Plot of Ωχ01h2 vs. R for a random scan over the parameter range listed in Eq. (2.6).

We take µ > 0 and mt = 171 GeV.

29

2.2.2 Three proposals to reconcile Yukawa-unified models withdark matter relic density

Dark matter solution via neutralino decay to axino

We see from Fig. 2.4 that models generated from the random scan with R ∼ 1.0 all have

Ωχ01h2 ∼ 30 − 30, 000 – far beyond the WMAP-measured result of ΩCDMh

2 ∼ 0.1. One

possible solution to reconcile the predicted and measured dark matter density is to assume

that the lightest neutralino χ01 is in fact not the LSP but is unstable. Some alternative LSP

candidates consist of the gravitino G or the axino a. In gravity-mediated SUSY breaking

models, the gravitino mass m3/2 arises due to the superHiggs mechanism and is expected

to set the scale for all the soft SUSY breaking terms. Usually it is assumed the gravitino

is heavier than the lightest neutralino m3/2 > mχ01, in which case the gravitino essentially

decouples from phenomenology. However, if m3/2 < mχ01, then the χ0

1 becomes unstable and

can decay via modes such as χ01 → γG. The χ0

1 lifetime is expected to be very long – of

order 104− 1012 sec – so the neutralino still escapes detection at collider experiments, but is

susceptible to constraints from Big Bang nucleosynthesis (BBN) and CMB anisotropies [23].

The relic density of gravitinos is expected to be simply ΩGh2 =

m3/2

mχ01

Ωχ01h2, since the

gravitinos “inherit” the thermally produced neutralino relic number density. Thus, a scenario

with a G superWIMP as LSP in SUGRA-type models can reduce the relic density by typically

factors of a few – which is not enough in the case of Yukawa-unified models, where relic

density suppression factors of 102 − 105 are needed.

A better option occurs if we hypothesize an axino a LSP. If indeed there is a Peccei-Quinn

solution to the strong CP problem, then one expects the existence of axions, typically with

mass below the eV scale. While axions can themselves form cold dark matter, it is also easily

possible that they contribute little to the CDM relic density. However, in models with SUSY

and axions, then the axion is just one element of an axion superfield, the superpartner of

the axion being a spin-12

axino a. The axino mass can be far different from the typical soft

SUSY breaking scale, and the range ma ∼ eV−GeV is allowed.

Axinos can be produced in the early universe both thermally or non-thermally from NLSP

decay. From the latter source, we expect roughly [24]

Ωah2 ∼ ma

mχ01

Ωχ01h2. (2.8)

30

Thus, for Ωχ01h2 ∼ 103 and with mχ0

1∼ 50 GeV as in Yukawa-unified models, an axino mass

of ma<∼ 5 MeV is required.

In this mass range, the axinos from χ01 decay are expected to give a hot/warm component

to the dark matter [92]. However, thermally produced axinos in this mass range could yield

the required cold dark matter. Thus, if an unstable neutralino decay χ01 → aγ is to reconcile

Yukawa-unified models with the relic density, then we would expect the dark matter to be

predominantly cold axinos produced thermally, but with a re-heat temperature TR < Tf ,

where Tf is the temperature where axinos decouple from the thermal plasma in the early

universe. This scenario admits a dark matter abundance that can be in accord with WMAP

measurements, and would be primarily CDM, but with a warm dark matter component

arising non-thermally from χ01 decays. For a bino-like neutralino, as in Yukawa-unified

models, the χ01 lifetime is given by [26]

τ ' 3.3× 10−2sec1

C2aY Y

(fa/N

1011GeV

)2(

50 GeV

mχ01

)3

, (2.9)

where the model-dependent constant CaY Y is of order 1, fa is the Peccei-Quinn breaking

scale, and N is a model dependent factor (N = 1(6) for the KSVZ (DFSZ) axion model).

Thus, for reasonable choices of model parameters, we expect the neutralino lifetime to be

of order 3× 10−2 sec. This is short enough so that photon injection into the early universe

from χ01 → aγ decay occurs before nucleosynthesis, thus avoiding constraints from BBN.

For illustration, we adopt a point A listed in Table 2.1 of Yukawa-unified benchmark

models. The point has m16 = 9202.9 GeV, m10 = 10966.1 GeV, MD = 3504.4 GeV,

m1/2 = 62.5 GeV, A0 = −19964.5 GeV, tan β = 49.1 GeV with µ > 0 and mt = 171

GeV. It has mχ01

= 55.6 GeV and Ωχ01h2 = 423 (IsaReD result). Thus, χ0

1 → aγ with ma<∼ 1

MeV would allow for a mixed warm/cold axino dark matter solution to the problem of relic

density in Yukawa-unified models.

Dark matter solution via non-universal gaugino masses

An alternative solution to reconciling the dark matter abundance with Yukawa-unified

models is to consider the possibility of non-universal gaugino masses. If we adopt any of the

Yukawa unified models from the random scan and vary the SU(2) (SU(3)) gaugino masses

M2 (M3), then the Yukawa coupling unification will be destroyed via the effect of tiWj (gq)

31

loops. However, if M1 is varied, Yukawa coupling unification is preserved since contributions

to fermion masses from loops containing χ01 are small.

By raising the GUT scale value of M1 to values higher than m1/2, the weak scale value

of M1 is also increased. If M1 is increased enough, then mχ01

(which is nearly equal to M1

since χ01 is largely bino-like) becomes close to mfW1

. When this happens, the χ01 becomes

more wino-like, with an increased annihilation cross section to WW pairs if mχ01> MW [27].

In our case, usually mχ01< MW . Then raising M1 still lowers the relic density, but now via

bino-wino co-annihilation (BWCA) [17].

In Fig. 2.5, we show the variation in Ωχ01h2 versus M1(MGUT ) for benchmark point A

in Table 2.1. The location of M1 for point A is marked by the arrow. The double dips at

low M1 are due to neutralino annihilation through the Z and h poles. Once M1(MGUT ) is

increased to ∼ 195 GeV, then we reach a relic density in accord with WMAP measurements.

Since mfW1' mχ0

1, and mfW1

∼ mχ01, the χ0

2 − χ01 mass gap is small, of order 10–20 GeV. We

list the raised M1(MGUT ) = 195 GeV point as point B in benchmark Table 2.1.

Figure 2.5: Plot of variation in Ωχ01h2 versus non-universal GUT scale gaugino mass M1 for

benchmark point A in Table 2.1.

Dark matter solution via generational non-universality

Another possibility for reconciling the neutralino relic density with the measured value is

to lower the first/second generation scalar masses m16(1, 2), while keeping m16(3) fixed

32

at m16. The Bagger et al. inverted hierarchy solution depends only on third generation

scalar masses, while the effects of the first two generations decouple. Ordinarily, solutions

with m16(1, 2) = m16(3) are taken to enforce the super-GIM mechanism for suppression of

flavor changing neutral current (FCNC) processes. Limits from FCNCs mainly require near

degeneracy between the first two generations, while limits on third generation universality

are much less severe [28]. Lowering m16(1, 2) works to lower the relic density because of the

large S term in the scalar mass RGEs:

S = m2Hu−m2

Hd+ Tr

[m2

Q −m2L − 2m2

U + m2D + m2

E

]. (2.10)

In models with universality, like mSUGRA, S = 0 to one-loop at all energy scales; in models

with non-universal Higgs scalars, like the HS model, this term can be large and have a major

influence on scalar mass running. The large S term helps suppress right-squark masses. If

m16(1, 2) is taken light enough, then muR' mcR

' mχ01, and neutralino-pair annihilation

into quarks and neutralino-squark co-annihilation can act to reduce the relic density.

In Fig. 2.6, we show the variation in Ωχ01h2 versusm16(1, 2) where we takem16(3) = 5018.8

GeV, m1/2 = 160 GeV, A0 = −10624.2 GeV, tan β = 47.8 and µ > 0. When m16(1, 2) is

lowered to 603.8 GeV, then muR' mcR

= 98.3 GeV, and we have neutralino annihilation

via light t-channel squark exchange and also neutralino-squark co-annihilation.4 IsaReD

and Micromegas give Ωχ01h2 ∼ 0.1 at this point, which we adopt as benchmark point C in

Table 2.1. The two light squarks are just at the limit of LEP2 exclusion. They may possibly

be excludable by Tevatron analyses, but the squark neutralino mass gap is quite small, so

the energy release from uR → uχ01 is low. So far, no such study has been made, and so the

possibility cannot yet be definitively excluded.

2.3 Discussion of Markov Chain Monte Carlo analysis

The Markov Chain Monte Carlo (MCMC) technique is an improvement over Random

Scanning (RS) in that it searches more efficiently for parameter space regions of good Yukawa

unification and WMAP-compatible DM relic density. A Markov Chain [29] is a discrete-time,

random process having the Markov property, which is defined such that given the present

state, the future state only depends on the present state, but not on the past states. That

4 A bug fix is needed in the Isajet 7.75 IsaReD subroutine in order to obtain the correct relic density.This bug has been rectified in all later versions of Isajet.

33

Figure 2.6: Plot of variation in Ωχ01h2 versus non-universal GUT scale first/second generation

scalar mass m16(1, 2) for benchmark point C in Table 2.1.

is:

P (X t+1 = x|X t = xt, ..., X1 = x1) = P (X t+1 = x|X t = xt). (2.11)

An MCMC constructs a Markov Chain through sampling from a parameter space with

the help of a specified algorithm. In this study, we have applied the Metropolis-Hastings

algorithm [30], which generates a candidate state xc from the present state xt using a proposal

density Q(xt;xc). The candidate state is accepted to be the next state xt+1 if the ratio

p =P (xc)Q(xt;xc)

P (xt)Q(xc;xt), (2.12)

(where P (x) is the probability calculated for the state x) is greater than a uniform random

number a = U(0, 1). If the candidate is not accepted, the present state xt is retained and a

new candidate state is generated. For the proposal density we use a Gaussian distribution

that is centered at xt and has a width σ. This simplifies the p ratio to P (xc)/P (xt).

Once taking off from a starting point, Markov chains are aimed to converge at a target

distribution P (x) around a point with the highest probability. The time needed for a Markov

chain to converge depends on the width of the Gaussian distribution used as the proposal

density. This width can be adjusted during the run to achieve a more efficient convergence.

In our search in the SO(10) parameter space, we assume flat priors and we approximate

34

the likelihood of a state to be e−χ2(x). We define the χ2 for R as

χ2R =

(R(x)−Runification

σR

)2

(2.13)

where Runification = 1 and σR is the discrepancy we allow from absolute Yukawa unification

which, in this case, we take to be 0.05. On the other hand, for Ωh2 we define

χ2Ωh2 =

1, (0.094 ≤ Ωh2 ≤ 0.136)(

Ωh2(x)−Ωh2mean

σΩh2

)2

, (Ωh2 < 0.094 or Ωh2 > 0.136)(2.14)

where Ωh2mean = 0.115 is the mean value of the range 0.094 < Ωh2 < 0.136 proposed

in [31], and σΩh2 = 0.021. This way, the MCMC primarily searches for regions of Yukawa-

unifications, and within these regions for solutions with a good relic density.

For each search, we select a set of ∼ 10 starting points in order to ensure a more thorough

investigation of the parameter space. Then we run the MCMC, aiming to maximize the

likelihood of either R alone, or R and Ωh2 simultaneously. For the case of simultaneous

maximization, we compute the p ratios for R and Ωh2 individually, requiring both pR > a and

pΩh2 > a separately. We do not strictly seek convergence to an absolute maximal likelihood,

but we rather use the MCMC as a tool to reach compatible regions and to investigate the

amount of their extension in the SO(10) parameter space.

2.3.1 HS model: neutralino annihilation via h resonance

The MCMC scans were initiated by selecting 10 starting points “pseudorandomly” – that

is, selecting them from different m16 regions to cover a wider range of the parameter space –

and imposing some loose limits (defined by previous works and random scans) on the rest of

their parameters to achieve a more efficient convergence. Our initial scan is directed to look

for points only with R as close to 1.0 as possible by maximizing solely the likelihood of R.

Based on the results of the first MCMC scan, we then pick a new set of 10 starting points

with low R and also low Ωχ01h2, and direct the second scan to look for points with both

R = 1.0 and Ωχ01h2 < 0.136 by maximizing the likelihoods of R and Ωh2 simultaneously. For

MCMC scans, the code is interfaced to the micrOMEGAs [32] package to evaluate the relic

density and low-energy constraints.

Figure 2.7 shows the Yukawa-unified region found by the MCMC results as a projection

in the plane of m16 versus m10. The light-blue dots are points which have R < 1.1, while

35

dark blue dots have R < 1.05. In addition, we show in orange (red) the points which satisfy

R < 1.1 (1.05) and Ωχ01h2 < 0.136. The points with low R are narrowly correlated along the

line m10 ' 1.2m16. While the low R points range over m16 values from 3 to over 12 TeV (in

agreement with the results from the random scans) the MCMC has also identified a range

of points with both R ' 1 and Ωχ01h2 < 0.136, but only for m16 values of about 3–4 TeV!

Figure 2.7: Plot of MCMC results in the m16 vs. m10 plane; the light-blue (dark-blue) pointshave R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05) and Ωχ0

1h2 < 0.136.

Fig. 2.8 shows the MCMC scan results in the m16 vs. A0/m16 plane. Again, we see that

points with low R populate the region with A0 ∼ (2–2.1)m16 over a wide range of m16 values.

The plot includes the Ωχ01h2 < 0.136 points around m16 ∼ 3–4 TeV.

In Fig. 2.9, we show MCMC results in the m16 vs. m1/2 plane. Here, we see the very

lowest R points select out the lowest possiblem1/2 values allowed for a given value ofm16, and

that the minimum m1/2 value allowed steadily decreases with increasing m16 – the boundary

being determined by the LEP2 limit on chargino masses. The points with a “good” relic

density are clustered around m1/2 ∼ 100 GeV.

We also show in Fig. 2.10 the individual GUT-scale values of Higgs soft terms mHu

(lower branch) and mHd(upper branch). This plot displays the required Higgs splitting and

confirms that mHd> mHu .

36

Figure 2.8: Plot of MCMC results in the m16 vs. A0/m16 plane; the light-blue (dark-blue) points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05) andΩχ0

1h2 < 0.136.

Figure 2.9: Plot of MCMC results in the m16 vs. m1/2 plane; the light-blue (dark-blue) pointshave R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05) and Ωχ0

1h2 < 0.136.

37

Figure 2.10: Plot of MCMC results in the m16 vs. mHd,uplane; the light-blue (dark-

blue) points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05) andΩχ0

1h2 < 0.136.

In Fig. 2.11, we show points with low R in the mh − 2mχ01

vs. mA − 2mχ01

plane. In

these solutions, mA is usually far greater than 2mχ01, indicating the neutralino annihilation

through the A-resonance is not the cause of the reduced relic density orange and red points.

However, the low Ωχ01h2 points all do lie along the mh ' 2mχ0

1line, indicating that h-

resonance annihilation is the mechanism at work to reduce the relic density in the early

universe. In Fig. 2.12, we show R vs. Ωχ01h2 for the MCMC scan. In this frame, we see that

the points with high relic density extend down to R = 1, while the low relic density points

reach below R = 1.05 but can reach no lower than R = 1.03.

In summary, what we learn from this set of scans is that the search for low R pushes

m16 to very high, multi-TeV values. Meanwhile, in order for h-resonance annihilation to

reduce the relic density to the WMAP-allowed range, m16 cannot be too large. The region

around m16 ∼ 3–4 TeV offers a compromise between these two tendencies: for m16 not too

large, the dip in relic density due to the h-resonance annihilation is sufficient to bring the

relic density into the desired range. But since m16 can’t be too large, the Yukawa unification

38

Figure 2.11: Plot of MCMC results in the mh − 2mχ01

vs. mA − 2mχ01

plane; the light-blue(dark-blue) points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05) andΩχ0

1h2 < 0.136.

Figure 2.12: Plot of MCMC results in the R vs. Ωχ01h2 plane; the light-blue (dark-blue) points

have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05) and Ωχ01h2 < 0.136.

39

is limited to a couple of percent at best. This new class of solutions was difficult to reach

using a random scan, since the h-resonance is so narrow. The necessary value of mχ01

has to

be just right – with 2mχ01

slightly below mh – so that the thermal averaging of neutralino

energies convolutes with the resonant cross section with enough strength to give substantial

neutralino annihilation in the early universe.

We adopt point D in Table 2.1 as being representative of the light Higgs h-resonance

annihilation compromise solutions. The relic density computed with micrOMEGAs (Ωχ01h2 =

0.06) is below the preferred range, while IsaReD gives Ωχ01h2 = 0.1. Yukawa couplings are

unified at the 9% level. We note here that we could have adopted a solution with even better

Yukawa coupling unification at the 4–5% level. These solutions tend to give light Higgs mass

mh<∼ 110 GeV (as can be seen by the red dots in Fig. 2.13) which are more likely to be

excluded by LEP2 Higgs search results.

Figure 2.13: Plot of MCMC results in the mχ02− mχ0

1vs. mh plane; the light-blue (dark-

blue) points have R < 1.1 (1.05), while for the orange (red) points R < 1.1 (1.05) andΩχ0

1h2 < 0.136.

The SO(10) model parameters leading to low R and good relic density occur only over

a very narrow range of m1/2 ∼ 100 GeV and m16 ∼ 3 TeV. This means the Yukawa-unified

40

h-resonance annihilation points have very specific mass spectra predictions. Implications for

collider searches within this model is discussed in Chapter 4 of this thesis.

2.3.2 Solutions using weak scale Higgs boundary conditions

In the analysis put forth by BDR [46], Yukawa-unified solutions are found with low values

of both µ and mA in the 100–200 GeV range, while m16 and m10 are typically at the 2–3

TeV scale. We have seen from our results so far that µ and mA are typically in the TeV

regime. Some low µ solutions were generated using Isajet in Table 2 of Ref. [76], but these

had R ∼ 1.25.

We find here that we can generate small µ and small mA solutions using Isajet by using

the pre-programmed non-universal Higgs model (NUHM).5 The approach is to start with a

set of GSH soft term boundary conditions and evolve the soft SUSY breaking Higgs masses

m2Hu

and m2Hd

down to the weak scale MSUSY . At Q = MSUSY , re-calculate what m2Hu

and

m2Hd

should have been in order to get the input values of mA and µ using the two electroweak

symmetry breaking minimization conditions (in practice, we use 1-loop relations):

B =(m2

Hu+m2

Hd+ 2µ2) sin 2β

2µand (2.15)

µ2 =m2

Hd−m2

Hutan2 β

(tan2 β − 1)− M2

Z

2, (2.16)

then run back up to the GUT scale using these new WSH boundary conditions. At each

iteration, the weak scale values of m2Hu

and m2Hd

have to be re-computed so as to maintain

the input value of µ and mA; in this case, the GUT scale values of m2Hu

and m2Hd

are outputs

instead of inputs. For this class of solutions, both GSH and WSH boundary conditions must

be used in Isajet. The GSH boundary conditions are needed just to get an acceptable EWSB

on the first iteration so that a spectrum can be computed, then later modified to yield the

input values of mA and µ. Using default universal GSH soft terms will usually fail to give

appropriate EWSB on any iteration where Yukawa couplings are unified.

We implement an MCMC scan over the modified parameter space

m16, m1/2, A0, tan β, mA, µ (2.17)

(effectively trading the GUT scale inputs m2Hu

and m2Hd

(or alternatively m10 and M2D) for

weak scale inputs mA and µ). We begin with 10 starting points selected pseudorandomly

5 This is model line 8 of the Isajet non-universal supergravity models (NUSUG).

41

from different regions of the above parameter space, and implement two MCMC scans on

them, one searching for points with lowest R values by maximizing the likelihood of R and

the other for solutions with R = 1 and Ωχ01h2 < 0.136 by maximizing likelihoods of R and

Ωh2 simultaneously.

Our first results are shown in Fig. 2.14 for the m16 vs. A0/m16 plane, where we plot points

with R < 1.1 (1.05) using dark blue (light blue) dots, and solutions with Ωχ01h2 < 0.136 for

R < 1.1 (1.05) using orange (red) dots. While we again get good Yukawa-unified solutions

over a wide range of multi-TeV values of m16, this time we pick up additional dark matter

allowed solutions for m16 : 3–6 TeV. The solutions again respect the Bagger et al. boundary

condition A0 ' −2m16.

Figure 2.14: Plot of MCMC results using WSH boundary conditions in the m16 vs. A0/m16

plane; the light-blue (dark-blue) points have R < 1.1 (1.05), while for the orange (red) pointsR < 1.1 (1.05) and Ωχ0

1h2 < 0.136.

Using these boundary conditions, while we again get good Yukawa-unified solutions over

a wide range of multi-TeV values of m16, this time we pick up additional dark matter allowed

solutions form16 : 3–6 TeV. The solutions again respect the Bagger et al. boundary condition

A0 ' −2m16. An additional scan (not included here) shows that the minimum in allowed

42

m1/2 values again decreases with increasing m16. We see that for the WSH class of solutions,

much larger values of m1/2 ranging up to 300− 500 GeV are DM-allowed.

In Fig. 2.15, the bulk of the DM-allowed solutions occur at relatively low values of

mA ∼ 130–250 GeV. These low mA solutions were extremely difficult to generate with the

top-down approach, and indicate that they have a high degree of fine-tuning.6 A scattering of

DM-allowed dots occur with high mA values. These turn out to be the h-resonance solutions

as generated with the GSH boundary conditions in Sec. 2.3.1. This is seen more clearly by

plotting in the mh − 2mχ01

vs. mA − 2mχ01

plane, Fig. 2.16 where we see a narrow strip at

mh − 2mχ01

= 0 corresponding to h-resonance annihilation solutions, while we also have a

wider band of solutions at mA − 2mχ01

= 0, which indicate neutralino annihilation through

the A-resonance. The width of the latter band is due to the fact that the A width can be

quite wide – typically a few GeV, while the h-width is much narrower, of order 50 MeV.

Figure 2.15: Plot of MCMC results using WSH boundary conditions in the mA vs. µ plane;the light-blue (dark-blue) points have R < 1.1 (1.05), while for the orange (red) pointsR < 1.1 (1.05) and Ωχ0

1h2 < 0.136.

The A-resonance solutions occur at tan β ∼ 50 and relatively low mA values. This can

6 The TW paper (Ref. [11]) remarks that there must be considerable fine-tuning as well to reconcileBF (b → sγ) with Yukawa unification and the dark matter relic abundance.

43

Figure 2.16: Plot of MCMC results using WSH boundary conditions in the mh − 2mχ01

vs.mA−2mχ0

1plane; the light-blue (dark-blue) points have R < 1.1 (1.05), while for the orange

(red) points R < 1.1 (1.05) and Ωχ01h2 < 0.136.

signal dangerously high branching fractions for Bs → µ+µ− decay [33] since the branching

fraction goes like tan6 β/m4A. We plot the BF (Bs → µ+µ−) vs. mh in Fig. 2.17. The recent

experimental limit from the CDF collaboration is that BF (Bs → µ+µ−) < 5.8 × 10−8 [34].

Thus, the entire band of A-resonance annihilation solutions becomes excluded! The

smattering of DM-allowed dots below the CDF limit all occur with DM annihilation via

the h-resonance. However, we may still want to consider A-resonance solutions in case they

are somehow allowed perhaps by additional flavor-violating soft terms. This was done by

Baer et al [79].

At this point, it is useful to compare the Isajet SUSY spectral solutions to those generated

by Dermisek et al. in Ref. [13] and [14]. In Fig. 2.18, we plot the Isajet 7.75 solutions in

the m1/2 vs. µ plane for m16 = 3 TeV, m10/m16 = 1.3, A0/m16 = −1.85, tan β = 50.9 and

∆m2H = 0.14, with mA = 500 GeV: i.e. corresponding closely to Fig. 1 of [13]. We plot

contours of R from 1.15 to 1.3. Also, the green-shaded regions give the WMAP-measured

relic density, while white-shaded regions give Ωχ01h2 < 0.095, and pink-shaded regions give

44

Figure 2.17: Plot of MCMC results using WSH boundary conditions in the mh vs.BF (Bs → µ+µ−) plane; the light-blue (dark-blue) points have R < 1.1 (1.05), while forthe orange (red) points R < 1.1 (1.05) and Ωχ0

1h2 < 0.136.

Ωχ01h2 > 0.13 (as in Dermisek et al.). The LEP2 constraint on mfW1

is indicated by the solid

contour at low m1/2 and low µ. We see qualitatively the same shape to the DM-allowed

regions as generated by Dermisek et al.: the thick green regions are DM-allowed either by

A-resonance annihilation at large µ, or by mixed higgsino DM annihilation at low µ. There

is also a light Higgs h-resonance solution at m1/2 ∼ 120 GeV.

A notable feature of Fig. 2.18 is that over much of the DM-allowed region, the Yukawa

unification has R > 1.2.7 As we move to larger µ values and lower m1/2 values, the Yukawa

unification gets better and better. Most of the region with R < 1.15 is DM-forbidden,

save for the upper part of the light h-resonance solution. In fact, now we can see why

our compromise solution (point D) works and why it is so hard to find using a top-down

approach: only the very narrow upper tip is both DM-allowed, and has a low R value.

7 Note that although the general features in Fig. 2.18 here and Fig. 1 of [13] are similar, the latter resultswere obtained in a top-down fit to low energy obervables assuming exact Yukawa unification, which is adifferent approach then the one followed here. Moreover, there are several important differences in the levelof sophistication of the spectrum computations between Ref. [46, 13, 14] and the study presented here. Forinstance, Ref. [46, 13, 14] has only 1-loop RGE running of the SUSY-breaking parameters, takes sparticlemasses to be running masses at scale Q = MZ ; ISAJET 7.75 applies full 2-loop running plus 1-loop thresholdcorrections.

45

Figure 2.18: Contours of R and DM-allowed regions in the m1/2 vs. µ parameter space form16 = 3 TeV, m10/m16 = 1.3, A0/m16 = −1.85, ∆mH = 0.14, tan β = 50.9, mA = 500 GeVand mt = 173.9 GeV, as in Dermisek et al., but using Isajet 7.75 for mass spectra generation.

2.4 Yukawa-unified benchmark scenarios and LHCsignatures

We have assembled in Table 2.1 five Yukawa-unified benchmark scenarios that yield the

correct relic abundance of dark matter in five different ways. With the LHC turn-on being

imminent, it is fruitful to examine what each of these five scenarios implies for new physics

signatures.

At the bottom of Table 2.1 we list Ωχ01h2, BF (b→ sγ), BF (Bs → µ+µ−), ∆aµ and spin-

independent neutralino-proton direct DM detection cross section σ(χ01p). For the first four

of these numbers, we list output from IsaReD/Isatools (upper) and micrOMEGAs (lower).

While the results for the low-energy constraints agree fairly well, there is almost a factor of

2 difference in the relic density when the neutralino dominantly annihilates through h or A

exchange (points A, D, E). This is due to differences in the treatment of the Higgs resonance.

For example, IsaReD in Isajet 7.75 uses Yukawa couplings evaluated at scale Q =√mtL

mtR

for annihilation through the A resonance and for evaluation of the heavy Higgs widths, while

micrOMEGAs uses an effective Lagrangian approach and Q = 2mχ01.8 This section serves

8 A complete discussion of the details of the calculations in the two programs is beyond the scope of this

46

as a discussion of a few various scenarios. Actual collider studies of these scenarios will be

saved for Chapter 4

Point A

Point A of Table 2.1 is a generic Yukawa-unified model with first and second generation

scalar masses ∼ 9 TeV, so they essentially decouple from LHC physics. Third generation

and heavy Higgs scalars have masses at the 2–3 TeV level, while the lightest charginos,

neutralinos and gluinos all have masses in the range 100–400 GeV. Since Ωχ01h2 ∼ 400, we

postulate that the neutralino χ01 is in fact an NLSP, decaying to aγ with a lifetime of order

0.03 seconds. In this case, the mean decay distance of a χ01 will be of order 104 km. Thus,

the χ01 will still escape the LHC detectors, leading to missing energy signatures (although it

is conceivable some may decay occassionally within the detector).

The LHC SUSY events will consist of a hard and soft component [70]. The hard

component comes from pair production of ∼ 400 GeV gluinos. The gluinos decay via 3-

body modes dominantly via g → tbW1, bbχ01 and especially bbχ0

2 [35]. The gg production

cross section is of order 105 fb at LHC, so we might expect 107 gluino pair events per 100

fb−1 of integrated luminosity. After cascade decays, we expect an assortment of events

with high jet and b-jet multiplicity, plus an assortment of isolated leptons. The χ02 → χ0

1ee

branching fraction is at 2.2% , which should be enough to reconstruct the dilepton mass

edge at mχ02−mχ0

1' 73 GeV. Correct pairing of b-jets and/or b-jets with isolated leptons,

plus the total event rate, should allow for an extraction of the gluino mass.

The soft component of signal will come from χ+1 χ

−1 , W1χ

02 and W1χ

01 production. These

events will be followed by 3-body decays to various final states, but since the visible

components of the signal are much softer than that from gluino pair production, these events

will be harder to see above SM background levels. With judicious cuts, the soft component

might also be visible at some level (e.g. W1χ02 → 3`+ Emiss

T ) [36].

Point B

Point B is the same as point A, except that in this case the gaugino mass M1 has been raised

to 195 GeV so that the W1 − χ01 mass gap shrinks to only 13 GeV. Since µ is quite large,

dissertation; we refer the interested reader to the respective manuals.

47

Table 2.1: Masses and parameters in GeV units for five benchmark Yukawa unified pointsusing Isajet 7.75 and mt = 171.0 GeV. The upper entry for the Ωχ0

1h2 etc. come from

IsaReD/Isatools, while the lower entry comes from micrOMEGAs; σ(χ01p) is computed with

Isatools.

parameter A B C D Em16 9202.9 9202.9 5018.8 2976.5 5877.3m1/2 62.5 62.5 160 107.0 113.6A0 −19964.5 −19964.5 −10624.2 −6060.3 −12052.6m10 10966.1 10966.1 6082.1 3787.9 —tan β 49.1 49.1 47.8 49.05 47.4MD 3504.4 3504.4 1530.1 1020.8 —M1 — 195 — — —m16(1, 2) — — 603.8 — —ft 0.51 0.51 0.49 0.48 0.49fb 0.51 0.51 0.41 0.47 0.49fτ 0.52 0.52 0.47 0.52 0.49µ 4179.8 4186.3 1882.6 331.0 865.3mg 395.6 395.4 495.5 387.7 466.6muL

9185.4 9185.4 622.1 2970.8 5863.0muR

9104.1 9104.2 98.3 2951.4 5819.2mt1 2315.1 2310.5 1048.4 434.5 944.7mb1

2723.1 2714.9 1894.0 849.3 1452.7meL

9131.9 9132.0 311.9 2955.8 5833.6meR

9323.7 9323.9 891.8 3009.0 5945.8mfW1

128.8 128.8 165.7 105.7 141.3

mχ02

128.6 128.1 165.1 105.1 140.9

mχ01

55.6 115.9 80.2 52.6 65.7

mA 3273.6 3266.0 1939.9 776.8 177.8mh 125.4 125.4 123.2 111.1 113.4

Ωχ01h2 423

2200.090.08

0.110.11

0.100.06

0.150.08

BF (b→ sγ) 3.0×10−4

3.3×10−43.0×10−4

3.3×10−46.2×10−4

3.7×10−41.9×10−4

4.0×10−42.5×10−4

2.2×10−4

∆aµ5.0×10−12

5.1×10−125.0×10−12

5.0×10−123.0×10−10

2.8×10−102.2×10−10

2.2×10−104.1×10−11

4.1×10−11

BF (Bs → µ+µ−) 5.0×10−9

4.4×10−95.0×10−9

4.4×10−911.8×10−9

6.9×10−95.8×10−8

6.2×10−82.0×10−5

2.0×10−5

σsc(χ01p) [pb] 1.3× 10−15 1.9× 10−17 1.5× 10−6 2.7× 10−9 5.3× 10−8

48

the χ01 remains nearly pure bino-like, but the relic density problem is solved via bino-wino

co-annihilation. This case will again give a hard component to the LHC new physics signal

from gluino pair production, but this time the m(`+`−) distribution will have an edge only at

13 GeV. When compared to any gluino mass reconstructions, this would indicate a violation

of gaugino mass unification at the GUT scale. In addition, the small χ02 − χ0

1 mass gap

suppresses 3-body decays such as χ02 → χ0

1qq and χ01`

¯ relative to any kinematically-allowed

2-body decays such as the loop-induced process χ02 → χ0

1γ [37]. Thus, the radiative χ02 decay

to photon χ02 → χ0

1γ can become large [17]: in this case, it reaches 10%. The final state γ

will be somewhat soft if the χ02 is at low velocity. But if χ0

2 is moving fast as a result of

production from cascade decays, then hard, isolated photons should occasionally be present

in the SUSY collider events.

Point C

The point C parameters are listed in Table 2.1. In this case, m16(1, 2) has been lowered

far below m16(3) so that the first two generations of scalars degenerate, but with a lower

mass than third generation scalars. The Higgs mass splitting leads to a large RGE S-term,

which drives uR and cR to very low masses ' 98.3 GeV. The relic density problem is solved

because χ01χ

01 → qq via qR exchange and neutralino-squark co-annihilation act to reduce the

relic density. The cross section for production of two flavors of extremely light squarks is

extremely large at LHC. Normally one would expect characteristic dijet+EmissT events since

qR → qχ01. However, in this case the mass gap muR

− mχ01∼ 18 GeV, so both the jets

and EmissT will be very soft. Gluinos and other squarks will also at large rates, although the

g → uuR, ccR decays are dominant. While left-squarks may decay wih large rates to χ02 and

W1, we note that χ02 → uuR and ccR is also large, leading again to relatively soft jet activity.

In spite of the soft jet activity, the scenario should be easily seen at LHC, since qL → q′W1

occurs at a large rate, and W1 → eνeχ01 occurs at 43% branching fraction. This can lead to a

large same-sign dilepton rate from pp→ uLuL production, along with a large asymmetry in

++ SS dileptons over −− SS dileptons (which occur from dLdL production). This scenario

may also be subject to exclusion by analysis of Fermilab Tevatron data. We further note that

point C is naively excluded by direct dark matter search limits. These latter limits depend

on an assumed standard local relic density mass and velocity distribution, so that the limits

can be avoided if one postulates that we live in a local underdensity of dark matter.

49

Point D

Point D is an example of a compromise solution, where we allow m16 as low as 3 TeV at

some expense to Yukawa unification (here, Yukawa unification is good to only ∼ 10%) in

order to allow for neutralino annihilation through the light Higgs h-resonance (neutralinos

can still annihilate through the light h resonance for higher m16 values; it is just that the relic

density can’t be pushed as low as Ωχ01h2 ∼ 0.1). This scenario is extremely predictive, with

gluinos around 350–450 GeV, so again we expect LHC events to be dominated by gluino pair

production. As in the case of point A, the gg events will be followed by 3-body decays to

b-jet rich final states. A dilepton mass edge at mχ02−mχ0

1' 53 GeV should be visible since

χ02 → χ0

1e+e− at 3.3% branching fraction. The t1 weighs only 434 GeV in this case, and b1

is at 849 GeV, so it may be possible to detect some third generation squark pair production

events. The top squark decays to bW1 with a 50% branching fraction, and also has significant

branching fractions to tχ01, tχ

02 and bW2 final states. The b1 dominantly decays to bg and

Wt1 final states. Moreover, the heavy Higgs bosons A0, H0 and H± have masses around 780

GeV and should be detectable at LHC [38].

Point E

Point E is a Yukawa-unified solution that solves the DM abundance problem via neutralino

annihilation through a 178 GeV pseudoscalar A resonance. The combination of light A

and large tan β leads to a branching fraction Bs → µ+µ− which is excluded by recent

CDF analyses. If we are allowed to somehow ignore this (possibly via other flavor-violating

interactions), then the scenario would be at the edge of observability via Tevatron searches

for A, H → τ+τ− and bb, which at present exclude mA<∼ 170 GeV [39]. The LHC (and

possibly soon also the Tevatron) would easily see the rather light spectrum of Higgs bosons.

Gluinos can be somewhat heavier in this case compared to points A and D, ranging to over

a TeV. However, in point E as listed, with a 467 GeV gluino, the gluino pair production

signatures will be rather similar to those of point A: rich in b-jets, with a visible dilepton

mass edge at 75 GeV.

50

CHAPTER 3

SUSY Cosmology

3.1 Dark Matter

An abundance of astrophysical evidence points to the conclusion that the bulk of the matter

in the universe is composed not of Standard Model (SM) particles, but of some unknown

non-relativistic elementary particle known as cold dark matter (CDM)[?]. An analysis of the

five-year WMAP and galaxy survey data sets[49] implies that the ratio of cold dark matter

density to critical density is

ΩCDMh2 ≡ ρCDM/ρc = 0.111+0.011

−0.015 (2σ), (3.1)

where h = 0.74±0.03 is the scaled Hubble constant. While the density of CDM is becoming

precisely known, the identity of the CDM particle (or particles) is still a complete mystery.

Although numerous candidate CDM particles populate the theoretical literature, the WIMPs

(weakly interacting massive particles) stand out in that their thermal abundance can be

calculated, and is found to be in rough accord with Eq. (3.1) provided the WIMP mass is

of order 100−1000 GeV. Of the numerous WIMP candidates in the literature, the lightest

neutralino of supersymmetric (SUSY) theories is especially popular because SUSY solves a

host of theoretical problems associated with the SM, and also receives some (albeit indirect)

support from data (in the form of the measured gauge couplings unifying at Q = MGUT

under MSSM RG evolution and also from other precision electroweak measurements[?]).

There is at present a multi-pronged effort aimed at identifying WIMP dark matter

particles and measuring their properties[?]. The most direct approach is to try to detect relic

WIMPs left over from the Big Bang by observing WIMP-nucleon collisions in experiments

located deep underground. Limits from the CDMS[?] experiment and more recently from

the Xenon-10[?] experiment have begun probing the upper limits of SUSY model parameter

51

space.

WIMP particles can also be searched for at collider experiments such as those at the

CERN LHC, especially if the dark matter particle is but one of a whole family of particles,

some of which can be produced via strong and electromagnetic interactions. The dark matter

particle would then be produced by cascade decays of heavier particles, and would lead to

missing transverse energy in collider events. Such is the case of theories such as R-parity

conserving supersymmetry[63], KK-parity conserving universal extra dimensions (UED)[?]

and little Higgs models with T -parity[?].

Dark matter may also be searched for indirectly. For instance, the sun can sweep up

WIMP particles as it traverses its galactic orbit, so that WIMPs accumulate at a high

density in the solar core. Then WIMP-WIMP annihilation to SM particles can occur at high

rates in the solar core. While most SM particles would be absorbed by the surrounding solar

medium, multi-GeV scale νµs would escape and later convert to muons in neutrino telescopes

such as Amanda/IceCube or Antares.

3.2 SO(10) SUSY GUTs and Yukawa unification

Simple SUSY GUT models based on the gauge group SO(10) require t − b − τ Yukawa

coupling unification, in addition to gauge coupling and matter unification. The Yukawa

coupling unification places strong constraints on the expected superparticle mass spectrum,

with scalar masses ∼ 10 TeV while gaugino masses are quite light. A problem generic

to all supergravity models comes from overproduction of gravitinos in the early universe:

if gravitinos are unstable, then their late decays may destroy the predictions of Big Bang

nucleosynthesis. We present a Yukawa-unified SO(10) SUSY GUT scenario which avoids the

gravitino problem, gives rise to the correct matter-antimatter asymmetry via non-thermal

leptogenesis, and is consistent with the WMAP-measured abundance of cold dark matter

due to the presence of an axino LSP. To maintain a consistent cosmology for Yukawa-unified

SUSY models, we require a re-heat temperature TR ∼ 106− 107 GeV, an axino mass around

∼ 0.1− 10 MeV, and a PQ breaking scale fa ∼ 1012 GeV.

The method adopted in this chapter using the Isajet 7.75 program for calculation of the

SUSY mass spectrum and mixings[83] and IsaReD[44] for the neutralino relic density has

been explained in Chapter ??. What has been learned from the work presented in that

52

chapter is that t− b− τ Yukawa coupling unification does occur in the MSSM for µ > 0 (as

preferred by the (g − 2)µ anomaly), but only if certain conditions are satisfied.

• The scalar mass parameter m16 should be very heavy: in the range 5-20 TeV.

• The gaugino mass parameter m1/2 should be as small as possible.

• The SSB terms should be related as A20 = 2m2

10 = 4m216, with A0 = −2m16 (in our sign

convention). This combination was found to yield a radiatively induced inverted scalar

mass hierarchy (IMH) by Bagger et al.[82] for MSSM+right hand neutrino (RHN)

models with Yukawa coupling unification.

• tan β ∼ 50.

• EWSB can be reconciled with Yukawa unification only if the Higgs SSB masses are

split at MGUT such that m2Hu

< m2Hd

. The HS prescription ends up working better

than DT splitting[46, 45].

In the case where the above conditions are satisfied, then Yukawa coupling unification

to within a few percent can be achieved. The resulting sparticle mass spectrum has some

notable features.

• First and second generation matter scalars have masses of order m16 ∼ 5− 20 TeV.

• Third generation scalars, mA and µ are suppressed relative to m16 by the IMH

mechanism: they have masses on the 1 − 2 TeV scale. This reduces the amount

of fine-tuning one might otherwise expect in such models.

• Gaugino masses are quite light, with mg ∼ 350 − 500 GeV, mχ01∼ 50 − 80 GeV and

mfW1∼ 100− 150 GeV.

The sparticle mass spectra from SO(10) SUSY GUTs shares some features with spectra

generated in “large cutoff supergravity” or LCSUGRA, investigated in Ref. [48]. LCSUGRA

also has high mass scalars – typically with mass around 5 TeV – and low mass gauginos.

The SO(10) SUSY GUT models are different from LCSUGRA in that they have a large A0,

with A0 ∼ −2m16, and a µ term of around 1-2 TeV. This means SO(10) SUSY GUTs have a

dominantly bino-like χ01 state, whereas the LCSUGRA authors adopt the mSUGRA model

53

focus point region, which has a mixed higgsino-bino χ01 state. The latter can easily give the

measured abundance of cold dark matter (CDM) in the form of lightest neutralinos.

Since the lightest neutralino of SO(10) SUSY GUTs is nearly a pure bino state, it turns

out the neutralino relic density Ωχ01h2 is calculated to be extremely high, of order 102− 104.

This conflicts with the WMAP-measured value given above.

Several solutions to the SO(10) SUSY GUT dark matter problem have been proposed

in Refs. [50, 79]. Here, we will concentrate on the most attractive one: that the dark

matter particle is in fact not the neutralino, but the axino a. Axino dark matter occurs in

models where the MSSM is extended via the Peccei-Quinn (PQ) solution to the strong CP

problem[51]. The PQ solution introduces a spin-0 axion field into the model; if the model is

supersymmetric, then a spin-12

axino is also required. It has been shown that the a state can

be an excellent candidate for cold dark matter in the universe[52]. In this chapter, we will

find that SO(10) SUSY GUT models with an axino DM candidate can (1) yield the correct

abundance of CDM in the universe, (2) avoid the gravitino/BBN problem, and (3) have an

compelling mechanism for generating the matter-antimatter asymmmetry of the universe via

non-thermal leptogenesis.

3.2.1 The gravitino problem

An affliction common to all models with gravity-mediated SUSY breaking (supergravity or

SUGRA) models is known as the gravitino problem. In realistic SUGRA models (those that

include the SM as their sub-weak-scale effective theory), SUGRA is broken in a hidden sector

by the superHiggs mechanism, which induces a mass for the gravitino G, commonly taken

to be of order the weak scale. The gravitino mass mG ends up setting the mass scale for all

the soft breaking terms, so then all SSB terms end up also being of order the weak scale.

The coupling of the gravitino to matter is strongly suppressed by the Planck mass, so the

G is never in thermal equilibrium with the thermal bath in the early universe. Nonetheless, it

does get produced by scatterings of particles that do partake of thermal equilibrium. Thermal

production of gravitinos in the early universe has been calculated in Refs. [53], where the

abundance is found to depend naturally on mG and on the re-heat temperature TR at the

end of inflation. Once produced, the Gs decay into all varieties of particle-sparticle pairs,

but with a lifetime that can exceed ∼ 1 sec, the time scale where Big Bang nucleosynthesis

(BBN) begins. The energy injection from G decays is a threat to dis-associate the light

54

element nuclei which are created in BBN. Thus, the long-lived Gs can destroy the successful

predictions of the light element abundances as calculated by nuclear thermodynamics.

The BBN constraints on gravitino production in the early universe have been calculated

by several groups [54]. The recent results from Ref. [55] give an upper limit on the re-heat

temperature as a function of mG. See Fig. 3.1 by Kohri et al. displaying the gravitino lifetime

and reheating temperature in the mSUGRA model space. The results depend on how long-

lived the G is (at what stage of BBN the energy is injected), and what its dominant decay

modes are. Qualitatively, for mG

<∼ 5 TeV, TR<∼ 106 GeV is required; if this is violated, then

too many G are produced in the early universe, which detroy the 3He, 6Li and D abundance

calculations. For mG ∼ 5−50 TeV, the re-heat upper bound is much less: TR<∼ 5×107−109

GeV (depending on the 4He abundance) due to overproduction of 4He arising from n ↔ p

conversions. For mG

>∼ 50 TeV, there is an upper bound of TR<∼ 5 × 109 GeV due to

overproduction of χ01 LSPs due to G decays.

Figure 3.1: The gravitino problem in generic SUGRA models: an overproduction ofgravitinos followed by late gravitino decay can destroy successful BBN predictions ⇒ upperbound on reheating temperature.

Solutions to the gravitino BBN problem then include: (1) having mG

>∼ 50 TeV but with

an unstable χ01 (no TR bound), (2) having a gravitino LSP so that G is stable or (3) keep

the re-heat temperature below the BBN bounds. We will here adopt solution number (3).

55

In the case of SO(10) SUSY GUT models, with mG ∼ m16 ∼ 5 − 20 TeV, this means we

need a re-heat temperature TR<∼ 108 − 109 GeV.

3.2.2 Non-thermal leptogenesis

The data gleaned on neutrino masses during the past decade has lead credence to a particular

mechanism of generating the baryon asymmetry of the universe known as leptogenesis[56].

Leptogenesis requires the presence of heavy gauge singlet Majorana right handed neutrino

states ψNci(≡ Ni) with mass MNi

(i = 1 − 3 is a generation index). The Ni states may be

produced thermally in the early universe, or perhaps non-thermally, as suggested in Ref.

[57] via inflaton φ→ NiNi decay. The Ni may then decay asymmetrically to elements of the

doublets – for instance Γ(N1 → h+u e

−) 6= Γ(N1 → h−u e+) – owing to the contribution of CP

violating phases in the tree/loop decay interference terms. Focussing on just one species of

heavy neutrino N1, the asymmetry is calculated to be[58]

ε ≡ Γ(N1 → `+)− Γ(N1 → `−)

ΓN1

' − 3

8π

MN1

v2u

mν3δeff , (3.2)

where mν3 is the heaviest active neutrino, vu is the up-Higgs vev, and δeff is an effective

CP -violating phase factor which may be of order 1. The ultimate baryon asymmetry of the

universe is proportional to ε, so larger values of MN1 lead to a higher baryon asymmetry.

To find the baryon asymmetry, one may first assume that the N1 is thermally produced

in the early universe, and then solve the Boltzmann equations for the B−L asymmetry. The

ultimate baryon asymmetry of the universe arises from the lepton asymmetry via sphaleron

effects. The final answer[59], compared against the WMAP-measured result nB

s' 0.9×10−10

for the baryon-to-entropy ratio, requires MN1

>∼ 1010 GeV, and thus a re-heat temperature

TR>∼ 1010 GeV. This high a value of reheat temperature is in conflict with the upper bound

on TR discussed in Sec. 3.2.1. In this way, it is found that generic SUGRA models are

apparently in conflict with leptogenesis as a means to generate the baryon asymmetry of the

universe.

If one instead looks to non-thermal leptogenesis, then it is possible to have lower reheat

temperatures, since the N1 may be generated via inflaton decay. The Boltzmann equations

for the B−L asymmetry have been solved numerically in Ref. [60]. The B−L asymmetry is

then converted to a baryon asymmetry via sphaleron effects as usual. The baryon-to-entropy

56

ratio is calculated in [60], wherein it is found

nB

s' 8.2× 10−11 ×

(TR

106 GeV

)(2MN1

mφ

)( mν3

0.05 eV

)δeff , (3.3)

where mφ is the inflaton mass. Comparing calculation with data, a lower bound TR>∼ 106

GeV may be inferred for viable non-thermal leptogenesis via inflaton decay.

3.2.3 Axino dark matter

The sparticle mass spectrum described in Sec. ?? is characterized by 5−20 GeV scalars, but

very light gauginos, with a µ parameter of order 1-2 TeV. As a consequence, the neutralino χ01

ends up being nearly pure bino. Since all the scalars are quite heavy, the predicted neutralino

relic abundance ends up being very high: the calculation of Refs. [50, 79] find values in the

range Ωχ01h2 ∼ 102 − 104, which is 3− 4 orders of magnitude beyond the WMAP-measured

abundance.

A solution was advocated in Ref. [79] that in fact the χ01 state is not the LSP, but

instead the axino a makes up the CDM of the universe. The axino is the spin-12

element of

the axion supermultiplet which is needed to solve the strong CP problem in supersymmetric

models. The axino is characterized by a mass in the range of keV−GeV. Its couplings are

of sub-weak interaction strength, since they are suppressed by the Peccei-Quinn symmetry

breaking scale fa, which itself has a viable mass range 1010 − 1012 GeV. While the axino

interacts very feebly, it does interact more strongly than the gravitino.

If the a is the lightest SUSY particle, then the χ01 will no longer be stable, and can decay

via χ01 → aγ. The relic abundance of axinos from neutralino decay (non-thermal production,

or NTP ) is given simply by

ΩNTPa h2 =

ma

mχ01

Ωχ01h2, (3.4)

since in this case the axinos inherit the thermally produced neutralino number density. Notice

that neutralino-to-axino decay offers a mechanism to shed large factors of relic density. For

a case where mχ01∼ 50 GeV and Ωχ0

1h2 ∼ 1000, as can occur in SO(10) SUSY GUTs, an

axino mass of less than 5 MeV reduces the DM abundance to below WMAP-measured levels.

The lifetime for these decays has been calculated, and it is typically in the range of

τ(χ01 → aγ) ∼ 0.03 s [52]. The photon energy injection from χ0

1 → aγ decay into the cosmic

soup occurs well before BBN, thus avoiding the constraints that plague the case of a gravitino

57

LSP [61]. The axino DM arising from neutralino decay is generally considered warm or even

hot dark matter for cases with ma<∼ 1 GeV [92].

Even though they are not in thermal equilibrium, axinos can still be produced thermally

in the early universe via scattering processes. The axino thermally produced (TP) relic

abundance has been calculated in Refs. [52, 62], and is given by

ΩTPa h2 ' 5.5g6

s ln

(1.108

gs

)(1011 GeV

fa/N

)2 ( ma

0.1 GeV

)( TR

104 GeV

), (3.5)

where gs is the strong coupling evaluated at Q = TR and N is the model dependent color

anomaly of the PQ symmetry, of order 1. The thermally produced axinos qualify as cold

dark matter as long as ma>∼ 0.1 MeV [52, 62].

3.2.4 A consistent cosmology for axino DM from SO(10) SUSYGUTs

At this point, we are able to check if we can implement a consistent cosmology for SO(10)

SUSY GUTs with axino dark matter. Our first step is to select points from the SO(10)

parameter space Eq. 2.2 that are very nearly Yukawa-unified. In Ref. [79], Yukawa unified

solutions were searched for by looking for R values as close to 1 as possible, where recall

R =max(ft, fb, fτ )

min(ft, fb, fτ )(3.6)

and the ft, fb and fτ Yukawa couplings were evaluated at MGUT . Thus, a solution with

R = 1.05 gives Yukawa unification to 5%.

We would like solutions where the axino DM is dominantly CDM. For definiteness, we

will insist on ΩNTPa h2 ∼ 0.01, while ΩTP

a h2 = 0.1. Thus, in step 1, we select models from

the random scan of Ref. [79] that have R < 1.05, and m16 : 5 − 20 TeV. In step 2, from

the known value of mχ01

and Ωχ01h2, we next calculate the axino mass needed to generate

ΩNTPa h2 = 0.01 according to Eq. 3.4. In step 3, we plug ma into Eq. 3.5, where we also

take gs = 0.915 (the running gs value at ∼ 106 GeV), and PQ scale fa/N = 1012 GeV. By

insisting that ΩTPa = 0.1, we may calculate the value of TR that is needed.

Our results are plotted in the ma vs. TR plane in Fig. 3.2 and occupy the upper band

of solutions. In this plane, solutions with TR<∼ 3 × 107 − 5 × 108 GeV are allowed by the

gravitino constraint (with mG ∼ 5 − 20 TeV) and BBN. Solutions with TR>∼ 106 GeV can

generate the matter-antimatter asymmetry correctly via non-thermal leptogenesis. Solutions

58

with ma>∼ 10−4 GeV give dominantly cold DM from TP of axinos. Solutions with m16 > 15

TeV are denoted by filled (turquoise) symbols, while solutions with m16 < 15 TeV have open

(dark blue) symbols.

1e-05 0.0001 0.001 0.01ma~ (GeV)

1e+06

1e+07

1e+08

1e+09T R (G

eV) BBN/gravitino

NT leptogenesis

warm a~DM

Figure 3.2: Plot of Yukawa unified solutions with R < 1.05 and 5 TeV < m16 < 20 TeV in thema vs.TR plane. The upper band of solutions has ΩNTP

a h2 = 0.01, ΩTPa h2 = 0.10 and fa/N = 1012

GeV, while the lower band of solutions has ΩNTPa h2 = 0.03, ΩTP

a h2 = 0.08 and fa/N = 5 × 1011

GeV.

We see that a variety of points fall in the allowed region. These points give rise to a

consistent cosmology for SO(10) SUSY GUT models! Of course, there is some uncertainty

in these results. We can take higher or lower values of the PQ breaking scale, higher or

lower fractions of ΩNTPa , and the TR upper (and lower) bounds have some variability built

into them. As an example, the lower band of solutions is obtained with ΩNTPa = 0.03,

ΩTPa h2 = 0.08 and fa/N = 5 × 1011 GeV. In this case, some of the previously excluded

solutions migrate into the allowed region to give a consistent cosmology with somewhat

different parameters.

59

CHAPTER 4

Collider Searches for New Physics

It is expected that the CERN Large Hadron Collider (LHC), a√s = 14 TeV pp collider,

will begin operation in late 2008 or early 2009. It is not unreasonable to expect of order 0.1

fb−1 of integrated luminosity in the first full year of running. One of the main goals of the

LHC is to either discover or exclude the existence of weak scale supersymmetry[63].

Most theories of weak scale SUSY have an added R-parity invariance which is necessary

to stabilize the proton against rapid decay through R-violating interactions. A consequence

of R-parity conservation is that superpartners of SM particles must decay to other superpart-

ners. In this case, the lightest SUSY particle (LSP) must be absolutely stable. If produced

in the early universe, then there should exist relic LSPs in the universe today, and in fact it

is popular to conjecture that these might make up the required cold dark matter (CDM) in

the universe. Null searches for massive charged or colored relics from the Big Bang indicate

that the LSP must be electrically and color neutral. In many models, the lightest neutralino

(χ01 or χ0

1) turns out to be the LSP, and is an excellent candidate CDM particle. A neutralino

LSP, if produced in a collider experiment, would escape detection and thus provide a signal

characterized by an apparent non-conservation of (transverse) energy.

It was recognized early on that perhaps the classic signature for production of SUSY

particles in collider events is the presence of an excess of EmissT + jets events above SM

background1. Thus, most studies of sparticle discovery at collider experiments rely on the

presence of large EmissT in the events in order to reject SM backgrounds such as multi-jet

production in QCD. At LHC, many analyses require for instance EmissT

>∼ 100 GeV as a

minimum requirement[65].

1Indeed, it is suggested in Ref. [64] that SUSY gives rise to so-called “zen” events: jets balanced byEmiss

T , which correspond to the sound of one hand clapping.

60

Even if the neutralino is not the LSP but is instead perhaps the axino, the SUSY

signatures would still be characterized by missing ET , and the above description still holds.

If the neutralino in the neutralino decay to axino plus photon were not so long lived, there

would be a possibility of the electromagnetic calorimeters of the LHC to capture these and

an implied discovery via direct detection could be made.

4.1 Early SUSY Discovery using Multi-leptons

From the experimental side, the requirement of large EmissT can be problematic, especially if

an early discovery of SUSY is desired. Missing transverse energy can arise not only from the

presence of weakly interacting neutral particles such as neutrinos or the lightest neutralinos,

but also from a variety of other sources, including:

• energy loss from cracks and un-instrumented regions of the detector,

• energy loss from dead cells,

• hot cells in the calorimeter that report an energy deposition even if there isn’t one,

• mis-measurement in the electromagnetic calorimeters, hadronic calorimeters or muon

detectors and

• the presence of mis-identified cosmic rays in events.

Thus, in order to have a solid grasp of expected EmissT from SM background processes, it will

be necessary to have detailed knowledge of the complete detector performance. As experience

at the Tevatron suggests, this complicated task may well take some time to complete. The

same is likely to be true at the LHC, as many SM processes will have to be scrutinized first

in order to properly calibrate the detector. For this reason, SUSY searches using EmissT as a

crucial requirement may well take rather longer than a year to provide reliable results.

On the other hand, if SUSY particles are relatively light, then production cross sections

can be huge, and many new physics events may be generated in the first few months of

running. For instance, for mg ∼ 400 GeV and heavy squarks, the expected gluino pair cross

sections are in the 105 fb range. If mg ∼ mq ∼ 400 GeV, then production cross sections are

even higher: of order 106 fb! Thus, with just 0.1 fb−1 of integrated luminosity, we might

61

expect of order 104 − 105 new physics events to be recorded on tape if the gluino is in the

400 GeV range.

In this chapter, we wish to examine if an early SUSY discovery might be made without

using EmissT cuts. The key is to take advantage of the large production cross sections of

strongly interacting SUSY particles (the gluinos and squarks) and their complex cascade

decays. Gluinos and squarks generally decay through a multi-step cascade of decays[98]

until the LSP state is reached, so that SUSY signal events are expected to be rich in

jet multiplicity, b-jet multiplicity, isolated lepton multiplicity and sometimes large tau-jet

multiplicity. In addition, gauge mediated SUSY can lead to collider events with high isolated

photon multiplicity. Thus, we would like to be able to use detected objects such as jets, b-jets

and isolated leptons to maximize signal over background, rather than inferred objects like

EmissT which requires a complete detector knowledge. Our main result in this chapter is that

we find a substantial reach for SUSY at the LHC by requiring multi-jet plus multi-lepton

events, without requiring the presence of EmissT .2 By searching in this channel, one may be

able to discover SUSY even before the detectors are fully calibrated such that EmissT is a

useful variable for background rejection.

We use Isajet 7.76[83] for the simulation of signal and background events at the LHC.

A toy detector simulation is employed with calorimeter cell size ∆η × ∆φ = 0.05 × 0.05

and −5 < η < 5. The HCAL energy resolution is taken to be 80%/√E + 3% for |η| < 2.6

and FCAL is 100%/√E + 5% for |η| > 2.6. The ECAL energy resolution is assumed to be

3%/√E + 0.5%. We use a UA1-like jet finding algorithm with jet cone size R = 0.4 and

require that ET (jet) > 50 GeV and |η(jet)| < 3.0. Leptons are considered isolated if they

have pT (e or µ) > 20 GeV and |η| < 2.5 with visible activity within a cone of ∆R < 0.2 of

ΣEcellsT < 5 GeV. The strict isolation criterion helps reduce multi-lepton backgrounds from

heavy quark (cc and bb) production.

We identify a hadronic cluster with ET > 50 GeV and |η(j)| < 1.5 as a b-jet if it contains

a B hadron with pT (B) > 15 GeV and |η(B)| < 3 within a cone of ∆R < 0.5 about the jet

axis. We adopt a b-jet tagging efficiency of 60%, and assume that light quark and gluon jets

can be mis-tagged as b-jets with a probability 1/150 for ET ≤ 100 GeV, 1/50 for ET ≥ 250

GeV, with a linear interpolation for 100 GeV< ET < 250 GeV[101].

2 Similar signal calculations for models with a charged stable LSP have been performed in Ref. [67].

62

For our initial analysis, we adopt the well-studied SPS1a′ benchmark point[69], which

occurs in the minimal supergravity (mSUGRA) model with parameters m0 = 70 GeV,

m1/2 = 250 GeV, A0 = −300 GeV, tan β = 10, µ > 0 and mt = 171 GeV. Here, m0 is a

common GUT scale scalar soft breaking mass, m1/2 is a common GUT scale gaugino mass,

A0 is a common GUT scale trilinear soft term and tan β is the ratio of Higgs vevs. The

parameter µ occurs in the superpotential; its magnitude, but not its sign, is determined

by requiring a radiative breakdown of electroweak symmetry. The sparticle mass spectrum

is generated by the Isajet 7.76 program, which adopts an iterative approach to solving the

MSSM RGEs using two-loop RGEs and complete 1-loop sparticle mass radiative corrections.

The SPS1a′ point leads to a spectrum with mg = 608 GeV, while squark masses tend to be

in the 550 GeV range. The gluinos and squarks then cascade decay via a multitude of modes

leading to events with high jet, b-jet, isolated lepton and tau lepton multiplicity.

In addition, we have generated background events using Isajet for QCD jet production

(jet-types include g, u, d, s, c and b quarks) over five pT ranges as shown in Table 4.3.

Additional jets are generated via parton showering from the initial and final state hard

scattering subprocesses. We have also generated backgrounds in the W + jets, Z + jets,

tt(171) and WW, WZ, ZZ channels at the rates shown in Table 4.3. The W + jets and

Z + jets backgrounds use exact matrix elements for one parton emission, but rely on the

parton shower for subsequent emissions.

We begin by applying a set of pre-cuts to our event samples. These cuts, known as set

C1 in Ref. [70], were used for studying gluino mass determination in the focus point region

of mSUGRA. Here, we abandon the EmissT > (100 GeV, 0.2Meff ) cut and call the new set of

cuts C1′.

C1′ cuts:

n(jets) ≥ 4, (4.1)

ET (j1, j2, j3, j4) ≥ 100, 50, 50, 50 GeV, (4.2)

ST ≥ 0.2. (4.3)

We will also make use of the augmented effective mass AT = EmissT +

∑jetsET (j) +∑

leptonsET (`). Here, ` stands for either e or µ. If we remove EmissT from AT , we will

63

Table 4.1: Events generated and cross sections for various SM background processes plusthe SPS1a′ case study. The C1′ cuts are specified in Eqns. (1− 3).

process events σ (fb) cuts C1′+ ≥ 3` (fb)QCD (pT : 50− 100 GeV) 106 2.6× 1010 –QCD (pT : 100− 200 GeV) 106 1.5× 109 –QCD (pT : 200− 400 GeV) 106 7.3× 107 –QCD (pT : 400− 1000 GeV) 106 2.7× 106 –QCD (pT : 1000− 2400 GeV) 106 1.5× 104 –W + jets;W → e, µ, τ (pT (W ) : 100− 4000 GeV) 5× 105 3.9× 105 0.8Z + jets;Z → τ τ , νs (pT (Z) : 100− 3000 GeV) 5× 105 1.4× 105 0.3tt 3× 106 5.1× 105 5.1WW,ZZ,WZ 5× 105 8.0× 104 –signal (SPS1a′: mg = 608 GeV) 2.5× 105 4.7× 104 46.6

call the new variable A′T . ST is transverse sphericity3. The event rates in fb are listed before

cuts in column 3 of Table 4.3.

In Fig. 4.1, we plot the resulting jet multiplicity nj after cuts C1′ for the SPS1a′

benchmark (orange histogram) along with the various SM backgrounds. The gray histogram

gives the sum of all SM backgrounds. We see immediately that SM background, dominated

by QCD multi-jet production, dominates out to very high jet multiplicity.

In Fig. 4.2, we plot the augmented effective mass (minus the EmissT component) A′

T . When

the EmissT cut is used in cut set C1, then signal generally emerges from background at some

large value of AT which is somewhat correlated with the values of mg and mq[71]. In this

case, with no EmissT cut, the signal is hopelessly below the summed BG distribution.

3 Sphericity is defined, e.g. in Collider Physics, V. Barger and R. J. N. Phillips (Addison Wesley, 1987).Here, we restrict its construction to using only transverse quantities, as is appropriate for a hadron collider.

64

5 10 15nj

0.01

0.1

1

10

100

1000

10000

1e+05

1e+06

1e+07

σ (fb

)

SPS1a’(70,250,-300,10,1,171)QCD JetsttW+jetsZ+jetsWW, WZ, ZZSum of Backgrounds

No. of JetsCuts C1’

Figure 4.1: Plot of jet multiplicity from SUSY collider events from SPS1a′ after cuts C1′.We also plot the histograms of various SM backgrounds, plus the total SM background (grayhistogram).

0 1000 2000 3000AT

’ (GeV)

0.01

0.1

1

10

100

1000

10000

1e+05

dσ/d

AT’ (

fb/G

eV)

SPS1a’(70,250,-300,10,1,171)QCD jetsttW+jetsZ+jetsWW, WZ, ZZSum of Backgrounds

Augmented Transverse Meff’

Cuts C1’

Figure 4.2: Plot of augmented effective mass A′T (without Emiss

T ) from SUSY collider eventsfrom SPS1a′ after cuts C1′. We also plot the histograms of various SM backgrounds, plusthe total SM background (gray histogram).

65

0 5nb

0.1

1

10

100

1000

10000

1e+05

1e+06

1e+07

σ (fb

)

SPS1a’(70,250,-300,10,1,171)QCD JetsttW+jetsZ+jetsWW, WZ, ZZSum of Backgrounds

No. of b-jetsCuts C1’

Figure 4.3: Plot of b-jet multiplicity nb from LHC SUSY events from SPS1a′ after cuts C1′.We also plot the histograms of various SM backgrounds, plus the total SM background (grayhistogram).

In Fig. 4.3, we plot the multiplicity of tagged b-jets nb in events after cuts C1′. We see

that out to nb = 5, SM background from QCD jet production – including both bb production,

parton shower production from g → bb and also jets faking a b-jet – dominates the signal.

In Fig. 4.4, we plot the multiplicity of isolated leptons n` for benchmark point SPS1a′

and SM background. Here we see that at low values of n` = 0 or 1, signal is dominated by

BG. However, at n` = 2, signal is above QCD BG, and only below tt BG. By the time we

require n` = 3, SM background is well below signal. In this case, it is clear that we can gain

good BG rejection by requiring the cut set C1′ , plus n` ≥ 3. The remaining signal is at the

40-50 fb level, which should be adequate for discovery if 0.1-1 fb−1 of integrated luminosity

is obtained. The dominant background comes from tt production. An early verification of tt

production via its one and two lepton signatures should allow for a solid calibration of this

most important background.

To gain an estimate of the LHC reach using cuts C1′ plus≥ 3`, in Fig. 4.5 we setm0 = 200

GeV (lighter squarks) and m0 = 1000 GeV (heavy squarks) and vary m1/2 from 170 to 500

GeV. We also take A0 = 0, tan β = 10 and µ > 0. We plot the resulting signal cross section

66

0 5nl

0.1

1

10

100

1000

10000

1e+05

1e+06

1e+07

σ (fb

)

SPS1a’(70,250,-300,10,1,171)QCD JetsttW+jetsZ+jetsWW, WZ, ZZSum of Backgrounds

No. of Isolated LeptonsCuts C1’

Figure 4.4: Plot of isolated lepton multiplicity n` from LHC SUSY events from SPS1a′

after cuts C1′. We also plot the histograms of various SM backgrounds, plus the total SMbackground (gray histogram).

as a function of mg rather than m1/2 in order to explicitly show the reach in terms of a

measureable parameter. We also plot the model line of m0 = 70 GeV and A0 = −300 GeV

(with tan β = 10 and µ > 0) which contains the SPS1a′ point. The 5σ BG level is shown

for 0.1 and 1 fb−1 of integrated luminosity. (The 5σ level for 0.1 fb−1 is ∼ 40 fb, so would

only correspond to four events.) While the total signal cross section for the red curve (with

m0 = 200 GeV) is larger than that for the blue curve (for m0 = 1000 GeV), the cross section

after cuts is actually larger for the large m0 case at low m1/2. This is because in this region,

around m0 = m1/2 ∼ 200 GeV, the χ02 branching fraction to leptons χ0

2 → χ01`

¯ is suppressed

due to destructive interference in the Z and slepton mediated decay processes. In the high

m0 case, 3-body decay of χ02 via the Z∗ is always dominant. However, in all cases, we see

the 5σ reach extends to mg ∼ 700− 750 GeV for 0.1 fb−1 of integrated luminosity, and out

to mg ∼ 1 TeV for 1 fb−1 of integrated luminosity. This would represent a significant leap

in experimental sensitivity to mg which could be obtained at relatively low LHC integrated

luminosity, while not using EmissT cuts.

67

400 600 800 1000 1200mg~ (GeV)

0

30

60

90

120

σ ≥3l (f

b)

m0 = 200 GeVm0 = 1000 GeVm0 = 70, A0 = -300

Cuts C1amt=171.0, A0=0, tanβ=10, sgnµ>0

5σ level at 0.1 fb-1

5σ level at 1 fb-1

Figure 4.5: Plot of signal cross section from mSUGRA model versus mg after cuts C1′ andn` ≥ 3, for m0 = 200 and 1000 GeV. We also take A0 = 0, tan β = 10, µ > 0 and mt = 171GeV. We also plot the 5σ background level for 0.1 and 1 fb−1 of integrated luminosity.

One possible criticism of our results so far is that we use only leading order cross sections

as calculated by Isajet. However, we expect that NLO total cross sections for both signal

and background to be somewhat enhanced beyond the LO Isajet results, so we would expect

our overall conclusion to remain valid qualitatively. Indeed, it is expected that the major

SM processes will be measured to high accuracy at LHC already at low luminosity, so that a

good background calibration should be at hand. A second criticism could be that there are

additional background processes to be checked. These would include 2 → n processes such as

tttt, ttV , ttV V , V V V and V V V V production, where V = W± or Z. While these processes

occur at higher order in perturbation theory, they do offer the possibility to generate multi-

lepton final states rather efficiently. We hope to address these in a future work. A third

criticism might be that we have not taken any “jet faking a lepton” probability into account.

This possibility is detector and lepton flavor dependent. However, if it turns out to be a

problem, we can only note that our cuts so far have been rather minimal, and can easily be

extended. For instance, requiring the presence of one b-jet in each event will severely reduce

W+jets and Z+jets BG. Then, plotting the distribution in A′T will allow signal to emerge

68

from tt and other backgrounds. This is illustrated in Fig. 4.6.

500 1000 1500 2000AT

’ (GeV)

0.001

0.01dσ

/dA

T’ (fb

/GeV

)

SPS1a’(70,250,-300,10,1,171)QCD jetsttW+jetsZ+jetsWW, WZ, ZZSum of Backgrounds

Augmented Transverse Meff’

Cuts C1’ + ≥3l + 1b-jet

Figure 4.6: Distribution in variable A′T from SUSY events from SPS1a′ after cuts C1′ plus

≥ 3` plus ≥ 1 b-jet. We also plot the remaining SM backgrounds (gray histogram).

We also note here that if a SUSY signal is found in the ≥ 4 jets plus ≥ 3` sample,

then the resulting event sample may be used for precision sparticle mass measurements

just as in the case where one requires jets +EmissT . As an example, we examine all events

passing cuts C1′ and ≥ 3` for benchmark SPS1a′ and plot the invariant mass of all opposite

sign/same flavor (OS/SF) dilepton pairs in Fig. 4.7. In this case, we expect a mass edge[72]

at m(`¯) = mχ02

√1− m2

˜

m2χ02

√1−

m2χ01

m2˜

= 82.3 GeV (since here mχ02

= 183.0, m˜R

= 123.3 GeV

and mχ01

= 97.8 GeV). The mass edge is evident from the plot, and serves as a starting point

for further sparticle mass reconstruction.

69

50 100 150mll (GeV)

0

1

2

3

4

dσ/d

mll (f

b/G

eV)

SPS1a’(70,250,-300,10,1,171)QCD JetsttW+jetsZ+jetsWW, WZ, ZZSum of Backgrounds

Cuts C1’ + ≥3l

mllmax

Figure 4.7: Plot of OS/SF dilepton invariant mass from SUSY events from SPS1a′ after cutsC1′ plus ≥ 3`. We also plot the remaining SM backgrounds (gray histogram).

0 50 100 150mll (GeV)

0

10

20

30

dσ/d

mll (f

b/G

eV)

SPS1a’ + BGQCD JetsttW+jetsZ+jetsWW, WZ, ZZSum of Backgrounds

Cuts C1a

Figure 4.8: Plot of OS/SF dilepton invariant mass from SUSY events from benchmark SPS1a′

after cuts C1′ plus a OS/SF pair of leptons. We also plot the remaining SM backgrounds(gray histogram).

70

While the requirement of jets + ≥ 3` works to see mg<∼ 750 GeV with just 0.1 fb−1 of

integrated luminosity, it is possible to see SUSY signals with even lower lepton multiplicities.

To illustrate, we examine SUSY models which give rise to the distinctive dilepton invariant

mass edge from χ02 decay to `¯χ0

1. In this case, we require cuts C1′ plus an OS/SF lepton

pair. We plot out in Fig. 4.8a) the resultant distribution in m(`¯) for both SPS1a′ and SM

BG. We see a continuum of background, along with a Z peak. The BG Z peak arises because

Isajet includes W and Z radiation in its parton shower algorithm. The orange histogram

shows the sum of signal plus BG, and the OS/SF dilepton mass edge clearly stands out

below the Z peak. A further example comes from SO(10) benchmark point A suggested in

Ref. [79], which includes a ∼ 400 GeV gluino. I n this case, the large gg cross section allows

signal to stand out even more abruptly from SM background.

4.2 Yukawa-unified SO(10) at the Cern LHC

In this section, we wish the explore consequences of Yukawa-unified SUSY models for

sparticle detection at the LHC. We focus most of our attention on the two cases presented

in Table 1 of Ref. [79]: 1. point A with m16 ∼ 9 GeV and an axino LSP, and 2. point D with

m16 ∼ 3 TeV and a neutralino LSP. While our studies here focus on just two cases, we feel

the qualitative features of the LHC signatures should be rather similar to these two cases.

In fact, the collider phenomenology of these cases is rather similar between the two, since in

the first case the neutralino decays to an axino far beyond the detector boundaries. Thus,

in both cases the lightest neutralino χ01 leads to missing ET at collider experiments.

For the benefit of the reader, we present in Table 4.2 the two case studies we examine. We

present the parameter space values, sparticle mass spectrum, and in addition the total tree-

level LHC sparticle production cross section. We also list as percentages some contributing

2 → 2 subprocess reactions.

4.2.1 Cross sections and branching fractions for sparticles inYukawa-unified models

Given the characteristic spectrum of superpartners obtained in Yukawa-unified SUSY models,

it is useful to examine what sort of new physics signals we would expect at the LHC.

Obviously, first/second generation squarks and sleptons in the multi-TeV mass range will

71

Table 4.2: Masses and parameters in GeV units for two cases studies points A and D ofRef. [79] using Isajet 7.75 with mt = 171.0 GeV. We also list the total tree level sparticleproduction cross section in fb at the LHC, plus the percent for several two-body final states.

parameter Pt. A Pt. Dm16 9202.9 2976.5m1/2 62.5 107.0A0 −19964.5 −6060.3m10 10966.1 3787.9tan β 49.1 49.05MD 3504.4 1020.8ft 0.51 0.48fb 0.51 0.47fτ 0.52 0.52µ 4179.8 331.0mg 395.6 387.7muL

9185.4 2970.8mt1 2315.1 434.5mb1

2723.1 849.3meL

9131.9 2955.8mχ±1

128.8 105.7

mχ02

128.6 105.1

mχ01

55.6 52.6

mA 3273.6 776.8mh 125.4 111.1σ [fb] 75579.1 89666.1% (gg) 86.8 80.5% (χ±1 χ

02) 8.8 12.8

% (t1¯t1) 0 1.1

essentially decouple from LHC physics. Gluinos – in the 350-500 GeV range – will be

produced in abundance via qq and gg fusion subprocesses. Charginos and neutralinos, being

in the 100–160 GeV range, may also be produced with observable cross sections.

As noted above, we list the tree-level total sparticle production cross sections obtained

from Isajet for cases A and D in Table 4.2. In case A, we find σ(tot) ∼ 8×104 fb, so that 8000

sparticle pair events are expected at LHC with just 0.1 fb−1 of integrated luminosity. Of this

total, 86.7% comes from gluino pair production, while 8.8% comes from W1χ02 production

and 4.5% comes from W+1 W

−1 production. In case D, σ(tot) ∼ 9.6 × 104 fb, with 80.4%

72

from gg production, 12.8% from W1χ02 production, 6.4% from W+

1 W−1 production while

top-squark pair production yields just 1.1% of the total. Given these production cross

sections, we expect Yukawa-unified SUSY to yield primarily gg events at the LHC. Gluino

pair production typically leads to events with hard jets, hard EmissT and isolated leptons from

the gluino cascade decays[98]. We also expect a soft component coming from W+1 W

−1 and

χ±1 χ02 production. While both these reactions lead to events with rather soft jets, leptons

and EmissT , the latter reaction can also yield clean trilepton events[?], which might be visible

at LHC above SM backgrounds.

For the case of gluino masses other than those listed in Table 4.2, we show in Fig. 4.9

the total gluino pair production rate versus mg at the LHC at tree level (solid) and next-

to-leading-order (NLO) using the Prospino program[100]. The scale choice is taken to be

Q = mg. We take mq to be 3 TeV (blue) and 9 TeV (red). As can be seen, the results

hardly vary between this range of squark masses. The tree level results agree well with

Isajet, but the NLO results typically show an enhancement by a factor ∼ 1.6. Thus, we

expect Yukawa-unified SUSY models to yield pp→ ggX events at a 30-150 pb level at LHC.

In Fig. 4.10, we show the total -ino pair production cross sections versus chargino mass

mfW1. While W±

1 χ02 and W+

1 W−1 production dominate, and have rates around 103−104 fb over

the range of interest, there exists a sub-dominant rate for W±1 χ

01 and also χ0

1χ02 production.

Now that we see that Yukawa-unified SUSY will yield dominantly gluino pair production

events at the LHC, we next turn to the gluino branching fractions in order to understand their

event signatures. All sparticle branching fractions are calculated with Isajet 7.75. In Fig.

4.11, we show various gluino branching fractions for points A and D. We see immediately that

in both cases, BF (g → bbχ02) dominates at around 56%. This is followed by BF (g → bbχ0

1) at

∼ 16%, and BF (g → btW+1 ) and BF (g → tbW−

1 ) each at ∼ 10%. Decays to first and second

generation quarks are much suppressed due to the large first and second generation squark

masses. From these results, we expect gluino pair production events to be rich in b-jets,

EmissT and occassional isolated leptons from the leptonic decays W1 → `ν`χ

01 and χ0

2 → `¯χ01,

where ` = e or µ.

73

360 380 400 420 440 460 480 500mg

50

100

150

200

Σ Hpp®

g g L HpbL

Figure 4.9: Plot of σ(pp→ ggX) in pb at√s = 14 TeV versus mg. We use Prospino with

scale choice Q = mg, and show LO (solid) and NLO (dashes) predictions in the vicinity ofpoint A (red) and point D (blue) from Table 4.2.

100 150 200 250mχ~1

± (GeV)

10-2

10-1

100

101

102

103

104

σ (fb

)

χ~1+χ~1

−

χ~1±χ~1

0

χ~1±χ~2

0

χ~10χ~2

0

Figure 4.10: Plot of various -ino pair production processes in fb at√s = 14 TeV versus

mχ±1, for mq = 3 TeV and µ = mg, with tan β = 49 and µ > 0.

74

0.0001

0.001

0.01

0.1

1g~ B

ranc

hing

Fra

ctio

ns

Point A Point D

bbχ∼20

gχ∼20

uuχ∼20

bbχ∼10

uuχ∼10

ddχ∼10

gχ∼10

btχ∼1±

duχ∼1±

bbχ∼20

bbχ∼10

btχ∼1±

gχ∼20

gχ∼10

uuχ∼10

ddχ∼10

uuχ∼20 ddχ∼2

0

duχ∼1±

gχ∼30

gχ∼40

Figure 4.11: Plot of various sparticle branching fractions taken from Isajet for points A andD from Table 4.2.

4.2.2 Gluino pair production signals at the LHC

To examine collider signals from Yukawa-unified SUSY at the LHC in more detail, we

generate 106 sparticle pair production events for points A and D, corresponding to 13 and

11 fb−1 of integrated luminosities. We use Isajet 7.75[83] for the simulation of signal and

background events at the LHC. A toy detector simulation is employed with calorimeter cell

size ∆η ×∆φ = 0.05 × 0.05 and −5 < η < 5. The HCAL energy resolution is taken to be

80%/√E + 3% for |η| < 2.6 and FCAL is 100%/

√E + 5% for |η| > 2.6. The ECAL energy

resolution is assumed to be 3%/√E + 0.5%. We use a UA1-like jet finding algorithm with

jet cone size R = 0.4 and require that ET (jet) > 50 GeV and |η(jet)| < 3.0. Leptons are

considered isolated if they have pT (e or µ) > 20 GeV and |η| < 2.5 with visible activity

within a cone of ∆R < 0.2 of ΣEcellsT < 5 GeV. The strict isolation criterion helps reduce

multi-lepton backgrounds from heavy quark (cc and bb) production. We also invoke a lepton

identification efficiency of 75% for leptons with 20 GeV< pT (`) < 50 GeV, and 85% for

75

Table 4.3: Events generated and cross sections (in fb) for various SM background and signalprocesses before and after cuts. The C1′ and Emiss

T cuts are specified in the text. TheW+jetsand Z + jets background has been computed within the restriction pT (W,Z) > 100 GeV.

process events σ (fb) C1′ C1′ + EmissT

QCD (pT : 0.05− 0.1 TeV) 106 2.6× 1010 4.1× 105 –QCD (pT : 0.1− 0.2 TeV) 106 1.5× 109 1.4× 107 –QCD (pT : 0.2− 0.4 TeV) 106 7.3× 107 6.5× 106 2199QCD (pT : 0.4− 1.0 TeV) 106 2.7× 106 2.8× 105 1157QCD (pT : 1− 2.4 TeV) 106 1.5× 104 1082 25W → `ν` + jets 5× 105 3.9× 105 3850 1275Z → τ τ + jets 5× 105 1.4× 105 1358 652tt 3× 106 4.9× 105 8.2× 104 2873WW,ZZ,WZ 5× 105 8.0× 104 197 7Total BG 9.5× 106 2.76× 1010 2.13× 107 8188Point A: 106 7.6× 104 3.6× 104 8914

S/B → – – 0.002 1.09S/√S +B (1 fb−1) → – – – 68

Point D: 106 9.0× 104 3.7× 104 10843S/B → – – 0.002 1.32

S/√S +B (1 fb−1) → – – – 78

leptons with pT (`) > 50 GeV.

We identify a hadronic cluster with ET > 50 GeV and |η(j)| < 1.5 as a b-jet if it contains

a B hadron with pT (B) > 15 GeV and |η(B)| < 3 within a cone of ∆R < 0.5 about the jet

axis. We adopt a b-jet tagging efficiency of 60%, and assume that light quark and gluon jets

can be mis-tagged as b-jets with a probability 1/150 for ET ≤ 100 GeV, 1/50 for ET ≥ 250

GeV, with a linear interpolation for 100 GeV< ET < 250 GeV[101].

In addition to signal, we have generated background events using Isajet for QCD jet

production (jet-types include g, u, d, s, c and b quarks) over five pT ranges as shown in

Table 4.3. Additional jets are generated via parton showering from the initial and final state

hard scattering subprocesses. We have also generated backgrounds in the W +jets, Z+jets,

tt (with mt = 171 GeV) and WW, WZ, ZZ channels at the rates shown in Table 4.3. The

W + jets and Z + jets backgrounds use exact matrix elements for one parton emission, but

rely on the parton shower for subsequent emissions.

First we require modest cuts: n(jets) ≥ 4. Also, SUSY events are expected to spray large

76

ET throughout the calorimeter, while QCD dijet events are expected to be typically back-to-

back. Thus, we expect QCD background to be peaked at transverse sphericity ST ∼ 0, while

SUSY events have larger values of ST .4 The actual ST distribution for point A is shown in

Fig. 4.12 (the ST distribution for point D is almost identical to that of point A). Motivated

by this, we require ST > 0.2 to reject QCD-like events.

0 0.2 0.4 0.6 0.8 1ST

1e+03

1e+04

1e+05

1e+06

1e+07

1e+08

dσ/d

S T (fb)

Point ACuts C1’

Figure 4.12: Plot of distribution in transverse sphericity ST for events with cuts C1′ frombenchmark point A and the summed SM background; point D leads to practically the samedistribution.

We plot the jet ET distributions of the four highest ET jets from Pt. A (in color) and

the total SM background (gray histogram) in Fig. 4.13, ordered from highest to lowest ET ,

with jets labelled as j1 − j4. The histograms are normalized to unity in order to clearly

see the differences in distribution shapes. Again, the distributions for point D just look the

same. We find that the highest ET jet distribution peaks around ET ∼ 150 GeV with a

long tail extending to higher ET values, while for the background it peaks at a lower value

4 Here, ST is the usual sphericity variable, restricted to the transverse plane, as is appropriate for hadroncolliders. Sphericity matrix is given as

S =( ∑

p2x

∑pxpy∑

pxpy

∑p2

y

)(4.4)

from which ST is defined as 2λ1/(λ1 + λ2), where λ1,2 are the larger and smaller eigenvalues of S.

77

of ET ∼ 100 GeV. Jet 2 and jet 3 have peak distributions around 100 GeV both for the

signal and backgrounds, while the jet 4 distribution backs up against the minimum jet ET

requirement that ET (jet) > 50 GeV. Thus, at little cost to signal but with large background

(BG) rejection, we require ET (j1) > 100 GeV.

The collection of cuts so far is dubbed C1′[102]:

C1′ cuts:

n(jets) ≥ 4, (4.5)

ET (j1, j2, j3, j4) ≥ 100, 50, 50, 50 GeV, (4.6)

ST ≥ 0.2. (4.7)

0

0.002

0.004

0.006

0.008

0.01

Jet 1BG

0

0.002

0.004

0.006

0.008

0.01Jet 2BG

100 200 300 400 5000

0.004

0.008

0.012

0.016Jet 3BG

100 200 300 400 5000

0.006

0.012

0.018

0.024

Jet 4BG

ET(ji) (GeV)

1/σ

dσ/E

T(j i) (fb

/GeV

)

Figure 4.13: Plot of jet ET distributions for events with ≥ 4 jets after requiring just ST > 0.2,from benchmark point A; distributions for point D are the same.

78

The classic signature for SUSY collider events is the presence of jets plus large EmissT [103].

In Fig. 4.14, we show the expected distribution of EmissT from points A and D, along with

SM BG. We do see that signal becomes comparable to BG around EmissT ∼ 150 GeV. We list

cross sections from the two signal cases plus SM backgrounds in Table 4.3 after cuts C1′ plus

EmissT > 150 GeV. While signal S is somewhat higher than the summed BG B, the signal

and BG rates are rather comparable in this case: S/B = 1.09 for pt. A while S/B = 1.32

for pt. D.

0 100 200 300 400

ETmiss (GeV)

1

10

100

1000

10000

1e+05

1e+06

dσ/d

E Tmiss

(fb/

GeV

)

Point APoint DBackground

Cuts C1’

Figure 4.14: Plot of missing ET for events with ≥ 4 jets after cuts C1′, from benchmarkpoints A (full red line) and D (dashed blue line).

Even so, it has been noted in Ref. [102] that EmissT may be a difficult variable to reliably

construct during the early stages of LHC running. The reason is that missing transverse

energy can arise not only from the presence of weakly interacting neutral particles such as

neutrinos or the lightest neutralinos, but also from a variety of other sources, including:

• energy loss from cracks and un-instrumented regions of the detector,

• energy loss from dead cells,

• hot cells in the calorimeter that report an energy deposition even if there is not one,

• mis-measurement in the electromagnetic calorimeters, hadronic calorimeters or muon

detectors,

79

• real missing transverse energy produced in jets due to semi-leptonic decays of heavy

flavors,

• muons and

• the presence of mis-identified cosmic rays in events.

Thus, in order to have a solid grasp of expected EmissT from SM background processes, it will

be necessary to have detailed knowledge of the complete detector performance. As experience

from the Tevatron suggests, this complicated task may well take some time to complete. The

same may also be true at the LHC, as many SM processes will have to be scrutinized first

in order to properly calibrate the detector[104]. For this reason, SUSY searches using EmissT

as a crucial requirement may well take rather longer than a year to provide reliable results.

For this reason, Ref. [102] advocated to look for SUSY signal events by searching for a

high multiplicity of detected objects, rather than inferred undetected objects, such as EmissT .

In this vein, we show in Fig. 4.15 the jet multiplicity from SUSY signal (Pts. A and D) along

with SM BG after cuts C1′, i.e with no EmissT cut. We see that at low jet multiplicity, SM

BG dominates the SUSY signal. However, signal/background increases with n(jets) until at

n(jets) ∼ 15 finally signal overtakes BG in raw rate.

One can do better in detected b-jet multiplicity, nb, as shown in Fig. 4.16. Since each

gluino is expected to decay to two b-jets, we expect a high nb multiplicity in signal. In this

case, BG dominates signal at low nb, but signal overtakes BG around nb ' 4.

The isolated lepton multiplicity n` is shown in Fig. 4.17 for signal and SM BG after cuts

C1′. In this case, isolated leptons should be relatively common in gluino cascade decays. We

see that signal exceeds BG already at n` = 2, and far exceeds BG at n` = 3. In fact, high

isolated lepton multiplicity was advocated in Ref. [102] in lieu of an EmissT cut to search for

SUSY with integrated luminosities of around 1 fb−1 at LHC.

We also point out here that gg production can lead to large rates for same-sign (SS)

isolated dilepton production[105], while SM BG for this topology is expected to be small.

We plot in Fig. 4.18 the rate of events from signal and SM BG for cuts C1′ plus a pair

of isolated SS dileptons, versus jet multiplicity. While BG is large at low n(jets), signal

emerges from and dominates BG at higher jet multiplicities.

80

4 5 6 7 8 9 10 11 12 13 14 15 16 17nj

0.1

1

10

100

1000

10000

1e+05

1e+06

1e+07

σ (f

b)

Point APoint DBackground

Cuts C1’

Figure 4.15: Plot of jet multiplicity from benchmark points A (full red line) and D (dashedblue line) after cuts C1′ along with SM backgrounds.

0 1 2 3 4 5 6 7 8nb

1

10

100

1000

10000

1e+05

1e+06

1e+07

σ (f

b)

Point APoint DBackground

Cuts C1’

Figure 4.16: Plot of b-jet multiplicity from benchmark points A (full red line) and D (dashedblue line) after cuts C1′ along with SM backgrounds.

81

0 1 2 3 4nl

1

10

100

1000

10000

1e+05

1e+06

1e+07

σ (f

b)

Point APoint DBackground

Cuts C1’

Figure 4.17: Plot of isolated lepton multiplicity from benchmark points A (full red line) andD (dashed blue line) after cuts C1′ along with SM backgrounds.

0 2 4 6 8 10 12nj

0

5

10

15

20

25

30

35

dσ/d

n j (fb/

coun

t)

Point APoint DQCD JetsttW+jetsZ+jetsWW,WZ,ZZSum of Backgrounds

Cuts ST ≥ 0.2 + 2 SS dileptons

Figure 4.18: Plot of jet multiplicity in events with isolated SS dileptons from benchmarkpoints A (full red line) and D (dashed blue line) after cut ST > 0.2 along with SMbackgrounds.

82

4.2.3 Sparticle masses from gluino pair production

There exists good prospects for sparticle mass measurements in Yukawa-unified SUSY models

at the LHC. One reason is that sparticle pair production is dominated by a single reaction:

gluino pair production. The other propitious circumstance is that the mass difference

mχ02−mχ0

1is highly favored to be bounded by MZ . This means that χ0

2 decays dominantly

into three body modes such as χ02 → χ0

1`¯ with a significant branching fraction, while the

so-called “spoiler decay modes” χ02 → χ0

1Z and χ02 → χ0

1h are kinematically closed. The three

body decay mode is important in that it yields a continuous distribution in m(`¯) which is

bounded by mχ02−mχ0

1: this kinematic edge can serve as the starting point for sparticle mass

reconstruction in cascade decay events[21, 99].

As an example, we require cuts set C1′ plus the presence of a pair of opposite-sign/same

flavor (OS/SF) isolated leptons. The remaining number of events after this selection is

shown in Table 4.3. The resulting dilepton invariant mass distributions for points A and

D are shown in Fig. 4.19. Furthermore, in Fig. 4.20, we plot the different-flavor subtracted

distributions: dσ/dm(`+`−) − dσ/dm(`+`−′), which allow for a subtraction of e+µ− and

e−µ+ pairs from processes like chargino pair production in cascade decay events. A clear

peak at m(`¯) = MZ is seen in the BG distribution. This comes mainly from QCD jet

production events, since Isajet includes W and Z radiation in its parton shower algorithm

(in the effective W approximation). The signal displays a histogram easily visible above SM

BG with a distinct cut-off at mχ02−mχ0

1= 73 GeV. Isajet contains the exact decay matrix

elements in 3-body decay processes, and in this case we see a distribution that differs from

pure phase space, and yields a distribution skewed to higher m(`¯) values. This actually

shows the influence of the virtual Z in the decay diagrams, since the decay distribution

is dominated by Z∗ exchange. The closer mχ02− mχ0

1gets to MZ , the more the Z-boson

propagator pulls the dilepton mass distribution towards MZ [106]. The dilepton mass edge

should be measureable to a precision of ∼ 50 MeV according to Ref. [107].

For Yukawa-unified SUSY models, the branching fraction BF (g → bbχ02) dominates at

around 56%. If one can identify events with a clean χ02 → `¯χ0

1 decay, then one might also

try to extract the invariant mass of the associated two b-jets coming from the gluino decay,

which should have a kinematic upper edge at mg−mχ02' 267 (283) GeV for point A (D). A

second, less pronounced endpoint is expected at mg−mχ01' 340 (335) GeV due to g → bbχ0

1

83

0 50 100 150m(ll) (GeV)

0

5

10

15

20

25

dσ/d

m(ll

) (fb

/GeV

)

Point APoint DBackground

Cuts C1’ + 2 SF/OS

mχ∼20 − mχ∼1

0

mχ∼20 − mχ∼1

0

Figure 4.19: SF/OS dilepton invariant mass distribution after cuts C1′ from benchmarkpoints A (full red line) and D (dashed blue line) along with SM backgrounds.

decays which have ∼ 16% branching ratio. A third endpoint can also occur from χ02 → χ0

1bb

decay where mχ02−mχ0

1= 73 (52.5) GeV, respectively.

The high multiplicity of b-jets (typically two from each gluino decay), however, poses

a serious combinatorics problem in extracting the bb invariant-mass distribution. In a first

attempt, we required at least two tagged b-jets along with cuts C1′ and SF/OS dileptons, and

plotted the minimum invariant mass of all the b-jets in the event. The resulting distributions

for points A and D peaked around m(bb) ∼ 100 GeV, with a distribution tail extending well

beyond the above-mentioned kinematic endpoints. It was clear that we were frequently

picking up wrong b-jet pairs, with either each b originating from a different gluino, or one

or both b’s originating from a χ02 → χ0

1bb decay. A more sophisticated procedure to pair the

correct b-jets is required for the bb invariant-mass distribution.

84

0 50 100 150m(ll) (GeV)

0

4

8

12

16

20

24

28

dσ/d

m(ll

) − d

σ/dm

(ll´)

(fb/G

eV)

Point APoint DBG

Cuts C1’ + 2 OS dileptons

mχ∼20 − mχ∼1

0

mχ∼20 − mχ∼1

0

Figure 4.20: Same as Fig. 4.19 but for same-flavor minus different-flavor subtracted invariant-mass.

85

Parton-level Monte Carlo simulations revealed that the two hardest (highest ET ) b-jets

almost always originated from different gluinos. Thus, we require events with at least

four tagged b-jets (along with cuts C1′) and combine the hardest b-jet with either the 3rd

or 4th hardest b-jet, creating an object X1(bb). Moreover, we combine the 2nd hardest

b-jet with the 4th or 3rd hardest b-jet, creating an object X2(bb). Next, we calculate

∆m(X1−X2) ≡ |m(X1)−m(X2)|/(m(X1) +m(X2)). We select the set of bb clusters which

has the minimum value of ∆m(X1 −X2), and plot the invariant mass of both the clusters.

This procedure produces a sharp kinematic edge in m(bb) in parton level simulations.5 The

resulting distribution from Isajet is shown in Fig. 4.21 for points A and D. The distribution

peaks at a higher value of m(Xi) (i = 1, 2), and is largely bounded by the kinematic

endpoints, although a tail still extends to high m(Xi). Part of the high m(Xi) tail is due

to the presence of g → χ01bb decays, which have a higher kinematic endpoint than the

g → χ02bb decays, and of χ0

2 → χ01bb decays. In addition, there is a non-negligible background

contribution, indicated by the gray histogram.

We can do much better, albeit with reduced statistics, by requiring in addition the

presence of a pair of SF/OS dileptons. Applying the same procedure as described above, we

arrive at the distribution shown in Fig. 4.22. In this case, the SM BG is greatly reduced,

and two mass edges begin to appear.

It should also be possible to combine the invariant mass of the SF/OS dilepton pair

with a bb pair. Requiring cuts C1′ plus ≥ 2 b-jets plus a pair of SF/OS dileptons (with

m(`¯) < mχ02−mχ0

1), we reconstruct m(bb`¯). The result is shown in Fig. 4.23. While the

distribution peaks at m(`¯bb) ∼ 300 GeV, a kinematic edge at mg −mχ01∼ 340 GeV is also

visible (along with a mis-identification tail extending to higher invarant masses).

We try to do better, again with a loss of statistics, by requiring ≥ 4 b-jets instead of ≥ 2,

and combining the bb clusters into objects X1 and X2 as described above. We again choose

the set of clusters which give the minimum of ∆m(X1 − X2) and combine each of these

clusters with the `¯ pair. We take the minimum of the two m(Xi`¯) values, and plot the

distribution in Fig. 4.24. In this case, the SM BG is even more reduced, and the mg −mχ01

mass edge seems somewhat more apparent.

5 We have also tried other methods such as picking the X1 − X2 pair with maximum ∆φ(X1 − X2),maximum ∆R(X1−X2), minimum δpT (X1−X2) and minimum average invariant mass avg(m(X1),m(X2)).We also tried to separate the resulting b jets into hemispheres. In the end, the best amount of correctassignment was achieved with the choice of the ∆m(X1 −X2)min.

86

0 100 200 300 400 500 600m(Xi)(∆m(X1−X2))min

(GeV)0

2

4

6

8

dσ/d

m(X

i) (∆m

(X1−X

2)) min (f

b/G

eV)

mg~ - mχ∼20

mg~ - mχ∼20 mg~ - mχ∼1

0

mg~ - mχ∼10

Point APoint DBackground

Cuts C1´ + ≥ 4 b-jets

Figure 4.21: Plot of m(X1,2) from benchmark points A and D along with SM backgroundsin events with cuts C1′ plus ≥ 4 b-jets and minimizing ∆m(X1 −X2); see text for details.

0 100 200 300 400 500 600m(Xi)(∆m(X1−X2))min

(GeV)0

0.02

0.04

0.06

0.08

0.1

0.12

dσ/d

m(X

i) (∆m

(X1−X

2)) min (f

b/G

eV)

mg~ - mχ∼10

mg~ - mχ∼10

mg~ - mχ∼20

mg~ - mχ∼20

Point APoint DBackground

Cuts C1´ + ≥ 4 b-jets + 2 SF/OS leptons

Figure 4.22: Same as Fig. 4.21 but requiring in addition a pair of SF/OS leptons.

87

According to [107], measurements of hadronic mass edges can be made with a precision

of roughly 10%. Nevertheless, from the kinematic distributions discussed above we can only

determine mass differences. There is still not enough information to extract absolute masses,

i.e., each of mg, mχ02

and mχ01. However, it is pointed out in Ref. [70] that in cases (such as

the focus point region of minimal supergravity) where sparticle pair production occurs nearly

purely from gg production, and when the dominant g branching fractions are known (from

a combination of theory and experiment), then the total gg production cross section after

cuts allows for an absolute measurement of mg to about an 8% accuracy. These conditions

should apply to our Yukawa-unified SUSY cases, if we assume the ∼ 56% branching fraction

for g → bbχ02 decay (from theory). The study of Ref. [70] required that one fulfill the cuts

C2 which gave robust gluino pair production signal along with small SM backgrounds:

88

0 200 400 600 800 1000m(ll+2b-jets)min (GeV)

0

0.2

0.4

0.6

0.8

dσ/d

m(ll

+2b

-jets)

min

(fb/

GeV

)

Point APoint DBackground

Cuts C1’ + 2 SF/OS

mg~ - mχ∼10

mg~ - mχ∼10

Figure 4.23: Plot of m(bb`+`−)min from points A and D along with SM backgrounds.

0 100 200 300 400 500 600 700 800m(Xill)(∆m(X1−X2))min

(GeV)0

0.02

0.04

0.06

dσ/d

m(X

ill)(∆

m(X

1−X2)) m

in (f

b/G

eV)

mg~ - mχ∼10 mg~ - mχ∼1

0

Point APoint DBackground

Cuts C1´ + ≥ 4 b-jets + 2 SF/OS leptons

Figure 4.24: Plot of m(X1,2`+`−)min from points A and D, minimizing ∆m(X1 − X2) as

explained in the text, along with SM backgrounds.

89

C2 cuts:

EmissT > (max(100 GeV, 0.2Meff ), (4.8)

n(jets) ≥ 7, (4.9)

n(b− jets) ≥ 2, (4.10)

ET (j1, j2− j7) > 100, 50 GeV, (4.11)

AT > 1400 GeV, (4.12)

ST ≥ 0.2 , (4.13)

where AT is the augmented effective mass AT = EmissT +

∑leptonsET +

∑jetsET . In this

case, the summed SM background was about 1.6 fb, while signal rate for Point A (D) is 57.3

(66.2) fb. The total cross section after cuts varies strongly with mg, allowing an extraction

of mg to about 8% for 100 fb−1 integrated luminosity, after factoring in QCD and branching

fraction uncertainties in the total rate. Once an absolute value of mg is known, then mχ02

and mχ01

can be extracted to about 10% accuracy from the invariant mass edge information.

4.2.4 Trilepton signal from W1χ02 production

While the signal from gluino pair production at the LHC from Yukawa-unified SUSY models

will be very robust, it will be useful to have a confirming SUSY signal in an alternative

channel. From Fig. 4.10, we see that there also exists substantial cross sections for W±1 χ

01,

W+1 W

−1 and W±

1 χ02 production. The χ±1 → χ0

1ff′ and χ0

2 → χ01ff decays (here f stands

for any of the SM fermions) are dominated by W and Z exchange, respectively, so that

in this case the branching fractions BF (χ±1 → χ01ff

′) are similar to BF (W± → ff ′) and

BF (χ02 → χ0

1ff) is similar to Z → ff .

The W±1 χ

01 → χ0

1qq′ + χ0

1 process will be difficult to observe at LHC since the final state

jets and EmissT will be relatively soft, and likely buried under SM background. Likewise, the

W−1 χ

01 → χ0

1`ν`χ01 signal will be buried under a huge BG from W → `ν` production. The

W+1 W

−1 production reaction will also be difficult to see at LHC. The purely hadronic final

state will likely be buried under QCD and Z+ jets BG, while the lepton plus jets final state

will be buried under W + jets BG. The dilepton final state will be difficult to extract from

W+W− and tt production.

90

The remaining reaction, W±1 χ

02 production, yields a trilepton final state from χ±1 → χ0

1`ν`

and χ02 → χ0

1`¯ decays which in many cases is observable above SM BG. The LHC reach for

χ±1 χ02 → 3` + Emiss

T production was mapped out in Ref. [108], and the reach was extended

into the hyperbolic branch/focus point (HB/FP) region in Ref. [109]. The method was to

use the cut set SC2 from Ref. [106] but as applied to the LHC. For the clean trilepton signal

from W±1 χ

02 → 3`+ Emiss

T production, we require:

• three isolated leptons with pT (`) > 20 GeV and |η`| < 2.5,

• SF/OS dilepton mass 20 GeV < m(`+`−) < 81 GeV, to avoid BG from photon and Z

poles in the 2 → 4 process qq′ → ` ¯ ′ν` ,

• a transverse mass veto 60 GeV < MT (`, EmissT ) < 85 GeV to reject on-shell W

contributions,

• EmissT > 25 GeV and,

• veto events with n(jets) ≥ 1.

The resulting BG levels and signal rates for points A and D are listed in Table 4.4. The

2 → 2 processes are calculated with Isajet, while the 2 → 4 processes are calculated at parton

level using Madgraph1[110]. The combination of hard lepton pT cuts and the requirement

that n(jets) = 0 leaves us with no 2 → 2 background, while the parton level 2 → 4 BG

remains at 0.7 fb. Here, we see that signal from the two Yukawa-unified points well exceeds

background.

In the clean 3` channel, since two of the leptons ought to come from χ02 → χ0

1`¯ decay,

they should display a confirmatory dilepton mass edge at mχ02− mχ0

1as is evident in the

gluino pair production events, where the dileptons are accompanied by high jet multiplicity.

The distribution in m(`+`−) is shown in Fig. 4.25. Event rates are seen to be lower than

those from gg production. Integrated luminosity needed for a discovery with 5σ significance

would be 2.83 fb−1 for pt. A and 1.5 fb−1 for pt. D6.

6 Significance is defined as S/√

(S + B)

91

Table 4.4: Clean trilepton signal after cuts listed in the text.

process events σ (fb) after cuts (fb)tt 3× 106 4.9× 105 –WW,ZZ,WZ 5× 105 8.0× 104 –W ∗Z∗, W ∗γ∗ → ` ¯ ′ν`′ 106 – 0.7Total BG 4.5× 105 – 0.7Point A: – 106 7.6× 104 3.4

S/B → – – 4.86S/√S +B (10 fb−1) → – – 5.31

Point D: – 106 9.0× 104 4.1S/B → – – 5.86

S/√S +B (10 fb−1) → – – 5.92

0 20 40 60 80 100m(ll) (GeV)

0

0.05

0.1

0.15

0.2

dσ/d

m(ll

) (fb

/GeV

)

Point APoint D

Cuts SC2 Clean TrileptonZero Jets Only

Figure 4.25: Plot of m(`+`−) in the clean trilepton channel from points A and D along withSM backgrounds.

92

CHAPTER 5

CONCLUSIONS

In chapter 2, we have presented a number of new results.

1. First, we verified former results presented in Ref. [76] that Yukawa unified models can

be generated with updated Isajet spectra code and an updated value of the top quark

mass mt = 171 GeV. Using both random scans and the more efficient MCMC scans,

we find that models with excellent Yukawa coupling unification can be generated in

the HS model if scalar masses are in the multi-TeV range, while gaugino masses are

quite light, and the W1 is slightly above the current LEP2 limit. The models require

the Bagger et al. boundary conditions if µ > 0 such that A20 = 2m2

10 = 4m216, and

A0 < 0 in our convention. The spectra generated is characterized by three mass scales:

multi-TeV first and second generation matter scalars, TeV scale third generation and

Higgs scalars and 100–200 GeV light charginos and gluinos of order 350–450 GeV.

The relic density is typically 30–30,000 times above the WMAP measured value. As

a solution, we propose i). hypothesizing an unstable neutralino χ01 which decays to

axino plus photon, ii). raising the GUT scale gaugino mass M1 so that bino-wino co-

annihilation reduces the relic density or iii). lowering the first/second generation scalar

masses relative to the third so that neutralinos can annihilate via light qR exchange

and neutralino-squark co-annihilation. We regard the first of these solutions as the

most attractive, and the third is actually susceptable to possible exclusion by analyses

of Fermilab Tevatron signals in the case of just two light squarks.

2. Using an MCMC analysis, we find a new class of solutions with m16 ∼ 3 TeV, where

neutralinos annihilate through the light higgs h resonance. This low a value of m16

typically leads to Yukawa unification at the 5–10% level at best.

93

3. We find we are able to generate solutions with low µ and low mA as did the BDR

group. The solutions generated by the Isajet code with low µ, low mA and m16 ∼ 3

TeV tend to have Yukawa unification in the 20% range or greater. We were able to

generate a class of solutions with excellent Yukawa unification and m16 ranging up to

6 TeV, where the DM problem is solved by neutralino annihilation through a 150–250

GeV A resonance. The combination of large tan β and low mA gives a Bs → µ+µ−

branching fraction at levels beyond those allowed by the CDF collaboration.

We also present a Table of five benchmark solutions suitable for event generation, and for

examination of collider signals expected at the LHC from DM-allowed Yukawa-unified SUSY

models. Based on this work, we are able to make several predictions, if the Yukawa-unified

MSSM is the correct effective field theory between MGUT and Mweak. We would expect the

following:

• New physics events at the CERN LHC to be dominated by gluino pair production

with mg ∼ 350–450 GeV. Since tan β is large, the final states are rich in b-jets, and

the OS/SF isolated dilepton invariant mass distribution should have a visible edge at

mχ02−mχ0

1∼ 50–75 GeV because the χ0

2 always decays via 3-body modes. Squarks and

sleptons are likely to be very heavy, and may decouple from LHC physics signatures.

• We would predict in this scenario that the (g − 2)µ anomaly is false, since in Yukawa-

unified SUSY models with large m16, the SUSY contribution to the muon QED vertex

is always highly suppressed.

• While SUSY should be easily visible at the LHC for Yukawa unified models, we would

predict a dearth of direct and indirect dark matter detection signals. This is because the

typically large values of µ and scalar masses tend to suppress such signals. However,

in the CDF-excluded case of point E, the direct and indirect DM signals may be

observable. Point D also has a low but observable direct DM detection rates, since

scalars are not too heavy. Point C, with its anomalously low uR and cR squark masses,

is already excluded by direct detection searches unless one appeals to a non-standard

local density of dark matter.

Chapter 3 finds that for Yukawa-unified supersymmetric models, as expected in SO(10)

SUSY GUT models, we find one can implement a consistent cosmology including the

94

following: (1) BBN safe mass spectra owing to the multi-TeV value of m16, which arises

in SUGRA models from a multi-TeV mG (2) a WMAP-allowed relic density of CDM that

consists dominantly of thermally produced axinos, and (3) the re-heat temperature needed

to fulfill the relic density falls above the lower bound required by non-thermal leptogenesis,

and below the upper bound coming from gravitino/BBN constraints.

We feel that the fact that Yukawa unified SO(10) SUSY GUT models pass these several

cosmological tests makes them even more compelling than they were based on pure particle

physics reasons. In any case, with a spectrum of light gluinos, charginos and neutralinos,

they should soon be tested by experiments at the CERN LHC[102].

Finally, in Chapter 4, we make an argument for using the multi-lepton channel instead of

missing ET and show collider results for Yukawa-unified SO(10) SUSY GUT models. In the

very early run of the LHC pp collider, it may not be possible to use EmissT as a discrimination

variable due to detector calibration issues. We show here that a substantial reach for gluino

and squark production followed by cascade decays can be gained by requiring events with

large jet and isolated lepton multiplicity, but with no requirement on EmissT . In the mSUGRA

model with a low and high value of m0, an LHC reach for mg of 750 (1000) GeV is found

with 0.1 (1) fb−1 of integrated luminosity by requiring ≥4 jets plus ≥ 3 isolated leptons.

If enough signal events are found, then some kinematic reconstruction of sparticle masses

should be possible as in the cases where large EmissT is required. SUSY signal can also be

seen above SM BG if just two OS/SF leptons are required, especially in the case where there

is a distinctive kinematic dilepton invariant mass edge.

Simple SUSY grand unified models based on the gauge group SO(10) may have t− b− τYukawa coupling unification in addition to gauge group and matter unification. By assuming

the MSSM is the effective field theory valid below MGUT , we can, starting with weak scale

fermion masses as boundary conditions, check whether or not these third generation Yukawa

couplings actually unify. The calculation depends sensitively on the entire SUSY particle

mass spectrum, mainly through radiative corrections to the b, t and τ masses. It was

found in previous works that t− b− τ Yukawa coupling unification can occur, but only for

very restrictive soft SUSY breaking parameter boundary conditions valid at the GUT scale,

leading to a radiatively induced inverted mass hierarchy amongst the sfermion masses. While

squarks and sleptons are expected to be quite heavy, gluinos, winos and binos are expected

to be quite light, and will be produced at large rates at the CERN LHC.

95

We expect LHC collider events from Yukawa-unified SUSY models to be dominated by

gluino pair production at rates of (30− 150)× 103 fb. The gs decay via 3-body modes into

bbχ02, bbχ

01 and tbχ±1 , followed by leptonic or hadronic 3-body decays of the χ0

2 and χ±1 . A

detailed simulation of signal and SM BG processes shows that signal should be easily visible

above SM BG in the ≥ 4 jets plus ≥ 3` channel, even without using the EmissT variable, with

about 1 fb−1 of integrated luminosity.

If Yukawa-unified signals from gg production are present, then at higher integrated

luminosities, mass edges in the m(`+`−), m(bb) and m(bb`+`−) channels along with total

cross section rates (which depend sensitively on the value of mg) should allow for sparticle

mass reconstruction of mg, mχ02

and mχ01

to O(10%) accuracy for ∼ 100 fb−1 of integrated

luminosity. The gluino pair production signal can be corroborated by another signal in the

clean trilepton channel from W1χ02 → 3` + Emiss

T , which should also be visible at higher

integrated luminosities. Thus, based on the study presented here, we expect LHC to either

discover or rule out t − b − τ Yukawa-unified SUSY models within the first year or two of

operation.

96

REFERENCES

[1] L. Girardello and M. Grisaru, Nucl. Phys. B 194 (1982) 65. 1.2

[2] H. Georgi, in Proceedings of the American Institue of Physics, edited by C. Carlson(1974); H. Fritzsch and P. Minkowski, Ann. Phys. 93, 193 (1975); M. Gell-Mann, P.Ramond and R. Slansky, Rev. Mod. Phys. 50, 721 (1978). For recent reviews, see R.Mohapatra, hep-ph/9911272 (1999) and S. Raby, in Rept. Prog. Phys. 67 (2004) 755.

[3] CDF and D0 collaborations, hep-ph/0703034 (2007). .

[4] B. Ananthanarayan, G. Lazarides and Q. Shafi, Phys. Rev. D 44 (1991) 1613 and Phys.Lett. B 300 (1993) 245; G. Anderson et al. Phys. Rev. D 47 (1993) 3702 and Phys. Rev.D 49 (1994) 3660; V. Barger, M. Berger and P. Ohmann, Phys. Rev. D 49 (1994) 4908;M. Carena, M. Olechowski, S. Pokorski and C. Wagner, Ref. [6]; B. Ananthanarayan,Q. Shafi and X. Wang, Phys. Rev. D 50 (1994) 5980; R. Rattazzi and U. Sarid, Phys.Rev. D 53 (1996) 1553; T. Blazek, M. Carena, S. Raby and C. Wagner, Phys. Rev. D56 (1997) 6919; T. Blazek and S. Raby, Phys. Lett. B 392 (1997) 371; T. Blazek andS. Raby, Phys. Rev. D 59 (1999) 095002; T. Blazek, S. Raby and K. Tobe, Phys. Rev.D 60 (1999) 113001 and Phys. Rev. D 62 (2000) 055001; see also [5]. 2, 2.2

[5] S. Profumo, Phys. Rev. D 68 (2003) 015006; C. Pallis, Nucl. Phys. B 678 (2004) 398;M. Gomez, G. Lazarides and C. Pallis, Phys. Rev. D 61 (2000) 123512, Nucl. Phys. B638 (2002) 165 and Phys. Rev. D 67 (2003) 097701; U. Chattopadhyay, A. Corsettiand P. Nath, Phys. Rev. D 66 (2002) 035003; M. Gomez, T. Ibrahim, P. Nath and S.Skadhauge, Phys. Rev. D 72 (2005) 095008.

[6] R. Hempfling, Phys. Rev. D 49 (1994) 6168; L. J. Hall, R. Rattazzi and U. Sarid, Phys.Rev. D 50 (1994) 7048; M. Carena et al., Nucl. Phys. B 426 (1994) 269; D. Pierce, J.Bagger, K. Matchev and R. Zhang, Nucl. Phys. B 491 (1997) 3. 4, 42

[7] M. Drees, Phys. Lett. B 181 (1986) 279; J.S. Hagelin and S. Kelley, Nucl. Phys. B342 (1990) 95; A.E. Faraggi, et al., Phys. Rev. D 45 (1992) 3272; Y. Kawamura andM. Tanaka, Prog. Theor. Phys. 91 (1994) 949; Y. Kawamura, et al., Phys. Lett. B 324(1994) 52 and Phys. Rev. D 51 (1995) 1337; N. Polonsky and A. Pomarol, Phys. Rev.D 51 (1994) 6532; H.-C. Cheng and L.J. Hall, Phys. Rev. D 51 (1995) 5289; C. Koldaand S.P. Martin, Phys. Rev. D 53 (1996) 3871. 4, 42

[8] H. Baer, M. Diaz, J. Ferrandis and X. Tata, Phys. Rev. D 61 (2000) 111701. 2.1

97

[9] H. Baer, M. Brhlik, M. Diaz, J. Ferrandis, P. Mercadante, P. Quintana and X. Tata,Phys. Rev. D 63 (2001) 015007. 2.1

[10] H. Baer,P. Mercadante and X. Tata, Phys. Lett. B 475 (2000) 289; H. Baer, C. Balazs,M. Brhlik, P. Mercadante, X. Tata and Y. Wang, Phys. Rev. D 64 (2001) 015002. 2.1

[11] K. Tobe and J. D. Wells, Nucl. Phys. B 663 (2003) 123. 2.1

[12] H. Baer, M. Diaz, P. Quintana and X. Tata, J. High Energy Phys. 0004 (2000) 016. 1,6

[13] R. Dermisek, S. Raby, L. Roszkowski and R. Ruiz de Austri, J. High Energy Phys. 0304(2003) 037.

[14] R. Dermisek, S. Raby, L. Roszkowski and R. Ruiz de Austri, J. High Energy Phys. 0509(2005) 029. 2.1, 2.3.2, 7

[15] D. Auto, H. Baer, A. Belyaev and T. Krupovnickas, J. High Energy Phys. 0410 (2004)066. 2.1, 2.3.2, 7

[16] H. Baer, A. Belyaev, T. Krupovnickas and A. Mustafayev, J. High Energy Phys. 0406(2004) 044. 2.1

[17] H. Baer, T. Krupovnickas, A. Mustafayev, E. K. Park, S. Profumo and X. Tata, J. HighEnergy Phys. 0512 (2005) 011. 2.1

[18] H. Baer, J. Ferrandis, K. melnikov and X. Tata, Phys. Rev. D 66 (2002) 074007. 2.1,2.2.2, 2.4

[19] H. Haber and R. Hempfling, Phys. Rev. D 48 (1993) 4280. 2

[20] B.C. Allanach, S. Kraml and W. Porod, J. High Energy Phys. 03 (2003) 016; G.Belanger, S. Kraml and A. Pukhov, Phys. Rev. D 72 (2005) 015003; S. Kraml andS. Sekmen in: Physics at TeV Colliders 2007, BSM working group report, in prep.. 2.2

[21] H. Baer, K. Hagiwara and X. Tata, Phys. Rev. D 35 (1987) 1598; H. Baer, D. Dzialo-Karatas and X. Tata, Phys. Rev. D 42 (1990) 2259; H. Baer, C. Kao and X. Tata,Phys. Rev. D 48 (1993) 5175; H. Baer, C. H. Chen, F. Paige and X. Tata, Phys. Rev.D 50 (1994) 4508; I. Hinchliffe et al., Phys. Rev. D 55 (1997) 5520 and Phys. Rev.D 60 (1999) 095002; H. Bachacou, I. Hinchliffe and F. Paige, Phys. Rev. D 62 (2000)015009; Atlas Collaboration, LHCC 99-14/15; C. Lester, M. Parker and M. White, J.High Energy Phys. 0601 (2006) 080. 2.2

[22] See I. Hinchliffe et al., Ref. [21]. 2.2.1, 4.2.3, 22

[23] J. Feng, A. Rajaraman and F. Takayama, Phys. Rev. Lett. 91 (2003) 011302 and Phys.Rev. D 68 (2003) 063504. 2.2.1

[24] L. Covi, H. B. Kim, J. E. Kim and L. Roszkowski, J. High Energy Phys. 0105 (2001)033. 2.2.2

98

[25] K. Jedamzik, M. Lemoine and G. Moultaka, JCAP0607 (2006) 010. 2.2.2

[26] L. Covi, J. E. Kim and L. Roszkowski, Phys. Rev. Lett. 82 (1999) 4180. 2.2.2, 3.2.3

[27] A. Birkedal-Hansen and B. Nelson, Phys. Rev. D 64 (2001) 015008 and Phys. Rev. D67 (2003) 095006; H. Baer, A. Mustafayev, E. K. Park and S. Profumo, J. High EnergyPhys. 0507 (2005) 046. 2.2.2

[28] F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, Nucl. Phys. B 477 (1996) 321.2.2.2

[29] A. A. Markov, Extension of the limit theorems of probability theory to a sum of variablesconnected in a chain, reprinted in Appendix B of: R. Howard, Dynamic ProbabilisticSystems, volume 1: Markov Chains, John Wiley and Sons, 1971. 2.2.2

[30] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, ”Equa-tions of State Calculations by Fast Computing Machines”. Journal of Chemical Physics,21(6):1087-1092, 1953; W.K. Hastings, Biometrika, 57(1):97–109, 1970;see also C.G. Lester, M.A. Parker, M.J. White, J. High Energy Phys. 0601 (2006) 080.2.3

[31] J. Hamann, S. Hannestad, M. S. Sloth, Y. Y. Y. Wong, Phys. Rev. D 75 (2007) 023522.2.3

[32] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 174(2006) 577; Comput. Phys. Commun. 176 (2007) 367. 2.3

[33] K. Babu and C. Kolda, Phys. Rev. Lett. 84 (2000) 228; A. Dedes, H. Dreiner and U.Nierste, Phys. Rev. Lett. 87 (2001) 251804; J. K. Mizukoshi, X. Tata and Y. Wang,Phys. Rev. D 66 (2002) 115003. 2.3.1

[34] K. Tollefson, talk given at Lepton Photon 07, August 2007, Daegu, Korea.2.3.2

[35] H. Baer, C. H. Chen, M. Drees, F. Paige and X. Tata, Phys. Rev. Lett. 79 (1997) 986.2.3.2

[36] H. Baer, C. H. Chen, F. Paige and X. Tata, Phys. Rev. D 50 (1994) 4508. 2.4

[37] H. Komatsu and H. Kubo, Phys. Lett. B 157 (1985) 90 and Nucl. Phys. B 263 (1986)265; H. E. Haber, G. Kane and M. Quiros, Phys. Lett. B 160 (1985) 297 and Nucl.Phys. B 273 (1986) 333; G. Gamberini, Z. Physik C 30 (1986) 605; R. Barbieri, G.Gamberini, G. F. Giudice and G. Ridolfi, Nucl. Phys. B 296 (1988) 75; H. E. Haberand D. Wyler, Nucl. Phys. B 323 (1989) 267; S. Ambrosanio and B. Mele, Phys. Rev.D 53 (1996) 2541 and Phys. Rev. D 55 (1997) 1399 [Erratum-ibid. D56, 3157 (1997)];H. Baer and T. Krupovnickas, J. High Energy Phys. 0209 (2002) 038. 2.4

[38] ATLAS Collaboration, LHCC 99-14/15 (1999); CMS Collaboration, J. Phys. G: Nucl.Part. Phys. 34 (2007) 1. 2.4

99

[39] A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 96 (2006) 011802; V. Abazovet al. (D0 Collaboration), Phys. Rev. Lett. 97 (2006) 121802. 2.4

[40] H. Georgi and S. Glashow, Phys. Rev. Lett. 32 (1974) 438; H. Georgi, H. Quinn and S.Weinberg, Phys. Rev. Lett. 33 (1974) 451; A. Buras, J. Ellis, M. K. Gaillard and D. V.Nanopoulos, Nucl. Phys. B 135 (1978) 66. 2.4

[41] H. Georgi, in Proceedings of the American Institue of Physics, edited by C. Carlson(1974); H. Fritzsch and P. Minkowski, Ann. Phys. 93, 193 (1975); M. Gell-Mann, P.Ramond and R. Slansky, Rev. Mod. Phys. 50, 721 (1978). For recent reviews, see R.Mohapatra, hep-ph/9911272 (1999) and S. Raby, in Rept. Prog. Phys. 67 (2004) 755.

[42] B. Ananthanarayan, G. Lazarides and Q. Shafi, Phys. Rev. D 44 (1991) 1613 and Phys.Lett. B 300 (1993) 245; G. Anderson et al. Phys. Rev. D 47 (1993) 3702 and Phys. Rev.D 49 (1994) 3660; V. Barger, M. Berger and P. Ohmann, Phys. Rev. D 49 (1994) 4908;M. Carena, M. Olechowski, S. Pokorski and C. Wagner, Ref. [6]; B. Ananthanarayan,Q. Shafi and X. Wang, Phys. Rev. D 50 (1994) 5980; R. Rattazzi and U. Sarid, Phys.Rev. D 53 (1996) 1553; T. Blazek, M. Carena, S. Raby and C. Wagner, Phys. Rev. D56 (1997) 6919; T. Blazek and S. Raby, Phys. Lett. B 392 (1997) 371; T. Blazek andS. Raby, Phys. Rev. D 59 (1999) 095002; T. Blazek, S. Raby and K. Tobe, Phys. Rev.D 60 (1999) 113001 and Phys. Rev. D 62 (2000) 055001; see also [5].

[43] Y. Kawamura, Prog. Theor. Phys. 105 (2001) 999; G. Altarelli and F. Feruglio, Phys.Lett. B 511 (2001) 257; L. Hall and Y. Nomura, Phys. Rev. D 64 (2001) 055003; A.Hebecker and J. March-Russell, Nucl. Phys. B 613 (2001) 3; A. Kobakhidze, Phys.Lett. B 514 (2001) 131.

[44] H. Baer, C. Balazs and A. Belyaev, J. High Energy Phys. 0203 (2002) 042; H. Baer, C.Balazs, A. Belyaev, J. K. Mizukoshi, X. Tata and Y. Wang, J. High Energy Phys. 0207(2002) 050.

[45] H. Baer and J. Ferrandis, Phys. Rev. Lett. 87 (2001) 211803; D. Auto, H. Baer, C.Balazs, A. Belyaev, J. Ferrandis and X. Tata, J. High Energy Phys. 0306 (2003) 023.3.2

[46] T. Blazek, R. Dermisek and S. Raby, Phys. Rev. Lett. 88 (2002) 111804; T. Blazek,R. Dermisek and S. Raby, Phys. Rev. D 65 (2002) 115004; R. Dermisek, S. Raby, L.Roszkowski and R. Ruiz de Austri, J. High Energy Phys. 0304 (2003) 037; R. Dermisek,S. Raby, L. Roszkowski and R. Ruiz de Austri, J. High Energy Phys. 0509 (2005) 029.2.1, 2.1, 3.2

[47] H. Baer, S. Kraml, S. Sekmen and H. Summy, arXiv:0801.1831 (Phys. Rev. D, in press).2, 2.1, 2.1, 1, 2.3.2, 7, 3.2

[48] M. Ibe, K.-I. Izawa and T. Yanagida, Phys. Rev. D 71 (2005) 035005. (document),2.3.2, 3.2, 3.2.3, 3.2.4, 3.2.4, 4.1, 4.2, 4.2

[49] D. N. Spergel et al. (WMAP Collaboration), astro-ph/0603449 (2006). 3.2

100

[50] D. Auto, H. Baer, A. Belyaev and T. Krupovnickas, J. High Energy Phys. 0410 (2004)066. 2.1, 3.1

[51] H. P. Nilles and S. Raby, Nucl. Phys. B 198 (1982) 102; J. E. Kim and H. P. Nilles,Phys. Lett. B 138 (1984) 150; J. E. Kim, Phys. Lett. B 136 (1984) 378. 3.2, 3.2.3

[52] L. Covi, J. E. Kim and L. Roszkowski, Phys. Rev. Lett. 82 (1999) 4180; L. Covi, H. B.Kim, J. E. Kim and L.Roszkowski, J. High Energy Phys. 0105 (2001) 033. 3.2

[53] M. Bolz, A. Brandenburg and W. Buchmuller, Nucl. Phys. B 606 (2001) 518; J. Pradlerand F. D. Steffen, Phys. Rev. D 75 (2007) 023509. 3.2, 3.2.3, 3.2.3

[54] S. Weinberg, Phys. Rev. Lett. 48 (1982) 1303; R. H. Cyburt, J. Ellis, B. D. Fields andK. A. Olive, Phys. Rev. D 67 (2003) 103521; K. Jedamzik, Phys. Rev. D 70 (2004)063524; M. Kawasaki, K. Kohri and T. Moroi, Phys. Lett. B 625 (2005) 7 and Phys.Rev. D 71 (2005) 083502. 3.2.1

[55] K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 73 (2006) 123511. 3.2.1

[56] M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45; M. Luty, Phys. Rev. D 45(1992) 455; W. Buchmuller and M. Plumacher, Phys. Lett. B 389 (1996) 73 and Int. J.Mod. Phys. A 15 (2000) 5047; R. Barbieri, P. Creminelli, A. Strumia and N. Tetradis,Nucl. Phys. B 575 (2000) 61; G. F. Giudice, A. Notari, M. Raidal, A. Riotto and A.Strumia, hep-ph/0310123; for a recent review, see W. Buchmuller, R. Peccei and T.Yanagida, Ann. Rev. Nucl. Part. Sci.55 (2005) 311. 3.2.1

[57] G. Lazarides and Q. Shafi, Phys. Lett. B 258 (1991) 305; K. Kumekawa, T. Moroi andT. Yanagida, Prog. Theor. Phys. 92 (1994) 437; T. Asaka, K. Hamaguchi, M. Kawasakiand T. Yanagida, Phys. Lett. B 464 (1999) 12. 3.2.2

[58] K. Hamaguchi, H. Murayama and T. Yanagida, Phys. Rev. D 65 (2002) 043512. 3.2.2

[59] W. Buchmuller, P. Di Bari and M. Plumacher, Nucl. Phys. B 643 (2002) 367 andErratum-ibid,B793 (2008) 362; Annals Phys. 315 (2005) 305 and New J. Phys. 6 (2004)105. 3.2.2

[60] M. Ibe, T. Moroi and T. Yanagida, Phys. Lett. B 620 (2005) 9. 3.2.2

[61] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. D 68 (2003) 063504. 3.2.2

[62] A. Brandenburg and F. Steffen, JCAP0408 (2004) 008. 3.2.3

[63] For an overview, see e.g. H. Baer and X. Tata, Weak Scale Supersymmetry, CambridgeUniversity Press (2006). 3.2.3, 3.2.3

[64] J. Ellis, J. Hagelin, D. V. Nanopoulos and M. Srednicki. Phys. Lett. B 127 (1983) 233.(document), 1.1, 3.1, 4

101

[65] H. Baer, C. H. Chen, F. Paige and X. Tata, Phys. Rev. D 52 (1995) 2746 and Phys.Rev. D 53 (1996) 6241; H. Baer, C. H. Chen, M. Drees, F. Paige and X. Tata, Phys.Rev. D 59 (1999) 055014; S. Abdullin and F. Charles, Nucl. Phys. B 547 (1999) 60;S. Abdullin et al. (CMS Collaboration), hep-ph/9806366; B. Allanach, J. Hetherington,A. Parker and B. Webber, J. High Energy Phys. 08 (2000) 017; H. Baer, C. Balazs,A. Belyaev, T. Krupovnickas and X. Tata, J. High Energy Phys. 0306 (2003) 054. 1

[66] H. Baer, J. Ellis, G. Gelmini, D. Nanopoulos and X. Tata, Phys. Lett. B 161 (1985)175; G. Gamberini, Z. Physik C 30 (1986) 605; H. Baer, V. Barger, D. Karatas and X.Tata, Phys. Rev. D 36 (1987) 96; R. M. Barnett, J. F. Gunion and H. Haber, Phys.Rev. D 37 (1988) 1892; H. Baer, X. Tata and J. Woodside, Phys. Rev. D 42 (1990)1568; A. Bartl, W. Majerotto, B. Mosslacher, N. Oshimo and S. Stippel, Phys. Rev. D43 (1991) 2214; H. Baer, M. Bisset, X. Tata and J. Woodside, Phys. Rev. D 46 (1992)303. A. Bartl, W. Majerotto and W. Porod, Z. Physik C 64 (1994) 499; A. Djouadi, Y.Mambrini and M. Muhlleitner, Eur. Phys. J. C 20 (2001) 563; J. Hisano, K. Kawagoeand M. Nojiri, Phys. Rev. D 68 (2003) 035007. 4

[67] S. Kumar Gupta, B. Mukhopadhyaya and S. Kumar Rai, Phys. Rev. D 75 (2007)075007. 4.1, 4.2.1

[68] P. G. Mercadante, J. K. Mizukoshi and X. Tata, Phys. Rev. D 72 (2005) 035009. 2

[69] B. C. Allanach et al., hep-ph/0202233 (2002). 4.1, 4.2.2

[70] H. Baer, V. Barger, G. Shaughnessy, H. Summy and L. T. Wang, Phys. Rev. D 75(2007) 095010. 4.1

[71] I. Hinchliffe, F. Paige, M. Shapiro, J. Soderqvist and W. Yao, Phys. Rev. D 55 (1997)5520. 2.4, 4.1, 4.2.3

[72] H. Baer, K. Hagiwara and X. Tata, Phys. Rev. D 35 (1987) 1598; H. Baer, D. Dzialo-Karatas and X. Tata, Phys. Rev. D 42 (1990) 2259; H. Baer, C. Kao and X. Tata,Phys. Rev. D 48 (1993) 5175; H. Baer, C. H. Chen, F. Paige and X. Tata, Phys. Rev.D 50 (1994) 4508; see also Ref. [71]. 4.1, 72

[73] H. Georgi, in Proceedings of the American Institue of Physics, edited by C. Carlson(1974); H. Fritzsch and P. Minkowski, Ann. Phys. 93, 193 (1975); M. Gell-Mann, P.Ramond and R. Slansky, Rev. Mod. Phys. 50, 721 (1978). For recent reviews, see R.Mohapatra, hep-ph/9911272 (1999) and S. Raby, in Rept. Prog. Phys. 67 (2004) 755.For additional perspective, see G. Altarelli and F. Feruglio, hep-ph/0405048. 4.1

[74] M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, Proceedings of the Workshop,Stony Brook, NY 1979 (North-Holland, Amsterdam); T. Yanagida, KEK Report No.79-18, 1979; R. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.

[75] Y. Kawamura, Prog. Theor. Phys. 105 (2001) 999; G. Altarelli and F. Feruglio, Phys.Lett. B 511 (2001) 257; L. Hall and Y. Nomura, Phys. Rev. D 64 (2001) 055003; A.Hebecker and J. March-Russell, Nucl. Phys. B 613 (2001) 3; A. Kobakhidze, Phys.Lett. B 514 (2001) 131.

102

[76] D. Auto, H. Baer, C. Balazs, A. Belyaev, J. Ferrandis and X. Tata, J. High EnergyPhys. 0306 (2003) 023. 1.3

[77] T. Blazek, R. Dermisek and S. Raby, Phys. Rev. D 65 (2002) 115004. 2.1, 2.2, 2, 2.2.1,2.3.2, 1

[78] The Tevatron Electroweak Working group (CDF and D0 Collaborations),arXiv:0803.1683 [hep-ex] (2007).

[79] H. Baer, S. Kraml, S. Sekmen and H. Summy, J. High Energy Phys. 0803 (2008) 056.2.2

[80] T. Blazek, R. Dermisek and S. Raby, Phys. Rev. Lett. 88 (2002) 111804. (document),2.3.2, 3.2, 3.2.3, 3.2.4, 3.2.4, 4.1, 4.2, 4.2

[81] R. Dermisek, S. Raby, L. Roszkowski and R. Ruiz de Austri, J. High Energy Phys. 0304(2003) 037 and J. High Energy Phys. 0509 (2005) 029; M. Albrecht, W. Altmannshofer,A. Buras, D. Guadagnoli and D. Straub, J. High Energy Phys. 0710 (2007) 055; W.Altmannshofer, D. Guadagnoli, S. Raby and D. Straub, arXiv:0801.4363 (2008).

[82] J. Feng, C. Kolda and N. Polonsky, Nucl. Phys. B 546 (1999) 3; J. Bagger, J. Fengand N. Polonsky, Nucl. Phys. B 563 (1999) 3; J. Bagger, J. Feng, N. Polonsky andR. Zhang, Phys. Lett. B 473 (2000) 264; H. Baer,P. Mercadante and X. Tata, Phys.Lett. B 475 (2000) 289; H. Baer, C. Balazs, M. Brhlik, P. Mercadante, X. Tata and Y.Wang, Phys. Rev. D 64 (2001) 015002; see also H. Baer, M. Diaz, P. Quintana and X.Tata, J. High Energy Phys. 0004 (2000) 016.

[83] ISAJET v7.74, by H. Baer, F. Paige, S. Protopopescu and X. Tata, hep-ph/0312045;http://www.hep.fsu.edu/∼isajet/; for details on the Isajet spectrum calculation, see H.Baer, J. Ferrandis, S. Kraml and W. Porod, Phys. Rev. D 73 (2006) 015010. 2.1, 2.2.1,3.2

[84] H. Baer, J. Ferrandis, S. Kraml and W. Porod, Phys. Rev. D 73 (2006) 015010. 2.1,2.2, 3.2, 4.1, 4.2.2

[85] S. P. Martin and M. Vaughn, Phys. Rev. D 50 (1994) 2282. 2.2

[86] H. E. Haber, R. Hempfling and A. Hoang, Z. Physik C 75 (1996) 539. 2.2

[87] D. Pierce, J. Bagger, K. Matchev and R. Zhang, Nucl. Phys. B 491 (1997) 3.

[88] D. Auto, H. Baer and A. Belyaev and T. Krupovnickas, J. High Energy Phys. 0410(2004) 066.

[89] D. N. Spergel et al. (WMAP Collaboration), Astrophys. J. Supp., 170 (2007) 377.

[90] L. Covi, J. E. Kim and L. Roszkowski, Phys. Rev. Lett. 82 (1999) 4180; L. Covi, H. B.Kim, J. E. Kim and L. Roszkowski, J. High Energy Phys. 0105 (2001) 033.

[91] E. J. Chun, J. E. Kim and H. P. Nilles, Phys. Lett. B 287 (1992) 123.

103

[92] K. Jedamzik, M. LeMoine and G. Moultaka, JCAP0607 (2006) 010.

[93] A. Brandenburg and F. Steffen, JCAP0408 (2004) 008. 2.2.2, 3.2.3

[94] K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 73 (2006) 123511.

[95] W. Buchmuller, P. Di Bari and M. Plumacher, Annal. Phys. 315 (2005) 305.

[96] G. Lazarides and Q. Shafi, Phys. Lett. B 258 (1991) 305; K. Kumekawa, T. Moroi andT. Yanagida, Prog. Theor. Phys. 92 (1994) 437; T. Asaka, K. Hamaguchi, M. Kawasakiand T. Yanagida, Phys. Lett. B 464 (1999) 12.

[97] H. Baer and H. Summy, Phys. Lett. B 666 (2008) 5.

[98] H. Baer, J. Ellis, G. Gelmini, D. Nanopoulos and X. Tata, Phys. Lett. B 161 (1985)175; G. Gamberini, Z. Physik C 30 (1986) 605; H. Baer, V. Barger, D. Karatas and X.Tata, Phys. Rev. D 36 (1987) 96; R. M. Barnett, J. F. Gunion and H. Haber, Phys.Rev. D 37 (1988) 1892; H. Baer, X. Tata and J. Woodside, Phys. Rev. D 42 (1990)1568; A. Bartl, W. Majerotto, B. Mosslacher, N. Oshimo and S. Stippel, Phys. Rev. D43 (1991) 2214; H. Baer, M. Bisset, X. Tata and J. Woodside, Phys. Rev. D 46 (1992)303; A. Bartl, W. Majerotto and W. Porod, Z. Physik C 64 (1994) 499; A. Djouadi, Y.Mambrini and M. Muhlleitner, Eur. Phys. J. C 20 (2001) 563; J. Hisano, K. Kawagoeand M. Nojiri, Phys. Rev. D 68 (2003) 035007.

[99] H. Baer, C. H. Chen, F. Paige and X. Tata, Phys. Rev. D 50 (1994) 4508. 4.1, 4.2.1

[100] W. Beenakker, R. Hopker and M. Spira, hep-ph/9611232 (1996). 4.2.3

[101] P. G. Mercadante, J. K. Mizukoshi and X. Tata, Phys. Rev. D 72 (2005) 035009. 4.2.1

[102] H. Baer, H. Prosper and H. Summy, Phys. Rev. D 77 (2008) 055017. 4.1, 4.2.2

[103] J. Ellis, J. Hagelin, D. V. Nanopoulos and M. Srednicki. Phys. Lett. B 127 (1983) 233.4.2.2, 4.2.2, 5

[104] F. Gianotti and M. L. Mangano, hep-ph/0504221; H. Baer, V. Barger and G.Shaughnessy, arXiv:0806.3745 (2008). 4.2.2

[105] R. M. Barnett, J. Gunion and H. Haber, Phys. Lett. B 315 (1993) 349; H. Baer, X.Tata and J. Woodside, Phys. Rev. D 41 (1990) 906 and Phys. Rev. D 45 (1992) 142.4.2.2

[106] H. Baer, M. Drees, F. Paige, P. Quintana and X. Tata, Phys. Rev. D 61 (2000) 095007.4.2.2

[107] H. Bachacou, I. Hichliffe, F. E. Paige, Phys. Rev. D 62 (2000) 015009; B.K. Gjelsten,E. Lytken, D.J. Miller, P. Osland and G. Polesello, ATLAS internal note ATL-PHYS-2004-007 (2004), published in: G. Weiglein et al. [LHC/LC Study Group], Phys. Rept.426 (2006) 47; M. M. Nojiri, G. Polesello, D. Tovey, J. High Energy Phys. 0805 (2008)

104

014; CMS Collaboration (G.L. Bayatian et al.), CMS technical design report, volumeII: Physics performance, CERN-LHCC-2006-021, CMS-TDR-008-2, J. Phys. G: Nucl.Part. Phys. 34 (2007) 995. 4.2.3, 4.2.4

[108] H. Baer, C. H. Chen, F. Paige and X. Tata, Phys. Rev. D 53 (1996) 6241. 4.2.3, 4.2.3

[109] H. Baer, T. Krupovnickas, S. Profumo and P. Ullio, J. High Energy Phys. 0510 (2005)020. 4.2.4

[110] W. F. Long and T. Stelzer, Comput. Phys. Commun. 81 (1994) 357. 4.2.4

4.2.4

105

BIOGRAPHICAL SKETCH

Heaya Ann Summy

EDUCATION

2001 – present Department of Physics, Florida State University (FSU)Degree: Ph.D., High Energy Physics, Theory

(expected 2008)Advisor: Professor Howard A. Baer

1999 – 2001 Department of Chemistry, Florida State UniversityProgram: Chemical Physics

1996 – 1999 Embry-Riddle Aeronautical University (ERAU),Daytona Beach, FL

Degree: B.S., Aerospace Engineeringwith minor in Mathematics

RESEARCH AND TEACHING POSITIONS

2006 – present Research/Teaching Assistant, Dept. of Physics, FSU

2003 – 2006 National Science Foundation (NSF) GK-12 Fellow

2001 – 2003 Research/Teaching Assistant, Dept. of Physics, FSU

1999 – 2001 Research/Teaching Assitant, Dept. of Chemistry, FSU

1996 – 1999 Teaching Assistant, Physical Sciences Dept., ERAUTeaching Assistant, Aerospace Engineering Dept., ERAU

106

HONORS, AWARDS, AND ACHIEVEMENTS

2007 – present FSU Women’s Table Tennis team, Team Captain2003 – 2006 NSF Graduate Teaching Fellowship in K-12 Education (GK-12)2003 FSU Student Star1999 ERAU Aerospace Engineering Student of the Year1999 Sigma Gamma Tau Honor Undergraduate Award Nominee1999 ERAU President’s Advisory Board1998 – 1999 American Institute of Aeronautics and Astronautics (AIAA),

Chairman, ERAU student chapterSigma Gamma Tau Aerospace Engineering Honor Society, Vice PresidentSociety of Automotive Engineers (SAE) Team XB-99 Intimidator Flying

Wing, Lead Fundraiser1998 ERAU academic scholarship

CONFERENCES AND SCHOOLS

• Theoretical Advanced Study Institute in Elementary Particle Physics (TASI): “The

Dawn of the LHC Era”. University of Colorado, Boulder, CO, June 2-27, 2008.

• Prospects in Theoretical Physics (PiTP): “The Standard Model and Beyond”. Institute

for Advanced Study, Princeton, NJ, July 16-27, 2007.

• International Workshop on the Interconnection Between Particle Physics and Cosmol-

ogy (PPC 2007). Cambridge-Mitchell (TAMU) Collaboration in Cosmology, Texas

A&M University, College Station, TX, May 14-18, 2007.

• Phenomenology 2006 Symposium (PHENO 06). University of Wisconsin-Madison,

Madison, WI, May 15-17, 2006.

• American Physical Society (APS) April Meeting. Tampa, FL, April 16-19, 2005.

• Gordon Research Conferences (GRC) Chemical Physics Summer School: “Many Body

Techniques in Chemical Physics”. Roger Williams University, Bristol, RI, June 16-28,

2002.

107

PUBLICATIONS

1. “Prospects for Yukawa Unified SO(10) SUSY GUTs at the CERN LHC” (with H. Baer,

S. Kraml and S. Sekmen), arXiv:0809.0710 [hep-ph].

2. “SO(10) SUSY GUTs, the gravitino problem, non-thermal leptogenesis and axino dark

matter” (with H. Baer), Phys. Lett. B 666, 5 (2008).

3. “SUSY interpretation of the EGRET GeV anomaly, Xenon-10 dark matter search

limits and the LHC” (with H. Baer and A. Belyaev), Phys. Rev. D 77, 095013 (2008).

4. “Early SUSY discovery at LHC without missing E(T): The Role of multi-leptons”

(with H. Baer and H. Prosper), Phys. Rev. D 77, 055017 (2008).

5. “Dark matter allowed scenarios for Yukawa-unified SO(10) SUSY GUTs” (with

H. Baer, S. Kraml and S. Sekmen), JHEP 03, 056 (2008).

6. “Mixed Higgsino dark matter from a large SU(2) gaugino mass” (with H. Baer,

A. Mustafayev and X. Tata), JHEP 10, 088 (2007).

7. “Precision gluino mass at the LHC in SUSY models with decoupled scalars” (with

H. Baer, V. Barger, G. Shaughnessy and L.-T. Wang), Phys. Rev. D 75, 095010

(2007).

8. “Science Graduate Students in K-8 Classrooms: Experiences and Reflections” (edited

by P.J. Gilmer, D.E. Granger and W. Butler), Southeast Eisenhower Regional Consor-

tium for Mathematics and Science Education @ SERVE, 2005.

108