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
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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
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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
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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
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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
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4.2.4
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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
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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.
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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.
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