the search for dark matter: from colliders to direct detection experiments …nm932yy4639/ml... ·...

THE SEARCH FOR DARK MATTER:

FROM COLLIDERS TO DIRECT DETECTION EXPERIMENTS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Mariangela Lisanti

July 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/nm932yy4639

© 2010 by Mariangela Lisanti. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/nm932yy4639

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Jay Wacker, Primary Adviser


Savas Dimopoulos


Michael Peskin

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Acknowledgements

My time in graduate school has been a very rewarding experience primarily because

of my research group and advisor. I have been very lucky to work with Jay Wacker,

whose mentorship and guidance has helped me grow as a research scientist. Our

collaboration has introduced me to new and exciting physics and has challenged

me to approach problems critically and deeply. Working in Jay’s group has always

been great fun: many thanks go out to Daniele Alves, Siavosh Behbahani, Anson

Hook, Eder Izaguirre, Martin Jankowiak, and Tomas Rube, for all the interesting

conversations and laughs.

I have had the opportunity to work with many brilliant and wonderful collabo-

rators during my time at Stanford: Daniele Alves, Johan Alwall, Roger Blandford,

Andy Haas, My Phuong Le, Louis Strigari, Jay Wacker, and Risa Wechsler. I have

learned new physics and research tools from each, and feel honored to have had the

opportunity to work with them. I would also like to thank my reading committee

(Savas Dimopoulos, Michael Peskin and Jay Wacker), as well as my oral committee

(Joanne Hewett, Michael Peskin, Jelena Vuckovic, Jay Wacker, and Risa Wechsler)

for their time in helping me meet the final requirements to graduate. I am especially

grateful to Michael Peskin, who pulled me aside one day after field theory class five

years ago and suggested I consider research in particle physics. That simple suggestion

gave me the courage to explore a new field of physics with great excitement.

During my time at Stanford, I have made wonderful friends with whom I have

shared great adventures. To all those who have shared in these experiences with me,

especially my friends, family, and Pete: thank you for your continual support and

encouragement.

v

Contents

Acknowledgements v

1 Introduction 1

1.1 The Standard Model and Beyond . . . . . . . . . . . . . . . . . . . . 3

1.2 Collider searches for dark matter . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Jets + ET6 Channel . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2 Discovery through the Higgs Sector . . . . . . . . . . . . . . . 13

1.3 Direct Detection Experiments . . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 CiDM and Directional Detection . . . . . . . . . . . . . . . . 21

1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Collider Searches for Jets+ET6 25

2.1 Event Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.2 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Projected Reach of Searches . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Model-Independent Jets+ET6 35

3.1 Overview of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Proposed Analysis Strategy . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Event Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

vi

3.4 Gluino Exclusion Limits . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.1 No Cascade Decays . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.2 Cascade Decays . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4.3 t-channel squarks . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.4 Monophoton Search . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.5 Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Dark Matter via the Higgs Sector 61

4.1 Light a0 Modifications to Higgs Phenomenology . . . . . . . . . . . . 63

4.2 h0 → a0a0 at Hadron Colliders . . . . . . . . . . . . . . . . . . . . . . 71

4.2.1 Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.3 Expected Sensitivity . . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Prospects for Inelastic Dark Matter 83

5.1 Inelastic Dark Matter at CRESST . . . . . . . . . . . . . . . . . . . . 86

5.2 XENON100 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Six Higgs Doublet Model 93

6.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 RG Influence on Low Energy Spectrum . . . . . . . . . . . . . . . . . 98

6.2.1 Gauge Unification . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2.2 Quartic Couplings . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.1 Relic abundance . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.2 Bounds from electroweak precision tests . . . . . . . . . . . . 104

6.4 Experimental Signatures . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4.1 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4.2 Indirect detection . . . . . . . . . . . . . . . . . . . . . . . . . 108

vii

6.4.3 Collider Signatures . . . . . . . . . . . . . . . . . . . . . . . . 110

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7 Parity Violation in CiDM Models 116

7.1 Models of CiDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.1.1 Axially Charged Quarks . . . . . . . . . . . . . . . . . . . . . 119

7.1.2 Vectorially Charged Quarks . . . . . . . . . . . . . . . . . . . 121

7.1.3 Parity Violation . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.2 Direct Detection Phenomenology . . . . . . . . . . . . . . . . . . . . 123

7.3 Searches for the Dark Photon . . . . . . . . . . . . . . . . . . . . . . 129

7.3.1 Limits from Direct Production . . . . . . . . . . . . . . . . . . 131

7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8 Directional Detection 137

8.1 Direct Detection Phenomenology . . . . . . . . . . . . . . . . . . . . 139

8.2 Directional Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Bibliography 152

viii

List of Tables

2.1 Selection criteria for jets +ET6 searches . . . . . . . . . . . . . . . . . 31

3.1 Differential cross section for background and signal . . . . . . . . . . 44

4.1 Relative and cumulative signal efficiencies . . . . . . . . . . . . . . . 73

4.2 Continuum backgrounds for low invariant mass muon pairs . . . . . . 75

7.1 Benchmark CiDM Models . . . . . . . . . . . . . . . . . . . . . . . . 128

ix

List of Figures

1.1 Rotation Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Schematic of dark matter freeze-out . . . . . . . . . . . . . . . . . . . 6

1.4 Radiative corrections to the Higgs mass . . . . . . . . . . . . . . . . . 8

1.5 MSSM particle content. . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Gluino decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Direct detection experiments for spin-independent dark matter. . . . 15

1.8 Recoil energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.9 DAMA’s modulated amplitude . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Boosted gluinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 The 95% gluino-bino exclusion curve for D06 at 4 fb−1 . . . . . . . . . 32

3.1 Comparison of D06 cuts and optimized cuts for a sample dijet signal . 41

3.2 Differential 0→ 1 jet rate for a matched sample of light gluino production 48

3.3 The importance of matching . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Differential cross section for theoretical model spectrum . . . . . . . . 52

3.5 The 95% exclusion region for D06 at 4 fb−1 . . . . . . . . . . . . . . . 53

3.6 Exclusion plot for cascading gluinos . . . . . . . . . . . . . . . . . . . 54

3.7 Gluino production cross section as a function of squark mass . . . . . 56

4.1 Branching fraction of Higgs to pseudoscalars . . . . . . . . . . . . . . 67

4.2 Values of 〈S〉/ sin 2β (GeV) excluded by LEP2 . . . . . . . . . . . . . 69

4.3 Parameter space excluded by CLEO . . . . . . . . . . . . . . . . . . . 70

x

4.4 Schematic of Higgs decay chain . . . . . . . . . . . . . . . . . . . . . 74

4.5 Muon invariant mass before and after cuts . . . . . . . . . . . . . . . 76

4.6 Total invariant mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.7 Expected sensitivity to Higgs production cross section . . . . . . . . . 79

5.1 CRESST recoil spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Average counts at CRESST . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Average counts at XENON100 . . . . . . . . . . . . . . . . . . . . . . 90

6.1 Allowed range for of quartic couplings . . . . . . . . . . . . . . . . . . 101

6.2 Allowed mass of the LSP in the six Higgs double model . . . . . . . . 105

6.3 Direct detection possibilities for six Higgs doublet model . . . . . . . 107

6.4 Flux from dark matter annihilation in six Higgs doublet model . . . . 109

6.5 Dark matter mass splitting . . . . . . . . . . . . . . . . . . . . . . . . 111

6.6 Width of the SM Higgs decay . . . . . . . . . . . . . . . . . . . . . . 112

6.7 LHC production cross section . . . . . . . . . . . . . . . . . . . . . . 114

7.1 Allowed regions in mπd− Λd parameter space . . . . . . . . . . . . . 125

7.2 Allowed regions in mπd− feff parameter space . . . . . . . . . . . . . 126

7.3 Limits on kinetic mixing . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.1 Modulation amplitude for FFeDM and iDM . . . . . . . . . . . . . . 139

8.2 Allowed m− δm parameter space . . . . . . . . . . . . . . . . . . . . 144

8.3 Estimated recoil spectrum at LUX . . . . . . . . . . . . . . . . . . . 146

8.4 Comparison of recoil angle spectra for FFeDM and iDM . . . . . . . . 148

8.5 Predicted rates at directional detection experiments . . . . . . . . . 149

xi

Chapter 1

Introduction

The presence of dark matter in the universe is one of the greatest mysteries of par-

ticle physics. While dark matter is believed to play an important role in structure

formation, very little is known about its properties. Because the dark matter emits

no detectable radiation, its presence has been deduced indirectly through gravita-

tional effects. Its existence was first inferred by the astronomer Fritz Zwicky in the

1930s. At the time, Zwicky noted that the predicted mass of the Coma Cluster from

luminosity measurements fell markedly short of dynamical estimates of the cluster’s

mass using the virial theorem [1]. In his paper “On the Masses of Nebulae and of

Clusters of Nebulae,” Zwicky wrote that “This discrepancy is so great that a further

analysis of the problem is in order,” however it was not until several decades later that

dark matter reemerged as a pressing scientific issue with the detailed measurements

of galactic rotation curves.

Rotation curves probe the gravitational potential of a galaxy by measuring its

orbital velocity as a function of radius. Measurements of rare stars and 21 cm emission

from neutral Hydrogen allow astronomers to extend these curves beyond where light

from the galactic disk ceases. Fig. 1.1 shows a sample rotation curve for the M33

galaxy. According to Newton’s gravitational law, one would expect that the velocity

should fall as v2 = GM/r far from the galactic disk. In sharp contrast, the data

shows that the velocity becomes constant at large radii. This indicates that the

galaxy’s mass is M(r) ∝ r, which implies the presence of an additional contribution

1

2 CHAPTER 1. INTRODUCTION

Figure 1.1: Rotation curve for the M33 galaxy. The solid line is the best-fit model.The different contributions to the total arise from the halo (dot-dashed), stellar disk(short dashed), and gas (long dashed) [2].

to the mass that extends beyond the galactic plane and that does not emit detectable

radiation.

Rotation curves indicate that dark halos account for ∼ 3− 10 times more mass in

a galaxy than visible matter does [3]. More precise estimates of the total baryon and

matter density exist, and all point to the fact that the amount of dark matter in the

universe is far greater than the amount of baryonic matter. There are three primary

methods for measuring Ωb, the ratio of the baryon density to the critical density

(see [4] for review). The first is through estimates of light element abundances, which

give baryon densities Ωbh2 = 0.0204 ± 0.0018, where h is the Hubble rate in units

of 100 km s−1 Mpc−1 [4]. The second is to measure how much light is absorbed by

distant quasars, which is an indicator of the amount of intervening hydrogen. This

measurement gives Ωbh1.5 ' 0.02 [5]. Finally, careful measurements of anisotropies in

the CMB give a remarkably precise estimate of Ωbh2 = 0.024+0.004

−0.003 [6, 7].

The total matter density of the universe can be estimated from galaxy surveys

and x-ray measurements [4]. Most recently, the matter density has been measured

from anisotropies in the CMB to be Ωmh2 = 0.16 ± 0.04 [6, 7]. Setting the Hubble

constant to h = 0.72 gives a matter density that is 30% of the critical density. In

comparison, baryons comprise only 5% of the critical density. Thus, the density of

1.1. THE STANDARD MODEL AND BEYOND 3

the dark matter is about five time the baryonic density.

Little is known about this additional matter component in the universe. Galactic

rotation curves tell us that it cannot emit radiation and that its dominant interaction

with baryons is gravitational. In addition, galaxy surveys indicate that dark matter

is required to explain the measured structure of the universe. In particular, power

spectrum used to measure the galaxy distribution is explained well with a model

that contains dark matter and a cosmological constant; a theory that only includes

baryons does not reproduce the measurements [4]. There are no candidates within the

Standard Model of particle physics that can simultaneously explain the results of the

rotation curves and the galaxy surveys. As a result, models of dark matter typically

involve Beyond the Standard Model (BSM) ingredients, introducing new elementary

particles that are relics of the Big Bang. To motivate the most popular BSM and

dark matter scenarios, let us begin with a brief review of the Standard Model.

1.1 The Standard Model and Beyond

The Standard Model (SM) describes all matter particles and the interactions between

them. It consists of two sets of spin-1/2 fermions (quarks and leptons) as well as three

spin-1 gauge bosons. The Standard Model gauge group is SU(3) × SU(2) × U(1),

which describes both the strong and electroweak interactions in the model. As shown

in Fig. 1.1, quarks carry the strong color charge, which is communicated by the gluon

force carrier. Both the quarks and leptons carry SU(2)×U(1) charge and can interact

via the electroweak bosons.

The Higgs boson is the only scalar in the SM and the only particle of the theory

that has yet to be discovered. It is an SU(2) doublet with the following Lagrangian:

LHiggs = (DµH)†(DµH) +m2H†H + λ(H†H)2. (1.1)

Dµ is the covariant derivative Dµ = ∂µ − ig2~σ · ~Aµ − ig

′

2Bµ, where ~σ are the Pauli

matrices and g, g′ are the SU(2) and U(1) coupling constants, respectively. The

Higgs boson is responsible for electroweak symmetry breaking, the process by which


Particle Spin SU(3) SU(2) U(1)

QL 1/2 3 2 1/3

uL 1/2 3 1 -4/3

dL 1/2 3 1 2/3

LL 1/2 1 2 -1

eL 1/2 1 1 2

Gµν 1 8 1 0

Aµν 1 1 3 0

Bµ 1 1 1 0

H 0 1 2 -1

Figure 1.2: The Standard Model of particle physics with spin, SU(3), SU(2), andU(1) charge assignments.

the W± and Z0 bosons acquire mass. The symmetry breaking arises from the fact

that the Higgs potential has a negative mass term, which leads to a ground state

energy with arbitrary phase (when λ > 0). When a particular value of the phase

is selected, the symmetry of the ground state is broken. In the Standard Model,

electroweak symmetry is broken when H → (0, (v + h0)/√

2), where v is the vacuum

expectation value (vev) of the Higgs field. The gauge boson masses arise from taking

the square of DµH and setting H to its vev. This results in three massive vector

bosons, W± and Z0:

W±µ =

1√2

(A1µ ∓ iA2

µ) Z0µ =

1√g2 + g′2

(gA3µ − g′Bµ) (1.2)

and one massless boson, the photon, which is orthogonal to Z0µ.

Neutrinos are the only natural candidate for dark matter in the SM. However,

studies of structure formation with neutrinos as the dominant dark matter do not

reproduce the results from galaxy surveys [4]. Therefore, neutrinos cannot contribute

all of the dark matter density, though they may still contribute a fraction of it.

There have been significant efforts at building models that add new, non-baryonic

1.1. THE STANDARD MODEL AND BEYOND 5

elementary particles to the SM to explain the dark matter. Axions and WIMPs are

currently the two leading dark matter candidates. Axions arise from the solution

of the strong CP problem in the Standard Model by the breaking of a new, chiral

symmetry referred to as the Peccei-Quinn symmetry [3]. With a mass ∼ 10−5 eV,

axions would populate the universe after the QCD phase transition [8]. The most

promising way to test for axions is to look for their conversion to photons in the

presence of strong magnetic fields.

WIMPs, or Weakly Interacting Massive Particles, are the second important class

of DM particles and are the main focus of this work. WIMPs are typically new heavy

particles in BSM models that are neutral and stable. They are strongly motivated be-

cause predictions for their cosmological abundance reproduce the dark matter density

observed today. To derive this explicitly, assume that the Standard Model is supple-

mented by an additional stable, weakly-interacting particle χ that is a potential dark

matter candidate (see [3, 8] for further detail). In the early universe, when T mχ,

the DM is in thermal equilibrium. Interactions of the form χχ↔ qq where the dark

matter annihilates into or is produced by quark/antiquark or lepton/antilepton pairs

maintain the equilibrium abundance. The equilibrium number density of the χ’s is

given by

neqχ =

g

(2π)3

∫f(p)d3p, (1.3)

where g is the number of internal degrees of freedom for the particle (i.e., g = 2 for

spin 1/2 state) and f(p) is the Fermi-Dirac or Bose-Einstein distribution. For the

temperatures of interest, quantum statistics can be ignored and f(E) → e(µ−E)/T .

When the temperature is much greater than the mass of the DM, then neqχ ∝ T 3.

However, as the temperature drops below mχ, the equilibrium abundance falls ex-

ponentially as neqχ ∝ T 3/2e−mχ/T . Because the reaction rate is proportional to the

DM abundance (Γ = nχ〈σv〉), a drop in number density corresponds to a drop in the

reaction rate. When the reaction rate falls below the expansion rate of the universe,

Γ . H, the dark matter cannot find annihilation partners and fall out of thermal

equilibrium. At this point, the DM number density “freezes-out” and remains con-

stant with time (Fig. 1.3). The larger the annihilation cross section at the time of


Increasing <!v>C

omov

ing

Num

ber D

ensi

ty

x=m"/T (time #)

Figure 1.3: Schematic of dark matter freeze-out

freeze-out, the smaller the DM number density today.

To estimate the WIMP number density today, one can use the fact that the

entropy per comoving volume in the universe is constant with time. As a result, nχ/s

is constant and its value is equal to that at freeze-out:(nχs

)today

=

(nχs

)f

, (1.4)

where s ' 0.4g∗T3 is the entropy density and g∗ is the effective number of rela-

tive degrees of freedom. At freeze-out, nχ = Γ/〈σv〉 = H/〈σv〉. Because the uni-

verse is still radiation-dominated at this time, the Hubble expansion rate is H(T ) =

1.66g1/2∗ T 2/Mpl, where Mpl ' 1019 GeV is the Planck mass. Estimating the temper-

ature at freeze-out to be Tf ' mχ/20, the comoving number density is(nχs

)today

' 100

mχMplg1/2∗ 〈σv〉

. (1.5)

1.2. COLLIDER SEARCHES FOR DARK MATTER 7

The entropy density and critical density today are s ' 4000 cm−3 and ρc ' 10−5h2

GeV/cm3, and the present mass density is therefore

Ωχh2 =

mχnχρc

' 0.04

(100

g∗

)1/2(α

0.01

)2(100 GeV

mχ

)2

, (1.6)

assuming that the annihilation cross section scales as 〈σv〉 ∼ α2/m2χ. The fact that

a weakly interacting DM with mχ ∼ 100 GeV gives a relic density very close to the

observed value is an incredible coincidence and is referred to as the “WIMP miracle.”

The WIMP miracle strongly motivates regions in mass-cross section parameter

space that experiments should focus on. Though it provides welcome guidance in

experimental searches, it does not provide any solid answers to the many open ques-

tions about dark matter. The mass, charge and spin of the DM is not known, nor do

we know with any certainty whether the dark sector consists of one or more particles.

In addition, astrophysical uncertainties concerning the density profile and velocity

distribution of the dark matter in the Milky Way further complicate predictions for

potential signals. Because there are so many unknowns about DM, a multifaceted

experimental approach is warranted. The three primary search strategies for dark

matter are to either observe it in the sky, create it in the lab, or observe it in the

lab. The first of these, often referred to as “indirect detection,” consists of satellite

experiments that search for the products of dark matter decay or annihilation into

gamma and/or cosmic rays in the Milky Way halo [8]. The focus of this work will

be on the alternate two search strategies that look for DM in lab-based experiments:

collider and direct-detection searches.

1.2 Collider searches for dark matter

Collider experiments, such as the Fermilab Tevatron in Illinois and the Large Hadron

Collider (LHC) in Switzerland, may produce the dark matter either as a direct product

of proton collisions or in the decays of other new particles produced in the collisions.

The Tevatron is a 2 TeV proton-antiproton collider that has accumulated ∼12 fb−1


of data combined from its two detectors, D06 and CDF. The LHC, which is currently

running at 7 TeV and will upgrade to 10 TeV, is a proton-proton collider that began

data collection this year. Both accelerators collide bunches of protons/antiprotons.

Several partons interact in each bunch collision, potentially producing new, massive

particles that are a signature for physics beyond the SM. New colored particles should

be produced most copiously at hadron machines such as the Tevatron and the LHC;

non-colored particles can also be directly produced, though their production cross

sections tend to be smaller. One of the most promising ways of producing DM in

colliders is through the decays of new colored particles. Because the dark matter

is neutral and stable, its presence is inferred by measuring the missing energy in a

collision event.

Searches for new physics at the Tevatron have been strongly guided by supersym-

metric model building. Supersymmetry is one of the most popular extensions of the

Standard Model because it provides a mechanism to keep the Higgs mass light, as well

as a potential dark matter candidate. To understand the motivation for current col-

lider searches for dark matter, it will be worthwhile to briefly review supersymmetry

and its particle spectrum [9,10].

The primary motivation for introducing supersymmetry is that it extends the

SM over many decades of energy. While the SM does an exceptionally good job

at explaining the visible particles of the universe and the forces that mediate their

interactions, it is an effective theory that breaks down at energies near the Planck

scale (EPlanck ' 1019 GeV). The Higgs boson is required to explain the hierarchy

of particle masses in the theory, but its own mass is divergent. In particular, the

largest contribution to the Higgs mass comes from the top quark loop, which leads

=

=

+

+

...

...+

Figure 1.4: Radiative corrections to the Higgs mass in the Standard Model (top) andin supersymmetry (bottom).


to quadratic and logarithmically divergent corrections.

Supersymmetric theories can eliminate these divergences. Supersymmetry posits

that every SM particle has a partner with the same SU(3)× SU(2)×U(1) quantum

numbers, but opposite spin. Thus, every SM fermion has a new bosonic partner, and

every SM boson has a new fermionic partner. As a result, the top loop in the Higgs

mass renormalization now has a partner diagram with a corresponding scalar loop.

Because fermionic loops contribute an additional factor of −1, the scalar and fermion

loops together cancel the quadratically-divergent corrections.

The minimal supersymmetric standard model (MSSM) contains all the particles of

the SM plus two Higgs bosons (one that couples to up quarks and another that couples

to down quarks), as well as their corresponding supersymmetric partners (Fig. 1.5).

Because no supersymmetric partner has ever been observed, supersymmetry must

be broken in the low energy theory. There are many different proposals for the

precise mechanism of supersymmetry breaking, all of which involve new particles

and interactions at high scales. To parametrize our ignorance of these interactions,

supersymmetry is broken softly in the effective MSSM Lagrangian via the following

Lagrangian:

Lsoft = −1

2

(M3gg +M2WW +M1BB + c.c.

)−(

˜uAuQHu − ˜dAdQHd − ˜eAeLHd + c.c.)

−Q†M2QQ− L†M2

LL− ˜uM2u˜u† − ˜dM2

d˜d† − ˜eM2

e˜e† (1.7)

−m2HuH

∗uHu −m2

HdH∗dHd − (bHuHd + c.c.).

Different high-energy mechanisms for susy breaking result in different relations be-

tween the soft-breaking parameters M3,2,1,Au,d,e,MQ,L,u,d,e,mHu,Hd , b. The CMMSM

(or mSUGRA) is one of the most commonly used susy-breaking schemes [11], and

requires common scalar masses, gaugino masses, and trilinear scalar soft couplings at

the unification scale, in addition to electroweak symmetry breaking, gauge coupling


SUSY Particle Symbol Partner

up squarks u, c, t u, c, t

down squarks d, s, b d, s, b

sleptons e, µ, τ e, µ, τ

sneutrinos νe, νµντ νe, νµ, ντ

gluinos g g

charginos C±1 , C±2 W±, H±u,d

neutralinos N01 , . . . , N

04 γ, Z0, h0, H0

u,d, A0

Figure 1.5: MSSM particle content.

unification, and R-parity conservation. As a result, the entire MSSM spectrum is de-

termined by just five parameters. A consequence of the CMSSM that will be relevant

to the discussion in Ch. 2- 3, is that it fixes the ratio between M3,2,1 to be

M3 : M2 : M1 ≈ 6 : 2 : 1, (1.8)

which implies that the gluino is always six times more massive than the bino.

To obtain a dark matter candidate in the MSSM, it is necessary to introduce

R-parity, a discrete symmetry defined as

PR = (−1)3(B−L)+2S, (1.9)

where B,L are the baryon and lepton numbers, respectively, and S is the particle

spin [9,10]. All SM particles have PR = 1, while all sparticles have PR = −1. R-parity

conservation requires that every interaction vertex must contain a (multiple of) two

sparticles. This means that any sparticle will always decay into another sparticle plus

a SM particle. As a result, the lightest supersymmetric particle (LSP) must be stable

because there are no other susy particles that it can decay into. If the LSP is also

neutral and weakly-interacting, it can be a good dark matter candidate.

Oftentimes, the LSP is the neutralino, which arises from the mixing of the neutral

higgsinos (Hu and Hd) and electoweakinos (B, W 0). In the gauge-eigenstate basis,


the neutralino mass term in the Lagrangian is

Lneutralino = −1

2(ψ)TMNψ

0 + c.c., (1.10)

where ψ0 = (B, W 0, H0d , H

0u) and

MN =

M1 0 −cβsWmZ sβsWmZ

0 M2 cβcWmZ −sβcWmZ

−cβsWmZ cβcWmZ 0 −µsβsWmZ −sβcWmZ −µ 0

where sβ = sin β, cβ = cos β, sW = sin θW , and cW = cos θW . The above mass

matrix can be diagonalized to obtain the neutralino mass eigenstates. In the limit

where mZ |µ ±M1|, |µ ±MW |, then the mass eigenstates are nearly “bino-like”

(N1 ≈ B1), “wino-like” (N2 ≈ W 0), and “higgsino-like” (N3, N4 ≈ (H0u ± H0

d)/√

2).

In this case, the bino is the lightest of the four states.

1.2.1 Jets + ET6 Channel

Because both the Tevatron and LHC are hadron colliders, they will copiously produce

colored particles. Therefore, the dominant production of susy particles will be through

pair-produced gluinos or squarks, and associated production of gluinos and squarks:

pp→ gg, pp→ qq, and pp→ qg, (1.11)

respectively (or pp in the case of the Tevatron). The gluinos and squarks will decay

down to the LSP, emitting many quarks along the decay chain. Figure 1.6 illustrates

two possible gluino decays. In the first case (left), the gluino decays directly into

the neutralino DM, while in the second case it cascades down through a squark, a

chargino, and a W± boson. The decays of gluinos and squarks will preferentially

produce quarks, which will shower and hadronize to form jets in the detector. The

decay chains will produce an LSP, which is stable and neutral, and thus manifest as


Figure 1.6: Two possible gluino decays within the MSSM: (left) direct decay intoa neutralino and two quarks through a squark and (right) cascade decay through achargino and a W± boson, into two quarks, two fermions, and a neutralino [10].

missing energy, ET6 , in the detector.∗ Supersymmetry is not the only model that leads

to a jets+ET6 signal; this signal can arise from Universal Extra Dimensions (UEDs) [12]

and Little Higgs models with T-parity [13]. As a result, such searches are powerful

probes for new physics signals, casting a wide net for potential new models.

The main challenge in any collider search lies in separating the signal from SM

background. For example, the dominant backgrounds for jets + ET6 include Z0 + nj

(the Z0 boson decays to neutrinos), W±+nj (the W± decays to a lepton and neutrino

and the lepton is missed), and tt+ nj (the tops decay to leptons and neutrinos, with

the leptons mistagged). To distinguish signal from background, one makes use of

differences in the distributions of observables such as the energy, pT , and the angular

distribution, η, of the jets, as well as the amount of missing energy, ET6 , in the event.

The pT and η of a massless particle are related to the four-momentum through

pµ ' (pT cosh η, ~pT , pT sinh η). (1.12)

The rapidity is related to the polar angle by cos θ = tanh η and describes the angular

distribution of the particles in the event.

Cuts on the pT and η of the jets are important for eliminating background events.

Ideally, these cuts should retain as many signal events as possible; this is a non-trivial

task, however, because the mass spectrum of the new physics signal is unknown and

the jet energies can vary significantly from model-to-model. Consider, for example,

∗In some cases, leptons will also be produced, in addition to the jets. Tagging on leptons istypically easier than tagging on jets, because the signals are cleaner and avoid the complicatedhadronic physics that affects the shapes and distributions of jets. This work focuses on the morechallenging scenario where leptons are not present, though a brief discussion of jets+ET6 +leptonsearches is saved for Ch. 3.4.5.


the scenario where the mass of the gluino is much heavier than the bino mass. In this

case, a lot of energy is emitted in the decay and the jets will have very large pT . This

is in direct contrast to the scenario where the gluino and bino are nearly degenerate,

leading to very soft (i.e., low pT ) jets. Requiring very hard pT jets would miss the

second of these models. Requiring only soft jets would save both potential signals,

but also retain significant amounts of background events.

Therefore, it is necessary to optimize the experimental cuts to maximize the

amount of signal relative to the amount of background. The approach taken at

the Tevatron and in all preliminary analyses for the LHC has been to select several

benchmark scenarios within the CMSSM and optimize cuts for these special cases.

However, the CMSSM is not indicative of all supersymmetry models, let alone all

possible models that lead to jets + ET6 signals. Additionally, the ratio between gluino

and bino mass is held fixed at 6 : 1 in the CMSSM; therefore, varying over different

CMSSM models does not cover the allowed range of kinematically-allowed parameter

space for the gluino and bino mass. In Ch. 2, I will show how the current experi-

mental analyses at the Tevatron can lead one to miss a new physics signal. In Ch.

3, I will propose a new search strategy that allows one to search for jets +ET6 in a

model-independent manner. This is a critical step in ensuring that no new physics

signal is missed and that we are searching for the full-range of possible dark matter

candidates in theories that lead to jets +ET6 signals.

1.2.2 Discovery through the Higgs Sector

The MSSM is one of the leading candidates for BSM physics and has thus played

an important role in guiding experimental searches. However, tension exists between

MSSM theories, which typically have light Higgs boson masses, and bounds from the

LEP experiment, which require mh0 & 114 GeV at 95% confidence [14]. This tension

has motivated additional model-building efforts that modify the higgs sector in the

MSSM. These new models can lead to new dark matter candidates and potentially

new signals at colliders.

As discussed above, supersymmetry regulates corrections to the Higgs mass in the


SM. In particular, the stops cancel the quadratic mass divergence that arises from the

top loop. Even after this cancellation, however, logarithmically-divergent corrections

remain at one-loop and are:

m2h0 ' m2

Z0 cos2 2β +3g2m4

t

8π2m2W

(log

mt1mt2

m2t

+ a2t

(1− a2

t

12

)), (1.13)

where at is the dimensionless trilinear coupling between the Higgs and top squarks.

In the case of moderate at and stop masses less than 1 TeV, the mass of the Higgs

is below 120 GeV [15, 16]. By taking the at to maximal mixing, the Higgs mass can

be pushed up to 130 GeV while still keeping the top squarks under 1 TeV. To avoid

problems with fine-tuning, the top squarks should not be significantly heavier than

the Higgs; even at 1 TeV, the fine-tuning is at the few-percent level. The amount of

fine-tuning can be decreased for stops below 400 GeV, but then the Higgs mass falls

below 120 Gev and is strongly constrained by LEP.

One way to resolve the tension between fine-tuning in the MSSM higgs sector and

the LEP bounds is to add an additional singlet to the theory. The Next-to-Minimal

Supersymmetric Standard Model (i.e., NMSSM) provides an example of this [10].

In this case, there is a new particle, the singlino S that can mix with the MSSM

neutralinos. In certain limits, the singlino may be the dark matter of the theory,

with phenomenological consequences. The presence of this new singlino can alter the

decay chain for the gluino; for example, it is possible that the cascade decays start

with the gluino, go to wino plus two jets, then to bino plus two additional jets, and

ultimately to wino plus two jets. These cascade decays will tend to have far more

jets and less missing energy than the cascade decays in the MSSM and one would

need to design searches where the cuts on pT and ET6 are flexible enough to be able

to cover this range. The model-independent searches discussed in Ch. 3 have broad

sensitivity, even to models such as this.

The additional singlet in the theory can provide novel signatures at the colliders.

In particular, if there is an approximate symmetry in the Higgs potential that is

explicitly broken, there will be additional pseudo-Goldstone bosons in the theory.

If these pseudoscalars have O(1) couplings to the Higgs, then the Higgs can have

1.3. DIRECT DETECTION EXPERIMENTS 15

Experiment Target Events Exposure (kg-day)

CDMS 73Ge 4 400

XENON10 132Xe 12 300

ZEPLIN2 132Xe 29 200

ZEPLIN3 132Xe 7 150

CRESST 184W 7 30

XENON100 132Xe 0 161

Figure 1.7: Direct detection experiments for spin-independent dark matter.

a substantial branching fraction to the pseudoscalars. In Ch. 4, we will consider

Higgs decays to a pair of light pseudoscalars that each decay to a pair of tau leptons.

Searching for these new decays can tell us a lot about the structure of the Higgs sector

in the model, which may provide hints as to the identity of the dark matter in the

theory.

1.3 Direct Detection Experiments

Direct detection experiments are complimentary to collider searches for dark matter.

These experiments search for dark matter interactions with the SM by looking for

nuclear recoils that might result from a DM collision in an underground detector. The

nuclear recoil can be detected using either ionization, photons, or phonons [17]. Most

experiments combine two of these three strategies to discriminate the signal from

the background. The current experiments for spin-independent DM interactions are

listed in Fig. 1.7. The CDMS [18, 19, 20, 21] and Edelweiss [22] experiments use Ge

targets and measure both ionization and phonon signals. CRESST [23,24,25], which

has a target of CaWO4, measures nuclear recoils through simultaneous detection of

phonons and scintillation light, while the liquid xenon experiments (ZEPLIN2 [26],

ZEPLIN3 [27, 28], XENON10 [29, 30], and XENON100 [31]) pair scintillation light

and ionization signals. The experiments in Fig. 1.7 have all performed blind analyses

and most have observed events in their signal window after unblinding. However,

because these events resemble background, no experiment has claimed discovery and


limits have been placed on WIMP mass and interaction cross section.

Unlike the null experiments listed in Fig. 1.7, the DAMA experiment has claimed

an 8.9σ discovery of dark matter [32,33,34,35]. The DAMA experiment, which uses a

NaI target, makes use of a unique property of DM to distinguish it from background

events: annual modulation of DM [36]. This modulation results from the relative

motion of the sun with respect to the center of the Milky Way halo, leading to a

“wind” of DM incident on the Earth. During the summer, when the Earth moves

towards the wimp wind, the DM can have velocities up to vesc + |~vE +~v|, where vesc

is the escape velocity in the local standard of rest, ~vE is the velocity of the Earth, and

~v is the velocity of the sun. During the winter, the Earth moves against the wind,

and the maximum velocity that can be measured is vesc + |~vE−~v|. This difference in

DM velocity as measured in the lab frame between summer and winter corresponds

to an oscillation in the flux, which is proportional to velocity (Φdm = ndmv). The

DAMA experiment has measured an annual modulation over a period of nearly eleven

years, however their signal is in contradiction with null results of all other direct

detection experiments. This section will overview how all these experiments might

be reconciled.

Determining the manner in which dark matter interacts with ordinary matter is

a critical step in understanding the properties of the dark sector. Such interactions

should exist for thermal dark matter because its abundance today is set by anni-

hilations to the SM in the early universe. DM annihilation into a quark-antiquark

pair (χχ → qq) is related to the scattering diagram for DM off a quark (χq → χq)

by crossing symmetry. To predict the scattering rate for DM off a nuclear target,

one must know how WIMPs interact with quarks and gluons. If the DM is a Majo-

rana fermion, then two types of interactions are allowed: spin-independent (SI) and

spin-dependent (SD), with the following Lagrangians:

LSI = χχqq LSD = χγµγ5χqγµγ5q. (1.14)

Because the dark matter couples coherently to all the quarks in the nucleus for SI

interactions, the cross section is typically larger than that for the SD case. As a


result, experiments searching for SI-interactions have set tighter limits than those

searching for SD-interactions. The focus of Chapters 5 - 8 will be on spin-independent

interactions.

Once the interaction of WIMPs with quarks and gluons is determined, it must

be translated into interactions with protons and neutrons. The coherence loss of

the scattering between the dark matter and the proton must also be accounted for

by including a momentum-dependent form factor, F (q2), where ~q is the momentum

transfer. This form factor accounts for the fact that the dark matter does not probe

the size of the nucleus at small momentum transfer (|~q| mN), leaving the scattering

cross section relatively unaffected. At large momentum transfer, however, the dark

matter is sensitive to the size of the nucleus and the cross section is diminished.

Putting it all together, the differential cross section for SI-scattering is

dσ

dER=mNσn2µ2

nv2

(Zfp + (A− Z)fn)2|F (q2)|2, (1.15)

where mN is the mass of the target nucleus, µn is the DM-nucleon reduced mass, v is

the DM velocity, A is the atomic number, Z is the nuclear charge, and σn is the cross

section per nucleon at zero momentum transfer [37]. The coupling to protons and

neutrons, fp and fn, respectively, are effectively equivalent in most circumstances.

However, because the u and d valence-quark densities differ between protons and

neutrons, fp,n may differ if the DM couples differently to u and d quarks.

The differential scattering rate per unit detector mass is

dR

dER=

ρ0

mχmN

∫ vesc

vmin

d3vf(v)vdσ

dER, (1.16)

where ρ0 is the local dark matter density, mχ is the dark matter mass, and f(v) is

the velocity distribution of the dark matter [37]. The maximum velocity is simply

the escape velocity, vesc, at our local standard of rest in the Milky Way (i.e., at 8 kpc

from the center). The local escape velocity has been measured by the RAVE survey

to fall within 480 . vesc . 650 km/s [38]. The minimum velocity, vmin, depends on

whether the scattering event is elastic or inelastic. In the case of inelastic scattering,


it is assumed that the dark matter consists of at least two states and up-scatters to

the higher-mass state when interacting with the target nucleus.

To derive a general expression for the minimum scattering velocity, let p1,2 be

the initial DM and nuclear momenta and p3,4 the final DM and nuclear momenta,

respectively. Energy conservation requires that

(p1 + p2)2 = (p3 + p4)2. (1.17)

The nucleus scatters elastically, p22 = p2

4 = m2N , and (1.17) simplifies to

p21 − p2

3 = 2(p3 · p4 − p1 · p2) (1.18)

Consider a general scenario where the dark matter consists of two nearly-degenerate

states, split in mass by δ. The elastic limit is recovered when δ → 0. Defining the

momentum transfer as q = p3 − p1 = p2 − p4, then (1.18) becomes

−mχδ = q(p1 − p2) + q2 (1.19)

to first-order in δ. The recoil energy of the nucleus, ER, is related to the momentum

transfer through q2 = 2mNER. In the frame where the nucleus is initially at rest and

the DM is incident along the x-axis,

p2 = (mN , 0, 0) (1.20)

p1 = (mχ +1

2mχv

2,mχv, 0)

q = (ER,√

2mNER cos θ,√

2mNER sin θ)

which assumes that the DM is highly non-relativistic, a reasonable assumption given

that v ∼ O(10−3). Substituting (1.21) into (1.19) and taking cos θ = 1 for the

minimum velocity yields

vmin =1√

2mNER

(ERmN

µN+ δ), (1.21)


Figure 1.8: The recoil energy spectrum for (a) elastically scattering and (b) inelasti-cally scattering dark matter.

where µN is the DM-nucleus reduced mass. In the elastic limit (δ = 0), the minimum

velocity is equal to√

ERmN2µ2N

. As a result, the differential rate is

dR

dER∝∫ vesc

vmin

d3vdσ

dERve−v

2/v20 ∼ e−ER/E0 , (1.22)

assuming a velocity distribution function that is Maxwell-Boltzmann with dispersion

v0 ' 220 km/s. The elastic recoil spectrum is a falling exponential, as illustrated in

Fig. 1.8. The number of scattered events dominates at energies below E0 =2µ2Nv0m∼ 30

keV. Most direct detection experiments focus on recoil energies from ∼ 10− 40 keV,

in order to maximize sensitivity to elastic DM.

The minimum scattering velocity for inelastic dark matter (iDM) is larger than

that for elastic dark matter. The main consequence of this large velocity threshold is

that an inelastic signal dominates at higher recoil energies than an elastic one. Unlike

the elastic recoil spectrum, which drops off exponentially, the inelastic spectrum has

a kinematic threshold below which no events are expected; the spectrum peaks above

this threshold (Fig. 1.8). The threshold exists because a minimum energy is required

to enable the DM to up-scatter to the more massive state.

The implications of iDM for current direct detection experiments are significant.

First, experiments must be sensitive to energies larger than what is typically expected

for elastic DM. Experiments such as XENON10 and ZEPLIN-III, which had only been

20 CHAPTER 1. INTRODUCTION3

S mcoun

tskgday

2 3 4 5 6 keVee

0.01

0.02

0.03

0.04

0.05

0.06

0.07

FIG. 2: We show the modulation spectra for the best fitpoint where scattering off iodine dominates, mχ = 77 GeV(dot-dashed orange), and three points where scattering off ofsodium dominates. The best fit point off sodium is mχ = 12GeV (solid red). We also show mχ = 2 GeV (dashed green)and mχ = 7 GeV (dotted blue). The points with error barsare the published DAMA/LIBRA data.

higher mass, say 20 GeV, this approach does not succeed.As the mass moves above 12 GeV, a contribution comingfrom iodine scattering begins to move into the low end ofthe observed energy region, spoiling the fit. Also plottedin Fig. 2 are spectra for WIMP masses of 2 and 7 GeVfor comparison.

The 68%, 90%, and 99% CL (∆χ2 < 2.3, 4.61, 9.21)contours consistent with our nine bin DAMA/LIBRA χ2

function are shown in Fig. 1. Both regions shrink dra-matically compared to the two bin χ2. In the left panel,the ∆χ2 is with respect to the global best fit point at 77GeV. In the right panel, we concentrate on the low massregion and have defined ∆χ2 relative to the low mass bestfit point of 12 GeV. Note this region is confined to massesabove 10 GeV. This can be understood from examiningthe recoil spectra for the light WIMPs in Fig. 2. For afixed overall modulation rate, sub-10 GeV WIMPs pre-dict too little modulation above a couple of keVee: theysimply do not have enough mass to cause recoils of thissize.

In Fig. 1, we have superimposed 90% limit contoursfrom both the CDMS [9] and XENON [10] experiments.We only show the CDMS contour relevant for low masses,corresponding to data taken with the silicon detectors inthe Soudan mine, which have a 7 keV threshold. We haverecalculated limits using the astrophysical parameters de-scribed here. For the XENON experiment, we accountfor the energy dependent efficiencies as described in [10]and to set the limit, apply the maximum gap method [17]to the energy recoil range of 4.5-26.9 keV. We see thatthe light SI DAMA/LIBRA mass region is excluded oncethe modulation spectrum is taken into account.

Another constraint can be imposed by looking atthe total (unmodulated) counts at low energies atDAMA/LIBRA. For a WIMP, the number of recoil eventsincreases with decreasing energy. Since DAMA/LIBRA

cannot distinguish background events from signal eventsin this sample, it is clear that the predicted number ofWIMP events should not excessively exceed their totalnumber of counts in any bin.

We require that the unmodulated rate in each bin from0.75 − 4 keVee not exceed the observed values withintheir 90% error [1]. We show this constraint in Fig. 1,labeled “DAMA-Total.” The allowed region lies belowthis curve. This constraint does not greatly impinge uponthe allowed region from our nine bin χ2 that accounts forthe spectral details. Its constraint is most striking if oneconsiders the allowed region of the two bin fit (see Fig. 1).

Since the modulation in the low (2−2.5 keV) bin seemsto be the most constraining, one can consider whether itis overly biasing our analysis. For instance, one couldworry if DAMA/LIBRA were to restate the efficiencyin the lowest bin, this might completely change our re-sults. We have explored such effects by various methods:tripling the error bar on the lowest bin, merging the en-tire range into a 2−3 keV bin, and discarding the lowestbin entirely. We find that only the last option (discard-ing the lowest bin) opens up a region of parameter space,with a point allowed with χ2 = 9.14 for 6 dof (p = 0.17).This point also has a unmodulated rate that is close tosaturating the observed rate.

IV. VARIATIONS FROM ASTROPHYSICS ANDPARTICLE PHYSICS

Thus far, we assumed a MB halo and an elastic, SI in-teraction. Relaxing these assumptions could enlarge theregion at light masses, so that modulation arises with anappropriate spectrum, consistent with other experiments.

Let us begin by considering astrophysical modificationsto the velocity distribution. Kinematics informs us ofwhat modifications are needed. To scatter with nuclearrecoil energy ER, a WIMP must have a minimum veloc-ity βmin =

MNER/2µ2, where µ is the reduced mass

of the WIMP-nucleus (not nucleon) system. Considerthe channeled possibility, for which the velocity require-ments are weakest: for scattering on sodium, with re-coil energy of 4.5 keV (the highest bin with significantmodulation), one finds βminc ≈ 1140, 790, 620 km/s formχ = 2, 3, 4 GeV. If halo particle velocities approxi-mately follow a MB distribution, the most significant de-viations naturally occur for the highest velocities, whererecent infall and streams may not have fully virialized.As such, the lightest particles are the most likely to haveallowed regions opened by such deviations from a MBdistribution.

One modification to the halo is to include streams[18, 19]. We investigated a wide range of streams, varyingits velocity −1200 km/s < vstr < 1200 km/s and disper-sion 10 km/s < σstr < 50 km/s. Our stream is such thatfor positive (negative) vstr, the stream is directly against(with) the Earth’s velocity as given in the sun’s restframe. Because we limit ourselves to small perturbations

2 GeV

7 GeV

77 GeV

12 GeV

Recoil Energy (keVee)2 3 4 5 6

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Rat

e (c

pd/k

g/ke

Vee)

Figure 1.9: DAMA’s measured modulated amplitude (grey points), compared to themodulated amplitude spectrum for a 2, 7, 12, and 77 GeV elastically scattering darkmatter (green, blue, orange, and red, respectively). Figure from [39].

looking below ∼ 40 keV, have expanded their signal region up to ∼ 75 − 100 keV

in search of iDM signals [28, 30]. Second, experiments with heavier target nuclei are

more optimal for iDM searches. The reason for this is that the minimum scattering

velocity (1.8) decreases as mN increases for inelastic scattering, and a larger region

of the velocity distribution is integrated over in the expression for the differential

rate. Consequently, experiments using xenon or tungsten targets, such as XENON10,

XENON100, ZEPLIN-III, and CRESST, should see larger scattering rates from iDM

than an experiment such as CDMS, which has a germanium target.

Inelastic dark matter has received increased attention recently because it provides

a means of reconciling DAMA’s positive signal with all other null experiments. The

spectrum of the modulated amplitude measured by DAMA is reproduced in Fig 1.9.

The modulated amplitude for a given recoil energy ER is defined as

MA(ER) =1

2

(RS(ER)−RW (ER)

), (1.23)

where RS,W (ER) is the rate in the summer and winter, respectively. In the case of


elastic DM, there is a modulation in the slope of the exponential. For light elastic

WIMPs, the rate in the summer is always larger than that in the winter, and so the

modulated amplitude spectrum is itself a falling exponential. As the mass of the DM

increases, the winter rate dominates over the summer rate, but only at low recoil

energies. In this case, the modulated amplitude does not resemble an exponential,

but rather has a peak at non-zero ER. Figure 1.9 illustrates that a 77 GeV elastic

dark matter fits the DAMA measurements. Such a heavy elastically scattering dark

matter is in conflict with the results of null experiments, however, and is thus ruled

out. The modulation amplitude for inelastic dark matter can reproduce DAMA’s

results. In particular, the data may be reproduced if there are two dark matter

weak-scale states that are nearly degenerate, with a splitting O(100 keV) [195]. The

threshold observed in the DAMA data corresponds to the kinematic threshold from

the inelastic transition. Inelastic dark matter challenges the standard WIMP picture,

where the dark sector consists of only one thermally produced relic. The presence of

a small mass splitting may indicate a non-minimal dark sector with novel dynamics

that give new experimental signals. Ch. 5 discusses the prospects for discovering

inealstic dark matter at upcoming direct detection experiments.

An example of a model that yields inelastic dark matter is presented in Ch. 6. In

this model, the SM is supplemented by an additional Higgs doublet in the 5 or 6-plet

of a new global discrete symmetry. The additional scalars in the theory lead to gauge

coupling unification, even though the model is not supersymmetric. The new Higgs

doublet results in a new charged scalar, as well as two neutral scalars. In certain

regions of parameter space, these two scalars may be nearly degenerate and scatter

inelastically off nuclei. Predictions for this model at direct and indirect detection

experiments, as well as at the LHC, are discussed in Ch. 6.

1.3.1 CiDM and Directional Detection

Chapter 7 presents another model of inelastically scattering dark matter [40]. In this

scenario, the dark sector consists of a spectrum of composite states that arises from

a new SU(Nc) gauge sector that confines at a scale Λd. The Lagrangian for this new


sector is

Ldark = −1

2TrG2

µν + ΨLiD6 ΨL + ΨHiD6 ΨH +mLΨLΨL +mHΨHΨH , (1.24)

where ΨH,L are fundamentals under the new gauge sector and have masses that satisfy

mH Λd,mL. As a result of the new SU(Nc) gauge sector, the “dark quarks” ΨH,L

form low energy stable states, in direct analogy to QCD. As a result, NH heavy quarks

can combine with NH anti-light quarks to form meson states or with Nc − NH light

quarks to form baryons.

The dark matter in composite inelastic dark matter models (CiDM) cannot arise

from thermal freeze-out if it is weak-scale. Instead, one must assume a primordial

cosmological dark quark asymmetry

(nH − nH) = −(nL − nL) 6= 0, (1.25)

where ni (i = H,L, H, L) is the number density of a quark of type i. This asymmetry

guarantees that when T Λd, the dark matter is the ΨHΨL meson state. Because

of the asymmetry, there are not enough anti-heavy quarks to screen the color charge

of the heavy quarks. As a result, the heavy quarks must be screened by anti-light

quarks, leading to ΨHΨL dark matter. Another possibility is that the dark matter

might consist of multi-core hadrons, however the process of forming such states tends

to be very slow. Indeed, the ratio of baryon to meson states is O(10−6), strongly

suggesting that the dark matter is in a meson state [41].

There are two possible spin combinations for the ΨHΨL state. The spin-0 con-

figuration is the dark pion, πd, and the spin-1 configuration is the dark-rho, ρd.

The hyperfine interaction between these two states results in a small mass splitting

∝ Λ2d/mH in the confinement limit. The mass splitting is suppressed because it is

inversely proportional to the mass of the heavy quark.

The dark sector in CiDM communicates with the SM via a new dark U(1)d that

is kinetically-mixed with hypercharge and is Higgsed near the electroweak scale. The


kinetic mixing in the Lagrangian arises from

Lkin = −F 2d − F 2

EM − εFdFEM +m2AA

2d + JEMAEM + JdAd, (1.26)

where mA is the mass of the new gauge photon Ad, JEM is the electromagnetic current,

and Jd is the dark current [42]. The kinetic mixing results in an effective interaction

between the dark sector and the SM. This can be shown explicitly by redefining the

SM photon such that AEM → AEM − εAd. This transformation diagonalizes the

kinetic terms and yields

L = −F 2d − F 2

EM +m2AA

2d + JEM(AEM − εAd) + JdAd. (1.27)

Therefore, the dark photon interacts with the electromagnetic current via Lint ∝εJµemAdµ, which is proportional to the kinetic mixing parameter.

When writing down the dark matter current, either a vector or axial-vector cou-

pling between the quarks and dark photon is allowed. In Ch. 7, it will be shown that

the latter leads to pure inelastic scattering while the former leads to pure charge-radius

elastic scattering. Charge-radius scattering arises when neutral composite states with

charged constituents interact with a background field, and gives an effective form fac-

tor in the dark matter-charge photon interaction

Fdm(q2) = r2cER, (1.28)

where rc is the charge radius [43]. This additional form factor must be included in

the scattering cross section, and results in additional powers of recoil energy that

suppress the scattering rate:dR

dER∝ (ER)e−ER . (1.29)

The recoil spectrum for form-factor elastic scattering therefore looks similar to that

for inelastic scattering (Fig. 1.8); it does not have a sharp kinematic threshold, but

is suppressed at low recoil energy.

Chapter 7 will also illustrate what happens when parity is violated in CiDM

models. In this case, both axial-vector and vector interactions are allowed and both


inelastic and charge-radius scattering are allowed. The non-trivial scattering mecha-

nisms in CiDM models have important phenomenological consequences. In particular,

because the recoil spectra for form-factor elastic and inelastic scattering are so simi-

lar, it may be challenging to distinguish the two using direct detection experiments.

Ch. 8 takes on this challenge and discusses how next-generation directional detection

experiments are key to solving this problem [44]. These experiments can measure

both the energy and direction of the recoiling nucleus. The great benefit of direc-

tional detection experiments is that they can observe a daily modulation in the DM

signal [45]. This modulation results from the fact that the wimp wind changes di-

rection every twelve hours due to the rotation of the Earth around its axis. This

daily modulation can have an amplitude as large as ∼ 100% and serves as an efficient

discriminator between signal and background.

1.4 Summary

This thesis is devoted to searches for dark matter at colliders and direct detection

experiments. The first half of the work focuses on the Tevatron and LHC. Ch. 2-3

discuss jets and missing energy, a signal that is sensitive to a broad class of BSM

models and dark matter candidates. I will show that current jets+ET6 analyses can

miss signals for new physics, and will suggest an alternative model-independent search

strategy that broadens the reach of these experiments. Ch. 4 discusses searches for

Higgs bosons that decay to light pseudoscalars. Information about the Higgs sector

obtained through such searches may hint at the nature of the dark matter. The second

half of the thesis turns to direct detection experiments, with a focus on inelastic dark

matter. Ch. 5 discusses prospects for discovering iDM at upcoming experiments,

while Ch. 6-7 present two BSM models with inelastically scattering dark matter. One

of these models posits composite dark matter and has unique phenomenology that

must be studied with directional detection experiments, discussed in Ch. 8.

Chapter 2

Collider Searches for Jets plus ET6

J. Alwall, M-P. Le, M. Lisanti, and J. G. Wacker, “Searching for Directly Decaying

Gluinos at the Tevatron,” Phys. Lett. B 666, 34 (2008).

In many theories beyond the Standard Model, there is a new color octet particle

that decays into jets plus a stable neutral singlet. This occurs, for example, in super-

symmetry [46] and Universal Extra Dimensions [12], as well as Randall-Sundrum [47]

and Little Higgs models [48]. As a result, jets plus missing transverse energy (ET6 ) is

a promising experimental signature for new phenomena [49,50,51,52,53].

At present, the jets + ET6 searches at the Fermilab Tevatron are based upon

the minimal supersymmetric standard model (MSSM) and look for production of

gluinos (g) and squarks (q), the supersymmetric partners of gluons and quarks, re-

spectively [51, 52]. Both gluinos and squarks can decay to jets and a bino (B), the

supersymmetric partner of the photon. The bino is stable, protected by a discrete

R-parity, and is manifest as missing energy in the detector. Different jet topologies

are expected, depending on the relative masses of the gluinos and squarks.

There are many parameters in the MSSM and setting mass bounds in a multidi-

mensional parameter space is difficult. This has lead to a great simplifying ansatz

known as the CMSSM (or mSUGRA) parameterization of supersymmetry break-

ing [11]. This ansatz sets all the gaugino masses equal at the grand unified scale and

25

26 CHAPTER 2. COLLIDER SEARCHES FOR JETS+ET6

runs them down to the weak scale, resulting in an approximately constant ratio be-

tween the gluino and bino masses (mg : mB = 6 : 1). Thus, the mass ratio between the

gluino and bino is never scanned when searching through CMSSM parameter space.

Since the bino is the LSP in most of the CMSSM parameter space, the restriction to

unified gaugino masses means that there is a large region of kinematically-accessible

gluinos where there are no known limits.

The CMSSM parametrization is not representative of all supersymmetric mod-

els. Other methods of supersymmetry breaking lead to different low-energy par-

ticle spectra. In anomaly mediation [54], the wino can be the LSP; for instance,

mg : mW ' 9 : 1. Mirage mediation [55], in contrast, has nearly degenerate

gauginos. A more comprehensive search strategy should be sensitive to all values of

mg and mB. Currently, the tightest model-independent bound on gluinos is 51 GeV

and comes from thrust data at ALEPH and OPAL [56].

In this paper, we describe how bounds can be placed on all kinematically-allowed

gluino and bino∗ masses. We will treat the gluino as the first new colored particle

and will assume that it only decays to the stable bino: g → q1q∗ → q1q2B. The spin

of the new color octet and singlet is not known a priori; the only selection rule we

impose is that the two have the same statistics. In practice, the spin dependence is

a rescaling of the entire production cross section. For our analysis, we will assume

that the octet has spin 1/2, and will show how the results vary with cross section

rescaling.

We show how a set of optimized cuts for ET6 and HT =∑

jetsET can discover

particles where the current Tevatron searches would not. In order to show this, we

model our searches on D06 ’s searches for monojets [53], squarks and gluinos [51].

In keeping the searches closely tied to existing searches, we hope that our projected

sensitivity is close to what is achievable and not swamped by unforeseen backgrounds.

∗Throughout this note, we will call the color octet a “gluino” and the neutral singlet the “bino,”though nothing more than the color and charge is denoted by these names.

2.1. EVENT GENERATION 27

BA

TE/

~g

B~q

_

~B

g~

q

_q

qq

j

q

_q

~B

g~ ~g

B~

q_

/ET

Figure 2.1: Boosted gluinos that are degenerate with the bino do not enhance themissing transverse energy when there is no hard initial- or final-state radiation. (A)illustrates the cancellation of the bino’s ET6 . (B) shows how initial- or final-stateradiation leads to a large amount of ET6 even if the gluino is degenerate with the bino.

2.1 Event Generation

2.1.1 Signal

The number of jets expected as a result of gluino production at the Tevatron depends

on the relative mass difference between the gluino and bino, mg−mB. When the mass

splitting is much larger than the bino mass, the search is not limited by phase space

and four or more well-separated jets are produced, as well as large missing transverse

energy. The situation is very different for light gluinos (mg . 200 GeV) that are

nearly degenerate with the bino. Such light gluinos can be copiously produced at the

Tevatron, with cross sections O(102 pb), as compared to O(10−2 pb) for their heavier

counterparts (mg & 400 GeV). Despite their large production cross sections, these

events are challenging to detect because the jets from the decay are soft, with modest

amounts of missing transverse energy. Even if the gluinos are strongly boosted, the

sum of the bino momenta will approximately cancel when reconstructing the missing

transverse energy (Fig. 2.1A). To discover a gluino degenerate with a bino, it is

necessary to look at events where the gluino pair is boosted by the emission of hard

QCD jets (Fig. 2.1B). Therefore, initial-state radiation (ISR) and final-state radiation

(FSR) must be properly accounted for.

The correct inclusion of ISR/FSR with parton showering requires generating

gluino events with matrix elements. We used MadGraph/MadEvent [57] to compute


processes of the form

pp→ gg +Nj, (2.1)

where N = 0, 1, 2 is the multiplicity of QCD jets. The decay of the gluino into a bino

plus a quark and an antiquark, as well as parton showering and hadronization of the

final-state partons, was done in PYTHIA 6.4 [58].

To ensure that no double counting of events occurs between the matrix-element

multi-parton events and the parton showers, a version of the MLM matching proce-

dure was used [59]. In this procedure, the matrix element multi-parton events and

the parton showers are constrained to occupy different kinematical regions, separated

using the k⊥ jet measure:

d2(i, j) = ∆R2ij min(p2

T i, p2Tj)

d2(i, beam) = p2T i, (2.2)

where ∆R2ij = 2(cosh ∆η − cos ∆φ) [60]. Matrix-element events are generated with

some minimum cut-off d(i, j) = QMEmin. After showering, the partons are clustered into

jets using the kT jet algorithm with a QPSmin > QME

min. The event is then discarded unless

all resulting jets are matched to partons in the matrix-element event, d(parton, jet) <

QPSmin. For events from the highest multiplicity sample, extra jets softer than the softest

matrix-element parton are allowed. This procedure avoids double-counting jets, and

results in continuous and smooth differential distributions for all jet observables.

The matching parameters (QMEmin and QPS

min) should be chosen resonably far below

the factorization scale of the process. For gluino production, the parameters were:

QMEmin = 20 GeV and QPS

min = 30 GeV. (2.3)

The simulations were done using the CTEQ6L1 PDF [61] and with the renor-

malization and factorization scales set to the gluino mass. The cross sections were

rescaled to the next-to-leading-order (NLO) cross sections obtained using Prospino

2.0 [62].

Finally, we used PGS [63] for detector simulation, with a cone jet algorithm with


∆R = 0.5. As a check on this procedure, we compared our results to the signal point

given in [51] and found that they agreed to within 10%.

2.1.2 Backgrounds

The three dominant Standard Model backgrounds that contribute to the jets plus

missing energy searches are: W±/Z0 + jets, tt, and QCD. There are several smaller

sources of missing energy that include single top and di-boson production, but these

make up a very small fraction of the background and are not included in this study.

The W±/Z0 +nj and tt backgrounds were generated using MadGraph/MadEvent

and then showered and hadronized using PYTHIA. PGS was used to reconstruct

the jets. MLM matching was applied up to three jets for the W±/Z0 background,

with the parameters QMEmin = 10 GeV and QPS

min = 15 GeV. The top background

was matched up to two jets with QMEmin = 14 GeV and QPS

min = 20 GeV. Events

containing isolated leptons with pT ≥ 10 GeV were vetoed to reduce background

contributions from leptonically decaying W± bosons. To reject cases of ET6 from jet

energy mismeasurement, a lower bound of 90 and 50 was placed on the azimuthal

angle between ET6 and the first and second hardest jets, respectively. An acoplanarity

cut of < 165 was applied to the two hardest jets. Because the D06 analysis did not

veto hadronically decaying tau leptons, all taus were treated as jets in this study.

Simulation of the missing energy background from QCD is beyond the scope of

PYTHIA and PGS, and was therefore not done in this work. However, to avoid the re-

gions where jet and calorimeter mismeasurements become the dominant background,

a lower limit of ET6 > 100 GeV was imposed. Additionally, in the dijet analysis, the

azimuthal angle between the ET6 and any jet with pT ≥ 15 GeV and |η| ≤ 2.5 was

bounded from below by 40. This cut was not placed on the threejet or multijet

samples because of the large jet multiplicities in these cases.

For each of the W±/Z0 + nj and tt backgrounds, 500K events were generated.

The results reproduce the shape and scale of the ET6 and HT distributions published

by the D06 collaboration in [51] for 1fb−1. For the dijet case, where the most statistics

are available, the correspondence with the D06 result is ±20%. With the threejet


and multijet cuts, the result for the tt background is similar, while the W±/Z0 + nj

backgrounds reproduce the D06 result to within 30−40% for the threejet and multijet

cases. The increased uncertainty may result from insufficient statistics to fully popu-

late the tails of the ET6 and HT distributions. The PGS probability of losing a lepton

may also contribute to the relative uncertainties for the W±+nj background. Heavy

flavor jet contributions were found to contribute 2% to the W±/Z0 backgrounds,

which is well below the uncertainties that arise from not having NLO calculations for

these processes and from using PGS.

2.2 Projected Reach of Searches

A gluino search should have broad acceptances over a wide range of kinematical

parameter space; it should be sensitive to cases where the gluino and bino are nearly

degenerate, as well as cases where the gluino is far heavier than the bino. As already

discussed, the number of jets and ET6 depend strongly on the mass differerence between

the gluino and bino. Because the signal changes dramatically as the masses of the

gluino and bino are varied, it is necessary to design searches that are general, but

not closely tied to the kinematics. We divided events into four mutually exclusive

searches for ET6 plus 1j, 2j, 3j and 4+j, respectively. For convenience, we keep the

nj + ET6 classification fixed for all gluino and bino masses (see Table 4.1). These

selection criteria were modeled after those used in D06 ’s existing search [51].† These

exclusive searches can be statistically combined to provide stronger constraints.

Two cuts are placed on each search: HminT and ET6 min. In the D06 analysis, the HT

and ET6 cuts are constant for each search. The signal (as a function of the gluino and

bino masses) and Standard Model background are very sensitive to these cuts. To

maximize the discovery potential, these two cuts should be optimized for all gluino

and bino masses. For a given gluino and bino mass, the significance (S/√S +B) is

maximized over HminT and ET6 min in each nj+ET6 search. Due to the uncertainty in the

background calculations, the S/B was not allowed to drop beneath the conservative

†It should be noted, however, that the D06 searches are inclusive because each is designed to lookfor separate gluino/squark production modes (i.e., pp→ qq, qg, gg).

2.2. PROJECTED REACH OF SEARCHES 31

1j + ET6 2j + ET6 3j + ET6 4+j + ET6ET j1 ≥ 150 ≥ 35 ≥ 35 ≥ 35ET j2 < 35 ≥ 35 ≥ 35 ≥ 35ET j3 < 35 < 35 ≥ 35 ≥ 35ET j4 < 20 < 20 < 20 ≥ 20

Table 2.1: Summary of the selection criteria for the four non-overlapping searches.The two hardest jets are required to be central (|η| ≤ 0.8). All other jets must have|η| ≤ 2.5.

limit of S/B > 1. More aggressive bounds on S/B may also be considered; D06 , for

instance, claims a systematic uncertainty of O(30%) in their background measure-

ments [51]. The resulting 95% sensitivity plot using the optimized HT and ET6 cuts

is shown in Fig. 2.2. The corresponding inset illustrates the effect of varying the

production cross section.

For light and degenerate gluinos, the 1j + ET6 and 2j + ET6 searches both have

good sensitivity. In an intermediate region, the 2j + ET6 , 3j + ET6 and 4+j + ET6 all

cover with some success, but there appears to be a coverage gap where no search does

particularly well. If one does not impose a S/B requirement, a lot of the gap can be

covered, but background calculations are probably not sufficiently precise to probe

small S/B. For massive, non-degenerate gluinos, the 3j+ET6 and 4+j+ET6 both give

good sensitivity, with the 4+j + ET6 giving slightly larger statistical significance.

In the exclusion plot, the ET6 and HT cuts were optimized for each point in gluino-

bino parameter space. However, for gluino masses 200 GeV <∼ mg<∼ 350 GeV, where

the monojet search gives no contribution, we found that the exclusion region does

not markedly change if the following set of generic cuts are placed:

(HT , ET6 ) ≥ (150, 100)2j+ET6 ,

(150, 100)3j+ET6 , (200, 100)4+j+ET6 . (2.4)


Out[129]=

XX

100 200 300 400 5000

50

100

150

200

Gluino Mass !GeV"

BinoMass!GeV

"Out[112]=

100 200 300 400 5000

50

100

150

200

Gluino Mass !GeV"

BinoMass!GeV

"

Figure 2.2: The 95% gluino-bino exclusion curve for D06 at 4 fb−1 for S/B > 1. Thedashed line shows the corresponding exclusion region using D06 ’s non-optimized cuts.The masses allowed in the CMSSM are represented by the dotted line; the “X” marksthe current D06 limit on the gluino mass at 2.1 fb−1 (see text for details) [51]. Theinset shows the effect of scaling the production cross section for the case of S/B > 1.The solid lines show the exclusion region for σ/3 (bottom) and 3σ (top).

As a comparison, the cuts used in the D06 analysis are

(HT , ET6 ) ≥ (325, 225)2j+ET6 ,

(375, 175)3j+ET6 , (400, 100)4+j+ET6 . (2.5)

The lowered cuts provide better coverage for intermediate mass gluinos, as indicated

in Fig. 2.2. For mg . 200 GeV, we place tighter cuts on the monojet and dijet samples

than D06 does. While D06 technically has statistical significance in this region with

their existing cuts, their signal-to-background ratio is low. Because of the admitted

difficulties in calculating the Standard Model backgrounds, setting exclusions with

a low signal-to-background should not be done and fortunately can be avoided by

tightening the HT and ET6 cuts. Similarly, for larger gluino masses, the generic cuts

are no longer effective and it is necessary to use the optimized cuts, which are tighter

than D06 ’s.

2.3. CONCLUSIONS AND OUTLOOK 33

2.3 Conclusions and Outlook

In this paper, we describe the sensitivity that D06 has in searching for gluinos away

from the CMSSM hypothesis in jets + ET6 searches. It was assumed that the gluino

only decayed to two jets and a stable bino. However, many variants of this decay are

possible and the search presented here can be generalized accordingly.

One might, for example, consider the case where the gluino decays dominantly

to bottom quarks and heavy flavor tagging can be used advantageously. Cascade

decays are another important possibility. Decay chains have a significant effect upon

the searches because they convert missing energy into visible energy. In this case,

additional parameters, such as the intermediate particle masses and the relevant

branching ratios, must be considered. In the CMSSM, the branching ratio of the

gluino into the wino is roughly 80%. This is the dominant decay affecting the D06gluino mass bound in CMSSM parameter space (see Fig. 2). While this cascade

decay may be representative of many models that have gluino-like objects, the fixed

mass ratio and branching ratio are again artifacts of the CMSSM. A more thorough

examination of cascade decays should be considered.

In addition to alternate decay routes for the gluino, alternate production modes

are important when there are additional particles that are kinematically accessible.

In this paper, it was assumed that the squarks are kinematically inaccessible at the

Tevatron; however, if the squarks are accessible, gq and qq production channels could

lead to additional discovery possibilities. For instance, a gluino that is degenerate

with the bino could be produced with a significantly heavier squark. The squark’s

subsequent cascade decay to the bino will produce a great deal of visible energy in

the event and may be more visible than gluino pair production.

Finally, in the degenerate gluino region, it may be beneficial to use a mono-photon

search rather than a monojet search [64]. Preliminary estimates of the reach of the

mono-photon search show that it is not as effective as the monojet search. This is

likely due to the absence of final-state photon radiation from the gluinos. However,

it may be possible to better optimize the mono-photon search, because the Standard

Model backgrounds are easier to understand in this case.


Ultimately, a model-independent search for jets plus missing energy would be

ideal. We believe that our exclusive nj + ET6 searches, with results presented in an

exclusion plot as a function of HT and ET6 , would provide significant coverage for

these alternate channels [?]. This analysis should be carried forward to the LHC to

ensure that the searches discover all possible supersymmetric spectra. The general

philosophy of parameterizing the kinematics of the decay can be easily carried over.

The main changes are in redefining the HT and ET6 cuts, as well as the hard jet energy

scale. We expect a similar shape to the sensitivity curve seen in Fig. 2.2, but at higher

values for the gluino and bino masses. Therefore, it is unlikely that there will be a

gap in gluino-bino masses where neither the Tevatron nor the LHC has sensitivity.

Chapter 3

Model-Independent Jets + ET6

J. Alwall, M-P. Le, M. Lisanti, and J. G. Wacker, “Model-Independent Jets plus

Missing Energy Searches,” Phys. Rev. D 79, 015005 (2009).

One of the most promising signatures for new physics at hadron colliders are events

with jets and large missing transverse energy (ET6 ). These searches are very general

and cover a wide breadth of potential new theories beyond the Standard Model. Jets

+ ET6 searches pose a significant challenge, however, because the Standard Model

background is difficult to calculate in this purely hadronic state. The general nature

of the signature motivates performing a search that only requires calculating the Stan-

dard Model background. The challenge, then, is to minimize the risk of missing new

physics while still accounting for our limited understanding of the background. All

experimental searches of jets + ET6 at hadron colliders have been model-dependent,

attempting to be sensitive to specific models [49,50,51,52,53,65]. Initial studies for the

Large Hadron Collider (LHC) have been dominantly model-dependent [66,67,68,69].

In this article, we explore how modest modifications to the existing jets and ET6 studies

can allow them to be model-independent, broadening the reach of the experimental

results in constraining theoretical models.

Currently, jets plus ET6 searches at the Tevatron are based on the Minimal Su-

persymmetric Standard Model (MSSM) [46] and look for production of gluinos (g)

and squarks (q), the supersymmetric partners of gluons and quarks, respectively

35

36 CHAPTER 3. MODEL-INDEPENDENT JETS+ET6

[50, 51, 52]. These particles subsequently decay into the stable, lightest supersym-

metric particle (LSP), which is frequently the bino, the supersymmetric partner of

the photon. The MSSM contains hundreds of parameters and it is challenging to

place mass bounds in such a multi-parameter space. To make this tractable, the

CMSSM (or mSUGRA) ansatz has been used [11]. The CMSSM requires common

scalar masses (m0), gaugino masses (m 12), and trilinear scalar soft couplings (A0) at

the unification scale, in addition to electroweak symmetry breaking, gauge coupling

unification, and R-parity conservation. The entire particle spectrum is determined

by five parameters.

One important consequence of this theory is that the ratio of gaugino masses is

fixed at approximately mg : mfW : m eB ' 6 : 2 : 1, where W refers to the triplet of

winos (W±, W 0), the supersymmetric partners of the electroweak gauge bosons. Due

to the number of constraints in the CMSSM, the bino is the LSP throughout the

range of parameter space that the Tevatron has access to. Furthermore, due to the

renormalization group running of the squark masses, the squarks are never signifi-

cantly lighter than the gluino. Thus, the ratio in masses between the lightest colored

particle and the LSP is essentially fixed. The CMSSM is certainly not representative

of all supersymmetric models (see, for example, [70, 71, 72, 73, 74, 54]), let alone the

wider class of beyond the Standard Model theories that jets and ET6 searches should

have sensitivity to. Verifying that a jets and ET6 search has sensitivity to the CMSSM

does not mean that the search is sensitive to a more generic MSSM.

Existing searches for gluinos and squarks make strong assumptions about the

spectrum and it is unclear what the existing limits on squark-like and gluino-like

particles are. Because squarks have electric charge, LEP can place limits of 92 GeV

on their mass [75]; however, gluinos do not couple to either the photon or Z0 and so

limits from LEP2 are not strong. Currently, the tightest model-independent bound

on color octet fermions (such as gluinos) comes from thrust data at ALEPH [76] and

OPAL [77]. New colored particles should contribute at loop-level to the running of the

strong coupling constant αs. To date, the theoretical uncertainties in the value of αs

have decreased its sensitivity to new particle thresholds. Advances in Soft-Collinear

Effective Theory, however, have been used to significantly reduce the uncertainties in

37

αs from LEP data. The current bound on color octet fermions is 51.0 GeV at 95%

confidence [56]; no limit can be set for scalar color octets.

There is no unique leading candidate for physics beyond the Standard Model;

therefore, searches for new physics need to be performed in many different channels.

Ideally, one should perform totally model-independent searches that only employ the

Standard Model production cross section for physics with the desired channels and

the correct kinematics. The goal is to be sensitive to a large number of different

models at the same time so that effort is not wasted in excluding the same parts of

Standard Model phase space multiple times.

Some progress on experimental model-independent searches has been made. In an

ambitious program, the CDF Collaboration at the Tevatron has looked at all possible

new channels simultaneously (i.e., Vista, Sleuth, Bumphunter) [78, 79, 80]; however,

these searches have some drawbacks over more traditional, channel-specific searches.

The most important drawback is that it is difficult, in the absence of a discovery, to

determine what parts of a given model’s parameter space are excluded.

On the theoretical front, MARMOSET [81] is a hybrid philosophy that attempts

to bridge model-independent and model-dependent searches with the use of On-Shell

Effective Theories (OSETs). OSETs parameterize the most experimentally relevant

details of a given model – i.e., the particle content, the masses of the particles, and

the branching ratios of the decays. By using an on-shell effective theory, it is possible

to easily search through all experimentally relevant parameters quickly. The on-shell

approximation is not applicable in all situations, but OSETs can still give a rough

idea of where new physics lies.

In this article, we will explore the discovery potential of jets and missing energy

channels. In previous work [82], we presented a simple effective field theory that can

be used to set limits on the most relevant parameters for jets and missing energy

searches: the masses of the particles. While this approach seems obvious, existing

searches at hadron colliders (Tevatron Run II, Tevatron Run I, UA2, UA1) are based

on CMSSM-parameterized supersymmetry breaking. The previous paper studied how

varying the decay kinematics changed the sensitivity of the searches and pointed out

regions of parameter space where sensitivity is particularly low due to kinematics.


However, this gluino-bino module was still a model-dependent analysis in that it

assumed pair-production of a new colored fermionic particle directly decaying to a

fermionic LSP.

This paper will extend the analysis in two ways. First, we propose a completely

model-independent analysis for jets and missing energy searches. This approach only

requires knowledge of the Standard Model and places limits on differential cross sec-

tions, from which it is possible to set model-dependent limits. In the second portion

of the paper, we use this approach to extend our previous analysis of a directly de-

caying colored particle to contain a single-step cascade and study how this altered

spectrum affects the final limits on the gluino’s mass.

3.1 Overview of Models

Before continuing with the main theme of the article, let us take a moment to describe

the class of models that jets + ET6 searches are sensitive to. There are two general

classes of particle spectra that will be covered by such searches, each of which has a

stable neutral particle at the bottom of the spectrum. Typically, the stability of these

neutral particles is protected by a discrete symmetry (e.g., R-parity, T-parity, or KK-

parity) and, consequently, these particles are good candidates for the dark matter. In

one class of models, the theory contains a new colored particle that cascade decays

into the dark matter. In the other class, new electroweak gauge bosons are produced.

The dark matter particle may either be produced along with the new bosons, or may

be the final step in their decays.

The first class can be thought of as being generally SUSY-like where the lightest

colored particle is dominantly produced through the Standard Model’s strong force.

The lightest colored particle then cascade decays down to the stable, neutral particle

at the bottom of that sector. These cascades will either be lepton-poor or lepton-

rich. Lepton-poor cascades occur when there is no state accessible in the cascades that

have explicit lepton number (e.g., sleptons) and frequently occur when the cascades

are mediated by W±, Z0, or Higgs bosons. A simple supersymmetric example of

a lepton-poor cascade decay is a theory where the scalar masses are made heavy

3.1. OVERVIEW OF MODELS 39

and only gauginos and Higgsinos are available in the decay chains. This occurs, for

instance, in PeV supersymmetry models, where the scalars are around 1000 TeV

and the fermions of the MSSM are in the 100 GeV to 1 TeV range. Producing the

color-neutral states of such a theory is difficult at hadron machines; consequently, the

production of new particles will occur primarily through the decay of the gluino.

One potential cascade decay of the gluino, which will be considered in further

detail in the second half of the paper, is

g → q1q2W → q1q2q3q4B. (3.1)

In this cascade, the W decays directly into the B and a W±, Z0 boson, which sub-

sequently decays to two jets. This single-step decay is the dominant cascade if the

gaugino masses are unified at high energies; in this case, the branching ratio of the

gluino into the wino is ∼ 80%. While these cascade decays are to some degree rep-

resentative, the precise mass ratio of mg : mfW : m eB makes a significant difference

in the searches. In the limit where mfW → m eB the energy from q3 and q4 is small,

while if mfW → mg the jets from q1 and q2 are soft. If mfW > mg, this cascade is

forbidden. Interestingly, spectra with unified gaugino masses are the most difficult to

see because all four jets are fairly hard and diminish the missing energy in the event

in comparison to the direct decay of the gluino, g → q1q2B.

Leptons from the decay of the W±, Z0 boson can be used in the analysis as well

(see Sec. 3.4.5). However, jets + ET6 + lepton studies are better suited for lepton-rich

cascades. The addition of leptons to the searches makes the experimental systematics

easier to control and improves trigger efficiencies. Not all spectra of new physics can

be probed with these types of searches, though, and they are thus complimentary to

the jets + ET6 search.

Other cascades may produce a greater number of jets as compared to (3.1). In

NMSSM theories where there is a new singlino at the bottom of the spectrum [83], it

is possible to have cascade decays that start with the gluino, go to wino plus two jets,

then bino plus two additional jets, and conclude with the singlino plus two more jets.

The additional step in the decay process further diminishes the amount of missing


energy in typical events, resulting in reduced limits on spectra. Other models, such as

Universal Extra Dimensions (UEDs) [12] and Little Higgs models with T-parity [48]

also have new colored particles that subsequently cascade decay. The details of the

exact spectra can alter the signal significantly as jets can become soft and missing

energy is turned into visible energy.

It is also possible that new electroweak gauge bosons are produced, which then

cascade decay, producing jets before ending with the neutral stable particle. Little

Higgs models with T-parity are one such example. In such models, the new heavy

bosons W±H and Z0

H are produced through s-channel processes. The W±H can decay

to the W± and the dark matter AH , while the Z0H can decay to the AH and higgs.

It is also possible to produce the W±H directly with the AH through an s-channel W±

boson. This vertex, however, is suppressed in comparison to the other two.

3.2 Proposed Analysis Strategy

At the Tevatron, the jets + ET6 channel is divided into four separate searches (monojet,

dijet, threejet, and multijet), with each search defined by jet cuts O(30 GeV). Cuts

on the missing transverse energy and total visible energy∗ HT of each event take

place during the final round of selection cuts. The ET6 and HT cuts are optimized for

“representative” points in CMSSM parameter space for each of the (inclusive) 1j−4+j

searches. However, these ET6 and HT cuts may not be appropriate for theories other

than the CMSSM. Indeed, considering the full range of kinematically allowed phase

space means accounting for many combinations of missing and visible energy. A set

of static cuts on ET6 and HT is overly-restrictive and excludes regions of phase space

that are kinematically allowed.

This is explicitly illustrated in Fig. 3.1, which shows the ET6 distribution of a dijet

sample passed through two different sets of ET6 and HT cuts. The signal, a 210 GeV

gluino directly decaying (i.e., no cascade) to a 100 GeV bino, is shown in white and

the Standard Model background, in gray. The plot on the left shows the events that

survive a 300 GeV HT cut. While the HT cut significantly reduces the background,

∗The total visible energy HT is defined as the scalar sum of the transverse momenta of each jet.

3.2. PROPOSED ANALYSIS STRATEGY 41

Figure 3.1: Comparison of D06 cuts and optimized cuts for a sample dijet signal formg = 210 GeV and m eB = 100 GeV. Background distribution is shown in gray andsignal distribution in white. (Left) Using the D06 cuts HT ≥ 300 GeV and ET6 ≥225 GeV (Right) Using the more optimal cuts HT ≥ 150 GeV and ET6 ≥ 100 GeV.The optimized cuts allow us to probe regions with larger S/B.

it also destroys the signal above the ET6 cut of 225 GeV. These cuts were used in the

D06 dijet search; they are optimized for a ∼ 400 GeV gluino, but are clearly not ideal

for the signal point shown here. A more optimal choice of cuts is shown on the right.

While the lower HT cut of 150 GeV keeps more background, it also keeps enough

signal for a reasonable S/B ratio at low ET6 . Therefore, with a ET6 cut of 100 GeV,

exclusion limits on this point in parameter space can be placed.

A model-independent search should have broad acceptances over a wide range

of kinematical parameter space. Ideally, searches should be sensitive to all possible

kinematics by considering all appropriate ET6 and HT cuts. This can be effectively

done by plotting the differential cross section as a function of ET6 and HT ,

d2σ

dHTdET6 ∆HT∆ET6 . (3.2)

In this case, the results of a search would be summarized in a grid, where each box

contains the measured cross section within a particular interval of ET6 and HT .

As an example, the differential cross section grids for exclusive 1j − 4+j searches

(see Table 4.1 for jet selection criteria) at the Tevatron are shown in Table 3.1.

The grids are made for the Standard Model background, which include W± + nj,


Z0 + nj, and tt+ nj. The QCD background was not simulated; we expect the QCD

contributions to be important for points in the lowest ET6 bin. For details concerning

the Monte Carlo generation of the backgrounds, see Sec. 3.3.2.

From these results, it is straightforward to obtain limits on the differential cross

section for any new physics signal. Consider a specific differential cross section mea-

surement that measures Nm events in an experiment. The Standard Model predicts

B events, while some specific theory predicts B+S events, where S is the number of

signal events.

The probability of measuring n events is given by the Poisson distribution with

mean µ = B + S. The mean µ is excluded to 84% such that

e−µexcl

Nm∑n=0

(µexcl)n

n!≤ 0.16. (3.3)

The solution to this equation gives the excluded number of signal events

Sexcl(Nm, B) = µexcl(Nm)−B. (3.4)

The expected limit on the signal is then given by

〈Sexcl(B)〉 =∞∑

Nm=0

Sexcl(Nm, B)e−BBNm

Nm!. (3.5)

In the limit of large B, the probability distribution approaches a Gaussian and we

expect that

limB→∞〈Sexcl(B)〉 =

√B. (3.6)

In the limit of small B, we expect that

limB→0〈Sexcl(B)〉 = − ln(0.16) ≈ 1.8. (3.7)

The right column of Table 3.1 shows the limit on the differential cross section for

any new physics process. When presented in this fashion, the experimental limits are

model-independent and versatile. With these limits on the differential cross section,

3.2. PROPOSED ANALYSIS STRATEGY 43

anyone can compute the cross section for a specific model and make exclusion plots

using just the signal limits shown in Table 3.1. For the comparison to be reliable, the

detector simulator should be properly calibrated.

In addition to the statistical uncertainty, systematic uncertainties can also be

important. Unlike the statistical uncertainties, the systematic uncertainties can be

correlated with each other. One important theoretical uncertainty is the higher-order

QCD correction to the backgrounds. These QCD uncertainties result in K-factors

that change the normalization of the background, but do not significantly alter the

background shapes with respect to HT and ET6 . Because this uncertainty is highly

correlated between different differential cross section measurements, treating the un-

certainty as uncorrelated reduces the sensitivity of the searches. If a signal changes the

shape of the differential cross section, e.g. causing a peak in the distribution, higher

order corrections would be unlikely to explain it. To make full use of the independent

differential cross section measurements, a complete error correlation matrix should

be used. In practice, because the backgrounds are steeply falling with respect to HT

and ET6 , assigning an uncorrelated systematic uncertainty does not significantly hurt

the resolving power of the experiment. In Table 3.1, we have assigned a systematic

uncertainty of εsys = 50% to each measurement, which should be added in quadrature

to the statistical uncertainty. This roughly corresponds to the requirement that the

total signal to background ratio is one.

The reduced chi-squared χ2N value for N measurements is

χ2N =

N∑j=1

S2j

(SLj)2 + (εsys ×Bj)2× 1

N, (3.8)

where Sj is the number of signal events and Bj is the number of background events

in the jth box of the grid. The statistical error SLj and the systematic error εsys×Bj

is read off from Table 3.1. In order to have a useful significance limit, it is necessary

to only include measurements where there is an expectation of statistical significance;

otherwise, the χ2N is diluted by a large number of irrelevant measurements. There

is no canonical way of dealing with this elementary statistical question, although

the CLS method is the most commonly used [84, 85]. In this article, we take a very


Simulated Background Signal Limits

Mon

oje

tD

ijet

Thre

ejet

Mult

ijet

Table 3.1: Differential cross section (in fb) for the Standard Model background isshown in the left column for exclusive 1j − 4+j searches. The expected signal sensi-tivity at 84% confidence is shown on the right (in fb). The statistical error is shownto the left of the ⊕ and the systematic error is on the right. For purposes of illus-tration, we assume a 50% systematic error on the background. The gray boxes arekinematically forbidden. These results are for 4 fb−1 luminosity at the Tevatron.


simple approach. If the expected significance for a single measurement is greater than

a critical number, Scrit, it is included in the χ2N , otherwise it is not. We tried several

values of Scrit and the experimental sensitivity to different theories was not altered

by the different choices. We chose Scrit = 0.5 for the exclusion plots. This method

does not maximize the reach in all cases, but because there are usually just a few

measurements that give large significance, we are relatively insensitive to the exact

statistical procedure.

In what follows, we will apply the general philosophy presented here to find the

exclusion region for gluinos that are pair-produced at the Tevatron.† In Sec. 3.3, we

will explain how the signal and background events have been generated. In Sec. 3.4,

we will show how mass bounds can be placed on the gluino and bino masses using

the proposed model-independent analysis and will discuss the challenges presented

by cascade decays. We conclude in Sec. 3.5.

3.3 Event Generation

3.3.1 Signal

In this section, we discuss the generation of signal events for the gluino cascade de-

cay shown in (3.1). The experimental signatures of this decay chain are determined

primarily by the spectrum of particle masses. In particular, the mass splittings de-

termine how much energy goes into the jets as opposed to the bino - i.e., the ratio

of the visible energy to missing transverse energy. Events with large HT and ET6 will

be the easiest to detect; this is expected, for example, when a heavy gluino decays

into a wino that is nearly degenerate with either the gluino or the bino. The reach of

the searches is degraded, however, when the wino is included as an intermediate state

in the decay chain. When the jets from the cascade decay are all hard, the missing

energy is significantly smaller than what it would be for the direct decay case. Picking

out signals with small missing transverse energy is challenging because they push us

†Throughout this article, “gluino” refers to a color octet fermion, “wino” to a charged SU(2)fermion, and “bino” to a neutral singlet. These names imply nothing more than a particle’s quantumnumbers.


closer to a region where the dominant background is coming from QCD and is poorly

understood. This happens, in particular, when the mass splitting between the gluino

and bino is large and the wino mass is sufficiently separated from both. When the

wino is nearly degenerate with either the gluino or the bino, then we expect to see

2 hard jets and 2 soft jets from the decay. This case begins to resemble the direct

decay scenario; there is more missing energy and, therefore, the signal is easier to see.

It is particularly challenging to probe regions of parameter space where the gluino

is nearly degenerate with the bino. For this case, even in the light-gluino region

(mg . 200 GeV), the benefit of the high production cross section for the gluinos is

overwhelmed by the small missing transverse momentum in each event; the jets in

these events are soft and the pT of the two binos approximately cancel when summed

together [82]. Even if the gluinos are produced at large invariant mass, the situation

is not markedly improved; in this case, the jets from each gluino are collinear and

aligned with the ET6 . Such events are easily mistaken as QCD events and eliminated

by the cuts that are implemented to reduce the QCD backgrounds.

The inclusion of hard initial-state jets significantly increases the exclusion reach

in this degenerate region of parameter space. The initial-state radiation boosts the

gluinos in the same direction, decreasing the angle between them, which in turn,

enhances the ET6 . Therefore, ISR jets allow us to capitalize on the high production

cross section of light gluinos to set bounds on their masses.

To properly account for initial-state radiation (ISR) and final-state radiation

(FSR), MadGraph/MadEvent [57] was used to generate events of the form

pp→ gg +Nj, (3.9)

where N = 0, 1, 2 is the multiplicity of QCD jets. Pythia 6.4 [58] was used for parton

showering and hadronization. Properly counting the number of events after parton

showering requires some care. In general, an (n+ 1)-jet event can be obtained in two

ways: by a (n+1) hard matrix-element, or by hard radiation emitted from an n-parton

event during showering. It is important to understand which of the two mechanisms

generates the (n+ 1)-jet final state to ensure that events are not double-counted.


In this article, a version of the so-called MLM matching procedure implemented

in MadGraph/MadEvent and Pythia [59] was used for properly merging the different

parton multiplicity samples. This matching has been implemented both for Standard

Model production and for beyond the Standard Model processes. In this procedure,

parton-level events are generated with a matrix element generator with a minimum

distance between partons characterized by the k⊥ jet measure:

d2(i, j) = ∆R2ij min(p2

T i, p2Tj)

d2(i, beam) = p2T i, (3.10)

where ∆R2ij = 2[cosh(∆η) − cos(∆φ)] [60]. The event is clustered using the kT

clustering algorithm, allowing only for clusterings consistent with diagrams in the

matrix element, which can be done since MadGraph generates all diagrams for the

process. The d2 values for the different clustered vertices are then used as scales in the

αs value corresponding to that vertex, i.e. the event weight is multiplied by∏

iαs(d2i )

αs(µ2R)

,

where the product is over the clustered vertices i. This is done in order to treat

radiation modeled by the matrix element as similarly as possible to that modeled

by the parton shower, as well as to correctly include a tower of next-to-leading log

terms. A minimum cutoff d(i, j) > QMEmin is placed on all the matrix-element multi-

parton events.

After showering, the partons are clustered into jets using the standard k⊥ al-

gorithm. Then, the jet closest to the hardest parton in (η, φ)-space is selected.

If the separation between the jet and parton is within some maximum distance,

d(parton, jet) < QPSmin, the jet is considered matched. The process is repeated for all

other jets in the event. In this way, each jet is matched to the parton it originated

from before showering. If an event contains unmatched jets, it is discarded, unless

it is the highest multiplicity sample. In this case, events with additional jets are

kept, provided the additional jets are softer than the softest parton, since there is no

higher-multiplicity matrix element that can produce such events. The matching pro-

cedure ensures that jets are not double-counted between different parton multiplicity

matrix elements, and should furthermore give smooth differential distributions for all


1!Differential Jet Rate 0

0 50 100 150 200 250

pb

/bin

-210

-110

1

10

Matched sum + 0-jet sampleg~g~

+ 1-jet sampleg~g~

+ 2-jet sampleg~g~

+ Pythia (unmatched)g~g~min

PSQ

Figure 3.2: Differential 0→ 1 jet rate for a matched sample of light gluino production.The full black curve shows the matched distribution, and the broken curves showthe contributions from different matrix element parton multiplicity samples. Thematching scale QPS

min is marked by the dashed line. The full red curve shows the resultusing Pythia only.

jet observables. The results should not be sensitive to the particular values of the

matching parameters, as long as they are chosen in a region where the parton shower

is a valid description. Typically, the matching parameters should be on the order of

the jet cuts employed and be far below the factorization scale of the process. For the

gluino production, the parameters were

QMEmin = 20 GeV QPS

min = 30 GeV. (3.11)

Figure 3.2 shows the differential jet rate going from zero to one jets D(1j → 0j),

which is the maximum k⊥ distance for which a 1j event is characterized as a 0j

event. Below QPSmin, all jets come from parton showering of the 0j multiplicity sample.

Above QPSmin, the jets come from initial-state radiation. The main contributions in this

region are from the 1j and 2j multiplicity samples. The sum of all the multiplicity

samples is a smooth distribution, eliminating double counting between the different


samples.

The simulations were done using the CTEQ6L1 PDF and with the renormalization

and factorization scales set to the gluino mass [61]. The matched cross-sections were

rescaled to the next-to-leading-order (NLO) cross sections obtained using Prospino

2.0. PGS was used for detector simulation [63], with jets being clustered according to

the cone algorithm, with ∆R = 0.5. As a check on this procedure, we compared our

results to the signal point given in [51] and found that they agreed to within 10%.

To emphasize the importance of properly accounting for initial-state radiation us-

ing matching, Fig. 3.3 compares the pT distribution for the hardest jet in a matched

(left) and unmatched (right) dijet sample for a 150 GeV gluino directly decaying to

a bino. The colors indicate the contributions from the different multiplicity samples:

0j (orange), 1j (blue), and 2j (cyan). When the gluino-bino mass splitting is large

enough to produce hard jets (top row), the 0j multiplicity sample is the main con-

tributor. ISR is not important in this case and there is little difference between the

matched and unmatched plots. The bottom row shows the results for a 130 GeV bino

that is nearly degenerate with the gluino. In this case, only soft jets are produced

in the decay and hard ISR jets are critical for having events pass the dijet cuts. In-

deed, we see the dominance of the 2j multiplicity sample in the histogram of matched

events. When ISR is important, the unmatched sample is clearly inadequate, with

nearly 60% fewer events than the matched sample.

3.3.2 Backgrounds

The dominant backgrounds for jets + ET6 searches are W±/Z0 + jets, tt, and QCD.

Additional background contributions come from single top and di-boson production

(WW, WZ, ZZ), but these contributions are sub-dominant, so we do not consider

them here. The missing transverse energy comes from Z0 → νν and W± → l±ν,

where the W± boson is produced directly or from the top quark. To reduce the W±

background, a veto was placed on isolated leptons with pT ≥ 10 GeV. However, these

cuts do not completely eliminate the W± background because it is possible to miss

either the electron or muon (or misidentify them). It should be noted that muon


isolation cuts were not placed by PGS, but were applied by our analysis software. If

the muon failed the isolation cut, then it was removed from the record and its four-

momentum was added to that of the nearest jet. Additionally, the W± can decay

into a hadronic τ , which is identified as a jet. Because the D06 analysis did not veto

on hadronic taus, we have treated all taus as jets in this study.

QCD backgrounds can provide a significant source of low missing energy events,

but are challenging to simulate. The backgrounds can arise from jet energy mismea-

surement due to poorly instrumented regions of the detector (i.e., dead/hot calorime-

ter cells, jet punch-through, etc.). Additionally, there are many theoretical uncer-

tainties - for example, in the PDFs, matrix elements, renormalisation, and factorisa-

tion/matching scales - that factor in the Monte Carlo simulations of the backgrounds.

For heavy-flavor jets, there is the additional ET6 contribution coming from leptonic

decays of the b-quarks. It is possible, for instance, to have the b-quark decay into a

lepton and a neutrino, with the neutrino taking away a good portion of the b-quark’s

energy. Simulation of the QCD background is beyond the scope of Pythia and PGS

and was not attempted in this work. To account for the QCD background, we imposed

a tight lower bound on the ET6 of 100 GeV. Jet energy mismeasurement was accounted

for by placing a lower bound of 90 and 50 on the azimuthal angle between the ET6and the first and second hardest jets, respectively. In addition, an acoplanarity cut of

165 was placed between the two hardest jets. For the dijet case, the azimuthal angle

between the ET6 and any jet with pT ≥ 15 GeV and |η| ≤ 2.5 was bounded from below

by 40. This cut was not placed on the threejet or multijet searches because of the

greater jet multiplicity in these cases. The W±/Z0 +nj and tt backgrounds were gen-

erated using MadGraph/MadEvent, with showering and hadronization in PYTHIA.

PGS was again used as the detector simulator for jet clustering. The W±/Z0 back-

grounds were matched up to 3 jets using the MLM matching procedure discussed in

the previous section, with matching parameters QMEmin = 10 GeV and QME

min = 15 GeV.

The tt backgrounds were matched up to 2 jets with parameters QMEmin = 14 GeV and

QMEmin = 20 GeV. For each of the separate backgrounds, 500K events were generated.

The results approximately reproduce the shape and scale of the ET6 and HT distri-

butions published by the D06 collaboration for 1 fb−1 [51]. In the dijet case, our

3.4. GLUINO EXCLUSION LIMITS 51

Matched Unmatched40

GeV

Bin

o13

0G

eVB

ino

Figure 3.3: Comparison of matched and unmatched events for a dijet sample of150 GeV gluinos directly decaying into 40 GeV (top) and 130 GeV (bottom) binos.The pT of the hardest jet is plotted in the histograms (1 fb−1 luminosity). Matchingis very important in the degenerate case when the contribution from initial stateradiation is critical. The different colors indicate the contributions from 0j (orange),1j (blue), and 2j (cyan).

results correspond to those of D06 within ±20%. The correspondence is similar for

the tt backgrounds in the threejet and multijet cases. For the W±/Z0 backgrounds,

the correspondence is within ±30 − 40%. It is possible that this discrepancy is due

to difficulties to fully populate the tails of the ET6 and HT distributions with good

statistics. In the case of the W± background, the modeling of the lepton detection

efficiency in PGS might also play a role. Heavy flavor jet contributions were found to

contribute 2% to the W±/Z0 backgrounds, which is well below the uncertainties that

arise from not having NLO calculations for these processes and from using PGS.


Sample Model

Mon

oje

t

Dij

et

Thre

ejet

Mult

ijet

Figure 3.4: Differential cross section (in fb) for the monojet, dijet, threejet, andmultijet samples of a theoretical model spectrum with a 340 GeV gluino decayingdirectly into a 100 GeV bino (4 fb−1). Some boxes show significant deviation fromthe signal limits shown in Table II: green indicates 0.5 < χi ≤ 2, blue indicates2 < χi ≤ 3, and red indicates χi > 3. All boxes with χi > 1/2 are included in thecalculation of the total χ2 value.

3.4 Gluino Exclusion Limits

3.4.1 No Cascade Decays

For the remainder of the paper, we will discuss how model-independent jets + ET6searches can be used to set limits on the parameters in a particular theory. We will

focus specifically on the case of pair-produced gluinos at the Tevatron and begin by

considering the simplified scenario of a direct decay to the bino. The expected number

of jets depends on the relative mass difference between the gluino and bino. When

the mass difference is small, the decay jets are very soft and initial-state radiation is

important; in this limit, the monojet search is best. When the mass difference is large,

the decay jets are hard and well-defined, so the multijet search is most effective. The

dijet and threejet searches are important in the transition between these two limits.

As an example, let us consider the model spectrum with a 340 GeV gluino decaying

directly into a 100 GeV bino. In this case, the gluino is heavy and its mass difference


Out[27]=

XX

100 200 300 400 5000

50

100

150

Gluino Mass !GeV"

BinoMass!GeV

"

Figure 3.5: The 95% exclusion region for D06 at 4 fb−1 assuming 50% systematic erroron background. The exclusion region for a directly decaying gluino is shown in lightblue; the worst case scenario for the cascade decay is shown in dark blue. The dashedline represents the CMSSM points and the “X” is the current D06 exclusion limit at2 fb−1.

with the bino is relatively large, so we expect the multijet search to be most effective.

Table 3.4 shows the differential cross section grids for the 1-4+ jet searches for this

simulated signal point. The colors indicate the significance of the signal over the

limits presented in Table II; the multijet search has the strongest excesses.

Previously [82], we obtained exclusion limits by optimizing the ET6 and HT cuts,

which involves simulating each mass point beforehand to determine which cuts are

most appropriate. This is effectively like dealing with a 1 × 1 grid, for which a

95% exclusion corresponds to χ2 = 4. The approach considered here considers the

significance of all such cuts, and only requires that a single n × n differential cross

section grid be produced for each search.

Fig. 3.5 shows the 95% exclusion limit for directly decaying gluinos at 4 fb−1

luminosity and 50% systematic uncertainty on the background. The results show

that such gluinos are completely excluded for masses below ∼ 130 GeV.


Figure 3.6: 95% exclusion region (purple) for a 240 GeV gluino decaying into a binothrough a wino. The dashed line is mfW = m eB +O(mZ0). The black dot at (m eB,mfW )= (60, 160), is the minimum bino mass for which a 240 GeV gluino is excluded for allwino masses. The inset shows the one-step cascade considered in the paper.

3.4.2 Cascade Decays

In this section, we will discuss the exclusion limits for the decay chain illustrated in

the inset of Fig. 3.6. In general, cascade decays are more challenging to see because

they convert missing energy to visible energy. The number of jets per event is greater

for cascading gluinos than directly decaying ones and the spectrum of jet energies

depends on the ratio of gaugino masses. When mg ∼ mfW , two hard jets are produced

in the decay of the wino to the bino. In the opposite limit, when mfW ∼ m eB, two hard

jets are produced in the decay of the gluino to the wino. When mg < mfW < m eB, four

fairly hard jets are produced, diminishing the ET6 and making this region of parameter

space the most challenging to see. In particular, the most difficult region to detect is

when

mfW = m eB +O(mZ0). (3.12)

In the region of parameter space, where mfW ∼ m eB, the jets from the wino to bino

decay become harder as the gauge bosons go on-shell.


Fig. 3.6 shows the values of mfW and m eB that are excluded up to 95% confidence

for a 240 GeV gluino (shaded region). The dark black dot, which represents the

minimum bino mass for which a 240 GeV gluino is excluded for all wino masses, falls

close to Eq. 3.12 (the dotted red line).

The exclusion region in Fig. 3.6 is not symmetric about the line mfW = m eB +

O(mZ0). The asymmetry is a result of the hard lepton cuts. When the gluino and

wino are nearly degenerate, the leptons from the gauge boson decays are energetic,

and these events are eliminated by the tight lepton cuts, reducing the significance

below the confidence limit. In the opposite limit, when the wino and bino are nearly

degenerate, much less energy is transfered to the leptons and fewer signal events are

cut. Additionally, the jets produced in this case are color octets and give rise to a

greater number of soft jets, as compared to the singlet jets emitted in the gauge boson

decays. The presence of many soft jets may decrease the lepton detection efficiency;

as a result, it may be that even fewer events than expected are being cut.

Figure 3.5 compares the 95% exclusion region for the cascade decay with that for

the direct decay case. The “worst-possible” cascade scenario is plotted; that is, it is

the maximum bino mass for which all wino masses are excluded. For the one-step

cascade considered here, gluinos are completely excluded up to masses of ∼ 125 GeV.

3.4.3 t-channel squarks

Thus far, it has been assumed that the squarks are heavy enough that they do not

affect the production cross section of gluinos. If the squarks are not completely decou-

pled, they can contribute to t-channel diagrams in gluino pair-production. Figure 3.7

shows the production cross section for a 120 GeV (red), 240 GeV (blue), and 360 GeV

(green) gluino, as a function of squark mass. When only one squark is light (and all

the others are ∼ 4−5 TeV), the production cross section is unaffected. However, when

the squark masses are brought down close to the gluino mass, the production cross

section decreases by as much as ∼ 25%, 60%, and 75% for 120, 240, and 360 GeV

gluinos, respectively. A reduction in the production cross section alters the exclusion

region in the gluino-bino mass plane; while the overall shape of the exclusion region


Out[27]=

1.0000.5000.1000.0500.0100.0050.001

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

m2gluino!m2

squark

!!! max

Figure 3.7: Gluino production cross section as a function of squark mass: (red)mg = 120 GeV, (blue) mg = 240 GeV, and (green) mg = 360 GeV.

remains the same, its size scales with the production cross section [82].

It is worthwhile to note, however, that while the inclusion of squarks reduces

the exclusion region for pair-produced gluinos by decreasing the production cross

section, it also provides alternate discovery channels through gq or qq production.

For example, if a gluino and squark are produced, with the gluino nearly degenerate

with the bino, the subsequent decay of the squark will produce more visible energy

than the gluino decay, thereby making the event more visible.

3.4.4 Monophoton Search

Initial-state QCD radiation is important for gaining sensitivity to degenerate gluinos.

Here, we will consider whether initial-state photon radiation may also be useful in

the degenerate limit. Such events are characterized by small ET6 and a hard photon.

The main benefit of the monophoton search is that the Standard Model back-

grounds are better understood; unlike the monojet case, QCD is no longer an im-

portant background. Instead, the primary backgrounds come from processes such as

Z0(→ νν) + γ, which is irreducible, and W± → e±ν where the electron is mistaken


as a photon or W±(→ l±ν) +γ, where the lepton is not detected. Other backgrounds

may come from W±/Z0 + jet, where the jet is misidentified as a photon, or situations

where muons or cosmic rays produce hard photons in the detector.

The D06 Collaboration recently published results for their monophoton study,

which searched for a Kaluza-Klein graviton produced along with a photon [64]. To

reduce the Standard Model background, they required all events to have one photon

with pT > 90 GeV and ET6 > 70 GeV. Events with muons or jets with pT > 15 GeV

were rejected. They estimate the total number of background events to be 22.4±2.5.

To investigate the sensitivity of monophoton searches to degenerate spectra, we

consider several points and compared them against D06 ’s background measurements.

We considered several benchmark values for gluino and bino masses and did a simple

cuts-based comparison between the monophoton search and an optimized monojet

search. For example, Figure 3.6 shows that the monojet search safely excludes the

case of a 140 GeV gluino and 130 GeV bino. A monophoton search (with the cuts

used in the D06 analysis) gives S/B = 0.48 and S/√B = 2.3 for this mass point; thus

the monophoton search is roughly as sensitive but has a lower S/B value. Similarly,

a 120 GeV gluino and 100 GeV bino is safely excluded by the monojet search, but

the monophoton search only gives S/B = 0.39 and S/√B = 1.86.

There are several reasons why the monophoton search is not as successful as the

monojet one. In the degenerate gluino region, the possibility of getting jets with

a pT above the 15 GeV threshold is significant (even though the mass difference is

O(10 GeV)) because the gluinos are boosted. The monophoton search vetoes many

events with such boosted decay jets. In addition, getting photon ISR is much more

difficult than getting QCD ISR for several reasons - most importantly, because αEM αs and because one is insensitive to the gluon-induced processes that contribute to

the cross section. Despite these challenges, the significance of the monophoton search

could still increase sensitivity. The monophoton does not fare significantly more

poorly than the monojet one with the current set of cuts. Thus, it is possible that a

more optimal set of cuts may increase the effectiveness of the search, especially given

that the backgrounds are better understood in this case. Finally, the above estimates

do not account for the photon detection rate in PGS, which may be different from


that used by D06 ’s full detector simulator, from which the background estimates were

taken.

3.4.5 Leptons

In this section, we address whether leptons from cascades can be used to augment the

sensitivity of jets + ET6 searches. In the gluino cascade decay considered in this paper,

it is possible to get leptons from the W± and Z0 boson decays. The 10 GeV lepton

veto, however, eliminates most of these events. The exclusion limit for the gluino

decay discussed in Sec. 3.4.2 is not improved by removing the lepton veto; most of

the irreducible backgrounds (W± + nj and tt+ nj) have a lepton and dominate over

the signal when the veto is removed. The exclusion limit is not improved even if we

require all events to have a certain number of leptons, or place cuts on lepton pT .

The question still remains as to whether there is any region in parameter space

where the jets + ET6 study places no exclusion, but a jets + ET6 + lepton study does.

The lepton signal is useful for light gluinos (. 250 GeV) that are nearly degenerate

with the wino. The signal point, a 210 GeV gluino decaying to a 50 GeV bino through

a 170 GeV wino, is not excluded by the ordinary jets + ET6 analysis. We find here,

though, that it has a significance‡ of ' 4.4 for a pT cut of 50 GeV, but with a

S/B ' 0.15.

For high-mass gluinos, inclusion of the lepton signal does not increase the sensi-

tivity of the search because the smaller production cross section decreases the signal

significance. It might however be possible that lepton signatures are effective for

high-mass gluinos in lepton-rich cascades that contain sleptons. Overall, though,

these results indicate that while jets + ET6 + lepton searches may be useful in cer-

tain regions of parameter space, they should be combined with jets + ET6 searches to

provide optimal coverage.

‡Here, the estimate of the significance only accounts for the statistical error; it does not includethe systematic uncertainty.

3.5. CONCLUSION 59

3.5 Conclusion

In this article, we discuss how model-independent bounds can be placed on the mass

of the lightest color octet particle that is pair-produced at the Tevatron. The main

aspects of the analysis focus on the advantage of running exclusive 1j− 4+j searches,

and placing limits using the measured differential cross section as a function of the

visible and missing energy. We show that the exclusion reach can be significantly

extended beyond those published by D06 because the ET6 and HT cuts used in their

analysis were only optimized for points in CMSSM parameter space. The proposed

analysis we present here opens up the searches to all regions of parameter-space,

allowing us to set limits on all kinematically-accessible gluinos. We also show how

the exclusion reach is degraded when gluino cascade decays are included, focusing on

the example of an intermediate wino, which decays to the dark matter candidate.

We have so far only focused on jet classification, ET6 , and HT as available handles

for increasing the reach of jets + ET6 searches. However, in certain special cases, other

techniques might be useful. For example, if the gluino decays dominantly to b jets,

heavy flavor tagging can be used advantageously.

In our analysis of the cascade decays, we often found that the regions of highest

significance in the differential cross section plot were pressed down against the 100

GeV cutoff in missing transverse energy. This lower limit was imposed to avoid regions

where the QCD background dominates. If the 100 GeV limit could be reduced, then

it would open up regions of high statistical significance that renders sensitivity to

a larger region of parameter space. The numerous uncertainties in the theory and

numerical generation of QCD events make it unlikely that precision QCD background

will be generated in the near future. However, it may still be possible to reduce the

cutoff by using event shape variables (i.e., sphericity).

Looking forward to the LHC, jets + ET6 searches are still promising discovery

channels for new physics. The general analysis presented in this paper can be taken

forward to the LHC without any significant changes. The primary modification will

be to optimize the jet ET used in the classification of the nj + ET6 searches. The

backgrounds for the LHC are dominantly the same; however tt will be significantly


larger and the size of the QCD background will also be different. Many of the existing

proposals for searches at the LHC focus primarily on 4+j + ET6 inclusive searches

and are insensitive to compressed spectra; see [86] for further discussion on MSSM-

specific compressed spectra at the LHC. By having exclusive searches over 1j + ET6to 4+j + ET6 , the LHC will be sensitive to most beyond the Standard Model spectra

that have viable dark matter candidates that appear in the decays of new strongly-

produced particles, regardless of the spectrum. Additionally, having the differential

cross section measurements will be useful in fitting models to any discoveries. Finally,

it is necessary to confirm that there are no gaps in coverage between the LHC and

Tevatron; in particular, if there is a light (∼125 GeV) gluino, finding signal-poor

control regions to measure the QCD background may be challenging.

Chapter 4

Dark Matter via the Higgs Sector

M. Lisanti and J. G. Wacker, “Discovering the Higgs with Low Mass Muon Pairs,”

Phys. Rev. D 79 115006 (2009).

The last unexplored frontier of the Standard Model is electroweak symmetry break-

ing, the process by which the Higgs field obtains a vacuum expectation value and

gives mass to the W± and Z0 gauge bosons. One of the major goals of current col-

liders is to discover the Higgs boson and understand the dynamics that give rise to

electroweak symmetry breaking. There have been direct and indirect searches for the

Standard Model (SM) Higgs at LEP and the Tevatron. The current lower bound on

the Higgs mass,

mh0 > 114.4 GeV (95% confidence),

comes from searches at LEP for e+e− → Z0h0, with the SM Higgs decaying to a pair of

taus or bottom quarks [14]. Recently, combined Higgs searches from the CDF and D06experiments at the Tevatron excluded a SM-like Higgs of 160 GeV ≤ mh0 ≤ 170 GeV

[87].

While direct searches for the Higgs point towards a heavy mass, indirect bounds

from electroweak constraints place a limit on how heavy the mass can be. In par-

ticular, the best fit for a SM Higgs mass is 77 GeV with a 95% upper bound of

167 GeV [88]. This limit comes from measurements of electroweak parameters that

61

62 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

depend logarithmically on the Higgs mass through radiative corrections. There is ten-

sion between the direct and indirect measurements; only a narrow window of masses

for the SM Higgs satisfies both results.

On the theoretical side, a light Higgs is preferred within the Minimal Supersym-

metric Standard Model (MSSM). Requiring a natural theory and minimizing fine

tuning drives the Higgs mass below the LEP direct bound. In the MSSM, there are

two new Higgs chiral superfields, Hu and Hd, that result in two CP-even scalars H0

and h0, the CP-odd scalar A0, and the charged Higgs H± after electroweak symmetry

breaking. Typically, the h0 has Standard Model-like couplings. At the one-loop level,

the Higgs boson mass is

m2h0 ' m2

Z0 cos2 2β

+3g2m4

t

8π2m2W

(log

mt1mt2

m2t

+ a2t

(1− a2

t

12

)),

where at is the dimensionless trilinear coupling between the Higgs and top squarks

at =At − µ cot β√12(m2

t1+m2

t2). (4.1)

For a moderate at <∼ 1 and top squarks lighter than 1 TeV, the Higgs mass is less than

120 GeV [15, 16]. By taking at to “maximal mixing,” where the contribution from

the A-terms gives the largest contribution to the Higgs mass, the Higgs can be as

heavy as 130 GeV while keeping the top squarks under 1 TeV. Two-loop corrections

can raise the Higgs mass by an additional . 6 GeV [16].

To avoid fine tuning, the top squarks should not be significantly heavier than the

Higgs. Even with masses at 1 TeV, the Higgs potential is tuned at the few percent

level. If the top squarks are at 400 GeV, the fine tuning of the Higgs potential

drops substantially; however, the upper limit on the Higgs mass falls to 120 GeV

even with maximal top squark mixing [71,89]. This has motivated studies giving the

Higgs quartic coupling additional contributions inside the supersymmetric Standard

Model [90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102], which usually leads to a less

4.1. LIGHT A0 MODIFICATIONS TO HIGGS PHENOMENOLOGY 63

minimal Higgs sector such as in the next-to-minimal supersymmetric Standard Model

(NMSSM).

Alternate models of electroweak symmetry breaking that can have naturally light

Higgs bosons are motivated by the indirect bounds coming from electroweak con-

straints and the desire to minimize fine tuning in the Higgs sector. These models,

which often have more elaborate Higgs potentials with additional scalar fields, allow

light Higgs masses, while simultaneously evading the LEP direct bound. Frequently,

these less minimal models of electroweak symmetry breaking have approximate global

symmetries and light pseudo-Goldstone bosons that can alter the phenomenology of

the Higgs. These light pseudo-Goldstone bosons can evade all existing limits because

they couple very weakly to light flavor fermions.

In this paper, we will focus on such non-minimal models of electroweak supersym-

metry breaking. We begin in Section II with a brief discussion of Higgs models that

contain light pseudo-Goldstone bosons, focusing primarily on current experimental

constraints. In Section III, we propose a new search for Higgs bosons that cascade

decay to pseudoscalars with masses below 10 GeV. In particular, we find that the

Higgs can be discovered in a subdominant decay mode where one pseudoscalar de-

cays to muons and the other to taus. We conclude with a discussion of the expected

sensitivity to the Higgs production cross section at the Tevatron and LHC. The pro-

posed search would allow possible discovery of a cascade-decaying Higgs with the

complete Tevatron data set or early data at the LHC.

4.1 Light a0 Modifications to Higgs Phenomenol-

ogy

In this section, we explore the couplings of a light pseudo-Goldstone boson, or “axion,”

to the Standard Model. We will describe how to analyze a general theory and we find

constraints on the maximal width for a Higgs decaying into a light axion. Finally, we

will analyze how CLEO limits on the direct coupling between pseudo-Goldstones and

Standard Model fermions set constraints on the Higgs width into axions. For Type II


two Higgs doublet models (i.e., MSSM), the CLEO and LEP results place important

limits on the light axion scenario.

The tension between the LEP limit on the Higgs mass and fine tuning can be

reduced if the Higgs branching fractions are altered from those of the Standard Model

[103, 104, 105, 106, 107, 108, 109] (see [110] for review). The LEP bound of 114 GeV

only applies when the Higgs decays dominantly to a bb or τ+τ− pair. While adding

new, invisible decay modes does not help because bounds for such processes are just

as strong [111], other nonstandard decays remain open possibilities. Consider the

case where the Higgs decays dominantly to two new scalars φ, which in turn decay

to SM particles:

h0 → φφ→ (XX)(XX). (4.2)

For an h0 with SM-like production cross section, this process is excluded formh0 < 110

GeV andX = b [112,113,114]. However, there is an 82 GeV model-independent bound

from LEP [115] and when X = g, c, τ , there are no limits for Higgs masses above 86

GeV [116].

The decay width of a 100 GeV Higgs into Standard Model particles is Γ(h0 →SM) ' 2.6 MeV; because the decay width is so small, the Standard Model decay

mode is easily suppressed by the presence of new decay modes. Any new light particle

with O(1) coupling to the Higgs will swamp the decay modes into SM particles.

Many theories, such as little Higgs models and non-supersymmetric two Higgs doublet

models, have light neutral states for the Higgs to decay into. This phenomenon arises

when there is an approximate symmetry of the Higgs potential that is explicitly broken

by a small term in the potential. There is a resulting light pseudo-Goldstone boson

that couples significantly to the Higgs boson. The Peccei-Quinn symmetry of a two

Higgs doublet model is one such example. In the MSSM, it is possible to have a light

A0, even with radiative corrections included [117]. More often, there is an additional

singlet, S, and an approximate symmetry that acts upon the Higgs boson doublets

as Hi → eiθqiHi with the singlet compensating by S → eiθqsS. During electroweak

symmetry breaking, S also acquires a vev, spontaneously breaking the symmetry; the


phase of S becomes a pseudo-Goldstone boson and has small interactions with the

Standard Model when 〈S〉 v.

For specificity, let us consider a two Higgs doublet model with an additional

complex singlet.∗ All three scalar fields acquire vacuum expectation values: vu =

v sin β, vd = v cos β, and 〈S〉. The interactions of the pseudo-Goldstone bosons can

be described in the exponential basis

Hu =

(ω+ sin β + h+ cos β

1√2(v sin β + hu)

)ei

auv sin β

Hd =

(1√2(v cos β + hd)

−ω− cos β + h− sin β

)ei

adv cos β

S =1√2

(〈S〉+ s0)eias〈S〉 . (4.3)

The pseudoscalar fields au, ad and as get interactions either through derivative

couplings from the kinetic terms or through explicit symmetry breaking. One linear

combination of the pseudoscalars,

ωZ0 = −au sin β + ad cos β,

becomes the longitudinal component of the Z0. The two other combinations are

physical fluctuations that get mass through symmetry breaking effects in the Higgs

potential. Any terms in the Higgs potential proportional to |Hu|2, |Hd|2, or |S|2do not affect the mass or interactions of the pseudo-Goldstones and there are only

a handful of possibilities for explicit symmetry breaking. As an example, consider

adding to the potential a sizeable coupling

V1 = λ1S2H†uH

†d + h.c. . (4.4)

∗A similar analysis was performed for a one Higgs doublet and a complex singlet in [118]. A oneHiggs doublet model with a light pseudoscalar typically does not have a large branching ratio of theHiggs into pseudoscalars.


This will give a weak-scale mass to the following linear combination of pseudo-

Goldstones:

A0 = cos θa(au cos β + ad sin β)− as sin θa. (4.5)

The singlet mixing component is given by

tan θa =v

〈S〉 sin 2β. (4.6)

The remaining linearly independent pseudo-Goldstone will be massless until the final

symmetry is broken. This linear combination is

a0 = sin θa(au cos β + ad sin β) + as cos θa (4.7)

and gets a mass through potentials such as

V2 = λ2S2HuHd + h.c. . (4.8)

There will be mixing between a0 and A0. However, when λ1 λ2, A0 and a0 are

nearly mass eigenstates with residual mixing proportional to m2a0/m2

A0 .

It is worth noting that as 〈S〉 v, the light pseudoscalar becomes the Peccei-

Quinn pseudoscalar and its couplings are independent of 〈S〉. This particular example

has the same symmetry structure as the NMSSM near the R-symmetric limit when

SHuHd A-term dominates the S3 A-term. We will couch our discussions in terms of

the R-symmetric NMSSM for comparison with the literature, but other realizations

of the symmetry breaking are just as applicable. To evade limits from LEP, the

Higgs needs a significant branching rate into the pseudoscalar (Fig. 1), and one

might worry that radiative corrections from the Higgs-pseudoscalar interaction might

induce a large radiative correction to the pseudoscalar mass. However, the interaction

that leads to the Higgs decay into pseudoscalars can occur even if the axion is an exact


500 1000 1500 2000

0.01

0.02

0.05

0.10

0.20

0.50

1.00

!s"!sin2b "GeV#

Br"h#"

aa#

5%

100 120 140 160 180 200

0.05

0.10

0.20

0.50

1.00

m_h !GeV"

1!MaxBr!h!"

aa"

10%

20%

50%

100%

Br(

h0!

a0a0

)

!S"/ sin 2! (GeV)

mh0 (GeV)100 120 140 160 180 200

mh0 = 100 GeV

5%

10%

20%

50%

Min

Br(

h0!

SM)

2%

1%

100%

Figure 4.1: The branching fraction of the Higgs into pseudoscalars as a function of〈S〉/ sin 2β for mh0 = 100 GeV when dh = 0 and 1 (solid and dashed lines, respec-tively). The inset shows the minimum value of the branching rate into the StandardModel as a function of mh0 .

Goldstone boson through the coupling

Lint = chv

〈S〉2h0∂µa

0∂µa0 − dhm2a0

vh0a0a0. (4.9)

The first interaction preserves the a0 → a0 + ε shift symmetry, where ch is an O(1)

constant and v = 246 GeV is the electroweak scale. Because this coupling exists in

the symmetry-preserving limit, ch can only depend on the vevs of the Higgs fields, the

particular charges of the approximate U(1) symmetry, and on the alignment of the

physical Higgs boson relative to the Higgs vev direction. When the physical Higgs

boson is in the direction of the Higgs vev, h0 = hu sin β+hd cos β, the example above

gives

ch = sin2 θa〈S〉2v2

=sin2 2β

1 + v2 sin2 2β〈S〉2

' 4

tan2 β. (4.10)

The second interaction breaks the shift symmetry and is proportional to m2a0 . This

term depends on the symmetry breaking that gives the axion a mass and is therefore

model-dependent. When the physical Higgs boson aligns with the Higgs vev, the


symmetry breaking coupling simplifies to

dh = 1 (4.11)

for the potential in Eq. 4.8. For a symmetry breaking potential

V2′ = λ2′S4 + h.c. , (4.12)

dh would be small, arising from the residual mixing between s0 and h0.

It is possible to increase dh by having multiple terms in the potential contribute

to m2a0 , with the pseudoscalar mass being less than either of the contributions. For

instance, with V2 and V2′

dh =1

1 +2λ2′ sin 2β

λ2 tan2 θa

>∼ 1 if λ2′ ' −λ2 tan2 θa2 sin 2β

. (4.13)

Of course, this is the technical definition of fine tuning and when dh >∼ 1, a0 has

been fine tuned to be light. We will discount this possibility in our discussion on the

expected sizes of couplings in this class of theories.

The partial width of the Higgs into pseudoscalars from these interactions is

Γh0→a0a0

mh0

=m2h0

16π

(chv

2〈S〉2 +dhm

2a0

vm2h0

)2(1 +O

(m2a0

m2h0

)). (4.14)

The symmetry-preserving interaction dominates when

〈S〉 ≤(ch

2dh

) 12mh0v

ma0

<∼ 1.5 TeV. (4.15)

When 〈S〉 is less than 1 TeV, the Higgs boson has an appreciable width into pseu-

doscalars (Fig. 1). This is precisely the region we are interested in, so we will set

dh = 0 for the rest of this discussion.

Figure 4.2 shows the values of 〈S〉/ sin 2β that are necessary to evade LEP2’s

search for a Standard Model Higgs [14]. For Higgs boson masses less than 100 GeV,


85 90 95 100 105 110 1150

200

400

600

800

Higgs Mass !GeV"

!S"!GeV

"

Higgs Mass (GeV)

!S"/

sin

2!(G

eV)

h0!

4!

h0 ! SM(LEP)

(LE

P)

Figure 4.2: Values of 〈S〉/ sin 2β (GeV) that have been excluded through LEP2’ssearch for a Standard Model Higgs [14]. The region below mh0 = 86 GeV is entirelyexcluded by the h0 → 4τ search [116].

the a0 has to be fairly strongly coupled to the Higgs boson, requiring 〈S〉 to be

small, which increases the size of the coupling to Standard Model fermions. However,

bounds from the recent CLEO results [119] are strongest for large fermion couplings.

The CLEO bounds place a 90% C.L. upper limit on Br(Υ → γa0) Br(a0 → τ+τ−).

For ma0 between 3.5 GeV and 9 GeV, this limit ranges from ∼ 10−5 − 10−4 for the

tau decays. The branching fraction for radiative Υ decays is [120]

Br(Υ→ a0γ)

Br(Υ→ µ+µ−)=GFm

2Υ

4√

2παg2d

(1− m2

a0

m2Υ

)F (4.16)

where F is a QCD correction factor ' 0.5, Br(Υ(1s)→ µ+µ−) = 2.5%, and gd is the

axion coupling to down quarks. For Type II two Higgs doublet models,† the axion

coupling to fermions is given by

Lint = igfmf

vfγ5fa

0, (4.17)

†This case applies to the MSSM and its extensions. For Type I two Higgs doublet models, whereall Standard Model fermions only couple to one Higgs doublet, there is no asymmetry between upand down-type quarks in the coupling to axions and this typically results in a cotβ suppression inthe coupling to axions.


4 5 6 7 8 90.0

0.5

1.0

1.5

2.0

m_a !GeV"

!!gd

ma0 (GeV)

!S"/ sin 2! # 1000 GeV

!S"/ sin 2! # 500 GeV

!S"/ sin 2! # 250 GeV

Figure 4.3: Region of ma0 − gd parameter space that has been excluded by CLEOto 90% C.L [119]. The dashed lines indicate values of 〈S〉/ sin 2β for tan β = 2. Theshaded region shows the minimum values of gd allowed by LEP for an 87-110 GeVHiggs.

where

gf = sin θa

cot β (up-type quarks)

tan β (down-type quarks/leptons)(4.18)

(see also [121, 122]). The coupling to up-type quarks is suppressed by two powers

of tan β. This means that, above the b-quark threshold, the axion will preferentially

decay to b-quarks. Below this threshold, it will preferentially decay to tau leptons,

rather than charm quarks.

The CLEO bound on the branching fraction of Υ sets a bound on gd, which can

be used to set limits on the allowed range of gd vs. ma0 (Fig. 4.3). The CLEO results

place the strongest constraints on small values of 〈S〉/ sin 2β. The value of the singlet

vev is a measure of the fine tuning of the theory because it induces a mass m2eff = λ〈S〉2

for the Higgs bosons in the scalar potential, and with an O(1) coupling, the singlet vev

should be less than a few TeV to avoid large fine tuning [103,104,105,106,107,108,109].

There is some tension, then, between keeping the coupling to fermions small and

keeping the coupling to the Higgs boson sufficiently large to evade LEP limits without

fine tuning the a0 to be light.

4.2. H0 → A0A0 AT HADRON COLLIDERS 71

It is also possible for LEP to have directly produced the Higgs through e+e− →Z0 → h0a0 [114]. The LEP searches place bounds on the product of the squared

Z0h0a0 coupling and the branching ratio of the Higgs into a Standard Model fermion

f :

ξ2 ' sin2 θa sin2 2β

1 + 112

m2h0

m2b

sin4 θa

Br(h0 → ff)SM ≤√

3mb

mh0

, (4.19)

where Br(h0 → ff)SM is the Standard Model’s branching ratio to fermion pairs.

There were searches for the (bb)(τ+τ−) final state at LEP, but there were no limits

for 75 GeV ≤ mh0 ≤ 125 GeV. For 125 GeV ≤ mh0 ≤ 165 GeV, the limits were

ξ2 <∼ O(0.4) which is automatically satisfied in these models. There were additional

searches for the (τ+τ−)(τ+τ−) final state, but these constraints are even weaker be-

cause the Higgs branching fraction into taus is a factor of ten smaller than that into

bottoms. LEP did search for e+e− → Z0 → h0a0 → (a0a0)a0, but the search for the

6τ final state was only performed at LEP1 and was thus not sensitive to Higgs masses

above 75 GeV.

4.2 h0 → a0a0 at Hadron Colliders

We will now discuss how the Higgs can be discovered if it decays into a light pseu-

doscalar a0, when 2mτ . ma0 . 2mb. In this range, the a0 decays predominantly

into taus and the signature of the Higgs is the appearance of 4τ events. All existing

searches for this decay channel have focused on the scenario where two or more taus

decay leptonically [123,124,125]. Currently, the ATLAS collaboration is exploring the

4µ8ν channel and CMS is analyzing (µ±τ∓h )(µ±τ∓h ) [124]. There are specific challenges

to the 4τ decay channel, however. The branching fraction of the taus to leptons is

only 33% and the pT spectrum of the events is soft because the visible lepton carries

less than half the momentum of the tau. Additionally, it is challenging to reconstruct

the Higgs and pseudoscalar masses from the final decay products.


4.2.1 Signal

The primary innovation of the search proposed in this paper is to use the subdominant

decay of the a0 into two muons, which exists because the a0 couples to the Standard

Model by mixing through the CP-odd Higgs. The relative branching ratio for the a0

into muons versus taus is

Γ(a0 → µ+µ−)

Γ(a0 → τ+τ−)=

m2µ

m2τ

√1− (2mτ/ma0)2

. (4.20)

The cross section of h0 → 2µ2τ depends upon the following product of branching

ratios:

εµτ = 2 Br(a0 → µ+µ−) Br(a0 → τ+τ−). (4.21)

For tan β >∼ 4, a 7 GeV pseudoscalar has a 0.4% branching ratio into muons and 98%

ratio into taus. As ma0 goes from the bottom threshold to the tau threshold, εµτ

varies from 0.8% to 1.5%. The remaining events go into hadrons and are divided

between the charm and glue-glue decay channels. For tan β = 2 the branching ratio

to charms becomes 15% and the branching ratio to taus and muons is reduced to 83%

and 0.3%, respectively, causing εµτ to fall to 0.2% for a 7 GeV a0 and 0.5% for an a0

just above the tau threshold. The events that go into hadrons do not typically have

significant missing energy and do not pass the missing energy cuts.

Due to the pseudoscalar’s small branching fraction into muons, this decay channel

has not been explored. However, the small branching fraction into muons need not be

a deterrent. The main contribution to the cross section for light (∼ 100 GeV) SM-like

Higgses comes from gluon-gluon fusion and can be as high as 2 pb at the Tevatron or

50 pb at the LHC [15]. As a result, it is still possible to get 300 events with 20 fb−1

at the Tevatron (combined D06 and CDF) and 250 events per experiment at the LHC

(at√s = 14 TeV) with 500 pb−1 luminosity, despite the small branching fraction

to muons. We ultimately find O(2%) cumulative efficiency for the signal (Table I),

resulting in 95% exclusion limits in certain mass windows at the Tevatron. At the

LHC, there is the possibility for discovery within the first year of running.


Signal EfficiencySelection Criteria Relative Cumulative

Pre-Selection Criteria 26% 26%Jet veto 99% 26%

Muon iso & tracking ∼ 50% 13%Mµµ < 10 GeV 98% 13%pµµT > 40 GeV 76% 9.8%ET6 > 30 GeV 29% 2.8%

∆φ(µ,ET6 ) > 140 73% 2.1%∆R(µ, µ) >0.26 63% 1.8%

Table 4.1: Relative and cumulative signal efficiencies due to the specified selectioncriteria. The signal point is a 100 GeV Higgs decaying to a 7 GeV a0 at the LHC.The pre-selection criteria include finding a pair of oppositely-signed muons, each with|η| < 2 and pT > 10 GeV.

When the Higgs boson decays to two light CP-odd scalars a0, the pseudoscalars are

highly boosted and back-to-back in the center-of-mass frame (Fig. 4.4). We consider

the case where there is a nearly-collinear pair of oppositely-signed muons on one side

of the event and a nearly-collinear pair of taus on the other, which we refer to as

a ditau (diτ). Each tau has a 66% hadronic branching fraction; consequently, there

is a 44% probability that both taus will decay into pions and neutrinos, which the

detector will see as jets and missing energy. Even if the taus do not both decay

hadronically, there is still missing energy, as well as a jet and a lepton, except when

both taus decay to muons, which occurs ∼ 3% of the time. The signal of interest is

pp→ µ+µ− + diτ + ET6 ,

where the missing energy comes from the boosted neutrinos and points in the direction

of the ditau. Because the taus are nearly collinear, the ditaus are often not resolved,

leading to a single jet-like object.

Signal events for a 7 GeV pseudoscalar decaying into 2µ2τ (εµτ = 0.8%) were

generated, showered, and hadronized using PYTHIA 6.4 [58].‡ Unlike at LEP, the

‡PYTHIA does not keep spin correlations in decays. This approximation does not affect the


µ

µ!!

a0 a0

h0ET!

Figure 4.4: Schematic of Higgs decay chain. The muons and taus will be highlyboosted and nearly collinear. It is likely that the taus will be reconstructed as onejet. Most of the ET6 in the event will be in the direction of this jet.

overall magnitude of the Standard Model Higgs production cross section is sensitive

to physics beyond the Standard Model and it is possible to increase the cross section

by an order of magnitude by adding new colored particles that couple to electroweak

symmetry breaking. In this study, the NNLO Standard Model production cross sec-

tion was used as the benchmark value [15].

PGS [63] was used as the detector simulator. Because the muons are adjacent,

standard isolation cannot be used. The muon isolation criteria must be modified

to remove the adjacent muon’s track and energy before estimating the amount of

hadronic activity nearby. As a result, we did not require standard muon isolation in

this study and instead reduced the overall efficiency by a factor of 50% to approx-

imate the loss of signal events from modified isolation. The approximate efficiency

for standard isolated muons is 50% and we do not expect that there should be an

additional cost for modified isolation [127].

4.2.2 Backgrounds

There are several backgrounds to this search: Drell-Yan muons recoiling against jets,

electroweak processes, and leptons from hadronic resonances (Table II). The Drell-

Yan background is the most important. The missing energy that results from the

tau decays is a critical feature in discriminating the signal from the background. In

addition, the fact that the missing energy is in the opposite direction as the muons

reduces the background from hadronic semileptonic decays.

The primary background arises from Drell-Yan muons recoiling against a jet. The

signal considered here because the taus are highly boosted in the direction of a0 and any kinematicdependence on spin is negligible. As verification of this, TAUOLA [126] was used to generate the fullspin correlated decays.


fb/GeV TeV LHC

DY+j 0.15 0.24W+W− 0.03 0.08

tt 0.02 0.14

bb <∼ 0.001 ∼ 0.03Υ + j 0.001 0.002

µµ+ττ 0.001 <∼ 0.001J/ψ + j 0.001 0.001

Total 0.20 0.49

Table 4.2: ]Continuum backgrounds for low invariant mass muons pairs with missing energy(dσ/dMµµ) for the h0 → a0a0 → (µ+µ−)(ττ) search at the Tevatron and LHC in

units of fb/GeV. The backgrounds are given for pµµT , ET6 , and ∆R cuts optimized fora 100 GeV Higgs.

missing energy is either due to mismeasurement of the jet’s energy or to neutrinos

from heavy flavor semi-leptonic decays in the jet. In the former instance, the analysis

is sensitive to how PGS fluctuates jet energies. While PGS does not parameterize

the jet energy mismeasurement tail correctly, the background only needs an O(30%)

fluctuation in the energy, which is within the Gaussian response of the detector. The

Drell-Yan background was generated using MadGraph/MadEvent, v.4.4.16§ [57] and

was matched up to 3j using an MLM matching scheme. It was then showered and

hadronized with PYTHIA. Again, the standard muon isolation criteria could not be

applied and we used the same 50% efficiency factor that was used for the signal.

All events are required to have a pair of oppositely-signed muons within |η| < 2.

Each muon must have a pT of at least 10 GeV. A jet veto is placed on all jets, except

the two hardest. The veto is 15 and 50 GeV for the Tevatron and LHC, respectively.

Lastly, it is required that the hardest muon and missing energy are separated by

∆φ ≥ 140. Table I shows the relative and cumulative cut efficiencies for the signal.

There are three higher-level cuts that further distinguish the signal from the back-

ground. These cuts are optimized as a function of the Higgs mass to maximize the

§This version of MadEvent does not apply the xqcut to leptons. We thank J. Alwall for alteringmatrix element-parton shower matching for this study.


4 5 6 7 8 9

0

5

10

15

20

25

Muon Invariant Mass !GeV"

Events

4 5 6 7 8 9

0

100

200

300

400

500

600

700

Muon Invariant Mass !GeV"Events

Eve

nts

Eve

nts

Muon Invariant Mass (GeV)

Muon Invariant Mass (GeV)

Figure 4.5: Muon invariant mass for 5 fb−1 at the LHC before (inset) and after thepµµT , ET6 , and ∆R cuts. The signal, a 100 GeV Higgs decaying to a pair of 7 GeVpseudoscalars, is shown in black and the Drell-Yan background is shown in gray.

significance of the signal. The first is a cut on the sum pT of the muons (pµµT ), and is

approximately

pµµT & 0.4mh0 . (4.22)

The second is a missing energy cut. There is a moderate amount of missing energy in

the signal events coming from the tau decays and this proves to be a very important

discriminant from the Standard Model background. The missing energy cut is

ET6 & (0.2− 0.25)×mh0 . (4.23)

For the LHC, the ET6 requirement is always held above 30 GeV. The last is a ∆R cut

on the muon pair, which depends on both the Higgs and pseudoscalar masses

∆R(µ, µ) &4ma0

mh0

. (4.24)

These cuts depend on the kinematics of the decays and the geometry of the events is

similar at both the Tevatron and LHC.

Figure 4.5 shows the invariant mass spectrum for the two oppositely-signed muons.

The inset shows the signal (black) and background (gray) before the pµµT , ET6 , and

∆R cuts. After these cuts are placed, the Drell-Yan background is mostly eliminated.


50 100 150 200 250 300 350 400

0

2

4

6

8

Total Invariant Mass !GeV"

Events

Total Invariant Mass (GeV)

Eve

nts

Figure 4.6: Total invariant mass for signal with mh0 = 100, 150, 200 GeV for 5 fb−1

at the LHC after pµµT , ET6 , and ∆R cuts. The ET6 is projected in the direction of thehardest jet.

The muon invariant mass reconstructs the mass of the pseudoscalar.

We used PGS to model the muon invariant mass resolution and used an ma0 ±80 MeV to exclude continuum backgrounds. The Drell-Yan background (dσ/dMµµ)

is O(0.15 fb/GeV) at the Tevatron and O(0.24 fb/GeV) at the LHC.¶

The other important kinematic handle in this analysis is the total invariant mass

of the event, which reconstructs the mass of the s-channel Higgs boson. The total

invariant mass of the signal is shown in Fig. 4.6 after all cuts have been applied.

The width of the peak is narrowed if the missing transverse energy is projected in

the direction of the jet. We expect that this should be the direction of the missing

energy because there will be boosted neutrinos from the hadronic tau decays.

In addition to Drell-Yan production, there are several electroweak production

mechanisms for muon pairs and ET6 . The most important one is W+W− production.

When the vector bosons are in a spin-0 configuration and decay leptonically, the

muons are nearly collinear and antiparallel to the neutrinos. When the W− decays

to µ−, the lepton momentum and spin are in the same direction as the gauge boson.

The antineutrino, however, is antialigned with the W−, and thus its momentum is

antiparallel to that of the muon. The situation is similar for the W+ decay, except

¶The background cross sections we quote in this section are for pµµT , ET6 , and ∆R cuts optimizedfor a 100 GeV Higgs.


that the directions of the muon and neutrino are reversed. The µ+µ−νν background

was generated with MadGraph and was found to be O(0.03 fb/GeV) at the Tevatron

and O(0.08 fb/GeV) at the LHC.

Top quark production is another important electroweak background. Using Madgraph

to generate µ+µ−ννbb, we estimate that this background is O(0.02 fb/GeV) at the

Tevatron and O(0.14 fb/GeV) at the LHC. The top background becomes nearly

comparable to the Drell-Yan background for larger Higgs masses due to the weaker

pµµT , ET6 , and ∆R cuts.

Electroweak production of µµττ has a production cross section on the order of

several attobarns when requiring low invariant-mass, high pT muons. Consequently,

it is subdominant to the W+W− and tt backgrounds, with O(8 × 10−5 fb/GeV) at

the Tevatron and O(2× 10−4 fb/GeV) at the LHC.

There are several other important backgrounds that arise from low-lying hadronic

spectroscopy that cannot be computed reliably with existing Monte Carlo generators.

These backgrounds come from (i) semi-leptonic decays (b → cµν), (ii) heavy-flavor

quarkonia, and (iii) leptonic decays of light mesons.

Double semileptonic decays (e.g. b→ c→ s/d) typically give rise to soft leptons

in jets but can occasionally fluctuate to give hard isolated leptons. It is challenging

to estimate this background contribution because the events are rare and we are

statistics-limited. However, we have attempted to estimate the relative magnitude

using PYTHIA. It was found that the total cross section for bb jets to produce two

muons is O(80 µb) at the LHC. Using a power law extrapolation from low pµµT , it was

estimated that a pµµT cut of 40 GeV reduces the cross section to O(10 pb). A missing

energy cut of 30 GeV is 0.6% efficient and requiring ∆φ(ET6 , µ) > 140 reduces the

cross section by an additional order of magnitude to O(5 fb). Placing a ∆R cut on

the muons and assuming that the muon isolation is 10% efficient, the cross section

becomes O(0.3 fb). The invariant mass of the muons in these events is distributed

over O(10 GeV), so the final background is approximately O(0.03 fb/GeV). This is

likely an overestimate because we have assumed that the cuts are uncorrelated. We

found that at the Tevatron, semileptonic decays do not produce enough high pT muon

pairs antialigned with the ET6 to be an important background.


100 120 140 160 180 2000.2

0.5

1.0

2.0

5.0

10.0

Higgs Mass !GeV"

!Br!h"#

aa"

Higgs Mass (GeV)

TeV

!pro

d!

Br(

h0"

a0a0

)(p

b)

250500LEP

Exclusion

10 fb!1

5 fb!1

20 fb!1

100 120 140 160 180 2001

2

5

10

20

50

100

Higgs Mass !GeV"

!Br!h"#

aa"

Higgs Mass (GeV)

!pro

d!

Br(

h0"

a0a0

)(p

b)

250

500

750

1000

LHC

.5 fb!1

5 fb!1

LHC

.5 fb!1

5 fb!1

LEP

Exclusion

Figure 4.7: Expected sensitivity to the Higgs production cross section at the Tevatron(left) and LHC (right) for ma0 = 7 Ge V. The contour lines indicate the cross sectionsfor several values of 〈S〉/ sin 2β (in GeV), which alters Br(h0 → a0a0). The StandardModel Higgs decay width and (NNLO) gluon fusion production cross sections wereobtained from [15]. An εµτ of 0.8% was used for the branching ratio of a0a0 → 2µ2τ .The region beneath the dashed line has been excluded by LEP.

Υs can decay into muon pairs, but their invariant mass is above the range we are

interested in. Υs can also decay into taus that can subsequently decay into muons

with a branching fraction of 3%. The invariant mass for these muon pairs will be in

the region of interest. There are, however, two factors that mitigate this background.

The first is that the muons will be soft unless the Υ has very high pΥT ∼ O(60 GeV).

The pT spectrum of Υs falls off rapidly. At the Tevatron, the differential cross section

is O(250 fb) at pΥT ∼ 20 GeV [128]. A naıve extrapolation to 60 GeV would place

this background at O(2 fb). Additionally, the missing energy in these events points


in the direction of the muon pairs rather than towards the recoiling jet and a cut

on the angle between the missing energy and the muon direction should reduce this

background by another order of magnitude. Accounting for this reduction, as well

as the 3% branching fraction of the taus to muons, we find that the cross section is

O(6×10−4 fb/GeV) at the Tevatron. We do not expect this background to dominate

Drell-Yan at the LHC, either; using NNLO predictions for the pT distribution of Υs

at the LHC [129], we estimate that this background will be O(2× 10−3 fb/GeV).

At the charm threshold, the J/ψ and ψ(2S) become important because the tails

of distributions arising from mismeasurement could spill over into higher invariant

mass bins. Again, the cross section for the decays of these particles drops sharply as a

function of pµµT and with pJ/ψT ≥ 40 GeV, the cross section is O(100 fb) [130]. Because

this peak is below the invariant mass of interest, only the tail of the Mµµ distribution

is a background. The dominant contribution comes either from the Lorentzian tail

of the decay width or from the non-Gaussian mismeasurement tail. The Lorentzian

tail suppresses the J/ψ contamination by O(10−9) for ma0 between 3.6 and 9 GeV.

The Gaussian tail of J/ψ mismeasurement goes out at least 5σ, meaning that the

contamination should be down by O(10−6). This gives a background cross section

smaller than O(10−5 fb/GeV) at the Tevatron and at the LHC. The contributions of

the ψ(2S) are subdominant to that of the J/ψ [131].

Resonances beneath the J/ψ are not a problem because they are far enough away

from the invariant mass window we are interested in. Peaks from fake muons may

arise from B → Kπ or similar decays where the kaons and pions punch through to

the muon chamber. These events are typically accompanied by significant hadronic

activity and tight muon isolation requirements (after removing the adjacent muon)

will reduce these backgrounds of fake muons [127]. Secondly, the ET6 from in-flight

decays is in the direction of the muons, but in the signal, it is back-to-back with the

muons. Placing a cut on the relative angle between the muons and the ET6 is effective

at eliminating these difficult backgrounds.

4.3. CONCLUSION 81

4.2.3 Expected Sensitivity

Figure 4.7 shows the expected 95% exclusion plot at the Tevatron and LHC. The

contours indicate the cross sections for values of 〈S〉/ sin 2β; this ratio affects the

partial width of the Higgs into the pseudoscalars (Eq. 14). The total projected

luminosity for the combined data sets at CDF and D06 is 20 fb−1; currently, each

experiment has ∼ 5 fb−1. With 10 fb−1 luminosity, the Tevatron will start probing

the interesting regime where 〈S〉/ sin 2β = 250 GeV. Once the benchmark luminosity

is reached, the Tevatron will have sensitivity up to 〈S〉/ sin 2β = 500 GeV.

With early data, the LHC has sensitivity to regions corresponding to 〈S〉/ sin 2β .

250 GeV. The sensitivity is weaker for Higgs masses below 100 GeV because the

backgrounds worsen due to a smaller pµµT cut. However, combined analyses by CDF

and D06 should be able to probe this region down to O(1 pb). By the time the LHC

reaches a luminosity of 5 fb−1, it will be sensitive to the most relevant region of

parameter space, with 〈S〉/ sin 2β . 1 TeV.

The sensitivity curves depend on the product of the pseudoscalar branching ratios

into muons and taus, εµτ . For Fig. 4.7, we assumed that the pseudoscalar was 7 GeV,

which corresponds to εµτ = 0.8%. For a lighter pseudoscalar (e.g., 4 GeV), εµτ is

nearly double this value. In this case, the signal limits can increase by as much as a

factor of two. To first order in m2a0/m2

h0 , the branching fraction of the Higgs into the

pseudoscalar is independent of ma0 and the contour lines in Fig. 4.7 are unaffected.

Therefore, if the pseudoscalar is near the tau threshold, the experiments are even

more sensitive to the Higgs production cross section than indicated in the figure.

4.3 Conclusion

We have shown that if the Higgs decays into a pair of light pseudoscalars that subse-

quently decay into taus, then the discovery of the Higgs boson is promising through

the subdominant channel where one pseudoscalar decays to a pair of muons. The

Tevatron has a chance of discovering this class of models if CDF and D06 perform

combined analyses with the full data sets. The Tevatron can begin to recover the


parameter space that LEP missed with their h0 → 4τ search, which was prematurely

stopped at 86 GeV. Assuming that the only new decay mode of the Higgs boson is

into a pair of pseudoscalars, the Tevatron is sensitive to mh0 ' 102 GeV with 10 fb−1,

and up to mh0 ' 110 GeV with 20 fb−1. When the Tevatron covers this ground, their

results, combined with the direct limits from LEP, will effectively establish a lower

limit on the Higgs mass regardless of the admixture of Higgs decays into light pseu-

doscalars or Standard Model fermions. With a 20 fb−1 cross section, the Tevatron

will be sensitive to Higgs bosons up to mh0 <∼ 150 GeV.

At the LHC, this search becomes a method of discovering the Higgs with early data

– potentially with sub-fb−1 data sets. With an integrated luminosity of O(1 fb−1),

the LHC will be able to recover the missing LEP limits. Eventually, the LHC will

be able to push this branching ratio down substantially, to the 3% level. A discovery

or even a limit on such a decay mode will be an important step in verifying the field

content and symmetry structure of the Higgs potential.

Chapter 5

Prospects for Inelastic Dark

Matter

D. Alves, M. Lisanti, and J. G. Wacker, “iDM’s Poker Face,” [arXiv: 1005.5421].

Predictions for direct detection experiments require a wide-range of assumptions con-

cerning the astrophysical properties of the dark matter, as well as its interactions with

the Standard Model (SM). These theoretical uncertainties are compounded by addi-

tional experimental challenges that arise from the nature of low energy experiments.

Ultimately, it is necessary to know the scattering rate for dark matter off SM nuclei in

detectors. There are many unknown physical quantities that go into this prediction

and they are often benchmarked to values in specific studies. However, in light of

a potential signal, the verification process requires a more systematic study of these

unknowns in order to have a complete picture of the range of consistent theories.

Inelastic Dark Matter (iDM) serves as a case study for this new treatment of

uncertainties and shows how marginalizing over astro and particle physics quantities

leads to at least an order of magnitude variation in detection prospects at upcoming

experiments.

Inelastic dark matter is an elegant explanation for DAMA’s on-going 8.9σ annual

modulation signal [34, 35, 132], resolving the inconsistency of this signal with the

plethora of null direct detection experiments [18,19,20,21,26,28,23,24,30,22]. In the

83

84 CHAPTER 5. PROSPECTS FOR INELASTIC DARK MATTER

iDM framework, a dark sector particle up-scatters off the detector’s target nucleus to

a higher mass state. To explain the DAMA anomaly, an O(100 keV) mass splitting

is required for weak-scale dark matter.

IDM requires a minimum velocity to up-scatter to the more massive state, which

depends on the mass of the target nucleus, mN, the reduced mass of the nucleus-dark

matter system, µN , the mass splitting, δ, and the recoil energy, ER, of the nucleus:

vmin =1√

2mNER

(mNERµN

+ δ). (5.1)

The detection rate [37] depends on vmin through

dR

dER=

ρ0

mdmmN

∫ vesc

vmin

f(~v + ~vE(t))vdσ

dERd3v, (5.2)

where f(~v) is the local velocity distribution function (vdf) for the dark matter halo in

the galactic frame, and ~vE(t) accounts for the boost to Earth’s rest frame [133]. The

differential scattering rate is larger for heavier target nuclei because vmin is reduced.

The spin-independent differential cross section can be parameterized as

dσ

dER=mNσN

2µ2Nv

2(fpZ + fn(A− Z))2|Fdm(q2)FN(q2)|2, (5.3)

where σN is the dark matter-nucleus cross section at zero momentum transfer and

q2 = 2mNER is the momentum transfer. The constants fp,n parameterize the cou-

pling to the proton and neutron, respectively, and are set to fp = fn = 1 throughout.

The dependence of the cross section on the nuclear recoil energy comes from the dark

matter and nuclear form factors, Fdm(q2) and FN(q2). Fdm(q2) describes non-trivial

behavior at low momentum transfer in models where higher dimensional operators

contribute to the scattering [134, 135, 136]. FN(q2) is the Helm/Lewin-Smith nuclear

form factor [37]. Analytic approximations to the nuclear form factor can have sub-

stantial errors, particularly for heavy nuclei such as 184W. The Helm/Lewin-Smith

form factor is better behaved than other Helm parameterizations, but can still give

errors of 25% for 184W in the range ER = 10 − 40 keVnr [137]. Around 100 keVnr,

85

these errors can be as large as 60%. The impact of nuclear form factor uncertain-

ties on predictions for direct detection has been addressed in the literature [138].

This work explores other sources of uncertainty, and for the rest of this paper the

Helm/Lewin-Smith form factor is adopted.

Typically, the vdf is taken to be Gaussian, isothermal and isotropic in the galactic

frame. This ‘Standard Halo Model’ (SHM) is parameterized as

f(v) ∝(e−v

2/v20 − e−v2esc/v20)

Θ(vesc − v), (5.4)

where vesc is the galactic escape velocity and v0 is the velocity dispersion. The range

of escape velocities is constrained by the RAVE stellar survey: 480 ≤ vesc ≤ 650 km/s

[38], and no constraints are placed on v0. The standard procedure when evaluating

direct detection rates is to assume a SHM distribution with benchmarked values for v0

and vesc. The solid blue curve in Fig. 5.1 shows the expected tungsten recoil spectrum

for this vdf; the bulk of events occur between 10-40 keVnr.

While previous studies have looked at the effect of varying v0 and vesc within

the SHM [139, 140], none have fully marginalized over both dark matter and halo

profile uncertainties. In addition, numerical N-body simulations indicate significant

departure of the vdf from the SHM hypothesis [141, 142, 143, 144], especially in the

high velocity tail. Because very little is known about either the vdf or the dark

matter model, experimental analyses should be designed to cover a wide range of

possibilities. In this paper, a scan over the parameter space for iDM is performed,

where we marginalize over the dark matter parameters (m, δ, σ) and halo velocity

parameters (v0, vesc, ~vstream), and set constraints by a global χ2 analysis [145].∗

The predicted number of events at CDMS [18,19,20,21], ZEPLIN-II [26], ZEPLIN-

III [28], CRESST-II [23,24], XENON10 [30], EDELWEISS [22], and the XENON100

calibration run [31] are included in the χ2 as well as the annual modulation ampli-

tude in the first twelve bins of DAMA (2-8 keVee) [132, 34, 35]. The high energy

∗In some circumstances, maximum gap techniques provide tighter limits than Poisson statistics fornull experiments [146,138]. However, Poisson statistics are used in this paper due to the complexityof combing a χ2 for DAMA with multiple max-gap tests to get a global limit.


bins from 8-14 keVee are combined into a single bin with modulation amplitude -

0.0002±0.0014 cpd/kg/keVee. Any model that over-predicts the number of events at

the null experiments by 2σ is excluded.

5.1 Inelastic Dark Matter at CRESST

The standard assumption is that the iDM interpretation of DAMA’s signal can be

confirmed or refuted by any experiment with a target nucleus heavy enough to cause

the inelastic transition [147]. For DAMA, the inelastic transition occurs through scat-

tering off the 127I nucleus in the NaI(Tl) target. Naively, any experiment with a target

mass greater than 127I could provide a sufficient test. Two upcoming experiments fall

into this category: XENON100 [31] and CRESST [25,148], which use 131Xe and 184W,

respectively. XENON100 has currently released results from a calibration run of 11.2

live days during Oct-Nov 2009. They report 161 kg-d effective exposure and have

observed no events in their acceptance region between 4.5 - 40 keVnr. CRESST,

which consists of nine detectors of CaWO4 and one detector of ZnWO4, has shown

preliminary results in the energy window from 10 - 40 keVnr obtained from summer

2009 until the present; however, the exposure was not reported [148].

CRESST provides a unique experimental environment for testing iDM because it

has the heaviest target nucleus of all current direct detection experiments. 184W is

expected to be highly sensitive to inelastic scattering because its velocity threshold is

a factor√mI/mW ∼ 0.83 lower than iodine. As a result, a larger fraction of the halo

can up-scatter off of 184W and one would expect a larger scattering rate compared

to lighter targets. However, an additional complication arises due to the large radius

RW of a 184W nucleus. In particular, when the momentum transfer is q ∼ 1/RW, the

dark matter probes the size of the nucleus and the scattering is no longer coherent.

Therefore, the scattering rate is suppressed at recoil energies ER ∝ 1/(2mNR2W). This

suppression occurs at lower recoil energies for 184W, as compared to 131Xe (55 keVnr

versus 90 keVnr, respectively).

The fact that the first zero of the 184W form factor occurs at such a low recoil

energy highlights an important challenge for CRESST. The typical recoil energy an

5.1. INELASTIC DARK MATTER AT CRESST 87

0 20 40 60 80 100

0.000

0.005

0.010

0.015

Recoil Energy !keVnr"

Rate!cpd#k

g#keVn

r"0 20 40 60 80 100

0.000

0.005

0.010

0.015

Figure 5.1: CRESST spectra for: regular iDM (blue), FFiDM with Fdm ∝ ER (green),DM stream (red) for QI = 0.085. Inset: QI = 0.07 (dashed), QI = 0.06 (dotted).

iDM particle deposits on a 184W nucleus is O(δ µ/mW) ∼ O(50 keVnr), where the

signal is suppressed due to loss of coherence. The form factor suppression is evident

in the recoil spectra of Fig. 5.1, which also illustrates the effects of three additional

sources of uncertainties: DAMA’s energy calibration, the dark matter interaction

with the SM, and the velocity distribution profile. Variations in any of these three

sources can significantly alter iDM’s rate at CRESST and ultimately affect the final

exposure that will be required for CRESST to exclude the DAMA iDM hypothesis.

DAMA’s Energy Calibration

DAMA only detects its nuclear recoil events with scintillation light. Nuclear recoils

typically deposit only a small fraction of their energy into scintillation. The quenching

factor for 127I , QI, relates the measured electron equivalent energy (given in keVee)

to the nuclear recoil energy (given in keVnr):

Eee = QI(Enr) Enr, (5.5)

where the energy dependence of the quenching factor is left explicit. Most studies

assume a constant quenching factor for iodine from ∼ 10 − 100 keVnr, with the

standard value taken to be QI = 0.085. However, there are large experimental uncer-

tainties in measurements of QI [32]. The four primary ones [149, 150, 151, 152] give


5 10 20 50

0.1

0.5

1.0

5.0

10.0

50.0

Events at CRESST H10-40 keVnrL

Eve

nts

atC

RE

SST

H40-

100

keV

nrL

Figure 5.2: Average counts at CRESST per 100 kg-d for regular iDM (blue), FFiDMwith Fdm(q) ∝ ER (green), and DM streams (red). The effect of lowering the quench-ing factor is illustrated for QI = 0.07 (dashed blue) and QI = 0.06 (dotted blue). Thecontours enclose all points with χ2 ≤ 18.

0.05 ≤ QI ≤ 0.10. The study in [151] gives the smallest error, however its measure-

ments are calibrated with 60 keV gamma rays, in contrast to the 3.2 keV electrons

that DAMA uses [149]. This difference reduces the central value of [151] by roughly

10% and induces larger systematic effects.

Lowering iodine’s quenching factor effectively shifts DAMA’s signal to higher

nuclear recoil energies, favoring slightly larger values for the iDM mass splitting

(100 . δ . 180 keVnr for QI = 0.06). Consequently, the predicted signal at other

experiments is also shifted to higher nuclear recoil energies. In addition, the spectral

shape is broadened, because DAMA’s reported rate is in units of cpd/kg/keVee.

Fig. 5.1 illustrates how the 184W recoil spectrum changes as QI is reduced from

0.085 to 0.06, and Fig. 5.2 shows the average annual rate for CRESST’s low and high

energy range, assuming a SHM profile. The shift of the iDM signal to higher recoils

translates into a significant reduction in CRESST’s average annual rate in the low

recoil window of 10− 40 keVnr and a substantial enhancement in its rate in the high

recoil range from 40− 100 keVnr.

5.1. INELASTIC DARK MATTER AT CRESST 89

Dark Matter Interaction

The identity of iDM is unknown and its interactions with the SM may not occur

through renormalizable operators. Non-renormalizable operators typically result in

matrix elements with non-trivial dependence on the momentum transfer q. These can

be parameterized by an effective DM form factor [43,153,154,137,134,135,140],

Fdm(q) =∑n,m

cn,m(q0)n|~q |m

Λn+m+ . . . (5.6)

where q0 = ER, |~q | =√

2mNER, and Λ is an arbitrary mass scale. Standard iDM

assumes that the constant n,m = 0, 0 term dominates the expansion. Models that

have an interaction mediator with mass lighter than O(|~q |) are dominated by c0,−2.

Composite iDM models have c0,1 6= 0 [40,155,156]. Form factors that are dominantly

n 6= 0 can be realized through dipole or other tensor interactions [153,154].

Standard iDM (i.e., n,m = 0, 0) and models with n = 0,m 6= 0 have comparable

rates at CRESST because the ratio of predicted events between these two scenar-

ios scales as N0,m/N0,0 ' (mWEW peak/mIEI peak)2m ' 1. DAMA’s spectrum peaks at

EI peak ' 35 keV while the tungsten spectrum at CRESST peaks at EW peak ' 25 keV.

In contrast, interactions with n 6= 0,m = 0 predict substantially smaller rates at

CRESST: Nn,0/N0,0 ' (EW peak/EI peak)2n ' (0.5)n. This effect is illustrated in

Fig. 5.2 for n,m = 1, 0.

Dark Matter Velocity Distribution

There is little direct observational evidence for the DM density profile, and the velocity

distribution is highly uncertain. While most studies assume a Maxwell-Boltzmann vdf

(5.4), N-body simulations indicate that this ansatz does not adequately parameterize

the vdf [144]. The iDM spectrum is particularly sensitive to changes in the tail of

the velocity distribution profile, which can arise from velocity anisotropies or from

dark matter substructure that has recently fallen into the galaxy [141,157]. A vdf in

which the high velocity tail is dominated by a stream of dark matter illustrates how

changes in the local vdf alter iDM predictions. This scenario can significantly lower


5 10 20 5010

100

50

20

200

30

300

15

150

70

Events at CRESST H10-40 keVnrL

Eve

nts

atX

EN

ON

100

H4.5

-10

0ke

Vnr

L

Figure 5.3: Average counts at CRESST (per 100 kg-d) versus XENON100 (per 1000kg-d) for regular iDM (blue), FFiDM with Fdm(q) ∝ ER (green), and DM streams(red). The effect of lowering the quenching factor is illustrated for QI = 0.07 (dashedblue) and QI = 0.06 (dotted blue). The contours enclose all points with χ2 ≤ 18.

the number of expected events; other possibilities for the velocity profile will result

in numbers of events between the SHM and stream expectations.

Streams of dark matter are characterized by low velocity dispersion [141]. Here,

streams will be parameterized as dispersionless vdfs that have an arbitrary incident

angle. The distribution profile is f(~v) = δ3(~v − ~vstream), with ~v and ~vstream given in

the frame of the sun. The differential scattering rate is obtained after boosting to the

Earth’s frame, and depends on the recoil energy through

dR

dER∝ Θ(|~vstream − ~vE(t)| − vmin)

|~vstream − ~vE(t)| |FN(q2)|2, (5.7)

where ~vE(t) is the Earth’s velocity in the frame of the solar system. The rate would

be constant if not for the nuclear form factor that shapes the distribution and yields

a highly peaked spectrum as illustrated for tungsten in Fig. 5.1.

The stream’s velocity and incident angle relative to the Earth are marginalized

over and constraints on the phase and higher harmonics of the annual modulated rate

are applied. These are set by the spectral decomposition of DAMA’s modulated rate,

which restricts the modulation to peak at May 24th ± 7.5 days and constrains the

power spectrum at a frequency ω = 2 yr−1 to be P (2 yr−1) <∼ 0.05P (yr−1) [34,35,132].

5.2. XENON100 PROSPECTS 91

Fig. 5.2 shows the iDM predictions for CRESST when the tail of the vdf is domi-

nated by a dispersionless DM stream. When the dark matter stream is nearly head-on

in the summer, very few events are expected in the fall and winter months, which is

consistent with the late-year running of the XENON100 calibration run. Marginal-

izing over stream parameters shows that velocities vstream ∼ 400 km/s are favored,

and that the DM incident angle with respect to the velocity of the Earth on June

2 can be as large as 75, although 90% of the consistent models have incident angle

θin < 50 and 75% have θin < 36. Fig. 5.2 shows that the annual average rate at

CRESST for dark matter streams can deviate dramatically from the SHM case and

highlights the importance of marginalizing over parameters and considering different

velocity profiles when making predictions for direct detection experiments.

5.2 XENON100 Prospects

The previous section discussed how uncertainties in energy calibration, dark matter

interactions, and vdfs significantly affect the range of predicted events at CRESST.

For O(100 kg-d) exposures, the number of events in the low energy window can vary

over an order of magnitude from 3− 30, while the number in the high energy window

ranges from 0.1 − 10. It is evident that having significant exposure over the full

nuclear recoil band where the iDM signal is expected to dominate (10-100 keVnr) is

essential for refuting or confirming a potential signal.

The XENON100 experiment will accumulate large exposuresO(3000 kg-d) in their

current data run. Compared to CRESST, it has better coverage of the relevant nuclear

recoil band because the 131Xe target is not affected by form factor suppression at

energies below 90 keVnr. The dominant variation in predictions for spin-independent

iDM scattering rates in 131Xe arises because DAMA’s modulation fraction is not

known and only constrained to be greater than 2% [34, 35, 132]. Larger modulation

fractions at DAMA imply a proportionately smaller rate at XENON100.

In their calibration run, XENON100 demonstrated the potential to be a “zero

background” experiment. The fact that no events were seen in their acceptance region

implies a lower bound on the modulation fraction for iDM of O(40%). Their next


data release, which will include summer data, will directly test the iDM parameter

space still allowed. Fig. 5.3 illustrates the average number of expected events for

XENON100 (per 1000 kg-d), including the uncertainties in QI, the DM form factor,

and the vdf. Again, there is an order of magnitude uncertainty, with the predictions

ranging from 20-200 counts per 1000 kg-d. For the expected exposure of XENON100’s

data release, the minimum number of events predicted by iDM is ∼ 60, which will be

enough to confirm or refute the spin-independent iDM scenario in a conclusive and

model-independent manner.

5.3 Conclusions

The process of testing the DAMA anomaly highlights many of the challenges inherent

to direct detection experiments. In addition to determining the properties of the

unknown dark matter particle, direct detection experiments must also consider the

unknown flux of the incident dark matter, as well as uncertainties in converting a

signal from one target nucleus to another.

The predictions for both the CRESST 2009 run and XENON100 2010 run show

an order of magnitude uncertainty. The nuclear form factor for 184W, when com-

bined with additional theoretical and experimental uncertainties, will likely prevent

CRESST from refuting the iDM hypothesis with an exposure of O(100 kg-d) in a

model-independent manner. XENON100, on the other hand, will be able to make

a definitive statement about a spin-independent, inelastically scattering dark matter

candidate. Still, the CRESST 2009 data can potentially confirm iDM for a large range

of parameter space. In case of a positive signal, the combined data from CRESST

and XENON100 will start probing the properties of the Milky Way DM profile and

the interaction of the SM with the dark matter.

Chapter 6

Six Higgs Doublet Model

M. Lisanti and J. G. Wacker, “Unification and Dark Matter in a Minimal Scalar Ex-

tension of the Standard Model,” [arXiv: 0704.2816].

The Standard Model (SM) of particle physics has enjoyed great success in ex-

plaining physics below the electroweak scale, but it is unlikely to remain the sole

description of nature up to the Planck scale. The SM does not contain a viable cold

dark matter candidate, such as a weakly interacting massive particle (WIMP), or sat-

isfactorily address the issue of gauge coupling unification. Both of these issues hint

at new physics beyond the electroweak scale. In addition, the SM must be fine-tuned

to regulate the large radiative corrections to the Higgs mass and the cosmological

constant. It is possible to obtain a natural Higgs from supersymmetry [46], large

extra dimensions [158], technicolor [159], Randall-Sundrum models [47], or the little

Higgs mechanism [13]. However, these models do not address the magnitude of the

cosmological constant, which appears to be more fine-tuned than the Higgs mass, and

leads us to question the central role of naturalness in motivating theories beyond the

Standard Model.

An alternative approach is to explore non-natural extensions of the SM in which

fine-tuning is explained by environmental selection criteria [160]. Weinberg noted

that if the cosmological constant was much larger than its observed value, galaxy

93

94 CHAPTER 6. SIX HIGGS DOUBLET MODEL

formation could not occur [161]. Additionally, the Higgs vacuum expectation value

cannot vary by more than a factor of a few before atoms become unstable [162,163]. In

the context of the string theory landscape populated by eternal inflation, there exists

a natural setting for environmental selection to play out [164, 165]. This motivates

considering models with parameters that may not be natural, but which are forced

to be small by environmental selection pressure.

Split supersymmetry is an example of a model that relaxes naturalness as a guiding

principle and focuses on unification and dark matter [163,166,167]. In this theory, the

scalar superpartners are ultra-heavy and the electroweak scale consists of one fine-

tuned Higgs and the fermionic superpartners, which are kept light by R-symmetry.

The additional fermions alter the running of the gauge couplings so that unification

occurs at high scales.

The “minimal model” was presented in [168,169], where a Dirac electroweak dou-

blet serves as a dark matter candidate and leads to gauge coupling unification. A

fermion singlet that mixes with the dark matter must also be introduced to avoid

conflicts with direct detection results. The additional fermion leads to richer phe-

nomenology, but at the cost of introducing a new mass scale.

In both these models, fermions serve as dark matter and technical naturalness

protects their masses from large radiative corrections. Split supersymmetry assumes

that the high energy dynamics are supersymmetric, but that high-scale susy break-

ing is preferred. If the susy breaking sector communicates R-symmetry breaking

inefficiently, the gauginos and Higgsinos end up much lighter than the typical super-

symmetric particles. This should be contrasted with models like those in [168], where

an ad hoc dynamical mechanism is invoked to make the fermionic dark matter much

lighter than the GUT scale. Without understanding how the “minimal model” fits

into a high energy theory, it may be that it requires fine-tunings of fermion masses to

get a viable dark matter particle. Similarly, without understanding how R-symmetry

breaking is explicitly communicated to the gauginos and Higgsinos, it may be that

split susy requires a tuning of a fermion mass to get weak-scale dark matter.

The use of technical naturalness to justify new light fermions may be particularly

6.1. THE MODEL 95

misleading for relevant couplings that determine the large-scale structure of the Uni-

verse. In [170,171], the formation of galactic structures with properties similar to the

Milky Way places bounds on the ratio of the dark matter density to baryon density.

Typically, these bounds are not as strong as those on the cosmological constant, but

they fix the dark matter mass to within an order of magnitude [170] (or three orders

of magnitude in [171]). This opens up the possibility that the mass of the dark matter

is unnatural and is set by environmental conditions so that the baryonic fraction of

matter is not over- or under-diluted.

A candidate for unnatural dark matter is the scalar WIMP. While there has been

some recent work on minimal models with scalar dark matter [172,173,169,174,175,

176], none provide a framework for gauge unification. In this paper, we will study a

minimal scalar unifon sector that also contains a viable dark matter candidate. An

additional Higgs doublet is added to the SM that is in a 5 or 6-plet of a new global

discrete symmetry.∗ This global symmetry remains unbroken, yielding a spectrum

of two five-plets of real neutral scalars and one five-plet of charged scalars. Relic

abundance calculations give the WIMP mass to either be ∼ 80 GeV or in the range

∼ 200-700 GeV. The model will be tested by next-generation direct and indirect

detection experiments, and may possibly have signatures at the LHC.

The model will be presented in greater detail in Sec. II. In Sec. III, the renor-

malization group equations are solved to illustrate gauge unification and the allowed

weak-scale values of the theory’s quartic couplings. The relic abundance calculation

is presented in Sec. IV and the predicted experimental signals, in Sec. V. The results

are summarized in Sec. VI.

6.1 The Model

The model proposed in this paper consists of the Standard Model Higgs, h, plus

an additional electroweak doublet, H5, in the 5 representation of a global discrete

symmetry group. The discrete symmetry is not necessary for maintaining the stability

of the dark matter; its purpose is to package the fine-tuning of the squared masses

∗Only the 5 option will be discussed here, but the results also hold for the 6 multiplet.


for the five additional doublets into a single tuning. This discrete symmetry also

reduces the number of quartic couplings in the potential to that of the two Higgs

doublet model. Any discrete symmetry group can be chosen, so long as it has a 5

representation (i.e., S6).

The scalar potential for the six Higgs doublet model is

V = −m20|h|2 +m2

5|Hi|2 + λ1(|h|2)2 + λ2(|Hi|2)2

+λ3|h|2|Hi|2 + λ4|h†Hi|2 + λ5((h†Hi)2 + h.c.)

+λ6cijk(hH†iHjH

†k + h.c.), (6.1)

where i, j, k = 1, . . . , 5. Depending on the choice of discrete symmetry, there may

be several couplings of the form |H5|4; one possibility is shown in the λ2 term. The

existence of the term proportional to λ5 is necessary for the phenomenological viability

of the model and forces the five-dimensional representation to be real. The term

proportional to λ6 is only allowed if the symmetry satisfies the following relation

5⊗ 5 = 1⊕ 5⊕ · · · . (6.2)

The couplings λ5 and λ6 lead to a physical phase and will induce CP violation in the

self-interaction of the 5-plets after electroweak symmetry breaking. This does not

alter the tree-level spectrum; because the self-coupling of the 5-plet is only affected

at loop-level, the CP violation does not significantly alter the experimental signatures

of the model.

The field h acquires a vev and gives masses to the gauge bosons. In contrast, the

field H5 does not acquire a vev and cannot have any Yukawa interactions with the

Standard Model fermions. Expanding about the minimum of the potential, 〈h〉 =

v/√

2, with v = 246 GeV. The Higgs 5-plets are

H5 =

(φ+

5

(s05 + ia0

5)/√

2

). (6.3)

6.1. THE MODEL 97

The physical masses of the particles at the minimum are

m2h0 = 2λ1v

2

m2φ± = m2

5 +1

2λ3v

2

m2s0 = m2

5 +1

2v2(λ3 + λ4 + 2|λ5|)

m2a0 = m2

5 +1

2v2(λ3 + λ4 − 2|λ5|) (6.4)

and must always be greater than zero. The lightest neutral particle, a0, serves as

the dark matter candidate; in order that it not become the charged φ± boson, the

quartics must satsify

λ4 − 2|λ5| < 0. (6.5)

The splitting between s0 and a0 is proportional to λ5 and breaks the accidental U(1)

symmetry. Results from direct detection experiments (see Sec. 6.4.1) require that

the mass splitting between s0 and a0 be more than O(100 keV), which sets the limit

|λ5| >∼ 10−6. (6.6)

The experimental lower bound on the Higgs mass [177] constrains the value of λ1 to

be

λ1>∼ 0.1. (6.7)

Additional constraints on the quartics come from the requirement of vacuum sta-

bility. In order that the potential (6.1) be bounded from below in all field directions,

the couplings must satisfy

λ1, λ2 > 0

λ3 > −2√λ1λ2

λ3 + λ4 − 2|λ5| > −2√λ1λ2. (6.8)

These conditions are for local stability of the potential at a given scale. If they are

satisfied at all scales, then they correspond to absolute stability.


6.2 RG Influence on Low Energy Spectrum

6.2.1 Gauge Unification

Unification is the key motivation for introducing the five-plet H5 and it is straightfor-

ward to check that the gauge couplings unify reasonably well with the addition of six

or seven scalars to the SM (see [178] for two-loop RGEs) . In particular, when two-

loop RGEs are evaluated, a threshold correction splitting a (fermionic and scalar)

5 + 5 by m2/m3 ' 30 is necessary to maintain unification for the six scalar case

(where m2 and m3 are, respectively, the masses of a doublet and triplet). This is

better than the case of seven scalars, where m2/m3 ' 300. As a point of comparison,

the threshold corrections for the MSSM require m3/m2 ' 20 [179].

The unification scale is given by

tGUT = 2πα−1

1 − α−12

b1 − b2

⇒MGUT ' 1014 GeV (6.9)

and the value of the gauge coupling at the GUT scale is

α−1GUT = α−1

2 −b2

2πtGUT ' 40. (6.10)

If this theory is embedded in a simple SU(5) GUT, the resulting six-dimensional

proton decay is

Γ(p→ e+π0) ' α2GUTm

5p

M4GUT

' 10−35 s−1, (6.11)

which is far too fast. This implies that GUT-scale physics is non-minimal and must

suppress gauge-mediated proton decay. Several approaches exist in the literature to

deal with this task. One possibility, discussed in [168], is to embed the theory in a

five-dimensional orbifold model. Proton decay is still allowed, but is highly suppressed

due to the configuration of the fields in the extra dimensions. Trinification, a GUT

based upon the group [SU(3)]3, provides another option because it completely forbids

proton decay via gauge bosons [180].

6.2. RG INFLUENCE ON LOW ENERGY SPECTRUM 99

6.2.2 Quartic Couplings

The ability to discover the six Higgs dark matter candidate depends on its mass and

its couplings to SM particles. The gauge interactions are fixed, but the couplings to

the SM Higgs are model-dependent. These couplings must satisfy two requirements:

perturbativity and vacuum stability. In addition, they must adhere to experimental

constraints from Higgs and dark matter searches (see Sec. 6.1). The model depen-

dence comes into play when choosing the value of the quartics at the GUT scale.

The most common understanding of how fine-tuning can give rise to Higgs and

dark matter candidates near the electroweak scale invokes a landscape of vacua, each

with its own values for couplings and masses. The string theory landscape allows for

a great range of possibilities for the physical parameters of the theory and naturally

leads to the question of what the typical values are in our neighborhood of vacua. The

distribution of couplings is clearly a UV sensitive question and cannot be obtained by

dimensional analysis because the quartics are dimensionless. Fortunately, there are

simple ansatze that lead to distinct weak-scale spectra.

This section will explore two possible GUT-scale distributions of the quartics:

parameter space democracy and susy. The couplings at the weak scale are obtained

by application of the renormalization group equations. The resulting differences in

weak-scale phenomenology for each of these distributions will be explored in Sec. 6.4.

Parameter Space Democracy

Perhaps the most obvious distribution of parameters is one where all couplings at the

GUT scale are equally probable – “parameter democracy.” This measure favors large

couplings of either sign. If all the couplings are positive, they quickly run down to

perturbative values and the initial boundary conditions of the quartics are not terribly

important. When the quartic couplings start off negative, they can become asymp-

totically free and may have Landau poles. Furthermore, negative quartic couplings

can lead to vacuum decay, especially when they have a large magnitude initially; thus,

most of the parameter space in the negative direction is ruled out.

Typically, the couplings approach a tracking solution rather rapidly [181]. Neither


λ5 nor λ6 has a significant affect on the fixed point values of the other couplings. The

gauge boson and top quark contributions also do not significantly affect the runnings

in this region. With these observations, the beta functions may be approximated as

16π2dλidt

= bλi , (6.12)

where

bλ1 ' 24λ21 + 2Nhλ

23 + 2Nhλ3λ4 +Nhλ

24

bλ2 ' 4(2Nh + 4)λ22 + 2λ2

3 + 2λ3λ4

bλ3 ' 4λ23 + 2λ2

4 + 4λ4(λ1 +Nhλ2)

+4λ3(3λ1 + (2Nh + 1)λ2)

bλ4 ' 4λ4(λ1 + λ2) + 4λ24 + 8λ3λ4

and Nh is the number of scalars added to the SM, in addition to the usual Higgs [182].

For the model considered here, Nh = 5.

When λ1 and λ2 are large at the GUT scale, the self-coupling terms in the beta

functions (6.12) dominate and the low energy values for these couplings are approxi-

mately

λmax1 ' 16π2

24tGUT

∼ 0.24 λmax2 ' 16π2

120tGUT

∼ 0.05. (6.13)

Figure 6.1 (gray points) shows the weak-scale distribution of λ1 and the effective

coupling

λeff = λ3 + λ4 − 2|λ5|, (6.14)

which parametrizes the interaction of the WIMP candidate a05 to the SM Higgs h0.

The values of the quartics at the GUT scale were randomly sampled within the range:

0 . λ1, λ2, |λ3| . O(4π), −1 . λ4 . 0, and |λ5| . 2.† They were then run down

to the electroweak scale by applying the renormalization group equations. Despite

the large range of possibilities at UV energies, the couplings are focused down to a

†Quartics outside this range either give the same result for the low-energy spectra or, as in thecase of λ5, cause the couplings to run down to non-perturbative values. This range was chosen tomaximize the sampling rate of the program.

6.2. RG INFLUENCE ON LOW ENERGY SPECTRUM 101

Figure 6.1: (Left) Distribution of λ1 and λeff = λ3 + λ4 − 2|λ5| at the electroweakscale obtained by solving the one-loop renormalization group equations for parameterspace democracy (gray) and susy (black) boundary conditions at the GUT-scale. Thecouplings are focused down to a small range at the weak scale. (Right) The distribu-tion of SM Higgs mass for the two sets of boundary conditions. The distribution ofallowed Higgs masses is smaller in the case of susy boundary conditions as opposedto parameter space democracy conditions.

narrow set at electroweak energies. Indeed, λ1 and λ2 do not vary much from the

values approximated in (6.13). The region of parameter space at the electroweak scale

corresponds to

0.1 <∼ λ1<∼ 0.3, 0 <∼ λ2

<∼ 0.1,

−0.2 <∼ λ3<∼ 0.4, −0.5 <∼ λ4

<∼ 0. (6.15)

λ5 renormalizes itself and thus remains small (|λ5| <∼ 0.1). With the assumption of

parameter democracy, the model comes close to saturating the upper values for λ1

and λ2, having a small λ3, and having a λ4 that is close to saturating the lower bound.

The acceptable range for the Higgs mass is

114 GeV . mh0 . 200 GeV. (6.16)

Higgs masses at the upper-end of this interval are preferred (see Fig. 6.1).


Minimal Susy Boundary Conditions

Another plausible set of boundary conditions are ones where supersymmetry is broken

at the GUT scale and the dominant quartic couplings are those arising from D-terms.

The simplest way of achieving the desired low-energy spectrum is if each low-energy

Higgs doublet comes from a vector-like chiral superfield: Φh and Φch for the Standard

Model Higgs and ΦH5 and ΦcH5

for the five-plet of scalar dark matter. Specifically,

Φh| = cβh− sβh Φch| = sβh

† + cβh†

ΦH5| = cβ5H5 − sβ5H5 ΦcH5| = sβ5H

†5 + cβ5H

†5,

where β and β5 are the orientation of the scalars inside the chiral superfields. The

resulting D-term potential has the following couplings

λ1 =2

5g2

GUTc22β λ2 =

2

5g2

GUTc22β5

λ3 = − 7

10g2

GUTc2βc2β5 λ4 = g2GUTc2βc2β5

λ5 = 0 λ6 = 0. (6.17)

A term must be added to the superpotential to generate λ5 = 0. In order that this not

alter the above relations significantly, it should be a small coupling. One possibility is

to take the minimum value allowed by direct detection experiments, λ5 ∼ 10−6. The

gauge couplings at the GUT scale are g2GUT = 0.32, so the susy boundary conditions

result in small couplings at the electroweak scale. In order to have neutral dark

matter, λ4 < 0, so cos 2β cos 2β5 < 0.

These couplings are a function of two angles and lead to a lighter Higgs and smaller

mass splittings for the scalars than the case of parameter space democracy. This is

apparent from Figure 1, where the black points show the allowed values of λ1 and λeff

at the electroweak scale obtained using the susy boundary conditions. In this case,

the Higgs mass falls within a much smaller range

147 GeV . mh0 . 159 GeV (6.18)

6.3. DARK MATTER 103

and is lighter than the most probable Higgs mass for parameter space democracy.

6.3 Dark Matter

6.3.1 Relic abundance

The lightest neutral component of the H5 doublet, a05, is a viable candidate for the

observed dark matter and its mass may be estimated from standard relic abundance

calculations. It is assumed that the a05 is in thermal equilibrium during the early

universe. When the annihilation rate of the a05 is on the order of the Hubble constant,

its number density ‘freezes out,’ resulting in the abundance seen today.

When the freeze-out temperature is on the order of the mass splittings ∆ms0a0

and ∆mφ±a0 , the presence of the additional scalars s05 and φ+

5 becomes relevant [183].

In this case, interactions involving the two other scalars as initial state particles

are important in determining the relic abundance of a05, which must fall within the

WMAP region 0.099 < Ωdmh2 < 0.113, where Ωdm is the dark matter fraction of the

critical density and h = 0.72 ± 0.05 is the Hubble constant in units of 100 km s−1

Mpc−1) [184]. Typically, coannihilation has a significant effect on the allowed mass

range of the relic.

The number density of a05 is given by

dn

dt= −3Hn−

∑i,j=a0,s0,φ±

〈σijvij〉(ninj − neqi neqj ), (6.19)

where σij is the sum of the annihilation cross sections of the new scalars Xi into

Standard Model particles X

σij =∑X

σ(XiXj → XX). (6.20)

The first term on the r.h.s. of equation (6.19) accounts for the decrease in the relic

density due to the expansion of the universe; the second term results from dilution

of the relic from interactions with other particles. The annihilation rate depends on


the number of scalars added to the theory in addition to the SM Higgs, Nh, through

the interaction cross sections σij. In general, σij ∝ N−1h m−2

a , so the mass of the dark

matter scales as

ma0 ∝ 1√Nh

. (6.21)

Thus, in the non-resonance regime, the dark matter mass decreases with the number

of electroweak doublets added to the SM. For this reason, the six Higgs doublet model

gives lighter dark matter than the inert doublet model [173].

When ma0 . 80 GeV, the only annihilation channel is to a pair of fermions.

Because these cross sections tend to be rather small, Ωdmh2 & 0.1. However, a

resonance due to s-channel SM Higgs exchange causes a sharp decrease in the relic

density ∼ 80 GeV, bringing it within the WMAP experimental range. For ma0 & 80

GeV, diboson production is the dominant annihilation mechanism and keeps the

abundance small. There is always another point in this large mass regime where

Ωdmh2 ∼ 0.1. Thus, the dark matter can take two possible mass values - one light

(∼ 80 GeV) and the other heavy (& 200 GeV).

The relic abundance calculation was performed numerically by scanning over the

parameter m2a0 for each set of randomly selected quartic couplings. Figure 6.2 is a

plot of the allowed mass of a05 as a function of the SM Higgs mass. A broad range of

values is allowed for the case of parameter space democracy, with ma0 falling between

∼ 200 − 700 GeV. For supersymmetric boundary conditions, the mass values range

from ∼ 200 − 400 GeV. An ∼ 80 GeV dark matter particle is also allowed for both

cases.

6.3.2 Bounds from electroweak precision tests

Electroweak precision tests place limits on the light mass range of the dark matter

[185]. The contribution of the new particles to the T parameter is given by

∆T =Nh

16π2αv2

[F (mφ± ,ma0) + F (mφ± ,ms0)

−F (ma0 ,ms0)], (6.22)

6.3. DARK MATTER 105

Figure 6.2: Allowed mass of the LSP a05 as a function of the Higgs mass for parameter

space democracy (gray) and susy (black) boundary conditions. All points includedin this plot fall within 1σ of the electroweak precision data (Sect. 6.3.2) and areconsistent with LEP results (Sect. 6.4.3). The SM Higgs can decay into a pair ofWIMPs if ma0 lies below the dashed red line.

where

F (m1,m2) =m2

1 +m22

2− m2

1m22

m21 −m2

2

logm2

1

m22

. (6.23)

The expression for F (m1,m2) can be simplified if one assumes that the mass splitting

∆m = m2 −m1 satisfies ∆m/m1 1. In this limit,

F (m,m+ ∆m) =2

3(∆m)2 +O

((∆m)4

m2

)(6.24)

and the expression for ∆T reduces to

∆T ' Nh

12π2αv2(mφ± −ma0)(mφ± −ms0)

' Nhv2

192π2αmams

(λ24 − 4λ2

5). (6.25)

Because λ5 is typically smaller than λ4, ∆T is always positive. In the minimal

SM, ∆T is driven more negative as the mass of the Higgs increases. The additional

scalar doublet H5 compensates for this change, driving T positive.


The S parameter also has contributions from the additional Higgs doublets [173]

and is easily generalized to the case of six Higgses

∆S =Nh

2π

∫ 1

0

x(1− x) log[xm2

s0 + (1− x)m2a0

mφ±

]dx. (6.26)

When the mass splittings are small,

∆S =Nh

12πma0

(∆ms0a0 − 2∆mφ±a0

)+O

(∆m2

m

)' Nhv

2λ4

24πm2a0

. (6.27)

For the heavy dark matter candidate, the corrections to the S and T parameters

fall well within the 1σ electroweak precision data [173]. Lighter dark matter can

make significant contributions to the S and T parameters. Couplings that give rise

to deviations in S and T that are more than 1σ away from the measured values have

not been used in the analysis of the experimental signatures of the model (Sect. 6.4).

6.4 Experimental Signatures

6.4.1 Direct detection

Direct detection experiments provide a means for observing the dark matter relic

when it scatters elastically off atomic nuclei [8]. The WIMP can either couple to the

spin of the nucleus or to its mass. The spin-independent contribution to the cross

section usually dominates and bounds on its value are being set by experiments such

as CDMS, DAMA, Edelweiss, ZEPLIN-I, and CRESST.

In the six Higgs doublet model, there are two contributions to the spin-independent

cross section. The first comes from an s-channel Higgs exchange described by the ef-

fective Lagrangian

Leff =∑q

(−iλeff

m2h0

)mqa

0a0qq. (6.28)

Experimental results are usually reported in terms of the cross section per nucleon,

6.4. EXPERIMENTAL SIGNATURES 107

Figure 6.3: Cross section per nucleon for the case of parameter space democracy(gray) and susy (black) boundary conditions. The lightest WIMP candidates willbe tested for at the current CDMS II run (upper dashed line). The third phase ofSuperCDMS (lower dashed line) will probe a greater region of the parameter space.

which in this case is

σn = 2× 10−9pb

(λeff

0.4

)2(350 GeV

ma0

)2(200 GeV

mh0

)4

. (6.29)

This cross section scales as Nh (see Eq. 6.21). Because σn ∝ m−2a0 , the lighter dark

matter candidate will have a stronger signal than its heavier counterpart. Figure 6.3

shows the cross section per nucleon for the case of parameter space democracy (gray)

and susy (black) boundary conditions. The current CDMS II run is sensitive to the

lightest WIMPs predicted by the model. A larger portion of the parameter space is

within the testable reach of the proposed SuperCDMS experiment [19]. The lower

dashed line on the plot is the expected limit from Phase C of SuperCDMS.

Another contribution to the spin-independent cross section comes from the inelas-

tic vector-like interaction a0 +p→ s0 +p, which is mediated by an off-shell Z0-boson.

In general, such inelastic transitions provide a means to reconcile DAMA’s detection

of relic-nucleon scattering, which conflicts with CDMS’s null result [33]. Consistency

with the experimental results requires a mass splitting ∆ms0a0 ' 100 keV between

the two lightest scalars [186].


6.4.2 Indirect detection

A concentration of WIMPs in the galactic halo increases the probability that they will

annihilate to produce high-energy gamma rays and positrons [187]. The gamma ray

signal is of particular interest because it is not scattered by the intergalactic medium;

thus, it should be possible to extract information about the WIMP mass from the

spectrum.

Monochromatic photons can be produced when the WIMP annihilates to produce

γγ and Z0γ. The dominant mechanisms that contribute to this annihilation depend

on the DM mass regime. The light dark matter, for example, annihilates primarily

through s-channel Higgs exchange with a one-loop h0γX vertex (X = γ, Z0). The

main contributions to the loop come from the W± boson, the top quark, and the φ±5

five-plet. Other box diagrams are suppressed. The WIMPs are highly non-relativistic

and their annihilation cross section in the light mass regime is nearly

σ(a0a0 → γX)u ' 1

Nh

v2λ2eff

(s−m2h0)2 +m2

h0Γ2h0

Γ(h0 → γX)√s

, (6.30)

where u is the relative velocity between the initial two WIMPs and s ≈ 4m2a0 . The

general expressions for the decay widths of the Higgs boson into a γγ and γZ0 final

state are found in [188,189].

The case of the heavy dark matter is significantly different [190]. In this regime,

the dominant contribution comes from the box diagram with three φ±5 and one W+

in the loop. When the a05 and φ±5 are nearly degenerate and ma0 mW± , there is an

effective long-range Yukawa force between the φ+5 φ−5 pair in the loop that is mediated

by the gauge boson:

V (r) ∼ −α2e−mW±r

r. (6.31)

As a result, the pair of charged scalars form a bound-state solution to the non-

relativistic Schrodinger equation. The optical theorem is used to obtain the s-wave


Figure 6.4: Approximate flux from dark matter annihilation in the galactic halovia a0

5a05 → γγ for parameter space democracy (gray) and susy (black) boundary

conditions. The dashed lines indicate the sensitivity of GLAST (red) and the ground-based detector HESS (green). The NFW profile was used.

production cross section for the bound state:

σ(a5a5 → φ+5 φ−5 )u ∼ 2α2

2m2a0

Nhm2W±

(1 +

√2m∆mφ±a0

m2W±

)−2

. (6.32)

Multiplying this by the decay width of the bound state to two photons (or, γZ0),

gives the total annihilation cross section

σ(a5a5 → γγ)u ∼ 2πα2α22

Nhm2W±

(1 +

√2m∆mφ±a0

m2W±

)−2

. (6.33)

This cross section does not depend onma0 (to zero-th order in the mass splittings) and,

as a result, is significantly enhanced in the heavy DM mass region. This enhancement

is critical; because of it, the heavy mass DM may be visible in gamma ray experiments.

Additionally, the only parameter dependence comes in through the mass-splittings,

which are small. Therefore, there is not much spread in the range of allowed cross

sections.

The monochromatic flux due to the gamma ray final states observed by a telescope


with a field of view ∆Ω and line of sight parametrized by Ψ = (θ, φ) is given by

Φ = CγX

(σγXu

1 pb

)(100 GeV

ma0

)2

J(Ψ,∆Ω)∆Ω, (6.34)

where

Cγγ = 1.1× 10−9cm−2s−1

CγZ0 = 5.5× 10−10cm−2s−1 (6.35)

and the function J includes the information about the dark matter distribution in

the halo. Note that the flux is independent of Nh. For the NFW profile, J ' 103

for ∆Ω = 10−3 [187]. Other profile models exist with either more mildly/strongly

cusped profiles at the galactic center [189]. Depending on which model is chosen, J

can be as small as 10 or as large as 105 for ∆Ω = 10−3. In this work, the moderate

NFW profile will be used, however the result can easily be scaled by two orders of

magnitude to get the predictions for other halo profiles.

The expected monochromatic flux for the γγ line is shown in Figure 6.4 (assuming

J∆Ω = 1). The estimated flux is right beneath the sensitivity limits of the ground-

based HESS detector (green line) and space-based GLAST telescope (red line) [189,

191]. The results for the γZ0 line are similar for low DM masses and are enhanced by

an order of magnitude for masses greater than 200 GeV, putting it within the reach

of HESS. Given the two order of magnitude uncertainty in the flux coming from the

details of the halo profile, the gamma ray line is an interesting signal for both current

and upcoming experiments.

6.4.3 Collider Signatures

It is possible that the scalars of the Higgs 5-plet were produced at the e+e− collider

LEP with√s ∼ 200 GeV via the processes

e+e− → φ+5 φ−5 and e+e− → a0

5s05. (6.36)


Figure 6.5: Mass splittings for the light dark matter (ma0 ∼ 80 GeV). LEP excludesthe region of intermediate s5 mass. The region on the upper left is excluded byelectroweak precision results, while that on the lower right is excluded by vacuumstability. Results are shown for parameter space democracy (gray) and susy (black)boundary conditions.

LEP placed limits on the production cross sections for the neutralino and chargino [77]

and these bounds can be directly translated to the processes in (6.36). By doing so,

approximate limits on the masses of the scalars a5, s5, and φ±5 can be deduced. The

charged Higgs 5-plet φ±5 is ruled out for masses below ∼ 90 GeV and the neutral

scalar s05 is ruled out for masses between ∼ 100 − 120 GeV (depending on the mass

of a5). Fig. 6.5 summarizes the important constraints on the mass splittings ∆mφ±5 a0

and ∆ms0a0 for the light dark matter.

The heavy dark matter candidate (ma0 & 200 GeV) could not be produced at

LEP. In addition, its contributions to the electroweak parameters always fall within

the 1σ experimental bounds. Thus, the main constraints on the mass splittings in this

region of parameter space come from vacuum stability and perturbativity. Typically,

∆mφ±5 a0 ,∆ms0a0 . 20 GeV in the heavy dark matter regime.

One of the most promising discovery channels for the 5-plet scalars at the Tevatron

and LHC is the width of the SM Higgs. The Higgs can decay into a5, s5, or φ±5 , in

addition to the SM modes. In Fig. 6.2, all points below the red dotted line satisfy

mh0 > 2ma0 ; here, the SM Higgs can decay into the dark matter. This decay channel

is open for significant portions of both the parameter space democracy and susy

boundary condition cases. For small mass splittings, decays to s5 and φ±5 are also


Figure 6.6: Width of the SM Higgs decay into the H5 scalars for the case of parameterspace democracy (gray) and susy (black) boundary conditions. The top line showsthe width of the SM decay modes, ΓSM. The bottom line is 0.1ΓSM .

possible, though they are subdominant.

The contribution of the new invisible decays to the width of the Higgs is

Γinv =Nhv

2

32πmh0

[λ2

eff

√1− 4m2

a0

m2h0

+ 2λ23

√1− 4m2

φ±

m2h0

+(λeff + 4|λ5|)2

√1− 4m2

s0

m2h0

]. (6.37)

Fig. 6.6 plots Γinv as a function of the SM Higgs mass. The (top) line is the width

due to the SM decay modes [192]. For points above the line, the invisible decays into

the 5-plet scalars are the dominant contribution. However, even when Γinv ∼ 0.1ΓSM,

it should be possible to detect the additional decay modes at the Tevatron or the

LHC.

The scalars may also be produced at the Large Hadron Collider via interactions


like

pp → a05s

05 → Z∗ + 6ET

pp → φ+5 φ−5 → W ∗W ∗ + 6ET

pp → s05φ±5 → Z∗W ∗ + 6ET

pp → a05φ±5 → W ∗ + 6ET . (6.38)

The vector bosons are always off-shell because the scalar mass splittings are less than

80 GeV (see Fig. 6.5) and, after using their leptonic branching fraction, it will be

challenging to detect this signal in the presence of a large background.

As an example, consider the first two processes in (6.38), which both result in

opposite-sign leptons plus 6ET after the decay of the gauge bosons. The production

cross section σprod for s5a5 and φ+5 φ−5 at the LHC was calculated using MadGraph [193]

for a sample point in parameter space and is plotted as a function of WIMP mass in

Fig. 6.7. The cross section for the SM background (thick line) is

σbackground = σ(pp→ WW )Br(W → lν)2

+σ(pp→ ZZ)Br(Z → l+l−)Br(Z → νν).

(6.39)

The signal cross section may be estimated as

σsignal ∼ Br(Z → l+l−)σprod, (6.40)

where the branching fraction is about 1% (dashed line). The ratio of signal to back-

ground is about 1:10 for the low mass dark matter. At higher mass, it is about 1:1000.

This estimate indicates that it may be possible to see the signal for the low-mass DM

region, if appropriate cuts are placed.


Figure 6.7: LHC production cross section σprod for a05s

05 and φ+

5 φ−5 , assuming ∆ms0a0

= 10 GeV, ∆mφ±a0 = 15 GeV, mh0 = 120 GeV, and λ3 = 0.3. The dotted line is thesignal cross section. The thick black line is the cross section for the SM background(see text).

6.5 Conclusions

In this paper, a minimal extension of the Standard Model was presented that lead to

gauge unification and a dark matter candidate. An electroweak doublet Hi in a 5 of a

global discrete symmetry was introduced. One of the new five-plet of particles is light

and neutral and serves as a good dark matter candidate. The addition of six scalars

to the SM leads to gauge coupling unification and fixes the number of electroweak

doublets.

The six Higgs doublet model has distinct signatures for direct detection, indirect

detection, and collider experiments. Typically, the light mass range (ma05∼ 80 GeV)

has the most promising signals, and will be tested for by GLAST and CDMS. In

addition, it can be produced by decays of the SM Higgs at the Tevatron or LHC.

The heavier candidates (ma05& 200 GeV) are more difficult to see, but lie within the

sensitivity of the HESS gamma ray detector and the next-generation direct detection

experiment, SuperCDMS. Direct production of these heavier candidates at colliders

is challenging due to large Standard Model backgrounds from di-boson production,

though further study is needed to determine whether appropriate cuts can reduce

these backgrounds.

6.5. CONCLUSIONS 115

Throughout this discussion, it has been assumed that an exact discrete symmetry

exists to keep the full five-plet of dark matter light under one fine-tuning. Discrete

symmetries can arise in string theoretic constructions (e.g., see [194]), and the ex-

istence of these symmetries is critical for viability of this particular model. If this

requisite symmetry is relatively common, then a single fine-tuning of the scalar mass

is comparable to the tuning necessary in the “minimal model” described in [168,169].

In general, this class of minimal models is not as economical in terms of fine-tuning

as split susy, where the desired mass spectrum is obtained by having the R-symmetry

breaking scale be small. However, these minimal models give rise to interesting phe-

nomenology that will be tested in upcoming experiments.

Chapter 7

Parity Violation in CiDM Models

M. Lisanti and J. G. Wacker, “Parity Violation in Composite Inelastic Dark Matter

Models,” [arXiv: 0911.4483].

Recent direct and indirect searches for dark matter hint that the dark sector may

have non-minimal structure and interactions. This is in sharp contrast with the

standard scenario of weakly interacting massive particles in which the dark matter

is the lightest neutral state in a spectrum and interacts elastically off of Standard

Model (SM) particles. The results of the DAMA experiment provide an example

of an anomaly that challenges the standard dark matter picture [34, 35]. In par-

ticular, if DAMA’s measured annual modulation arises from inelastic dark matter

(iDM), then DAMA can be reconciled with all other null results from direct detection

experiments [195, 147, 138]. The presence of multiple states that lead to inelastic in-

teractions may indicate novel dynamics in the dark sector. For example, iDM requires

an O(100 keV) splitting between the dark matter states, which may be a sign that

dark matter is composite [40, 155,196,197,198,199,200,201,202,156].

Composite inelastic dark matter (CiDM) is a recent proposal that provides a

dynamical origin for the 100 keV mass splitting [40]. CiDM models have a ground

state degeneracy that is split by the hyperfine interaction. In [40], a minimal CiDM

model was proposed where a new strong gauge group confines at low energies and

the quarks that are charged under the new strong gauge group form “dark hadrons”

116

7.1. MODELS OF CIDM 117

after confinement. The interactions between the Standard Model and the dark sector

are mediated by a kinetically-mixed U(1)d.

The parity of the U(1)d current determines how visible the dark matter is to

the Standard Model. The minimal CiDM model conserved parity and the axial-

vector current interaction led to only inelastic interactions. However, parity can be

explicitly broken by the dynamics of the new strong gauge group through a Θ term.

Parity violation leads to elastic interactions that arise from charge-radius scattering

and are phenomenologically different from typical elastic dark matter interactions

because the charge-radius scattering is suppressed at low nuclear recoil energies. The

recoil spectrum of this model looks strikingly similar to standard iDM models due

to the suppression from low recoil energy interactions. CiDM models provide a new

framework for studying kinematic scenarios where several types of scattering events

are allowed.

This article presents the theory of CiDM models with parity violation. Sec. 7.1

describes the low energy effective theory in terms of “dark mesons.” Sec. 8.1 com-

putes the direct detection phenomenology using a global fit while marginalizing over

the uncertainty in the dark matter velocity distribution function. Sec. 7.3 discusses

constraints arising from QED tests and summarizes prospects for collider searches.

7.1 Models of CiDM

In this section, the effective field theory for CiDM models is reviewed and the con-

sequences of parity violation in the new strong sector is explored. The high energy

theory is a two flavor SU(Nc) gauge theory with a Lagrangian given by

L = LSM + LCiDM (7.1)

LCiDM = −1

2Tr G2

dµν + ΨLiD6 ΨL + ΨHiD6 ΨH

+mLΨLΨL +mHΨHΨH

where Ψa, a = L,H, are Dirac fermions that are fundamentals under the strong

gauge sector and Gdµν is the SU(Nc) gauge field strength. In Atomic Inelastic Dark

118 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

Matter (AiDM) [155], the strong gauge sector is replaced by an Abelian gauge group

where Ψa are charge ±1, respectively. If Ψa have chiral gauge charges, then the ma

arise through a symmetry breaking interaction (i.e., ma = ya〈φ〉). If the theory is

asymptotically free as in [40], then the theory will confine at a scale Λd. The bound

states will be approximately Coulombic if mL>∼ Λd and the resulting spectrum is

qualitatively similar to AiDM.

To have the appropriate relic density, the dark matter must either be very heavy

( >∼ 30 TeV) or the dark matter density must be generated non-thermally, possi-

bly linked to baryogensis. O(30 TeV) inelastic dark matter is not compatible with

CDMS’s null results [18,19,20] if it fits the DAMA signal and therefore the relic abun-

dance of the dark matter needs to be generated non-thermally. Assuming that there

is a cosmological asymmetry generated early in the Universe between heavy quarks

and light anti-quarks,

nH − nH = −nL + nL 6= 0, (7.2)

the dominant component of the dark matter will be in dark mesons with a single

heavy quark [40].

The dark matter is the ground state of a ΨHΨL meson and will be denoted as

πd. The πd is a spin 0, complex scalar with parity −1. Dark matter scattering is

primarily a transition to the complex, spin 1 meson, ρd. The parity of this state is

Pρdµ = (−1)µρµ (−1)µ =

1 µ = 0

−1 µ = 1, 2, 3. (7.3)

The mass splitting between the πd and the ρd arises through the hyperfine interaction

and is suppressed when mH mL,Λd. In particular,

κΛd2

mH

, mL Λd

δm = mρd −mπd' (7.4)

λ4dm

2L

NcmH

mL Λd,


where λd = Ncg2d/4π is the ’t Hooft coupling with Nc = 1 applying to Abelian gauge

groups and κ is an O(1/Nc) constant. Note that mπd∼ mH in the heavy quark mass

limit. For mass splittings O(100 keV) and a dark matter mass near the weak scale,

the confinement scale is ∼ 100 MeV.

The dark matter mesons interact through a massive spin 1 gauge field Aµd that

kinetically mixes with U(1)Y [42, 203,204,205,206,207]

LGauge = −1

4F 2

d +ε

2F µν

d Bµν

LHiggs = |Dµφ|2 − λ(|φ|2 − 1

2f 2φ)2, (7.5)

where Dµφ = ∂µφ − 2igdAµdφ. After φ acquires a vev, fφ, Ad becomes massive and

the mixing between the dark sector and the Standard Model can be diagonalized.

The ε mixing between the hypercharge field strength and U(1)d is the source of the

interactions between the Standard Model fermions and the dark matter. Assuming

that

mAd= 2gdfφ mZ0 (7.6)

after electroweak symmetry breaking, the couplings can be diagonalized. The inter-

actions relevant for dark matter scattering are given by

LInt = (Jµd + εcθJµEM)Adµ, (7.7)

where cθ = cos θw, JEM is the electromagnetic current and Jd is the current of the dark

quarks. A more complete analysis of the interactions is given in Sec. 7.3. Anomaly

cancellation restricts the charge assignments of the two dark quarks leaving only three

anomaly-free possibilities for the current of the dark quark sector.

7.1.1 Axially Charged Quarks

The types of interactions that are allowed depend on whether Jd is an axial or vector

current. In [40] and [155], Jd is an axial vector current. The only anomaly-free axial


charge assignment in terms of Weyl spinors is

Axial ψH ψcH ψL ψcL

SU(Nc)

qU(1)d +1 +1 −1 −1

, (7.8)

where the Dirac spinors Ψa = (ψa, ψca). The masses of the quarks arise from U(1)d

breaking through the Higgs mechanism

LYuk = yHφψHψcH + yLφ

†ψLψcL + h.c. . (7.9)

Because both the mass of the quarks and the Ad arise from the vev of φ, there is a

hierarchy between the gauge couplings and the Yukawa couplings

mAd

mπd

=2gd

yH. (7.10)

Fitting DAMA requires mπd∼ 100 GeV, while mAd

can in principle take on a range

of values from 10 MeV to 100 GeV. Eq. 7.10 implies that the gauge coupling for

the axial sector may be small in comparison to the Standard Model gauge couplings

because yH is capped by perturbativity at O(1).

The effective operators describing the interactions of the πd − ρd system are

LAxial eff = dainmπd

πd†ρµdAdµ

+ca

in

Λd

πd†∂µρνdFdµν

+da

el

Λ2d

(πd†∂µπd + ρνd

†∂µρdν)∂νFµνd

+caelρd

†µρdνF

µνd + h.c. . (7.11)

The operators with coefficients denoted by d are suppressed by a factor of the relative

velocity vrel, while the ones denoted by c are not velocity suppressed. The elastic

scattering operator for the πd is dimension 6 and velocity suppressed, resulting in an

overall suppression of the elastic to inelastic scattering rate of v2rel.


7.1.2 Vectorially Charged Quarks

There are two anomaly-free charge assignments for vectorially charged dark matter:

one gives the composite dark matter a charge and the other leaves it neutral. Charged

dark matter will have an enormous scattering rate and will look qualitatively similar

to standard elastic dark matter. The charge assignment that leaves the dark matter

neutral will only scatter off higher moments of the charge distribution and will be

suppressed at low recoil energy. The charge assignments for the neutral dark matter

theory are

Neutral Vector ψH ψcH ψL ψcL

SU(Nc)

qU(1)d +1 −1 +1 −1

. (7.12)

With these charge assignments, Ad couples to a vector current and the allowed oper-

ators are

LVector eff =dv

in

Λd

πd†∂µρνdFdµν

+cv

el

Λ2d

(πd†∂µπd + ρνd

†∂µρdν)∂νFµνd

+dvelρd

†µρdνF

µνd + h.c. . (7.13)

d denotes operators that are velocity suppressed and c denotes unsuppressed opera-

tors. The leading operator that is not velocity suppressed is the elastic charge-radius

operator, but this is a dimension 6 operator. Recent work on form factor-suppressed

inelastic transitions indicates that this type of scattering may be an explanation for

DAMA [135,134]. The primary difference between form factor elastic scattering dark

matter and iDM is the existence of a threshold in iDM. The next section illustrates

that it is possible to have dark matter dominantly scatter inelastically and have a

residual form factor elastic contribution.


7.1.3 Parity Violation

In the two models above, parity determined the interactions of the dark meson fields;

however, parity is not a fundamental symmetry of nature. If parity is broken, both

charge-radius scattering and inelastic scattering are allowed without a velocity sup-

pression. This is quite natural in strongly coupled CiDM models because CP violation

arises from the dynamics of the strong sector through the term

LP6 = ΘdTr GdGd, (7.14)

and results in mixing between states of different parity. The size of Θd is not neces-

sarily related to the size of ΘQCD and in principle Θd could be O(1).

Because the Θd term is a total derivative, its effects only appear non-perturbatively.

The dominant effect of the CP violation is to cause a small mixing between states of

different parity. In QCD, for example, the π0 with IG(JP ) = 1−(0−) and the a0 with

IG(JP ) = 1−(0+) mix in the presence of a ΘQCD term. A similar process will happen

in the dark sector. When mL . Λd, the mixing angle between fields of opposite parity

is given by

sin θP6 ∼ ΘdmL

Λd

. (7.15)

The mixing vanishes in the limit where mL → 0 because the Θd term can be removed

by a chiral rotation of the ΨL. If mL Λd, the mesons form Coulombic bound states

and the mixing angle is given by

sin θP6 =〈πd|HP6 |a0 d〉ma0 d

−mπd

' ΘdΛd

λ2dmL

, (7.16)

where the matrix elements of the perturbing CP-violating Hamiltonian is set by the

non-perturbative scale where the effects of Θd are not exponentially suppressed. As

mL →∞, the CP-violating effects decouple and parity violation vanishes. Therefore,

even if Θd ∼ O(1), its effects on the interactions of the dark mesons might be small

if mL → 0,∞. Maximal parity violation occurs when mL ' Λd.

7.2. DIRECT DETECTION PHENOMENOLOGY 123

With an axially coupled U(1)d, the πd−a0 d interaction becomes an elastic charge-

radius operator with parity violation:

Lπda0 d=cael

Λ2d

π†d∂µa0 d∂νFµνd →

cael

2Λ2d

sin 2θP6 π†d∂µπd∂νF

µνd .

Therefore, the effects of Θd can be estimated by replacing the field strengths in Eq.

7.11 and Eq. 7.13 with∗

F µνd → cos 2θP6 F

µνd + sin 2θP6 F

µνd . (7.17)

Therefore, turning on parity violation in the strong sector allows admixtures of vector

and axial vector interactions. The ratio of elastic to inelastic cross sections becomes

a free parameter. The next section will show that an upper bound of θP6 <∼ 0.08 is

necessary to avoid direct detection constraints assuming that all c, d ∼ O(1) and

mN ∼ mπd.

7.2 Direct Detection Phenomenology

Novel features in the direct detection phenomenology of composite models arise be-

cause the dark matter has a finite size Λ−1d m−1

πd. The cross section is suppressed

by an effective form factor when a neutral bound state interacts with momentum

|~q | Λd [43]. States with nonzero spin have multipole interactions with the field.

These moments vanish for states with zero spin; scalar states that can only couple

through the charge-radius and polarizability interactions are the dominant scattering

mechanisms. For the dark pion scattering off the SM, the charge-radius interaction

dominates over the polarizability interaction, which is suppressed by an additional

factor of the mixing parameter ε2.

The charge-radius is the effective size of the πd probed by the dark photon. In

the limit of small momentum transfer |~q | Λd, the wavelength of the dark photon is

too long to probe the charged constituents of the composite state and the scattering

∗This only applies to the field strengths, Fµνd , not the gauge potentials, Aµd , whose interactionsare constrained by gauge invariance.


rate is suppressed. Elastic charge-radius scattering cannot be the sole contributor to

the direct detection signal due to constraints from current null experiments. However

see [135, 134] for examples on how form factors can reconcile DAMA with the null

experiments.

The dominant scattering is inelastic and there is a subdominant elastic component

that accounts for a fraction of the total scattering rate. Specifically, the differential

scattering cross section is

dσ

dER=

(θP6

2 4m2NE

2Rκ

(mπdδm)2

+mNER

2mπdδm

)dσ0

dER,

where mN is the mass of a nucleus with charge Z recoiling with energy ER, cain, cael = 1

of Sec. 7.1.3 and

dσ0

dER=

8Z2αmN

v2

1

f 4eff

|FHelm(ER)|2(1 + 2mNER/m2

Ad

)2 . (7.18)

The scattering operators couple the dark matter states coherently to the nuclear

charge, and the Helm form factor accounts for loss of the coherence at large recoil

|FHelm(ER)|2 =

(3j1(|q|r0)

|q|r0

)2

e−s2|q|2 , (7.19)

where s = 1 fm, r0 =√r2 − 5s2, and r = 1.2A1/3 fm [208].

The differential cross section depends on the confinement scale Λd =√mπd

δm/κ,

the mass of the dark photon mAd, and the couplings of the effective theory

f 2eff =

m2Ad

κgdε, (7.20)

where κ is the O(1/Nc) constant defining the mass difference from Eq. 7.4.

To determine the preferred region of parameter space for CiDM models, a global

χ2 analysis was performed that included the results from all current direction detec-

tion experiments. This procedure is outlined in [145] and is summarized here. The


160

180

200

220

240

240

Dark Matter Mass (GeV)

! (M

eV)

60 80 100 120 140 160 180 200

95% contours

60 keV

100 keV140 keV

150

200

250

300

350

400

450

100 200 300 400 500

! (M

eV)

Dark Matter Mass (GeV)

68% contours

Figure 7.1: 95% contours in mπd− Λd parameter space for θP6 = 0.00, 0.06, 0.07, 0.08.

For this figure, κ = 1/4 and mAd= 1 GeV. The dashed lines show contours of δm in

keV. The inset shows the 68% confidence regions for θP6 = 0.00, 0.04, 0.06 for the sameκ and mAd

. The colors correspond to θP6 = 0.00 (blue), 0.04 (teal), 0.06 (magenta),0.07 (yellow), 0.08 (green).

differential scattering rate per unit detector mass is

dR

dER=

ρ0

mπdmN

∫d3v f(~v + ~ve) v

dσ

dER, (7.21)

where ρ0 = 0.3 GeV/cm3 is the local dark matter density and ~ve is the velocity of

the Earth in the galactic rest frame. There are significant uncertainties in the dark

matter velocity distribution function f(v), and constraints on the particle physics

model can vary wildly depending on the particular choice of benchmark halo model.

To find the full scope of allowed CiDM models, we marginalize over a parameterized

velocity distribution function of the form:

f(v) ∝ exp

(v

v0

)2α

− exp

(vesc

v0

)2α

, (7.22)


95% contours400

600

800

1000

1200

0 100 200 300 400 500 600Dark Matter Mass (GeV)

f eff (G

eV)

68% contours

f eff (G

eV)

600

700

800

900

1000

1100

50 100 150 200Dark Matter Mass (GeV)

Figure 7.2: 95% confidence limit regions of mπd− feff for θP6 = 0.00, 0.06, 0.07, 0.08.

For this figure, κ = 0.25 and mAd= 1 GeV. The inset shows the 68% confidence

regions for θP6 = 0.00, 0.04, 0.06 for the same κ and mAd.

where the parameters are constrained to be within

200 km/s ≤ v0 ≤ 300 km/s

500 km/s ≤ vesc ≤ 600 km/s

0.8 ≤ α ≤ 1.25 . (7.23)

These values are motivated by observational constraints [37,38] and analytic approx-

imations to the Via Lactea results [141,143,142].

The global χ2 fit is performed by marginalizing over the six unknown parameters

of the dark matter and halo model: mπd, δm, feff, v0, vesc, and α. The measurements

used in the χ2 fit are the first twelve bins of DAMA’s modulation amplitude, as

well as a single high energy bin from 8 keVee to 12 keVee [34, 35, 39]. In addition to

DAMA’s signal, the dark matter predictions are required to not supersaturate any

observation from null experiments at the 95% confidence level. The null experiments

included in the analysis are: CDMS [18, 19, 20], ZEPLINII [26], ZEPLINIII [27],

CRESSTII [23, 24], and the new XENON10 inelastic dark matter analysis [29, 30].

The 1σ and 2σ allowed regions in the mπd− Λd and mπd

− feff spaces are shown


in Fig. 7.1 and 7.2, respectively. The minimal χ2 has a value of 4.61 and the corre-

sponding point is listed as model CiDM1 in Table 7.1. The 1σ and 2σ regions are

set by

χ2 < χ2min + ∆χ2, (7.24)

where ∆χ2(1σ) = 7.0 and ∆χ2(2σ) = 12.6. Therefore, the 68% and 95% regions

are set by requiring that χ2 ≤ 11.6, 17.2, respectively. As the fraction of form factor

elastic scattering increases relative to the inelastic contribution, the allowed regions

in Fig. 7.1 and 7.2 each separate into two. At 95% confidence, dark matter masses

with mπd& 200 GeV correspond to “slow” velocity distribution functions where

v0<∼ 225km/s and α >∼ 1.15. One benchmark model is shown in Table 7.1 as CiDM3.

However, the correlations between the dark matter mass and the velocity distribution

parameters are far weaker in the low mass region (mπd. 200 GeV).

The mass of the dark photon is related to the mixing parameter ε as

ε =m2Ad

gdf 2eff

=2√

2mAdmπd

yHf 2eff

. (7.25)

For theories with dominant inelastic scattering, feff ∼ O(700 GeV) and mπd∼

O(100 GeV) to satisfy both the DAMA and null experiments. Therefore, keeping

yH ' 1

ε = O(10−4)mAd

1 GeV, (7.26)

which corresponds well with the results of the χ2 global fit. Fig. 7.3 shows the 1σ

and 2σ regions in the mAd− ε parameter space allowed by all current direct detection

experiments. A benchmark value of yH = 1 is chosen; the contours shift to larger ε

for smaller Yukawa coupling.

Light Ad alter the fit to the DAMA spectrum because the propagator suppresses

high momentum scattering events. The momentum transfer needed to explain the

highest energy bin with a statistically signifiant annual modulation rate in the DAMA


CiDM1 CiDM2 CiDM3 CiDM4

mπd72 GeV 75 GeV 234 GeV 162 GeV

δm 109 keV 105 keV 91 keV 126 keVΛd 177 MeV 177 MeV 292 MeV 286 MeVfeff 738 GeV 846 GeV 563 GeV 268 GeVε 3.7× 10−4 3.0× 10−4 2.1× 10−3 3.8× 10−4

mAd1 GeV 1 GeV 1 GeV 60 MeV

θP6 0.00 0.04 0.04 0.06v0 272 km/s 273 km/s 202 km/s 280 km/svesc 510 km/s 501 km/s 558 km/s 501 km/sα 0.86 0.82 1.30 0.98

χ2 4.6 6.2 12.8 9.9

Table 7.1: Four benchmark models showing different regions of parameter space.CiDM1 corresponds to the best-fit point. CiDM2 shows a representative mixture ofinelastic and subdominant elastic scattering. CiDM3 shows the larger mass windowwith slow halo parameters. CiDM4 shows a light mAd

model.

spectrum (ER = 5 keVee) is

|~q | =√

2mIERqI

' 120 MeV. (7.27)

If the mass of the dark photon is less than 120 MeV, its propagator in (8.3) suppresses

the scattering rate in the high energy bins. The suppression of the high momentum

transfer events can be compensated if the mass splitting, δm, grows larger; however

this eventually forces f−1eff to become large, increasing the allowed values of ε. These

effects are shown in Fig. 7.3. A low mAdbenchmark model is shown as CiDM4

in Table. 7.1. The following section presents constraints on the allowed parameter

region arising from indirect and direct searches for the dark photon.

7.3. SEARCHES FOR THE DARK PHOTON 129

0.01 0.1 1 10 1001 ´ 10-5

5 ´ 10-5

1 ´ 10-4

5 ´ 10-4

0.001

0.005

0.010

Dark Photon Mass HGeVL

Ε

Figure 7.3: The 95% limits on mAd− ε with yH = 1 for θP6 = 0.00 (blue), 0.06

(magenta), 0.08 (green). The dark gray regions are excluded by limits on the g − 2of the electron and muon (top left) and fixed-target experiments (light mAd

andmoderate ε). The light gray region shows limits from the BABAR Υ(3S)→ γµ+µ−

search; the direct search limits are model-dependent and must be interpreted on acase-by-case basis.

7.3 Searches for the Dark Photon

The dark photon communicates with the Standard Model through kinetic mixing and

experimental bounds on these interactions arise from tests of QED. The most model-

independent bound comes from the virtual exchange of the Ad between SM fields.

The best limits arise from the constraints on the magnetic dipole moments of the µ

and e [209]. The constraints can be expressed as

ε2F( m2

e

m2Ad

)< 1.5×10−8 ε2F

( m2µ

m2Ad

)< 6.4×10−6, (7.28)

where

F (x) =

∫ 1

0

dz2z(1− z)2

(1− z)2 + z/x. (7.29)

Fig. 7.3 shows that g − 2 limits are most important at low mAdand large ε.

For larger values of mAd, constraints arise from precision electroweak interactions,

which depend on the gauge terms of the Lagrangian. The kinetically-mixed U(1)d


only alters the neutral currents and the Lagrangian for this sector is

LGauge = −1

4

(F 2

dµν +B2µν − 2εF µν

d Bµν +W 23µν

)+m2Z0

2

(1 +

2h0

v

)Z2 +

m2Ad

2

(1 +

√2φ0

fφ

)A2

d

+AdJd + AEMJEM + ZJZ . (7.30)

The precision electroweak constraints have not been performed for dark photons with

masses between 1 GeV and 100 GeV. In addition to oblique corrections, there is a

non-oblique correction coming from the contribution of the dark photon to precision

electromagnetic observables, such as differential Bhabha scattering. A full analysis

is beyond the scope of this paper and will be performed in [?]. This article uses a

bound on ε of [210,211]

ε <∼ 1× 10−2. (7.31)

This constraint becomes more important than the muon g−2 limit whenmAd>∼ 250 MeV.

The width of the Z0 is altered by the presence of the dark sector. The interaction

between the Z0 and the dark current in the canonically normalized mass eigenstate

basis is

LZd '(εsθ

m2Ad

m2Z0

)Z0µJ

µd , (7.32)

to lowest order in ε and m2Ad/m2

Z0 . The width of the Z0 decay into the dark sector is

Γ(Z0 → ΨLΨL) ' Ncg2dε

2s2θm

4Ad

12πm3Z0

' Ncs2θ

12π

m8Ad

m3Z0f 4

eff

. (7.33)

Any additional Z0 decay mode cannot have a branching ratio of more than 0.18%

[212]. This sets a limit on the dark photon mass of

ε <∼0.045

N12c

( mπd

70 GeV

)(100 GeV

mAd

)3

, (7.34)


which is never a constraint.

7.3.1 Limits from Direct Production

The allowed parameter space shown in Fig. 7.3 can be further constrained by searches

for the direct production of the Ad [213,214,215,216,217]. In this section, we outline

the prospects for such searches and the challenges of translating the experimental

bounds to theoretical constraints in composite models.

If the dark photon is the lightest state in the dark sector (mAd< Λd), then it

will decay directly to the SM. Such a light Ad will be dominantly produced from

e+e− → γAd and from the decays of the dark hadrons. Once produced, the dark

photon will decay promptly; for example,

Γ(Ad → `+`−) ' 1

3ε2αc2

θmAd' 30 eV (7.35)

for the benchmark model CiDM1. Hadronic decays are also allowed, but are subdom-

inant to the lepton decays, except near resonances [214].

When the dark photon is heavier than Λd, it can either decay to dark mesons

or directly to SM leptons. However, the coupling of the dark photon to the electro-

magnetic current is suppressed by a factor of ε relative to the coupling to the dark

quarks

Ldd + Ldem ' AdµJµd + εcθAdµJ

µem. (7.36)

In this limit, the branching fraction into SM leptons is negligible:

Br(Ad → `¯) ' ε2c2θg

2

Ncg2d

' 64g2c2θ

Nc

m4πd

f 4eff

' 4.0×10−4 (7.37)

for Nc = 4 and the benchmark model CiDM1. The dark photon preferentially decays

to the dark quarks, which first parton shower, then hadronize, and finally cascade

decay back to SM particles.

The most common mesons formed in the hadronization process will be the lightest

in the spectrum. There are no light pseudo Goldstone bosons in this theory because


there is only one light quark. Therefore, the lightest meson is η′d (0−+), in analogy

with QCD. Using the SM η′ as a prototype and the SM a0 meson as a typical hadronic

state, the mass of the η′d is estimated to be

mη′d'√

3√Nc

mη′

ma0

Λd ' 1.7Λd√Nc

. (7.38)

The η′d becomes light in the large Nc limit. For Nc & 10, the dark photon can decay

to the η′d. However, η′d is cosmologically stable because it has a chirality suppressed

decay to electrons and primarily decays via a loop-induced process to two leptons,

which dominates over the four body decay η′d → A∗dA∗d [214, 218]. The decay width

is suppressed by an additional factor of (Λd/fφ)2 relative to the φ decay mode of

[214, 218, 219] because the η′d has to mix with the φ to mediate the decay. The

resulting decay width is

Γ(η′d → e+e−) ' ε4α2m2emη′d

Λ2d

(4π)3f 4φ

1

1010years. (7.39)

The cosmological relic abundance of η′d is sufficiently small to make up a small fraction

of the matter density of the Universe.

Unlike the η′d, the next-lightest meson, ωd (1−−), can have prompt decays. The

mass of the ωd also becomes small in the large Nc limit. The ωd will decay to SM

leptons by mixing with the dark photon. Approximating the mixing angle by

θωd' m4

ωd

(m2ωd

+m2Ad

)2, (7.40)

the decay width is [218]

Γ(ωd → e+e−) ' ε2αg2

dc2θ

3

m5ωd

m4Ad

' αc2θ

3

Λ5d

f 4eff

' 1

20 m(7.41)

for the CiDM1 benchmark point in Table 7.1. Therefore, if the ωd is produced, it will

decay to two leptons with a long displaced vertex.

The dark photon will decay promptly to leptons if mAd Λd and both the


BABAR [220] and CLEO [221] searches for Υ(3S)→ γµ+µ− may be used to set

bounds. However, when the dark photon decay channels are closed, the muon decay

channels are often closed as well because Λd<∼ 2mµ. The estimated bounds from these

searches are shown in Fig. 7.3. It may also be possible to use the Υ(1S) → γ + X,

where X is invisible, when the Ad decays outside the detector [222] .

The best chance of discovering the dark sector is by directly producing the Ad

at low-energy lepton colliders. BABAR has recently searched for the Ad in the 4`

channel [223] and future work is being pursued at a myriad of experiments [224,225].

The searches are complicated because the decay of theAd back to leptons is suppressed

by the factor in Eq. 7.37. To gain efficacy, it is necessary to use the decay into the

dark quarks. For mAd/Λd

<∼ O(10), the number of dark hadrons is moderate. Because

the fluctuations in the number of hadrons is non-Gaussian, the cost of fragmenting to

two ωd mesons is not limiting and it is possible to set limits using the BABAR and

CLEO searches. When mAd/Λd 1, there can be a large number of dark hadrons

in the decay products of the Ad, and an inclusive, multi-lepton search is necessary.

When the dark photon decays through the hadronic channel, the analysis becomes

more challenging because it is necessary to know how the dark partons fragment into

dark hadrons and then decay down to the Standard Model. Setting limits on this

model is beyond the scope of this work because of the significant uncertainties in the

hadronic spectrum.

In addition to low energy searches for the decay products for the Ad, high energy

colliders provide a useful laboratory. LEP-I can search for rare decays down to the

Br(Z0) <∼ 10−5. From Eq. 7.33, this corresponds to masses of the Ad>∼ 4 GeV.

When the Ad decays with a mass mAd Λd, it decays into a pair of dark quarks,

ΨLΨL, and proceeds to shower and hadronize. Future studies of LEP2 are needed to

determine the relevant final states and the procedure necessary to set limits on the

hetrogeneous final states.


7.4 Discussion

This article introduced a composite inelastic dark matter model with dominant in-

elastic scattering off of nuclei and a subdominant elastic scattering component. The

subdominant elastic component is a signature of the symmetry structure of the model

and is a critical feature to measure. It was found that parity violating effects can be

nearly maximal, with

θP6 ' ΘdmL

Λd

<∼ 0.08. (7.42)

Discovering the elastic subcomponent will place a lower limit onmL and could sharpen

the Standard Model’s strong CP problem and flavor structure.

Directional detection experiments, which measure both the energy and direction

of recoiling nuclei in detectors [45, 226, 227], can be used to distinguish the elastic

and inelastic scattering components by looking for large-angle scattering events [145].

There is an upper bound on the allowed scattering angle, which depends on the types

of interactions that are allowed. In particular,

cos γmax =vesc − vmin

ve, (7.43)

where γ is the angle between the direction of the Earth’s velocity and the recoiling

nucleus in the lab frame and vmin is the minimum velocity to scatter at a given recoil

energy,

vmin(ER) =

√

mNER2µ2 elastic

1√2mNER

(mNERµ

+ δm)

inelastic, (7.44)

and µ is the reduced mass of the dark matter-nucleus system. Inelastic scattering

events have a much smaller cos γmax than elastic scattering events and these two

types of interactions can be distinguished with the next generation of directional

detection experiments [44, 228,229,230,231].

The best fit for CiDM had δm ∼ 100 keV and mπd∼ 70 GeV. This leads to an

7.4. DISCUSSION 135

estimate of

Λd '√δmmπd

= 150 MeV (7.45)

for the dark sector confining scale. The lower bound is Λd>∼ 70 MeV/

√κ and the

upper bound is Λd<∼ 240 MeV/

√κ (for Fig. 7.1, a value of κ = 0.25 is used). This

indicates that the dark matter form factor may be important in shaping the higher

energy bins of DAMA. Applying a dark matter form factor might therefore change

the allowed parameter range. For Coulombically-bound dark matter, the form factor

can be found by Fourier transforming the hydrogenic wave functions to get

FDM(q2) =1

1 + q2r2DM

, (7.46)

where r−1DM = λdmL is the Bohr radius of the Coulombically-bound state and λd is

the ’t Hooft coupling evaluated at the Bohr radius. The strongly interacting form

factors can be estimated by extrapolating r−1DM → Λd and behave similarly to having

mAd∼ r−1

DM.

CiDM models have sub-components to the dark matter that are not the πd. There

are roughly three classes of particles: the ρd, multiple heavy quark mesons (e.g.,

ΨHΨHΨLΨL states), and baryons (e.g., ΨH · · ·ΨH states). The relative populations of

these states is determined by the interactions in the early Universe [40,41]. Detecting

the latter two classes of particles would be a clear indication of composite inelastic

dark matter. The signature will be striking because the mass of the dark matter

subcomponents would be near-integer multiples of the πd mass, ending at Ncmπd.

The collider signatures of CiDM are challenging because many of the decay prod-

ucts have long lifetimes and give rise to extremely displaced vertices or missing energy.

To interpret the results from current e+e− colliders, it is necessary to have better es-

timates for the dark hadron multiplicity distributions from dark photon decays. If

there are dynamics that stabilize the vev of both the Standard Model Higgs and the

dark sector Higgs, then it is possible to produce dark sector states at the Tevatron

and LHC in the decays of electroweak scale particles. The phenomenology of these


events will be similar to that of hidden valley theories, with the majority of parameter

space giving rise to extremely displaced vertices [232, 233, 234]. Beam dump experi-

ments are ideally suited for identifying leptons from particles with a finite lifetime and

provide the best prospect for discovering the dark sector through direct production.

Proposals for these experiments are presently underway [225].

Chapter 8

Directional Detection

M. Lisanti and J. G. Wacker, “Disentangling Dark Matter Dynamics with Directional

Detection,” Phys. Rev. D 81 096005 (2010).

The annual modulation anomaly from the DAMA experiment is an intriguing hint

of the identity of dark matter [34, 35]. Direct detection experiments such as DAMA

measure the energy of nuclear recoils from incident dark matter particles. The signal

experiences an annual modulation because of the variation of the Earth’s relative mo-

tion with respect to the dark matter in the halo. While the DAMA experiment has

measured an 8.2σ modulation with the correct phase, its results are in conflict with

those from CDMS [18, 19, 20], XENON10 [29, 30], CRESST [23, 24], ZEPLIN-II [26],

and ZEPLIN-III [27], which measure the total, unmodulated scattering rate.

All current direct detection experiments are optimized to look for elastic scatter-

ing, which has an exponentially falling recoil energy spectrum [37]. A distinguishing

feature of DAMA’s measured modulation spectrum is that it is suppressed at ener-

gies below ∼ 25 keVnr, where elastic events should dominate. DAMA’s results, taken

together with the null results from all other direct detection experiments, can be el-

egantly explained if the dark matter scatters inelastically off of nuclei to an adjacent

state that is O(100 keV) more massive than the ground state [195, 147, 235, 138]. In

the case of inelastic dark matter (iDM), a minimum velocity is needed to upscatter

and the recoil spectrum is suppressed below this kinematic threshold.

137

138 CHAPTER 8. DIRECTIONAL DETECTION

Inelastic dark matter has three features that lead to the consistency of all current

experiments. First, iDM has a much larger dependence on the Earth’s velocity than

elastic dark matter and thus the annual modulation is often 25% or larger, in compar-

ison to 2.5% for elastic scattering. This decreases DAMA’s unmodulated cross section

by a factor of 10 or more and therefore reduces the expected signal by the same fac-

tor in all experiments looking for unmodulated scattering. Second, iDM dominantly

scatters off heavier nuclei and lighter nuclei may not have enough kinetic energy to

excite the transition. As a result, CDMS’ sensitivity is strongly suppressed due to in-

elastic kinematics. Lastly, inelastic scattering events have higher recoil energies than

elastic events. Recent experiments such as XENON10 and ZEPLIN-III have shrunk

their recoil energy window to eliminate higher energy scattering events, reducing the

acceptance for iDM. However, XENON10 has recently reanalyzed their data over a

larger signal window to be sensitive to iDM and their latest results are used in this

article [30].

A dynamical alternative to the inelastic hypothesis was recently discussed in [135,

134]; in this scenario, the interaction between the dark matter and Standard Model

(SM) is elastic, but is suppressed by a form factor

Fdm(q2) = c0 + c1q2 + c2q

4 + · · · . (8.1)

If the first term in this expansion vanishes, the elastic scattering rate is multiplied

by additional factors of q2 = 2mNER and goes to zero as ER → 0. Form factor-

suppressed scattering can arise from multipole, polarization, and charge-radius inter-

actions between the dark sector and the Standard Model [43]. The recoil spectrum

for form factor elastic dark matter (FFeDM) is suppressed at low energies and more

closely resembles the spectrum for inelastic events rather than standard elastic events,

which peaks at low energies. However, it is more challenging to reconcile DAMA and

the null experiments using only FFeDM because the form factor depends on the

product mNER and any suppression for light nuclei can be compensated by looking

at larger energies.

In this paper, we consider a scenario where inelastic scattering is complemented


0 2 4 6 8

0.00

0.01

0.02

0.03

Recoil Energy HkeVeeL

Mod

ulat

ion

Am

plitu

deHcp

dkg

keV

eeL

Figure 8.1: Comparison of the modulation amplitude for two scattering scenarios:completely inelastic scattering (solid) and inelastic scattering with a subdominantelastic charge-radius component (dashed). The points show the modulation amplitudemeasured by DAMA and DAMA/LIBRA [34]. The electron equivalent energy (keVee)is the recoil energy rescaled by the quenching factor for iodine (qI = 0.085).

by a form factor-suppressed elastic component. These scenarios arise, for example,

in composite dark matter models [136] and yield new challenges for direct detection

phenomenology. As is illustrated in Fig. 8.1, the modulation amplitude for inelastic

scattering with subdominant charge-radius scattering (dashed line) resembles that

for inelastic scattering (solid line) and is markedly different from the regular elastic

spectrum. Distinguishing the elastic and inelastic contributions is critical for under-

standing the underlying structure of the theory. This article addresses how directional

detection experiments, which can measure the direction of the nuclear recoil in addi-

tion to its energy [236,227], can differentiate the contributions to the scattering rate.

In the following section, we review the direct detection phenomenology. In Sect. 8.2,

we introduce directional detection experiments and show how they can distinguish

the dynamics in the dark sector. We conclude in Sec. 8.3.

8.1 Direct Detection Phenomenology

Direct detection experiments measure the energy of nuclear recoils in a detector. For

a detector consisting of nuclei with mass mN , the differential scattering rate per unit


detector mass is

dR

dER=

ρ0

mdmmN

⟨dσ

dERv

⟩, (8.2)

where ρ0 = 0.3 GeV/cm3 is the local dark matter density. The rate depends on the

dark matter mass mdm, as well as the differential cross section. When both elastic

and inelastic scattering are allowed, the differential cross section is parameterized by

dσ

dER=

[c2

el

(2mNERΛ2

)nel

+ c2in

(2mNERΛ2

)nin

]dσ0

dER,

(8.3)

where cel,in are the dimensionless couplings for the elastic and inelastic interactions,

Λ is the scale for new physics, and

dσ0

dER=mN

2v2

σN

µ2

(fpZ + fn(A− Z))2

f 2p

|FH(ER)|2 (8.4)

is the standard rate for elastic scattering off a nucleus with charge Z and atomic

number A. Here, µ is the reduced mass of the dark matter-nucleus system, σN is

the cross section for the dark matter-nucleus interaction at zero momentum transfer,

and fp,n are the couplings to the proton and neutron, respectively. Our results are

normalized to fp = 1 and fn = 0, and we assume that Λ = 1 GeV.

The scattering operators coherently couple the dark matter states to the nuclear

charge, and the Helm form factor accounts for loss of coherence at large momentum

transfer, q,

|FH(ER)|2 =

(3j1(|q|r0)

|q|r0

)2

e−s2|q|2 , (8.5)

where s = 1 fm, r0 =√r2 − 5s2, and r = 1.2A1/3 fm [208].

Eq. 8.3 captures the dependence of the cross section on the recoil energy ER.

In general, non-relativistic scattering only depends upon q2 and the relative velocity,

leading to a power series in q2 and vrel. Eq. 8.3 includes the leading non-vanishing term


of this series, which is typically a good approximation. The specific values of nin and

nel are model-dependent. For instance, regular iDM has nin = 0, while regular elastic

scattering corresponds to nel = 0. Form factor suppression in either the inelastic or

elastic scattering interactions results in additional powers of ER. Composite inelastic

dark matter models have nin = 1, though this does not make a substantive change to

the spectrum as long as Λ2 mNER [40,155]. In this paper, we focus on the scenario

where inelastic scattering (nin = 1) is supplemented by a form factor-suppressed

elastic component (nel = 2), although all the conclusions are general and apply to

nin = 0.

For a given dark matter velocity distribution function f(v) defined in the galactic

rest frame, ⟨dσ

dERv

⟩=

∫vmin

d3v f(~v + ~ve) vdσ

dER. (8.6)

The minimum velocity is set by the kinematics of the scattering process:

vmin(ER) =

√

mNER2µ2 elastic

1√2mNER

(mNERµ

+ δm), inelastic

(8.7)

where δm is the dark matter mass splitting. The Earth’s velocity in the galactic rest

frame, ~ve, is defined as

~ve = ~v + ~v⊕(t). (8.8)

In the coordinate system where x points towards the galactic center, y points in the

direction of the galactic rotation, and z points towards the galactic north pole, the

sum of the Sun’s local Keplerian velocity and its peculiar velocity is [237,238]

~v(t) ≈ (0, 220, 0) + (10, 5, 7) km/s. (8.9)


The velocity of the Earth in the sun’s rest frame is given by

~v⊕ ≈ v⊕

[ε1 cos

2π(t− t0)

yr+ ε2 sin

2π(t− t0)

yr

], (8.10)

where t0 is the spring equinox (≈ March 21), v⊕ = 29.8 km/s is the orbital speed of

the Earth [37], and

ε1 = (0.9931, 0.1170,−0.01032) (8.11)

ε2 = (−0.0670, 0.4927,−0.8676)

are the unit vectors in the direction of the Sun at the spring equinox and summer

solstice, respectively [239,240].

Typically, the velocity distribution function f(v) is assumed to be isothermal and

isotropic in the galactic frame. In the “Standard Halo Model” (SHM), the Keplerian

velocity is constant throughout the galaxy and the velocity dispersion is assumed to

be Gaussian. These assumptions are not consistent with recent N-body simulations

of dark matter particles; simulations such as Via Lactea [141] show that the dark

matter velocity dispersion is anisotropic and that the density falls off more steeply at

larger radii than in the isothermal scenario.

This article adopts the following ansatz for the dark matter halo velocity distri-

bution

f(v) = N(e−(v/v0)2α − e−(vesc/v0)2α

)Θ(vesc − v), (8.12)

which reproduces the SHM in the limit α → 1. This distribution function captures

the most important qualitative features of Via Lactea, but in an analytical form that

is easy to compute with. The parameter α changes the shape of the profile near

the escape velocity. Inelastic scattering events are particularly sensitive to the high-

velocity tail because their minimum velocity is larger in comparison to the elastic

case [142,143].

To find the regions of parameter space consistent with current direct detection

experiments, we perform a global chi-squared analysis and marginalize over the six

unknown parameters: v0, vesc, α,mdm, δm, and σp, the cross section per nucleon. The


χ2 function is

χ2(mdm, δm, σn, v0, vesc, α) =

Nexp∑i=1

(Xpredi −Xobs

i

σi

), (8.13)

where Nexp is the number of experiments included in the fit, Xpredi is the predicted

experimental result, Xobsi is the observed result and σi is the known error. The

velocity distribution parameters are constrained to be within

200 km/s ≤ v0 ≤ 300 km/s

500 km/s ≤ vesc ≤ 600 km/s

0.8 ≤ α ≤ 1.25. (8.14)

These values are motivated by rough observational constraints [38, 37] and the fact

that (8.12) is a spherically symmetric form of the Via Lactea fits by [143]. It has been

found that alternate statistical methods, such as the maximum gap test [146], can have

small effects on the allowed region of parameter space for inelastic dark matter [138].

We found that both the χ2 and maximum gap methods gave quantitatively similar

answers (though some points allowed by max-gap were excluded by Poisson and vice

versa). Because it is not straightforward to combine a χ2 fit for DAMA with multiplet

max-gap tests, we chose to use Poisson statistics to obtain a global fit.

The results are fit to the first twelve bins (2-8 keVee) of the recoil spectrum

measured by DAMA [34]. The modulation amplitude from 8-14 keVee is combined

into a single bin with amplitude −0.0002± .0014 cpd/kg/keVee. The contribution of

this last bin to the total χ2 is typically negligible. The DAMA results are quoted in

electron equivalent energy and must be rescaled by an appropriate quenching factor to

get the nuclear recoil energy (qI = Eee/Enr). For the results in this work, qI = 0.085.

However, the allowed region of parameter space is sensitive to this factor [142], which

can vary from 0.06 . qI . 0.09 [32].

The region of parameter space that is consistent with DAMA is constrained by the

null experiments, each of which has observed a number of events in its signal window.


Figure 8.2: The 95% contours in the mdm − δm parameter space corresponding tocel/cin = 0 (blue), 0.4 (magenta), 0.6 (yellow) and 0.8 (green) for the scenario withnin = 1 and nel = 2. The four regions overlap one another.

The results from CDMS, CRESST, ZEPLIN-II, and ZEPLIN-III are included in the

fit, in addition to XENON10’s recently updated analysis with an expanded signal

window [30]. We require that the theory not saturate the number of observed events

for each experiment to 95% confidence.

Figure 8.2 shows the allowed regions of mdm − δm parameter space for different

ratios of elastic to inelastic scattering. The contours are defined as

χ2(mdm, δm, σn, v0, vesc, α) = χ2min + ∆χ2(CL), (8.15)

where ∆χ2(95%) = 12.6 for six degrees of freedom. The minimal χ2 found was

χ2min = 4.1 for cel/cin= 0 and thus every prediction was forced to have a χ2 ≤ 16.7

corresponding to the 2σ band.

The best-fit point for the completely inelastic scenario is

(v0, vesc, α) = (279, 586, 0.80) (8.16)

(mdm, δm, σn) = (59 GeV, 121 keV, 5.9× 10−39 cm2),

where σn = σN(µ2N/µ

2n) and µN(n) is the reduced mass of the dark matter-nucleus

(nucleon) system. As the ratio of elastic to inelastic scattering increases, the null


experiments become more constraining and the minimum χ2 increases. The maximum

allowed ratio of elastic to inelastic scattering is cel/cin = 0.8. For this ratio, the best-fit

point has χ2 = 14.3 and corresponds to

(v0, vesc, α) = (297, 500, 0.82) (8.17)

(mdm, δm, σn) = (74 GeV, 98 keV, 1.2× 10−39 cm2).

There is a high degree of degeneracy in the allowed values of the halo parameters for

each mdm − δm point. The parameter points with the least tension have mdm and

δm near the best-fit points of (8.16) and (8.17), but the halo parameters and cross

sections can vary wildly. Some areas of the allowed regions are highly sensitive to the

exact range of the halo parameters. For instance, the large mass region corresponds to

exceedingly “slow” velocity distribution functions where v0<∼ 225 km/s and α >∼ 1.15.

In the near future, definitive experiments such as XENON100 [31] and LUX [241]

will confirm or refute the inelastic dark matter hypothesis. These two experiments

are the upgrades to the XENON10 detector. The LUX detector will consist of 300

kg of liquid xenon with a planned fiducial volume of 100 kg, and it will be sensitive

to recoil energies as large as ∼ 300 keVnr [242]. Figure 8.3 shows an estimated recoil

spectrum at LUX after 1000 kg-days and assuming 30% efficiency. The solid line

denotes the spectrum for the best-fit point in the pure inelastic scenario, while the

dashed line shows the spectrum for the best-fit point when cel = 0.8cin. LUX would

see many events in its signal window; for models with a small elastic subcomponent,

it could see as many as ∼ 40 − 50 events in the winter (gray) and ∼ 70 − 80 events

in the summer (black).

The inelastic and elastic scattering channels have remarkably similar nuclear re-

coil spectra because of the threshold behavior of the dominant inelastic channel and

the form factor suppression of low momentum transfer events. The shape of the

spectra differ only at small recoil energies (. 10 keVnr). Because this is near the

threshold energy of the experiment (∼ 5 keVnr), it is difficult to unambiguously dis-

tinguish the two contributions to the scattering rate with experiments such as LUX

and XENON100. In the following section, we show that additional information about


0 20 40 60 80

0.05

0.10

0.50

1.00

5.00

Recoil Energy HkeVnrL

Cou

ntsk

eV

Figure 8.3: Estimated recoil spectrum at LUX with 1000 kg-days and 30% efficiencyduring the summer (black) and winter (gray). The solid line corresponds to the best-fit point for completely inelastic scattering; the dashed line corresponds to the best-fitpoint when cel = 0.8cin.

the nuclear recoil direction is a critical component in understanding the dynamics of

the dark sector.

8.2 Directional Detection

Directional detection experiments take advantage of the daily modulation in the direc-

tion of the dark matter wind in the lab frame, which arises from the Earth’s rotation

around the galactic center [45, 44]. In particular, the direction of the dark matter

changes every twelve hours as the Earth rotates about its axis. For the case of elastic

scattering, the daily modulation amplitude can be nearly ∼100%, compared to the

∼2.5% change in the annual modulation amplitude.

Measuring the strong angular dependence of the nuclear recoil can be an important

tool for detecting dark matter [45]. Several directional detection experiments are

currently running: DMTPC [228], NewAge [229], DRIFT [230], and MIMAC [231].

To get reasonable angular resolution, the track left by the recoiling nucleus must be

sufficiently long (∼ 1 mm). This means that the detector material must be a gas,

because liquid and crystalline detectors have scattering lengths that are too long.

All the current detectors are using CF4, except for DRIFT, which uses CS2. These

8.2. DIRECTIONAL DETECTION 147

detectors are specifically designed to look for elastic spin-dependent interactions, and

so the use of atoms with high spin coupling, such as fluorine, is preferred.

It was recently pointed out that directional detection experiments can serve as

important tests of inelastic dark matter [227] if much heavier atoms are used in the

detectors. In this section, we show that such experiments are the key to distinguishing

the contributions from elastic and inelastic scattering components in the dark matter

sector. To begin, we will briefly review how the rate equation derived in the previous

section is generalized to include the angular dependence of the recoiling nucleus. A

more complete discussion of the theory can be found in [227,226].

Consider the lab frame where the incoming dark matter particle has a velocity

~v′ = vx and scatters off a nucleus at rest. The recoiling nucleus has velocity ~vR

and makes an angle θ with the x-axis. Energy and momentum conservation yield an

expression for the recoil velocity,

vR =2µv

mN

cos θ, (8.18)

which can be written in the frame-invariant form

v · vR − vmin

v= 0. (8.19)

The differential directional scattering rate per unit detector mass is

d2R

dERd cos γ=

ρ0

mdmmN

(v2 dσ

dER

)(8.20)

×∫d3vf(v)δ(~v · vR − ~ve · vR − vmin).

The integral expression for the differential rate is an example of a Radon transform,

the properties of which are reviewed in [226]. The Radon transform for the modified

halo distribution function we consider is

f(w) =

∫d3v f(v)δ(~v · vR − w) = 2π

∫ ∞w

dv vf(v), (8.21)


1.0 0.8 0.6 0.4 0.2 0.0 0.21104

2104

5104

0.001

0.002

0.005

0.010

0.020

cos Γ

dRdcosΓ

cpdkg

iDM

FFeDM

Figure 8.4: cos γ spectrum during the summer for the best-fit parameters whencel = 0.8cin, assuming a detector of CF3I. The spectrum is taken for Er = 50 keVnr.The dashed line is the total from form factor-suppressed and inelastic scattering.The shaded circle marks the “cross-over” point where the elastic scattering starts todominate over the inelastic scattering.

where w = ~ve · vR +vmin = ve cos γ+vmin. Note that γ is defined as the angle between

the recoiling nucleus and the direction of the Earth’s velocity ~ve. Evaluating the

integral for (8.12), we find

f(w) = πN

(w2 − v2

esc)e−(vesc/v0)2α

+v2

0

α

(Γ[ 1

α,(wv0

)2α]− Γ

[ 1

α,(vesc

v0

)2α])Θ(vesc − w).

There are several important features of this expression. Firstly, the differential rate is

peaked at cos γ = −1. this can also be seen explicitly from the rate equation, where

the largest fraction of the parameter space is allowed when ~ve · vR = −1 in the delta

function, which makes sense because the rate is maximized when the Earth is moving

in the same direction as the dark matter wind.

The second important feature arises from the theta function in f(w). In particular,

there is a maximum value of cos γ above which the rate is zero:

cos γmax =vesc − vmin

ve. (8.22)

8.2. DIRECTIONAL DETECTION 149

100 kg-yr

5 kg-yr

Figure 8.5: The detection rate corresponding to the value of cos γ at the “cross-over”point, where the elastic scattering starts to dominate over the inelastic scattering. Theshaded regions correspond to cel/cin = 0.2 (magenta), 0.4 (yellow), and 0.8 (green).The rate is calculated for summer at a recoil energy of Er = 50 keVnr, assuming adetector of CF3I. The dashed lines show the projected sensitivity for the DMTPCCF4 detector after 5 and 100 kg-yr [236]. These projected sensitivies should be evenstronger for a CF3I detector, which would use a heavier nucleus to test the inelastichypothesis.

This cutoff depends on the minimum velocity, which is larger for inelastic scattering

than elastic scattering. As a result, the recoil spectrum for any inelastic scattering

component will have a cut off below that of any elastic scattering component. This

is illustrated in Fig. 8.4, which shows the recoil spectrum for the best-fit parameters

in a cel/cin = 0.8 theory at a recoil energy of 50 keVnr, which is near the threshold of

most current experiments. The rate for the inelastic interaction becomes negligible

by cos γ ∼ −0.5. There is a tail at larger values of cos γ, which arises from the elastic

scattering component.

Whether the elastic and inelastic scattering components can be distinguished de-

pends on whether there are enough events that fall along the tail of the spectrum.

Fig. 8.5 shows the expected rate at the value of cos γ where the transition from inelas-

tic to elastic scattering occurs. The spread arises from varying over the six unknowns

of both the particle physics and halo profile models. Even given the uncertainties

of the halo profile distribution, one can obtain enough events on the cos γ tail for

discovery with approximately 5-10 kg years-worth of data. This is well within reach


of current experiments; the dashed lines in Fig. 8.5 show the projected sensitivity of

the DMTPC CF4 detector. If a heavier detector material is used, as would be needed

to test for inelastic dark matter, then the sensitivity should be even larger.

8.3 Discussion

We have shown that directional detection experiments can distinguish mixed inelastic-

elastic scattering scenarios that would otherwise be difficult to discover unambigu-

ously, even with next-generation experiments such as LUX and XENON100. In par-

ticular, directional detection experiments can detect elastic wide angle scatters that

are kinematically forbidden in inelastic transitions. To evade current null experi-

ments, a relatively large exposure of several kg-yrs is needed, but this is well within

reach of current detectors. However, these experiments are currently optimized to

look for spin-dependent elastic scattering and need to use heavier nuclei, such as

iodine or xenon, to have sensitivity to inelastic recoils.

Distinguishing different scattering mechanisms is critical for understanding the un-

derlying symmetry structure of dark matter theories. Many types of inelastic models

can give small elastic scattering contributions. For instance, nontrivial scattering

mechanisms arise in models where the dark matter has a finite size, such as compos-

ite [40, 196, 199, 201], atomic [155], mirror [202], and quirky [156] dark matter. All

these models have several scattering channels with a hierarchy of scales given by a

series of higher dimensional operators. Both elastic and inelastic operators may be

allowed, and approximate discrete symmetries may cause the elastic scattering rate

to be subdominant to the inelastic rate [136].

Another class of models that leads to form factor-suppressed elastic scattering is

characterized by a pseudo-Goldstone mediator between the dark sector and the SM

[135]. In this case, additional dimension six operators in the Lagrangian may no longer

be negligible in comparison to the standard spin-independent and spin-dependent

operators. These higher dimension operators lead to momentum suppression in the

scattering rate. There has been much interest recently in models with light mediators

[203,205,217,243,244,207], given the results from PAMELA [245], Fermi [246], ATIC

8.3. DISCUSSION 151

[247], and HESS [248].

If the dark sector does indeed have non-minimal structure, it may give rise to sev-

eral types of scattering mechanisms. Distinguishing these different scattering events is

of fundamental importance for understanding the symmetry structure and unraveling

the dynamics of the dark sector.

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