The Search for Dark Matter: from Colliders to Direct Detection Experiments

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

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

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

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

Jay Wacker, Primary Adviser

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

Savas Dimopoulos

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

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 in University Archives.

iii iv 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 + ET Channel ...... 11 6 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+ET 25 6 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+ET 35 6 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 +ET searches ...... 31 6 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 D0 at 4 fb−1 ...... 32 6 3.1 Comparison of D0 cuts and optimized cuts for a sample dijet signal . 41 6 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 D0 at 4 fb−1 ...... 53 6 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 h i 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 mπ Λd parameter space ...... 125 d − 7.2 Allowed regions in mπ feﬀ parameter space ...... 126 d − 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 xii Chapter 1

Introduction

The presence of dark matter in the universe is one of the greatest mysteries of particle 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 gravitational 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 2 give baryon densities Ωbh = 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 1.5 measurement gives Ωbh 0.02 [5]. Finally, careful measurements of anisotropies in ' 2 +0.004 the CMB give a remarkably precise estimate of Ωbh = 0.024−0.003 [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 2 from anisotropies in the CMB to be Ωmh = 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:

† µ 2 † † 2 Higgs = (DµH) (D H) + m H H + λ(H H) . (1.1) L

g ~ g0 Dµ is the covariant derivative Dµ = ∂µ i ~σ Aµ i Bµ, where ~σ are the Pauli − 2 · − 2 matrices and g, g0 are the SU(2) and U(1) coupling constants, respectively. The Higgs boson is responsible for electroweak symmetry breaking, the process by which 4 CHAPTER 1. INTRODUCTION

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

QL 1/2 3 2 1/3

u¯L 1/2 3¯ 1 -4/3 ¯ dL 1/2 3¯ 1 2/3

LL 1/2 1 2 -1

e¯L 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), and U(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:

± 1 1 2 0 1 3 0 W = (A iA ) Z = (gA g Bµ) (1.2) µ √2 µ ∓ µ µ g2 + g02 µ −

p 0 and one massless boson, the photon, which is orthogonal to Zµ. 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 because predictions for their cosmological abundance reproduce the dark matter density observed today. To derive this explicitly, assume that the Standard Model is supplemented 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 g neq = f(p)d3p, (1.3) χ (2π)3 Z 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 neq T 3. χ ∝ However, as the temperature drops below mχ, the equilibrium abundance falls exponentially as neq T 3/2e−mχ/T . Because the reaction rate is proportional to the χ ∝ DM abundance (Γ = nχ σv ), a drop in number density corresponds to a drop in the h i 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 constant with time (Fig. 1.3). The larger the annihilation cross section at the time of 6 CHAPTER 1. INTRODUCTION

Increasing Comoving Number Density

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:

n n χ = χ , (1.4) s s !today !f

3 where s 0.4g∗T 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- h i h i verse is still radiation-dominated at this time, the Hubble expansion rate is H(T ) = 1/2 2 19 1.66g∗ T /Mpl, where Mpl 10 GeV is the Planck mass. Estimating the temper- ' ature at freeze-out to be Tf mχ/20, the comoving number density is ' n 100 χ . (1.5) s 1/2 ! ' mχMplg∗ σv today h i 1.2. COLLIDER SEARCHES FOR DARK MATTER 7

−3 −5 2 The entropy density and critical density today are s 4000 cm and ρc 10 h ' ' GeV/cm3, and the present mass density is therefore

1/2 2 2 2 mχnχ 100 α 100 GeV Ωχh = 0.04 , (1.6) ρc ' g∗ ! 0.01! mχ ! assuming that the annihilation cross section scales as σv α2/m2 . The fact that h i ∼ χ 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 ∼ 8 CHAPTER 1. INTRODUCTION

of data combined from its two detectors, D0 and CDF. The LHC, which is currently 6 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 supersymmetric 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 collider 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 19 scale (EPlanck 10 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) and in supersymmetry (bottom). 1.2. COLLIDER SEARCHES FOR DARK MATTER 9

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:

1 soft = M3g˜g˜ + M2W W + M1BB + c.c. L −2 ¯˜ ˜ u¯˜AuQHu dAdQHd e¯˜AeLHd + c.c. − − ff − e e † 2 † 2 2 † ˜ 2 ˜† 2 † Q M Q L˜ M L˜ u¯˜M u¯˜ d¯M ¯d¯ e¯˜M e¯˜ (1.7) − Qe − L e− u¯ − d − e¯ 2 ∗ 2 ∗ m H Hu m H Hd (bHuHd + c.c.). − e Hu ue − Hd d −

Different high-energy mechanisms for susy breaking result in different relations between 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 10 CHAPTER 1. INTRODUCTION

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 ν˜e, ν˜µν˜τ νe, νµ, ντ gluinos g˜ g ± ± ± ± charginos C1 , C2 W ,Hu,d 0 0 0 0 0 0 neutralinos N1 ,..., N4 γ, Z , h ,Hu,d,A e e Figure 1.5: MSSMe particlee content.

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

3(B−L)+2S PR = ( 1) , (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 0 higgsinos (Hu and Hd) and electoweakinos (B, W ). In the gauge-eigenstate basis,

e e e f 1.2. COLLIDER SEARCHES FOR DARK MATTER 11

the neutralino mass term in the Lagrangian is

1 T 0 neutralino = (ψ) M ˜ ψ + c.c., (1.10) L −2 N

0 ˜ 0 ˜ 0 ˜ 0 where ψ = (B, W , Hd , Hu) and

f M1 0 cβsW mZ sβsW mZ −  0 M2 cβcW mZ sβcW mZ  MN˜ = −  cβsW mZ cβcW mZ 0 µ   − −   sβsW mZ sβcW mZ 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” | ± | | ± | 0 0 0 (N1 B1), “wino-like” (N2 W ), and “higgsino-like” (N3, N4 (H H )/√2). ≈ ≈ ≈ u ± d In this case, the bino is the lightest of the four states. e e e f e e e e

1.2.1 Jets + E Channel 6 T 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 g˜g,˜ pp q˜q,˜ and pp q˜g,˜ (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 12 CHAPTER 1. INTRODUCTION

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

∗ missing energy, ET, in the detector. Supersymmetry is not the only model that leads 6 to a jets+ET signal; this signal can arise from Universal Extra Dimensions (UEDs) [12] 6 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 0 background. For example, the dominant backgrounds for jets + ET include Z + nj 6 (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, ET, in the event. 6 The pT and η of a massless particle are related to the four-momentum through

µ p (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 is typically easier than tagging on jets, because the signals are cleaner and avoid the complicated hadronic physics that aﬀects the shapes and distributions of jets. This work focuses on the more challenging scenario where leptons are not present, though a brief discussion of jets+ET+lepton searches is saved for Ch. 3.4.5. 6 1.2. COLLIDER SEARCHES FOR DARK MATTER 13

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 + ET signals. Additionally, the ratio between gluino 6 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 experimental 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 +ET in a 6 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 +ET signals. 6

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 14 CHAPTER 1. INTRODUCTION

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:

2 4 2 2 2 2 3g mt mt˜1 mt˜2 2 at mh0 mZ0 cos 2β + 2 2 log 2 + at 1 , (1.13) ' 8π mW mt − 12

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 ET are ﬂexible enough to be able 6 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 (1) couplings to the Higgs, then the Higgs can have O 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 16 CHAPTER 1. INTRODUCTION

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:

µ 5 5 SI =χχ ¯ qq¯ SD =χγ ¯ γ χqγ¯ µγ q. (1.14) L L

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 1.3. DIRECT DETECTION EXPERIMENTS 17

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 mN σn 2 2 2 = 2 2 (Zfp + (A Z)fn) F (q ) , (1.15) dER 2µnv − | | 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 ρ vesc dσ = 0 d3vf(v)v , (1.16) dE m m dE R χ N Zvmin R 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, 18 CHAPTER 1. INTRODUCTION

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

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

2 2 2 The nucleus scatters elastically, p2 = p4 = mN , and (1.17) simpliﬁes to

2 2 p p = 2(p3 p4 p1 p2) (1.18) 1 − 3 · − ·

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 − −

2 mχδ = q(p1 p2) + q (1.19) − −

to ﬁrst-order in δ. The recoil energy of the nucleus, ER, is related to the momentum 2 transfer through q = 2mN ER. 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) 1 p = (m + m v2, m v, 0) 1 χ 2 χ χ

q = (ER, 2mN ER cos θ, 2mN ER sin θ) p p which assumes that the DM is highly non-relativistic, a reasonable assumption given that v (10−3). Substituting (1.21) into (1.19) and taking cos θ = 1 for the ∼ O minimum velocity yields

1 ERmN vmin = + δ , (1.21) √2mN ER µN 1.3. DIRECT DETECTION EXPERIMENTS 19

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

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

ERmN velocity is equal to 2 . As a result, the differential rate is 2µN q vesc dR 3 dσ −v2/v2 −E /E d v ve 0 e R 0 , (1.22) dE ∝ dE ∼ R Zvmin R 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 ' 2 2µN v0 Fig. 1.8. The number of scattered events dominates at energies below E0 = 30 m ∼ 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. INTRODUCTION 3

0.070.07 cannot distinguish background events from signal events in this sample, it is clear that the predicted number of 2 GeV 0.060.06 WIMP events should not excessively exceed their total 0.050.05 number of counts in any bin. day

7 GeV We require that the unmodulated rate in each bin from 0.040.04 kg 0.75 4 keVee not exceed the observed values within − 0.030.03 their 90% error [1]. We show this constraint in Fig. 1, labeled “DAMA-Total.” The allowed region lies below counts 77 GeV 0.020.02 m this curve. This constraint does not greatly impinge upon Rate (cpd/kg/keVee) S 12 GeV 2 0.010.01 the allowed region from our nine bin χ that accounts for the spectral details. Its constraint is most striking if one keVee 2 3 4 5 6 considers the allowed region of the two bin fit (see Fig. 1). Recoil Energy (keVee) Since the modulation in the low (2 2.5 keV) bin seems − FIG. 2: We show the modulation spectra for the best fit to be the most constraining, one can consider whether it point where scattering off iodine dominates, m =77GeV is overly biasing our analysis. For instance, one could Figure 1.9: DAMA’s measured modulated amplitude (grey points),χ compared to the (dot-dashed orange), and three points where scattering off of worry if DAMA/LIBRA were to restate the efficiency modulated amplitude spectrum for a 2, 7, 12, and 77 GeV elastically scattering dark sodium dominates. The best fit point off sodium is mχ =12 in the lowest bin, this might completely change our re- matter (green, blue, orange, and red, respectively). Figure from [39]. GeV (solid red). We also show mχ = 2 GeV (dashed green) sults. We have explored such effects by various methods: and mχ = 7 GeV (dotted blue). The points with error bars tripling the error bar on the lowest bin, merging the en- are the published DAMA/LIBRA data. tire range into a 2 3 keV bin, and discarding the lowest looking below 40 keV, have expanded their signal region up to 75 100 keV − ∼ ∼ − bin entirely. We find that only the last option (discard- in search of iDM signals [28, 30]. Second, experiments with heavier target nucleiing are the lowest bin) opens up a region of parameter space, more optimalhigher for iDM mass, searches. say 20 The GeV, reason this for approach this is that does the not minimum succeed. scatteringwith a point allowed with χ2 =9.14 for 6 dof (p =0.17). As the mass moves above 12 GeV, a contribution coming This point also has a unmodulated rate that is close to velocity (1.8) decreases as mN increases for inelastic scattering, and a larger region from iodine scattering begins to move into the low end of saturating the observed rate. of the velocitythe distribution observed is energy integrated region, over spoiling in the expressionthe fit. Also for plotted the differential rate. Consequently,in Fig. experiments 2 are spectra using for xenon WIMP or tungsten masses targets, of 2 and such 7 as GeV XENON10, XENON100, ZEPLIN-III,for comparison. and CRESST, should see larger scattering rates from iDMIV. VARIATIONS FROM ASTROPHYSICS AND 2 than an experimentThe such 68%, as CDMS,90%, and which 99% has CL a germanium (∆χ < 2.3, target. 4.61, 9.21) PARTICLE PHYSICS contours consistent with our nine bin DAMA/LIBRA χ2 Inelastic darkfunction matter are has shown received in increased Fig. 1. attention Both regions recently shrink because dra- it providesThus far, we assumed a MB halo and an elastic, SI in- 2 a means of reconcilingmatically DAMA’s compared positive to the signal two with bin allχ other. In the null left experiments. panel, Theteraction. Relaxing these assumptions could enlarge the 2 spectrum of thethe modulated∆χ is with amplitude respect measured to the global by DAMA best is fit reproduced point at 77in Fig 1.9.region at light masses, so that modulation arises with an GeV. In the right panel, we concentrate on the low mass appropriate spectrum, consistent with other experiments. The modulated amplitude for a given recoil2 energy ER is defined as region and have defined ∆χ relative to the low mass best Let us begin by considering astrophysical modifications fit point of 12 GeV. Note1 this region is confined to masses to the velocity distribution. Kinematics informs us of above 10MA GeV.(ER This) = canRS( beER understood) RW (ER) from, examining (1.23) 2 − what modifications are needed. To scatter with nuclear the recoil spectra for the light WIMPs in Fig. 2. For a recoil energy ER,aWIMPmusthaveaminimumveloc- fixed overall modulation rate, sub-10 GeV WIMPs pre- 2 where RS,W (ER) is the rate in the summer and winter, respectively. In the caseity of βmin = MN ER/2µ ,whereµ is the reduced mass dict too little modulation above a couple of keVee: they of the WIMP-nucleus (not nucleon) system. Consider simply do not have enough mass to cause recoils of this the channeled possibility, for which the velocity require- size. ments are weakest: for scattering on sodium, with re- In Fig. 1, we have superimposed 90% limit contours coil energy of 4.5 keV (the highest bin with significant from both the CDMS [9] and XENON [10] experiments. modulation), one finds β c 1140, 790, 620 km/s for min ≈ We only show the CDMS contour relevant for low masses, mχ =2, 3, 4 GeV. If halo particle velocities approxi- corresponding to data taken with the silicon detectors in mately follow a MB distribution, the most significant de- the Soudan mine, which have a 7 keV threshold. We have viations naturally occur for the highest velocities, where recalculated limits using the astrophysical parameters de- recent infall and streams may not have fully virialized. scribed here. For the XENON experiment, we account As such, the lightest particles are the most likely to have for the energy dependent efficiencies as described in [10] allowed regions opened by such deviations from a MB and to set the limit, apply the maximum gap method [17] distribution. to the energy recoil range of 4.5-26.9 keV. We see that One modification to the halo is to include streams the light SI DAMA/LIBRA mass region is excluded once [18, 19]. We investigated a wide range of streams, varying the modulation spectrum is taken into account. its velocity 1200 km/s

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 (100 keV) [195]. The O 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 22 CHAPTER 1. INTRODUCTION

sector is

1 2 ¯ ¯ ¯ ¯ dark = TrG + ΨLiDΨL + ΨH iDΨH + mLΨLΨL + mH ΨH ΨH , (1.24) L −2 µν 6 6 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 n ¯ ) = (nL n¯) = 0, (1.25) − H − − L 6 ¯ ¯ 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 ΨH ΨL 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 ΨH ΨL 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 (10−6), strongly O suggesting that the dark matter is in a meson state [41].

¯ There are two possible spin combinations for the ΨH ΨL 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 2 Λ /mH in the confinement limit. The mass splitting is suppressed because it is ∝ d 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 1.3. DIRECT DETECTION EXPERIMENTS 23

kinetic mixing in the Lagrangian arises from

2 2 2 2 kin = F F FdFEM + m A + JEM AEM + JdAd, (1.26) L − d − EM − A d 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 Ad. This transformation diagonalizes the → − kinetic terms and yields

2 2 2 2 = F F + m A + JEM (AEM Ad) + JdAd. (1.27) L − d − EM A d −

Therefore, the dark photon interacts with the electromagnetic current via int L ∝ µ JemAdµ, which is proportional to the kinetic mixing parameter. When writing down the dark matter current, either a vector or axial-vector coupling 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 factor in the dark matter-charge photon interaction

2 2 Fdm(q ) = rc ER, (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 −ER (ER)e . (1.29) dER ∝ 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 24 CHAPTER 1. INTRODUCTION

inelastic and charge-radius scattering are allowed. The non-trivial scattering mechanisms in CiDM models have important phenomenological consequences. In particular, because the recoil spectra for form-factor elastic and inelastic scattering are so similar, 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 directional 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 direction 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+ET analyses can 6 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 E 6 T

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 supersymmetry [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 (ET) is 6 a promising experimental signature for new phenomena [49,50,51,52,53].

At present, the jets + ET searches at the Fermilab Tevatron are based upon 6 the minimal supersymmetric standard model (MSSM) and look for production of gluinos (˜g) and squarks (˜q), the supersymmetric partners of gluons and quarks, respectively [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 ansätz known as the CMSSM (or mSUGRA) parameterization of supersymmetry breaking [11]. This ansätzsets all the gaugino masses equal at the grand unified scale and

25 26 CHAPTER 2. COLLIDER SEARCHES FOR JETS+ET 6 runs them down to the weak scale, resulting in an approximately constant ratio between 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 models. Other methods of supersymmetry breaking lead to diﬀerent low-energy particle spectra. In anomaly mediation [54], the wino can be the LSP; for instance, mg˜ : m ˜ 9 : 1. Mirage mediation [55], in contrast, has nearly degenerate W ' 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 ˜ q¯1q˜ q¯1q2B. 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 ET and HT = ET can discover 6 jets particles where the current Tevatron searches would not. In orderP to show this, we model our searches on D0 ’s searches for monojets [53], squarks and gluinos [51]. 6 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

A B j

_ _ q ~ ~ q ~ g g ~g g ~ ~ B B q E q /T _ _ q q E ~ ~ /T B B q q

Figure 2.1: Boosted gluinos that are degenerate with the bino do not enhance the missing transverse energy when there is no hard initial- or ﬁnal-state radiation. (A) illustrates the cancellation of the bino’s ET. (B) shows how initial- or ﬁnal-state 6 radiation leads to a large amount of ET even if the gluino is degenerate with the bino. 6 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˜ m ˜ . When the mass − B 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 (102 pb), as compared to (10−2 pb) for their heavier O O 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 28 CHAPTER 2. COLLIDER SEARCHES FOR JETS+ET 6 processes of the form pp¯ g˜g˜ + 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 procedure 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:

2 2 2 2 d (i, j) = ∆Rij min(pT i, pT j) 2 2 d (i, beam) = pT i, (2.2) where ∆R2 = 2(cosh ∆η cos ∆φ) [60]. Matrix-element events are generated with ij − ME some minimum cut-oﬀ d(i, j) = Qmin. After showering, the partons are clustered into PS ME jets using the kT jet algorithm with a Qmin > Qmin. The event is then discarded unless all resulting jets are matched to partons in the matrix-element event, d(parton, jet) < PS Qmin. 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.

ME PS The matching parameters (Qmin and Qmin) should be chosen resonably far below the factorization scale of the process. For gluino production, the parameters were:

ME PS Qmin = 20 GeV and Qmin = 30 GeV. (2.3)

The simulations were done using the CTEQ6L1 PDF [61] and with the renormalization 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 2.1. EVENT GENERATION 29

∆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, ME PS with the parameters Qmin = 10 GeV and Qmin = 15 GeV. The top background ME PS was matched up to two jets with Qmin = 14 GeV and Qmin = 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 ET from jet 6 energy mismeasurement, a lower bound of 90◦ and 50◦ was placed on the azimuthal angle between ET and the ﬁrst and second hardest jets, respectively. An acoplanarity 6 cut of < 165◦ was applied to the two hardest jets. Because the D0 analysis did not 6 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 regions where jet and calorimeter mismeasurements become the dominant background, a lower limit of ET > 100 GeV was imposed. Additionally, in the dijet analysis, the 6 azimuthal angle between the ET and any jet with pT 15 GeV and η 2.5 was 6 ≥ | | ≤ 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 ET and HT distributions published 6 by the D0 collaboration in [51] for 1fb−1. For the dijet case, where the most statistics 6 are available, the correspondence with the D0 result is 20%. With the threejet 6 ± 30 CHAPTER 2. COLLIDER SEARCHES FOR JETS+ET 6 and multijet cuts, the result for the tt¯ background is similar, while the W ±/Z0 + nj backgrounds reproduce the D0 result to within 30 40% for the threejet and multijet 6 − cases. The increased uncertainty may result from insuﬃcient statistics to fully populate the tails of the ET and HT distributions. The PGS probability of losing a lepton 6 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 ET depend strongly on the mass differerence between 6 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 ET plus 1j, 2j, 3j and 4 j, respectively. For convenience, we keep the 6 nj + ET classification fixed for all gluino and bino masses (see Table 4.1). These 6 selection criteria were modeled after those used in D0 ’s existing search [51].† These 6 exclusive searches can be statistically combined to provide stronger constraints. min min Two cuts are placed on each search: H and ET . In the D0 analysis, the HT T 6 6 and ET cuts are constant for each search. The signal (as a function of the gluino and 6 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 min min maximized over H and ET in each nj +ET search. Due to the uncertainty in the T 6 6 background calculations, the S/B was not allowed to drop beneath the conservative

†It should be noted, however, that the D0 searches are inclusive because each is designed to look for separate gluino/squark production modes6 (i.e., pp q˜q˜,q ˜g˜,g ˜g˜). → 2.2. PROJECTED REACH OF SEARCHES 31

+ 1j + ET 2j + ET 3j + ET 4 j + ET 6 6 6 6 ET j 150 35 35 35 1 ≥ ≥ ≥ ≥ ET j < 35 35 35 35 2 ≥ ≥ ≥ ET j < 35 < 35 35 35 3 ≥ ≥ ET j < 20 < 20 < 20 20 4 ≥ 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; D0 , for 6 instance, claims a systematic uncertainty of (30%) in their background measure- O ments [51]. The resulting 95% sensitivity plot using the optimized HT and ET cuts 6 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 + ET and 2j + ET searches both have 6 6 + good sensitivity. In an intermediate region, the 2j + ET, 3j + ET and 4 j + ET all 6 6 6 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 + ET and 4 j + ET both give 6 6 + good sensitivity, with the 4 j + ET giving slightly larger statistical signiﬁcance. 6

In the exclusion plot, the ET and HT cuts were optimized for each point in gluino- 6 < < 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 ,ET) (150, 100)2j+E6 , 6 ≥ T + (150, 100)3j+E6 T , (200, 100)4 j+E6 T . (2.4) 32 CHAPTER 2. COLLIDER SEARCHES FOR JETS+ET 6

200 200 150 " GeV ! 100 Out[112]= Mass Bino 150 50

" 0 100 200 300 400 500 Gluino Mass GeV GeV

! ! " 100 Out[129]= Mass Bino 50 X

0 100 200 300 400 500 Gluino Mass GeV

! " Figure 2.2: The 95% gluino-bino exclusion curve for D0 at 4 fb−1 for S/B > 1. The dashed line shows the corresponding exclusion region using6 D0 ’s non-optimized cuts. The masses allowed in the CMSSM are represented by the dotted6 line; the “X” marks the current D0 limit on the gluino mass at 2.1 fb−1 (see text for details) [51]. The inset shows the6 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 D0 analysis are 6

(HT ,ET) (325, 225)2j+E6 , 6 ≥ T + (375, 175)3j+E6 T , (400, 100)4 j+E6 T . (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 D0 does. While D0 technically has statistical signiﬁcance in this region with 6 6 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 ET cuts. Similarly, for larger gluino masses, the generic cuts 6 are no longer eﬀective and it is necessary to use the optimized cuts, which are tighter than D0 ’s. 6 2.3. CONCLUSIONS AND OUTLOOK 33

2.3 Conclusions and Outlook

In this paper, we describe the sensitivity that D0 has in searching for gluinos away 6 from the CMSSM hypothesis in jets + ET searches. It was assumed that the gluino 6 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 D0 6 gluino 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,g ˜q˜ andq ˜q˜ 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. 34 CHAPTER 2. COLLIDER SEARCHES FOR JETS+ET 6

Ultimately, a model-independent search for jets plus missing energy would be ideal. We believe that our exclusive nj + ET searches, with results presented in an 6 exclusion plot as a function of HT and ET, would provide signiﬁcant coverage for 6 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 ET cuts, as well as the hard jet energy 6 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 + E 6 T

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 (ET). These searches are very general 6 and cover a wide breadth of potential new theories beyond the Standard Model. Jets

+ ET searches pose a significant challenge, however, because the Standard Model 6 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 + ET at hadron colliders have been model-dependent, 6 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 ET studies 6 can allow them to be model-independent, broadening the reach of the experimental results in constraining theoretical models.

Currently, jets plus ET searches at the Tevatron are based on the Minimal Su- 6 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+ET 6

[50, 51, 52]. These particles subsequently decay into the stable, lightest supersymmetric 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 1 ), and trilinear scalar soft couplings (A0) at 2 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˜ : m : m 6 : 2 : 1, where W refers to the triplet of Wf Be ' winos (W ±, W 0), the supersymmetric partners of the electroweak gauge bosons. Due f to the number of constraints in the CMSSM, the bino is the LSP throughout the f f 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 significantly 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 ET searches should 6 have sensitivity to. Verifying that a jets and ET search has sensitivity to the CMSSM 6 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. 38 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

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 sections, 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 decaying 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 + ET searches are sensitive to. There are two general 6 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˜ q¯1q2W q¯1q2q¯3q4B. (3.1) → →

In this cascade, the W decays directlyf into the B ande a W ±,Z0 boson, which subsequently decays to two jets. This single-step decay is the dominant cascade if the f e 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 m : m : m makes a significant difference g˜ Wf Be in the searches. In the limit where m m the energy fromq ¯3 and q4 is small, Wf → Be while if m mg˜ the jets fromq ¯1 and q2 are soft. If m > mg˜, this cascade is Wf → Wf 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 ˜ q¯1q2B. → Leptons from the decay of the W ±, Z0 boson can be used in the analysis as well e (see Sec. 3.4.5). However, jets + ET + lepton studies are better suited for lepton-rich 6 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 + ET search. 6 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 40 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

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 ± 0 ± bosons WH and ZH are produced through s-channel processes. The WH can decay ± 0 to the W and the dark matter AH , while the ZH can decay to the AH and higgs. ± ± It is also possible to produce the WH 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 + ET channel is divided into four separate searches (monojet, 6 dijet, threejet, and multijet), with each search deﬁned by jet cuts (30 GeV). Cuts O ∗ on the missing transverse energy and total visible energy HT of each event take place during the ﬁnal round of selection cuts. The ET and HT cuts are optimized for 6 “representative” points in CMSSM parameter space for each of the (inclusive) 1j 4+j − searches. However, these ET and HT cuts may not be appropriate for theories other 6 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 ET and HT is overly-restrictive and excludes regions of phase space 6 that are kinematically allowed.

This is explicitly illustrated in Fig. 3.1, which shows the ET distribution of a dijet 6 sample passed through two diﬀerent sets of ET and HT cuts. The signal, a 210 GeV 6 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 D0 cuts and optimized cuts for a sample dijet signal for m = 210 GeV and m = 1006 GeV. Background distribution is shown in gray and g˜ Be signal distribution in white. (Left) Using the D0 cuts HT 300 GeV and ET 6 ≥ 6 ≥ 225 GeV (Right) Using the more optimal cuts HT 150 GeV and ET 100 GeV. The optimized cuts allow us to probe regions with larger≥ S/B. 6 ≥

it also destroys the signal above the ET cut of 225 GeV. These cuts were used in the 6 D0 dijet search; they are optimized for a 400 GeV gluino, but are clearly not ideal 6 ∼ 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 ET. Therefore, with a ET cut of 100 GeV, 6 6 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 ET and HT cuts. This can be effectively 6 done by plotting the differential cross section as a function of ET and HT , 6 d2σ ∆HT ∆ET. (3.2) dHT dET 6 6 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 ET and HT . 6 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, 42 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

Z0 + nj, and tt¯+ nj. The QCD background was not simulated; we expect the QCD

contributions to be important for points in the lowest ET bin. For details concerning 6 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

Nm excl n excl (µ ) e−µ 0.16. (3.3) n! n=0 ≤ X The solution to this equation gives the excluded number of signal events

excl excl S (Nm,B) = µ (Nm) B. (3.4) −

The expected limit on the signal is then given by

∞ −B Nm excl excl e B S (B) = S (Nm,B) . (3.5) h i Nm! N =0 Xm In the limit of large B, the probability distribution approaches a Gaussian and we expect that lim Sexcl(B) = √B. (3.6) B→∞h i In the limit of small B, we expect that

lim Sexcl(B) = ln(0.16) 1.8. (3.7) B→0h i − ≈

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 ET. Because this uncertainty is highly 6 correlated between different differential cross section measurements, treating the uncertainty 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 ET, assigning an uncorrelated systematic uncertainty does not significantly hurt 6 the resolving power of the experiment. In Table 3.1, we have assigned a systematic uncertainty of sys = 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. 2 The reduced chi-squared χN value for N measurements is

N S2 1 χ2 = j , (3.8) N (SL )2 + ( B )2 N j=1 j sys j × X × where Sj is the number of signal events and Bj is the number of background events th in the j box of the grid. The statistical error SLj and the systematic error sys 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; 2 otherwise, the χN 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 44 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

Simulated Background Signal Limits Monojet Dijet Threejet Multijet

Table 3.1: Diﬀerential cross section (in fb) for the Standard Model background is shown in the left column for exclusive 1j 4+j searches. The expected signal sensitivity at 84% conﬁdence is shown on the right− (in fb). The statistical error is shown to 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 are kinematically forbidden. These results are for 4 fb−1 luminosity at the Tevatron. 3.3. EVENT GENERATION 45

simple approach. If the expected significance for a single measurement is greater than crit 2 a critical number, S , it is included in the χN , 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 decay 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 determine 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 ET will 6 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 quantum numbers. 46 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6 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 ET. Such events are easily mistaken as QCD events and eliminated 6 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 ET. Therefore, ISR jets allow us to capitalize on the high production 6 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¯ g˜g˜ + 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. 3.3. EVENT GENERATION 47

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:

2 2 2 2 d (i, j) = ∆Rij min(pT i, pT j) 2 2 d (i, beam) = pT i, (3.10)

2 where ∆R = 2[cosh(∆η) cos(∆φ)] [60]. The event is clustered using the kT ij − 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 2 αs(di ) αs value corresponding to that vertex, i.e. the event weight is multiplied by i 2 , αs(µR) where the product is over the clustered vertices i. This is done in orderQ 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 ME terms. A minimum cutoﬀ d(i, j) > Qmin is placed on all the matrix-element multi- parton events.

After showering, the partons are clustered into jets using the standard k⊥ algorithm. 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, PS d(parton, jet) < Qmin, 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 procedure ensures that jets are not double-counted between diﬀerent parton multiplicity matrix elements, and should furthermore give smooth diﬀerential distributions for all 48 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

Matched sum ~g~g + 0-jet sample pb/bin 10 ~~ QPS gg + 1-jet sample min ~g~g + 2-jet sample ~g~g + Pythia (unmatched) 1

10-1

10-2

0 50 100 150 200 250 Differential Jet Rate 0 ! 1

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 show the contributions from diﬀerent matrix element parton multiplicity samples. The PS matching scale Qmin is marked by the dashed line. The full red curve shows the result using 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

ME PS Qmin = 20 GeV Qmin = 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 PS event. Below Qmin, all jets come from parton showering of the 0j multiplicity sample. PS Above Qmin, 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 3.3. EVENT GENERATION 49

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 using 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 contributor. 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

± 0 The dominant backgrounds for jets + ET searches are W /Z + jets, tt¯, and QCD. 6 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 50 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6 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 D0 analysis did not veto 6 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 mismeasurement due to poorly instrumented regions of the detector (i.e., dead/hot calorimeter cells, jet punch-through, etc.). Additionally, there are many theoretical uncertainties - 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 ET contribution coming from leptonic 6 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 ET of 100 GeV. Jet energy mismeasurement was accounted 6 ◦ ◦ for by placing a lower bound of 90 and 50 on the azimuthal angle between the ET 6 and 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 ET and any jet with pT 15 GeV and η 2.5 was bounded from below 6 ≥ | | ≤ 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 generated using MadGraph/MadEvent, with showering and hadronization in PYTHIA. PGS was again used as the detector simulator for jet clustering. The W ±/Z0 backgrounds were matched up to 3 jets using the MLM matching procedure discussed in ME ME the previous section, with matching parameters Qmin = 10 GeV and Qmin = 15 GeV. ¯ ME The tt backgrounds were matched up to 2 jets with parameters Qmin = 14 GeV and ME Qmin = 20 GeV. For each of the separate backgrounds, 500K events were generated.

The results approximately reproduce the shape and scale of the ET and HT distri- 6 butions published by the D0 collaboration for 1 fb−1 [51]. In the dijet case, our 6 3.4. GLUINO EXCLUSION LIMITS 51

Matched Unmatched 40 GeV Bino 130 GeV Bino

Figure 3.3: Comparison of matched and unmatched events for a dijet sample of 150 GeV gluinos directly decaying into 40 GeV (top) and 130 GeV (bottom) binos. −1 The pT of the hardest jet is plotted in the histograms (1 fb luminosity). Matching is very important in the degenerate case when the contribution from initial state radiation is critical. The diﬀerent colors indicate the contributions from 0j (orange), 1j (blue), and 2j (cyan).

results correspond to those of D0 within 20%. The correspondence is similar for 6 ± 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 ET and HT distributions with good 6 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. 52 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

Sample Model Dijet Monojet Multijet Threejet

Figure 3.4: Diﬀerential cross section (in fb) for the monojet, dijet, threejet, and multijet samples of a theoretical model spectrum with a 340 GeV gluino decaying directly into a 100 GeV bino (4 fb−1). Some boxes show signiﬁcant deviation from the signal limits shown in Table II: green indicates 0.5 < χi 2, blue indicates ≤ 2 < χi 3, and red indicates χi > 3. All boxes with χi > 1/2 are included in the calculation≤ 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 + ET 6 searches 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 3.4. GLUINO EXCLUSION LIMITS 53

150 " GeV ! 100

Out[27]= Mass Bino 50 X

0 100 200 300 400 500 Gluino Mass GeV

Figure 3.5: The 95% exclusion region for D0 at 4 fb−1 assuming 50% systematic error on background. The exclusion region for a6 directly! " decaying gluino is shown in light blue; the worst case scenario for the cascade decay is shown in dark blue. The dashed line represents the CMSSM points and the “X” is the current D0 exclusion limit at 2 fb−1. 6

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 ET and HT cuts, 6 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. ∼ 54 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

Figure 3.6: 95% exclusion region (purple) for a 240 GeV gluino decaying into a bino through a wino. The dashed line is m = m + (m 0 ). The black dot at (m , m ) Wf Be Z Be Wf = (60, 160), is the minimum bino mass for whichO a 240 GeV gluino is excluded for all wino 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˜ m , two hard jets are produced ∼ Wf in the decay of the wino to the bino. In the opposite limit, when m m , two hard Wf ∼ Be jets are produced in the decay of the gluino to the wino. When m < m < m , four g˜ Wf Be fairly hard jets are produced, diminishing the ET and making this region of parameter 6 space the most challenging to see. In particular, the most diﬃcult region to detect is when

m = m + (m 0 ). (3.12) Wf Be O Z In the region of parameter space, where m m , the jets from the wino to bino Wf ∼ Be decay become harder as the gauge bosons go on-shell. 3.4. GLUINO EXCLUSION LIMITS 55

Fig. 3.6 shows the values of m and m that are excluded up to 95% confidence Wf Be 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 m = m + Wf Be (m 0 ). The asymmetry is a result of the hard lepton cuts. When the gluino and O Z 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 56 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

1.0

0.9

0.8

0.7 max Σ Out[27]= ! 0.6 Σ

0.5

0.4

0.3

0.001 0.005 0.010 0.050 0.100 0.500 1.000 2 2 m gluino m squark

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 throughg ˜q˜ orq ˜q˜ 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 ET and a hard photon. 6 The main beneﬁt of the monophoton search is that the Standard Model backgrounds are better understood; unlike the monojet case, QCD is no longer an important 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 → → 3.4. GLUINO EXCLUSION LIMITS 57

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 D0 Collaboration recently published results for their monophoton study, 6 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 ET > 70 GeV. Events with muons or jets with pT > 15 GeV 6 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 D0 ’s background measurements. 6 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 D0 analysis) gives S/B = 0.48 and S/√B = 2.3 for this mass point; thus 6 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 (10 GeV)) because the gluinos are boosted. The monophoton search vetoes many O 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 58 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6

that used by D0 ’s full detector simulator, from which the background estimates were 6 taken.

3.4.5 Leptons

In this section, we address whether leptons from cascades can be used to augment the

sensitivity of jets + ET searches. In the gluino cascade decay considered in this paper, 6 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 + ET study places no exclusion, but a jets + ET + lepton study does. 6 6 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 + ET analysis. We find here, 6 ‡ 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 sensitivity 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 + ET + lepton searches may be useful in cer- 6 tain regions of parameter space, they should be combined with jets + ET searches to 6 provide optimal coverage.

‡Here, the estimate of the signiﬁcance only accounts for the statistical error; it does not include the 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 D0 because the ET and HT cuts used in their 6 6 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, ET, and HT as available handles 6 for increasing the reach of jets + ET searches. However, in certain special cases, other 6 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 + ET searches are still promising discovery 6 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 + ET searches. The 6 backgrounds for the LHC are dominantly the same; however tt¯ will be significantly 60 CHAPTER 3. MODEL-INDEPENDENT JETS+ET 6 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 + ET inclusive searches 6 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 + ET 6 + to 4 j + ET, the LHC will be sensitive to most beyond the Standard Model spectra 6 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 breaking, 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 colliders 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 D0 6 experiments at the Tevatron excluded a SM-like Higgs of 160 GeV m 0 170 GeV ≤ h ≤ [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 particular, 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 tension 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 0 two new Higgs chiral superﬁelds, Hu and Hd, that result in two CP-even scalars H 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

2 2 2 mh0 mZ0 cos 2β ' 2 4 2 3g m m˜ m˜ a + t log t1 t2 + a2 1 t , 8π2m2 m2 t − 12 W t

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

At µ cot β at = − . (4.1) 1 (m2 + m2 ) 2 t˜1 t˜2 q < 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 constraints 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 supersymmetry 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 decays 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 proposed 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 64 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

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 b¯b 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) → →

0 For an h with SM-like production cross section, this process is excluded for mh0 < 110 GeV and X = 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 (1) coupling to the Higgs will swamp the decay modes into SM particles. O 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

iθq iθqs as Hi e i Hi with the singlet compensating by S e S. During electroweak → → symmetry breaking, S also acquires a vev, spontaneously breaking the symmetry; the 4.1. LIGHT A0 MODIFICATIONS TO HIGGS PHENOMENOLOGY 65

phase of S becomes a pseudo-Goldstone boson and has small interactions with the Standard Model when S v. h i 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 h i be described in the exponential basis

+ + ω sin β + h cos β i au Hu = e v sin β √1 (v sin β + h ) 2 u ! 1 √ (v cos β + h ) ad 2 d i Hd = e v cos β ω− cos β + h− sin β ! − 1 0 i as S = ( S + s )e hSi . (4.3) √2 h i

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,

ω 0 = au sin β + ad cos β, Z − becomes the longitudinal component of the Z0. The two other combinations are physical fluctuations that get mass through symmetry breaking effects in the Higgs 2 2 2 potential. Any terms in the Higgs potential proportional to Hu , Hd , or S | | | | | | do 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

2 † † V1 = λ1S HuHd + h.c. . (4.4)

∗A similar analysis was performed for a one Higgs doublet and a complex singlet in [118]. A one Higgs doublet model with a light pseudoscalar typically does not have a large branching ratio of the Higgs into pseudoscalars. 66 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

This will give a weak-scale mass to the following linear combination of pseudo- Goldstones:

0 A = cos θa(au cos β + ad sin β) as sin θa. (4.5) −

The singlet mixing component is given by

v tan θ = sin 2β. (4.6) a S h i The remaining linearly independent pseudo-Goldstone will be massless until the ﬁnal symmetry is broken. This linear combination is

0 a = sin θa(au cos β + ad sin β) + as cos θa (4.7) and gets a mass through potentials such as

2 V2 = λ2S HuHd + h.c. . (4.8)

0 0 0 0 There will be mixing between a and A . However, when λ1 λ2, A and a are 2 2 nearly mass eigenstates with residual mixing proportional to ma0 /mA0 .

It is worth noting that as S v, the light pseudoscalar becomes the Peccei- h i Quinn pseudoscalar and its couplings are independent of S . This particular example h i has the same symmetry structure as the NMSSM near the R-symmetric limit when 3 SHuHd A-term dominates the S 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 4.1. LIGHT A0 MODIFICATIONS TO HIGGS PHENOMENOLOGY 67

100%1.00 ) 0.50 M 50% S 100%1.00 " aa " → !

h 0.20 !

0 20% Br h

50%0.50 ( r 10%Max 0.10 ! B 1 ) n i 0 5%0.05 a 20%0.20 M 0 # a 100 120 140 160 180 200 100 120 140 160 180 200 aa m_h GeV

" 10%0.10 → mh0 (GeV) # h 0 ! " " h

Br 0.05

( 5% r B 0.02 2%

0.01 1% mh0 = 100 GeV

500 1000 1500 2000 S!s/"ssin2bin 2βGeV(GeV) ! "

! " # 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- htively).i The inset shows the minimum value of the branching rate into the Standard Model as a function of mh0 .

Goldstone boson through the coupling

m2 v 0 0 µ 0 ˜ a0 0 0 0 int =c ˜h h ∂µa ∂ a dh h a a . (4.9) L S 2 − v h i 0 0 The ﬁrst interaction preserves the a a + shift symmetry, wherec ˜h is an (1) → O constant and v = 246 GeV is the electroweak scale. Because this coupling exists in the symmetry-preserving limit,c ˜h 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 0 boson is in the direction of the Higgs vev, h = hu sin β + hd cos β, the example above gives

2 2 2 S sin 2β 4 c˜h = sin θa h i = . (4.10) v2 v2 sin2 2β ' tan2 β 1 + hSi2

2 The second interaction breaks the shift symmetry and is proportional to ma0 . 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 68 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

symmetry breaking coupling simpliﬁes to

˜ dh = 1 (4.11) for the potential in Eq. 4.8. For a symmetry breaking potential

4 V20 = λ20 S + h.c. , (4.12)

˜ 0 0 dh would be small, arising from the residual mixing between s and h . ˜ It is possible to increase dh by having multiple terms in the potential contribute 2 to ma0 , with the pseudoscalar mass being less than either of the contributions. For instance, with V2 and V20

2 ˜ 1 > λ2 tan θa dh = 1 if λ20 . (4.13) 2λ20 sin 2β 1 + 2 ' − 2 sin 2β λ2 tan θa ∼

˜ > 0 Of course, this is the technical definition of fine tuning and when dh 1, a 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

2 2 ˜ 2 2 Γh0→a0a0 mh0 c˜hv dhma0 ma0 = 2 + 2 1 + 2 . (4.14) m 0 16π 2 S vm 0 O m 0 h h i h ! h !! The symmetry-preserving interaction dominates when

1 c˜ 2 m 0 v S h h < 1.5 TeV. (4.15) h i ≤ 2d˜ ma0 h ∼ When S is less than 1 TeV, the Higgs boson has an appreciable width into pseu- h i 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 h i search for a Standard Model Higgs [14]. For Higgs boson masses less than 100 GeV, 4.1. LIGHT A0 MODIFICATIONS TO HIGGS PHENOMENOLOGY 69

800

0 ) h SM )

V → 600 P

e (LEP) " E L G ( ( GeV ! β τ 2 4 " 400 n S i → ! s 0 / h " S ! 200

0 85 90 95 100 105 110 115 HigHiggsgs MMassass (GGeVeV)

Figure 4.2: Values of S / sin 2β (GeV) that! have" been excluded through LEP2’s h i search for a Standard Model Higgs [14]. The region below mh0 = 86 GeV is entirely excluded by the h0 4τ search [116]. → the a0 has to be fairly strongly coupled to the Higgs boson, requiring S to be h i 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 −). → → −5 −4 For m 0 between 3.5 GeV and 9 GeV, this limit ranges from 10 10 for the a ∼ − tau decays. The branching fraction for radiative Υ decays is [120]

0 2 2 Br(Υ a γ) GF m m 0 → = Υ g2 1 a F (4.16) Br(Υ µ+µ−) 4√2πα d − m2 → Υ !

+ − 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

mf ¯ 0 int = igf fγ5fa , (4.17) L v

†This case applies to the MSSM and its extensions. For Type I two Higgs doublet models, where all Standard Model fermions only couple to one Higgs doublet, there is no asymmetry between up and down-type quarks in the coupling to axions and this typically results in a cot β suppression in the coupling to axions. 70 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

2.0

1.5 S / sin 2β 250 GeV ! " ∼

g ∆ 1.0 − d S / sin 2β 500 GeV ! " ∼

0.5 S / sin 2β 1000 GeV ! " ∼

0.0 4 5 6 7 8 9 m_a GeV ma0 (GeV)

! " Figure 4.3: Region of ma0 gd parameter space that has been excluded by CLEO to 90% C.L [119]. The dashed− lines indicate values of S / sin 2β for tan β = 2. The h i shaded region shows the minimum values of gd allowed by LEP for an 87-110 GeV Higgs.

where

cot β (up-type quarks) gf = sin θa (4.18) tan β (down-type quarks/leptons) 

(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 h i vev is a measure of the fine tuning of the theory because it induces a mass m2 = λ S 2 eff h i for the Higgs bosons in the scalar potential, and with an (1) coupling, the singlet vev O 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:

sin2 θ sin2 2β √3m ξ2 a Br(h0 ff¯) b , (4.19) m2 SM ' 1 h0 4 → ≤ mh0 1 + 12 2 sin θa mb

0 ¯ where Br(h ff)SM is the Standard Model’s branching ratio to fermion pairs. → There were searches for the (b¯b)(τ +τ −) final state at LEP, but there were no limits for 75 GeV m 0 125 GeV. For 125 GeV m 0 165 GeV, the limits were ≤ h ≤ ≤ h ≤ ξ2 < (0.4) which is automatically satisfied in these models. There were additional ∼ O searches for the (τ +τ −)(τ +τ −) final state, but these constraints are even weaker because 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- 0 0 doscalar a , when 2mτ . ma0 . 2mb. In this range, the a 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 τh )(µ τh ) [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. 72 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

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

0 + − 2 Γ(a µ µ ) mµ → = . (4.20) 0 + − 2 2 Γ(a τ τ ) m 1 (2mτ /m 0 ) → τ − a p The cross section of h0 2µ2τ depends upon the following product of branching → ratios:

0 + − 0 + − µτ = 2 Br(a µ µ ) Br(a 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, µτ 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% 0 0 and 0.3%, respectively, causing µτ to fall to 0.2% for a 7 GeV a and 0.5% for an a 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 D0 and CDF) and 250 events per experiment at the LHC 6 (at √s = 14 TeV) with 500 pb−1 luminosity, despite the small branching fraction to muons. We ultimately find (2%) cumulative efficiency for the signal (Table I), O 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. 4.2. H0 A0A0 AT HADRON COLLIDERS 73 →

Signal Eﬃciency Selection Criteria Relative Cumulative Pre-Selection Criteria 26% 26% Jet veto 99% 26% Muon iso & tracking 50% 13% ∼ Mµµ < 10 GeV 98% 13% µµ pT > 40 GeV 76% 9.8% ET > 30 GeV 29% 2.8% 6 ◦ ∆φ(µ, ET) > 140 73% 2.1% ∆R(µ,6 µ) >0.26 63% 1.8%

Table 4.1: Relative and cumulative signal efficiencies due to the specified selection criteria. 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 + ET, → 6 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 (µτ = 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 74 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

0 0 µ E τ a a ! T τ µ h0

Figure 4.4: Schematic of Higgs decay chain. The muons and taus will be highly boosted and nearly collinear. It is likely that the taus will be reconstructed as one jet. Most of the ET in the event will be in the direction of this jet. 6 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 section 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 approximate 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 kinematic dependence on spin is negligible. As veriﬁcation of this, TAUOLA [126] was used to generate the full spin correlated decays. 4.2. H0 A0A0 AT HADRON COLLIDERS 75 →

fb/GeV TeV LHC DY+j 0.15 0.24 W +W − 0.03 0.08 tt¯ 0.02 0.14 b¯b < 0.001 0.03 Υ + j ∼ 0.001 ∼ 0.002 µµ+ττ 0.001 < 0.001 J/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 0 0 0 + − (dσ/dMµµ) for the h a a (µ µ )(ττ) search at the Tevatron and LHC in → → µµ units of fb/GeV. The backgrounds are given for pT ,ET, and ∆R cuts optimized for a 100 GeV Higgs. 6

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 (30%) O 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 background. 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 altering matrix element-parton shower matching for this study. 76 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

25 700 600

500 s t 400 n

20 e v Events 300 E 200

100

s 15

t 0 4 5 6 7 8 9 n

e Muon MuonInvaInvariantriantMassMasGeVs (GeV) v Events E 10 ! "

0 4 5 6 7 8 9 MMuonuon InvariantInvariantMassMass (GeVGeV)

−1 Figure 4.5: Muon invariant mass for 5 fb at the! LHC" before (inset) and after the µµ pT , ET, and ∆R cuts. The signal, a 100 GeV Higgs decaying to a pair of 7 GeV pseudoscalars,6 is shown in black and the Drell-Yan background is shown in gray.

µµ signiﬁcance of the signal. The ﬁrst is a cut on the sum pT of the muons (pT ), and is approximately µµ pT & 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

ET (0.2 0.25) m 0 . (4.23) 6 & − × h

For the LHC, the ET requirement is always held above 30 GeV. The last is a ∆R cut 6 on the muon pair, which depends on both the Higgs and pseudoscalar masses

4ma0 ∆R(µ, µ) & . (4.24) mh0

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. µµ The inset shows the signal (black) and background (gray) before the p , ET, and T 6 ∆R cuts. After these cuts are placed, the Drell-Yan background is mostly eliminated. 4.2. H0 A0A0 AT HADRON COLLIDERS 77 →

6 s t n

e 4 v Events E

0 50 100 150 200 250 300 350 400 TotTotalal InInvariantvariantMassMassGeV(GeV)

! " −1 Figure 4.6: Total invariant mass for signal with mh0 = 100, 150, 200 GeV for 5 fb µµ at the LHC after pT , ET, and ∆R cuts. The ET is projected in the direction of the hardest jet. 6 6

The muon invariant mass reconstructs the mass of the pseudoscalar.

We used PGS to model the muon invariant mass resolution and used an m 0 a ± 80 MeV to exclude continuum backgrounds. The Drell-Yan background (dσ/dMµµ) is (0.15 fb/GeV) at the Tevatron and (0.24 fb/GeV) at the LHC.¶ O O 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 ET. The most important one is W W production. 6 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

¶ µµ The background cross sections we quote in this section are for pT ,ET, and ∆R cuts optimized for a 100 GeV Higgs. 6 78 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

that the directions of the muon and neutrino are reversed. The µ+µ−νν background was generated with MadGraph and was found to be (0.03 fb/GeV) at the Tevatron O and (0.08 fb/GeV) at the LHC. O Top quark production is another important electroweak background. Using Madgraph to generate µ+µ−ννb¯b, we estimate that this background is (0.02 fb/GeV) at the O Tevatron and (0.14 fb/GeV) at the LHC. The top background becomes nearly O comparable to the Drell-Yan background for larger Higgs masses due to the weaker µµ p ,ET, and ∆R cuts. T 6 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 (8 10−5 fb/GeV) at O × the Tevatron and (2 10−4 fb/GeV) at the LHC. O × 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 b¯b jets to produce two muons is (80 µb) at the LHC. Using a power law extrapolation from low pµµ, it was O T estimated that a pµµ cut of 40 GeV reduces the cross section to (10 pb). A missing T O ◦ energy cut of 30 GeV is 0.6% efficient and requiring ∆φ(ET, µ) > 140 reduces the 6 cross section by an additional order of magnitude to (5 fb). Placing a ∆R cut on O the muons and assuming that the muon isolation is 10% efficient, the cross section becomes (0.3 fb). The invariant mass of the muons in these events is distributed O over (10 GeV), so the final background is approximately (0.03 fb/GeV). This is O O 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 ET to be an important background. 6 4.2. H0 A0A0 AT HADRON COLLIDERS 79 →

10.0 TeV )

b 1 p 5 fb− ( 5.0 1 10 fb− ) 0 1 a

" 20 fb− 0 a a

a 2.0 # → " h ! 0 h Br

( 1.0 r Σ B

× 0.5 d o r

p LEP σ 2 5 5 Exclusion 00 0 0.2 100 120 140 160 180 200 Higgs Mass GeV Higgs Mass (GeV) 100 ! " LHC )

b 1 p 50 .5 fb− ( 1 5 fb− ) 0 a

" 20 0 a a a # → " 10 h 0 ! h Br ( r Σ 5 B × d

o 2 r 50

p 2 LEP 1 σ 0 7 5 Exclusion 0 5 0 0 0 0 1 100 120 140 160 180 200 Higgs Mass GeV Higgs Mass (GeV)

! " 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 sections for several values of S / sin 2β (in GeV), which alters Br(h0 a0a0). The Standard Model Higgs decay widthh i and (NNLO) gluon fusion production→ cross sections were obtained from [15]. An µτ 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 pΥ (60 GeV). T ∼ O The pT spectrum of Υs falls off rapidly. At the Tevatron, the differential cross section is (250 fb) at pΥ 20 GeV [128]. A na¨ıve extrapolation to 60 GeV would place O T ∼ this background at (2 fb). Additionally, the missing energy in these events points O 80 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

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 (6 10−4 fb/GeV) at the Tevatron. We do not expect this background to dominate O × 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 (2 10−3 fb/GeV). O ×

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 pµµ and with pJ/ψ 40 GeV, the cross section is (100 fb) [130]. Because T T ≥ O 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 −9 tail suppresses the J/ψ contamination by (10 ) for m 0 between 3.6 and 9 GeV. O a The Gaussian tail of J/ψ mismeasurement goes out at least 5σ, meaning that the contamination should be down by (10−6). This gives a background cross section O smaller than (10−5 fb/GeV) at the Tevatron and at the LHC. The contributions of O 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 ET from in-flight 6 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 ET is effective 6 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 h i partial width of the Higgs into the pseudoscalars (Eq. 14). The total projected luminosity for the combined data sets at CDF and D0 is 20 fb−1; currently, each 6 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 h i is reached, the Tevatron will have sensitivity up to S / sin 2β = 500 GeV. h i With early data, the LHC has sensitivity to regions corresponding to S / sin 2β h i . 250 GeV. The sensitivity is weaker for Higgs masses below 100 GeV because the µµ backgrounds worsen due to a smaller pT cut. However, combined analyses by CDF and D0 should be able to probe this region down to (1 pb). By the time the LHC 6 O 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. h i . The sensitivity curves depend on the product of the pseudoscalar branching ratios into muons and taus, µτ . For Fig. 4.7, we assumed that the pseudoscalar was 7 GeV, which corresponds to µτ = 0.8%. For a lighter pseudoscalar (e.g., 4 GeV), µτ is nearly double this value. In this case, the signal limits can increase by as much as a 2 2 factor of two. To first order in ma0 /mh0 , 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 subsequently 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 D0 perform 6 combined analyses with the full data sets. The Tevatron can begin to recover the 82 CHAPTER 4. DARK MATTER VIA THE HIGGS SECTOR

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 −1 into a pair of pseudoscalars, the Tevatron is sensitive to mh0 102 GeV with 10 fb , −1 ' and up to m 0 110 GeV with 20 fb . When the Tevatron covers this ground, their h ' 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 pseudoscalars 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 (1 fb−1), O 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 concerning the astrophysical properties of the dark matter, as well as its interactions with the Standard Model (SM). These theoretical uncertainties are compounded by additional 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 (100 keV) mass splitting O 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:

1 mNER vmin = + δ . (5.1) √2mNER µN

The detection rate [37] depends on vmin through

dR ρ vesc dσ = 0 f(~v + ~v (t))v d3v, (5.2) dE m m E dE R dm N Zvmin R 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 mN σN 2 2 2 2 = 2 2 (fpZ + fn(A Z)) Fdm(q )FN (q ) , (5.3) dER 2µNv − | |

where σN is the dark matter-nucleus cross section at zero momentum transfer and 2 q = 2mN ER 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 2 2 2 matter and nuclear form factors, Fdm(q ) and FN (q ). Fdm(q ) describes non-trivial behavior at low momentum transfer in models where higher dimensional operators 2 contribute to the scattering [134, 135, 136]. FN(q ) is the Helm/Lewin-Smith nuclear form factor [37]. Analytic approximations to the nuclear form factor can have substantial 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 184 errors of 25% for W 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 uncertainties 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

2 2 2 2 −v /v −vesc/v f(v) e 0 e 0 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 2 ∗ parameters (v0, vesc,~vstream), and set constraints by a global χ 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 amplitude 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 for null experiments [146,138]. However, Poisson statistics are used in this paper due to the complexity of combing a χ2 for DAMA with multiple max-gap tests to get a global limit. 86 CHAPTER 5. PROSPECTS FOR INELASTIC DARK MATTER

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 scattering 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 ∼ 184 can up-scatterp oﬀ of W and one would expect a larger scattering rate compared to lighter targets. However, an additional complication arises due to the large radius 184 RW of a W 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. 2 Therefore, the scattering rate is suppressed at recoil energies ER 1/(2mNR ). This ∝ W 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

0.015 0.015 " 0.010 keVnr # 0.010 0.005 kg #

0.000

cpd 0 20 40 60 80 100 ! 0.005 Rate

0.000 0 20 40 60 80 100 Recoil Energy keVnr

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).

184 iDM particle deposits on a W nucleus is (δ µ/mW) (50 keVnr), where the O ∼ O 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 127 factor for I, 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 uncertainties in measurements of QI [32]. The four primary ones [149, 150, 151, 152] give 88 CHAPTER 5. PROSPECTS FOR INELASTIC DARK MATTER

50.0 L

keVnr 10.0

100 5.0 - 40 H

1.0

CRESST 0.5 at

Events 0.1

5 10 20 50 Events at CRESST H10-40 keVnrL

Figure 5.2: Average counts at CRESST per 100 kg-d for regular iDM (blue), FFiDM with 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). The contours 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.

184 Fig. 5.1 illustrates how the W 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],

(q0)n ~q m Fdm(q) = cn,m | | + ... (5.6) Λn+m n,m X

0 where q = ER, ~q = √2mN ER, 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 ( ~q ) are dominated by c0,−2. O | | Composite iDM models have c0,1 = 0 [40,155,156]. Form factors that are dominantly 6 n = 0 can be realized through dipole or other tensor interactions [153,154]. 6 Standard iDM (i.e., n, m = 0, 0) and models with n = 0, m = 0 have comparable 6 rates at CRESST because the ratio of predicted events between these two scenar- 2m ios scales as N0,m/N0,0 (mWEW peak/mIEI peak) 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 = 0, m = 0 predict substantially smaller rates at 6 2n n CRESST: Nn,0/N0,0 (EW peak/EI peak) (0.5) . 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 90 CHAPTER 5. PROSPECTS FOR INELASTIC DARK MATTER

300 L 200

keVnr 150 100

- 100 4.5 H 70 50

30 XENON100

at 20 15 Events 10 5 10 20 50 Events at CRESST H10-40 keVnrL

Figure 5.3: Average counts at CRESST (per 100 kg-d) versus XENON100 (per 1000 kg-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 (dashed 2 blue) and QI = 0.06 (dotted blue). The contours enclose all points with χ 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 3 angle. The distribution profile is f(~v) = δ (~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 Θ( ~vstream ~vE(t) vmin) 2 2 | − | − FN(q ) , (5.7) dER ∝ ~vstream ~vE(t) | | | − |

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 dominated 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 (100 kg-d) exposures, the number of events in the low energy window can vary O 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 exposures (3000 kg-d) in their O 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 (40%). Their next O 92 CHAPTER 5. PROSPECTS FOR INELASTIC DARK MATTER

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 combined with additional theoretical and experimental uncertainties, will likely prevent CRESST from refuting the iDM hypothesis with an exposure of (100 kg-d) in a O 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 explaining 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 doublet 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 phenomenology, 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 breaking is preferred. If the susy breaking sector communicates R-symmetry breaking inefficiently, the gauginos and Higgsinos end up much lighter than the typical supersymmetric 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 renormalization 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. 96 CHAPTER 6. SIX HIGGS DOUBLET MODEL

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

2 2 2 2 2 2 2 2 V = m h + m Hi + λ1( h ) + λ2( Hi ) − 0| | 5| | | | | | 2 2 † 2 † 2 +λ3 h Hi + λ4 h Hi + λ5((h Hi) + h.c.) | | | | | | † † +λ6cijk(hHi HjHk + h.c.), (6.1) where i, j, k = 1,..., 5. Depending on the choice of discrete symmetry, there may 4 be several couplings of the form H5 ; 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 = h i v/√2, with v = 246 GeV. The Higgs 5-plets are

+ φ5 H5 = . (6.3) 0 0 (s5 + ia5)/√2 ! 6.1. THE MODEL 97

The physical masses of the particles at the minimum are

2 2 mh0 = 2λ1v 2 2 1 2 m ± = m + λ v φ 5 2 3 2 2 1 2 m 0 = m + v (λ3 + λ4 + 2 λ5 ) s 5 2 | | 2 2 1 2 m 0 = m + v (λ3 + λ4 2 λ5 ) (6.4) a 5 2 − | | 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) − | | 0 0 The splitting between s and a 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 (100 keV), which sets the limit O

> −6 λ5 10 . (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 stability. 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 > 2pλ1λ2. (6.8) − | | − p 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. 98 CHAPTER 6. SIX HIGGS DOUBLET MODEL

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 straightforward 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

−1 −1 α1 α2 14 tGUT = 2π − MGUT 10 GeV (6.9) b1 b2 ⇒ ' − and the value of the gauge coupling at the GUT scale is

−1 −1 b2 α = α tGUT 40. (6.10) GUT 2 − 2π '

If this theory is embedded in a simple SU(5) GUT, the resulting six-dimensional proton decay is

α2 m5 + 0 GUT p −35 −1 Γ(p e π ) 4 10 s , (6.11) → ' MGUT '

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 dependence 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 asymptotically 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 100 CHAPTER 6. SIX HIGGS DOUBLET MODEL

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

dλ 16π2 i = b , (6.12) dt λi

where

2 2 2 bλ 24λ + 2Nhλ + 2Nhλ3λ4 + Nhλ 1 ' 1 3 4 2 2 bλ 4(2Nh + 4)λ + 2λ + 2λ3λ4 2 ' 2 3 2 2 bλ 4λ + 2λ + 4λ4(λ1 + Nhλ2) 3 ' 3 4 +4λ3(3λ1 + (2Nh + 1)λ2) 2 bλ 4λ4(λ1 + λ2) + 4λ + 8λ3λ4 4 ' 4

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 approximately 2 2 max 16π max 16π λ1 0.24 λ2 0.05. (6.13) ' 24tGUT ∼ ' 120tGUT ∼

Figure 6.1 (gray points) shows the weak-scale distribution of λ1 and the eﬀective coupling

λeﬀ = λ3 + λ4 2 λ5 , (6.14) − | | 0 0 which parametrizes the interaction of the WIMP candidate a5 to the SM Higgs h . The values of the quartics at the GUT scale were randomly sampled within the range: † 0 λ1, λ2, λ3 (4π), 1 λ4 0, and λ5 2. They were then run down . | | . O − . . | | . 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 the case of λ5, cause the couplings to run down to non-perturbative values. This range was chosen to maximize 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 electroweak scale obtained by solving the one-loop renormalization group− equations| | for parameter space democracy (gray) and susy (black) boundary conditions at the GUT-scale. The couplings are focused down to a small range at the weak scale. (Right) The distribution of SM Higgs mass for the two sets of boundary conditions. The distribution of allowed Higgs masses is smaller in the case of susy boundary conditions as opposed to 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). 102 CHAPTER 6. SIX HIGGS DOUBLET MODEL

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 c Higgs doublet comes from a vector-like chiral superfield: Φh and Φh for the Standard c Model Higgs and ΦH5 and ΦH5 for the five-plet of scalar dark matter. Specifically,

˜ c † ˜† Φh = cβh sβh Φ = sβh + cβh | − h| ˜ c † ˜ † ΦH = cβ H5 sβ H5 Φ = sβ H + cβ H , 5 | 5 − 5 H5 | 5 5 5 5

where β and β5 are the orientation of the scalars inside the chiral superﬁelds. The resulting D-term potential has the following couplings

2 2 λ = g2 c2 λ = g2 c2 1 5 GUT 2β 2 5 GUT 2β5 7 2 2 λ3 = g c2βc2β λ4 = g c2βc2β −10 GUT 5 GUT 5 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 −6 to take the minimum value allowed by direct detection experiments, λ5 10 . The ∼ 2 gauge couplings at the GUT scale are gGUT = 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

0 The lightest neutral component of the H5 doublet, a5, is a viable candidate for the observed dark matter and its mass may be estimated from standard relic abundance 0 calculations. It is assumed that the a5 is in thermal equilibrium during the early 0 universe. When the annihilation rate of the a5 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 0 + and ∆mφ±a0 , the presence of the additional scalars s5 and φ5 becomes relevant [183]. In this case, interactions involving the two other scalars as initial state particles 0 are important in determining the relic abundance of a5, which must fall within the 2 WMAP region 0.099 < Ωdmh < 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. 0 The number density of a5 is given by

dn = 3Hn σ v (n n neqneq), (6.19) dt ij ij i j i j − − 0 0 ±h i − i,j=Xa ,s ,φ where σij is the sum of the annihilation cross sections of the new scalars Xi into Standard Model particles X

σij = σ(XiXj XX). (6.20) → X X 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 104 CHAPTER 6. SIX HIGGS DOUBLET MODEL

the number of scalars added to the theory in addition to the SM Higgs, Nh, through −1 −2 the interaction cross sections σij. In general, σij N m , so the mass of the dark ∝ h a matter scales as 1 ma0 . (6.21) ∝ √Nh 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. 2 Because these cross sections tend to be rather small, Ωdmh & 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 m 0 80 ∼ a & GeV, diboson production is the dominant annihilation mechanism and keeps the abundance small. There is always another point in this large mass regime where 2 Ωdmh 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 2 parameter ma0 for each set of randomly selected quartic couplings. Figure 6.2 is a 0 plot of the allowed mass of a5 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

Nh ∆T = F (m ± , m 0 ) + F (m ± , m 0 ) 16π2αv2 φ a φ s h F (m 0 , m 0 ) , (6.22) − a s i 6.3. DARK MATTER 105

0 Figure 6.2: Allowed mass of the LSP a5 as a function of the Higgs mass for parameter space democracy (gray) and susy (black) boundary conditions. All points included in this plot fall within 1σ of the electroweak precision data (Sect. 6.3.2) and are consistent with LEP results (Sect. 6.4.3). The SM Higgs can decay into a pair of WIMPs if ma0 lies below the dashed red line. where

2 2 2 2 2 m1 + m2 m1m2 m1 F (m1, m2) = log . (6.23) 2 − m2 m2 m2 1 − 2 2

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, − 2 (∆m)4 F (m, m + ∆m) = (∆m)2 + (6.24) 3 O m2 and the expression for ∆T reduces to

Nh ∆T (m ± m 0 )(m ± m 0 ) ' 12π2αv2 φ − a φ − s 2 Nhv 2 2 2 (λ4 4λ5). (6.25) ' 192π αmams −

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. 106 CHAPTER 6. SIX HIGGS DOUBLET MODEL

The S parameter also has contributions from the additional Higgs doublets [173] and is easily generalized to the case of six Higgses

1 2 2 N xm 0 + (1 x)m 0 ∆S = h x(1 x) log s − a dx. (6.26) 2π 0 − mφ± Z h i When the mass splittings are small,

2 Nh ∆m ∆S = ∆ms0a0 2∆mφ±a0 + 12πma0 − O m 2 Nhv λ4 2 . (6.27) ' 24πma0

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 effective Lagrangian iλeff 0 0 Leff = − mqa a qq.¯ (6.28) m2 q h0 X 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 will be tested for at the current CDMS II run (upper dashed line). The third phase of SuperCDMS (lower dashed line) will probe a greater region of the parameter space.

which in this case is

2 2 4 −9 λeﬀ 350 GeV 200 GeV σn = 2 10 pb . (6.29) × 0.4 m 0 m 0 a h

−2 This cross section scales as Nh (see Eq. 6.21). Because σn m 0 , the lighter dark ∝ a 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 inelastic 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 ∆m 0 0 100 keV between s a ' the two lightest scalars [186]. 108 CHAPTER 6. SIX HIGGS DOUBLET MODEL

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 φ5 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

2 2 0 0 0 1 v λeﬀ Γ(h γX) σ(a a γX)u 2 2 2 2 → , (6.30) → ' Nh (s m 0 ) + m 0 Γ 0 √s − h h h where u is the relative velocity between the initial two WIMPs and s 4m2 . The ≈ a0 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 φ5 and one W 0 ± in the loop. When the a5 and φ5 are nearly degenerate and ma0 mW ± , there is an + − effective long-range Yukawa force between the φ5 φ5 pair in the loop that is mediated by the gauge boson: −m r e W ± V (r) α2 . (6.31) ∼ − r

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 6.4. EXPERIMENTAL SIGNATURES 109

Figure 6.4: Approximate ﬂux from dark matter annihilation in the galactic halo 0 0 via a5a5 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:

−2 2 2 + − 2α2ma0 2m∆mφ±a0 σ(a5a5 φ5 φ5 )u 2 1 + 2 . (6.32) → ∼ NhmW ± s mW ± !

Multiplying this by the decay width of the bound state to two photons (or, γZ0), gives the total annihilation cross section

−2 2 2 2πα α2 2m∆mφ±a0 σ(a5a5 γγ)u 2 1 + 2 . (6.33) → ∼ NhmW ± s mW ± !

This cross section does not depend on ma0 (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 110 CHAPTER 6. SIX HIGGS DOUBLET MODEL

with a ﬁeld of view ∆Ω and line of sight parametrized by Ψ = (θ, φ) is given by

2 σγX u 100 GeV ¯ Φ = CγX J(Ψ, ∆Ω)∆Ω, (6.34) 1 pb ma0 where

−9 −2 −1 Cγγ = 1.1 10 cm s × −10 −2 −1 C 0 = 5.5 10 cm s (6.35) γZ ×

and the function J¯ includes the information about the dark matter distribution in ¯ 3 the halo. Note that the flux is independent of Nh. For the NFW profile, J 10 ' 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+φ− and e+e− a0s0. (6.36) → 5 5 → 5 5 6.4. EXPERIMENTAL SIGNATURES 111

Figure 6.5: Mass splittings for the light dark matter (m 0 80 GeV). LEP excludes a ∼ the region of intermediate s5 mass. The region on the upper left is excluded by electroweak precision results, while that on the lower right is excluded by vacuum stability. 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 φ5 can be deduced. The charged Higgs 5-plet φ± is ruled out for masses below 90 GeV and the neutral 5 ∼ scalar s0 is ruled out for masses between 100 120 GeV (depending on the mass 5 ∼ − of a5). Fig. 6.5 summarizes the important constraints on the mass splittings ∆m ± 0 φ5 a 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,

∆m ± 0 , ∆ms0a0 20 GeV in the heavy dark matter regime. φ5 a . 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 φ5 , 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 φ5 are also 112 CHAPTER 6. SIX HIGGS DOUBLET MODEL

Figure 6.6: Width of the SM Higgs decay into the H5 scalars for the case of parameter space democracy (gray) and susy (black) boundary conditions. The top line shows the 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

2 2 4m2 Nhv 2 4ma0 2 φ± Γinv = λeﬀ 1 2 + 2λ3 1 2 32πmh0 " s − mh0 s − mh0 2 2 4ms0 +(λeﬀ + 4 λ5 ) 1 2 . (6.37) | | s − mh0 #

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 6.4. EXPERIMENTAL SIGNATURES 113

0 0 ∗ pp a s Z + ET → 5 5 → 6 + − ∗ ∗ pp φ φ W W + ET → 5 5 → 6 0 ± ∗ ∗ pp s φ Z W + ET → 5 5 → 6 0 ± ∗ pp a φ W + ET . (6.38) → 5 5 → 6

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 ET after the decay of the gauge bosons. The production 6 + − cross section σprod for s5a5 and φ5 φ5 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

2 σbackground = σ(pp WW )Br(W lν) → → +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 background 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. 114 CHAPTER 6. SIX HIGGS DOUBLET MODEL

0 0 + − Figure 6.7: LHC production cross section σprod for a5s5 and φ5 φ5 , assuming ∆ms0a0 = 10 GeV, ∆mφ±a0 = 15 GeV, mh0 = 120 GeV, and λ3 = 0.3. The dotted line is the signal 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 (ma0 80 GeV) 5 ∼ 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 (m 0 200 GeV) are more diﬃcult to see, but lie within the a5 & 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 existence 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 phenomenology 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 particular, 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 interactions may indicate novel dynamics in the dark sector. For example, iDM requires an (100 keV) splitting between the dark matter states, which may be a sign that O 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 consequences 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

= SM + CiDM (7.1) L L L 1 2 ¯ ¯ CiDM = Tr G + ΨLiDΨL + ΨH iDΨH L −2 d µν 6 6 ¯ ¯ +mLΨLΨL + mH ΨH ΨH

where Ψa, a = L, H, are Dirac fermions that are fundamentals under the strong gauge sector and Gd µν 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 h i 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. (30 TeV) inelastic dark matter is not compatible with O CDMS’s null results [18,19,20] if it ﬁts the DAMA signal and therefore the relic abundance 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 n ¯ = nL + n¯ = 0, (7.2) − H − L 6 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

1 µ = 0 µ µ P ρdµ = ( 1) ρµ ( 1) = . (7.3) − −  1 µ = 1, 2, 3 −  The mass splitting between the πd and the ρd arises through the hyperﬁne interaction and is suppressed when mH mL, Λd. In particular, 2 κΛd , mL Λd mH

δm = mρd mπd (7.4) − ' 4 2 λdmL mL Λd, NcmH 7.1. MODELS OF CIDM 119

2 where λd = Ncgd/4π is the ’t Hooft coupling with Nc = 1 applying to Abelian gauge groups and κ is an (1/Nc) constant. Note that mπ mH in the heavy quark mass O d ∼ limit. For mass splittings (100 keV) and a dark matter mass near the weak scale, O the conﬁnement scale is 100 MeV. ∼ µ The dark matter mesons interact through a massive spin 1 gauge ﬁeld Ad that kinetically mixes with U(1)Y [42,203,204,205,206,207]

1 2 µν Gauge = F + F Bµν L −4 d 2 d 2 2 1 2 2 Higgs = Dµφ λ( φ f ) , (7.5) L | | − | | − 2 φ

µ µ µ where D φ = ∂ φ 2igdA φ. After φ acquires a vev, fφ, Ad becomes massive and − d the mixing between the dark sector and the Standard Model can be diagonalized.

The 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

mA = 2gdfφ m 0 (7.6) d Z after electroweak symmetry breaking, the couplings can be diagonalized. The interactions relevant for dark matter scattering are given by

µ µ Int = (J + cθJ ) Ad , (7.7) L d EM µ 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 120 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

charge assignment in terms of Weyl spinors is

c c Axial ψH ψH ψL ψL SU(Nc) , (7.8)

qU(1) +1 +1 1 1 d − −

¯c where the Dirac spinors Ψa = (ψa, ψa). The masses of the quarks arise from U(1)d breaking through the Higgs mechanism

c † c Yuk = yH φψH ψ + yLφ ψLψ + h.c. . (7.9) L H L

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

m 2g Ad = d . (7.10) mπd yH

Fitting DAMA requires mπ 100 GeV, while mA can in principle take on a range d ∼ d 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 (1). O

The eﬀective operators describing the interactions of the πd ρd system are −

a † µ Axial eﬀ = dinmπd πd ρdAdµ L a cin † µ ν + πd ∂ ρdFd µν Λd a del † ν † ˜µν + 2 (πd ∂µπd + ρd ∂µρdν)∂νFd Λd a † ˜µν +celρdµρdνFd + 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 2 overall suppression of the elastic to inelastic scattering rate of vrel. 7.1. MODELS OF CIDM 121

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

c c Neutral Vector ψH ψH ψL ψL SU(Nc) . (7.12)

qU(1) +1 1 +1 1 d − −

With these charge assignments, Ad couples to a vector current and the allowed operators are

v din † µ ν ˜ Vector eﬀ = πd ∂ ρdFd µν L Λd v cel † ν † µν + 2 (πd ∂µπd + ρd ∂µρdν)∂νFd Λd v † µν +delρdµρdνFd + h.c. . (7.13) d denotes operators that are velocity suppressed and c denotes unsuppressed operators. 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. 122 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

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 suppression. This is quite natural in strongly coupled CiDM models because CP violation arises from the dynamics of the strong sector through the term

˜ 6P = ΘdTr GdGd, (7.14) L 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 (1). O 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 0 G P − − different parity. In QCD, for example, the π with I (J ) = 1 (0 ) and the a0 with G P − + I (J ) = 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

mL sin θ6P Θd . (7.15) ∼ Λd

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

πd H6P a0 d ΘdΛd sin θ6P = h | | i 2 , (7.16) ma mπ ' λ mL 0 d − d d 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 (1), its effects on the interactions of the dark mesons might be small ∼ O 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:

a a cel † µν cel † µν πda0 d = 2 πd∂µa0 d∂νFd 2 sin 2θ6P πd∂µπd∂νFd . L Λd → 2Λd

Therefore, the eﬀects of Θd can be estimated by replacing the ﬁeld strengths in Eq. 7.11 and Eq. 7.13 with∗

µν µν ˜µν F cos 2θ6PF + sin 2θ6PF . (7.17) d → d d

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 θ6P 0.08 is ∼ necessary to avoid direct detection constraints assuming that all c, d (1) and ∼ O mN mπ . ∼ d

7.2 Direct Detection Phenomenology

Novel features in the direct detection phenomenology of composite models arise because the dark matter has a ﬁnite size Λ−1 m−1. The cross section is suppressed d πd 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 2.

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

∗ µν µ This only applies to the ﬁeld strengths, Fd , not the gauge potentials, Ad , whose interactions are constrained by gauge invariance. 124 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

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 4m2 E2 κ m E dσ = θ 2 N R + N R 0 , dE 6P (m δm)2 2m δm dE R πd πd R a a where mN is the mass of a nucleus with charge Z recoiling with energy ER, cin, cel = 1 of Sec. 7.1.3 and

2 2 dσ0 8Z αmN 1 FHelm(ER) = | | . (7.18) 2 4 2 2 dER v feﬀ 1 + 2m E /m N R Ad 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

2 2 3j1( q r0) −s2|q|2 FHelm(ER) = | | e , (7.19) | | q r0 | |

2 2 1/3 where s = 1 fm, r0 = √r 5s , and r = 1.2A fm [208]. −

The differential cross section depends on the confinement scale Λd = mπd δm/κ, the mass of the dark photon m , and the couplings of the effective theory Ad p

m2 2 Ad feﬀ = , (7.20) κgd

where κ is the (1/Nc) constant deﬁning the mass diﬀerence from Eq. 7.4. O 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 detection experiments. This procedure is outlined in [145] and is summarized here. The 7.2. DIRECT DETECTION PHENOMENOLOGY 125

450 95% contours 140 keV 100 keV 400

350 60 keV

300 240 68% contours

(MeV) 240 ! 250 220 200 (MeV) ! 200 180 160

60 80 100 120 140 160 180 200 150 Dark Matter Mass (GeV) 100 200 300 400 500 Dark Matter Mass (GeV)

Figure 7.1: 95% contours in mπ Λd parameter space for θ6P = 0.00, 0.06, 0.07, 0.08. d − 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 θ6P = 0.00, 0.04, 0.06 for the same

κ and mAd . The colors correspond to θ6P = 0.00 (blue), 0.04 (teal), 0.06 (magenta), 0.07 (yellow), 0.08 (green).

diﬀerential scattering rate per unit detector mass is

dR ρ dσ = 0 d3v f(~v + ~v ) v , (7.21) dE m m e dE R πd N Z R 3 where ρ0 = 0.3 GeV/cm 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:

v 2α v 2α f(v) exp exp esc , (7.22) ∝ v − v 0 0 126 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

1100 1200 1000

900

1000 (GeV) 800 eff f

700

600 800 68% contours (GeV)

eff 50 100 150 200 f Dark Matter Mass (GeV) 600

400 95% contours 0 100 200 300 400 500 600 Dark Matter Mass (GeV)

Figure 7.2: 95% confidence limit regions of mπ feff for θ6P = 0.00, 0.06, 0.07, 0.08. d − For this figure, κ = 0.25 and mAd = 1 GeV. The inset shows the 68% confidence

regions for θ6P = 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 approximations 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 mπ Λd and mπ feﬀ spaces are shown d − d − 7.2. DIRECT DETECTION PHENOMENOLOGY 127

in Fig. 7.1 and 7.2, respectively. The minimal χ2 has a value of 4.61 and the corresponding point is listed as model CiDM1 in Table 7.1. The 1 σ and 2 σ regions are set by

2 2 2 χ < χmin + ∆χ , (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 as

m2 √ Ad 2 2mAd mπd = 2 = 2 . (7.25) gdfeﬀ yH feﬀ

For theories with dominant inelastic scattering, feﬀ (700 GeV) and mπ ∼ O d ∼ (100 GeV) to satisfy both the DAMA and null experiments. Therefore, keeping O yH 1 ' m = (10−4) Ad , (7.26) O 1 GeV which corresponds well with the results of the χ2 global ﬁt. Fig. 7.3 shows the 1σ and 2σ regions in the mA parameter space allowed by all current direct detection d − experiments. A benchmark value of yH = 1 is chosen; the contours shift to larger 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 128 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

CiDM1 CiDM2 CiDM3 CiDM4

mπd 72 GeV 75 GeV 234 GeV 162 GeV δm 109 keV 105 keV 91 keV 126 keV Λd 177 MeV 177 MeV 292 MeV 286 MeV feﬀ 738 GeV 846 GeV 563 GeV 268 GeV 3.7 10−4 3.0 10−4 2.1 10−3 3.8 10−4 × × × × mAd 1 GeV 1 GeV 1 GeV 60 MeV θ6P 0.00 0.04 0.04 0.06 v0 272 km/s 273 km/s 202 km/s 280 km/s vesc 510 km/s 501 km/s 558 km/s 501 km/s α 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 of inelastic and subdominant elastic scattering. CiDM3 shows the larger mass window with slow halo parameters. CiDM4 shows a light mAd model.

spectrum (ER = 5 keVee) is

2m E ~q = I R 120 MeV. (7.27) | | s qI '

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 −1 this eventually forces feﬀ to become large, increasing the allowed values of . These eﬀects are shown in Fig. 7.3. A low mAd benchmark 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

0.010 0.005

0.001 5´10-4 Ε

1´10-4 5´10-5

1´10-5 0.01 0.1 1 10 100 Dark Photon Mass HGeVL

Figure 7.3: The 95% limits on mAd with yH = 1 for θ6P = 0.00 (blue), 0.06 (magenta), 0.08 (green). The dark gray− regions are excluded by limits on the g 2 − of the electron and muon (top left) and ﬁxed-target experiments (light mAd and moderate ). 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 a case-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

2 2 2 me −8 2 mµ −6 F 2 < 1.5 10 F 2 < 6.4 10 , (7.28) mA × mA × d d where

1 2z(1 z)2 F (x) = dz − . (7.29) (1 z)2 + z/x Z0 −

Fig. 7.3 shows that g 2 limits are most important at low mA and large . − d

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 130 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

only alters the neutral currents and the Lagrangian for this sector is

1 2 2 µν 2 Gauge = F + B 2F Bµν + W L −4 dµν µν − d 3µν 2 0 2 0 m 0 2h m √2φ + Z 1 + Z2 + Ad 1 + A2 2 v 2 f d φ ! +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 of [210,211]

< 1 10−2. (7.31) ∼ × > This constraint becomes more important than the muon g 2 limit when mAd 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 m2 Ad 0 µ Zd sθ 2 ZµJd , (7.32) L ' mZ0 !

2 2 0 to lowest order in and m /m 0 . The width of the Z decay into the dark sector is Ad Z

2 2 2 4 2 8 Ncg s m N s m 0 ¯ d θ Ad c θ Ad Γ(Z ΨLΨL) 3 3 4 . (7.33) → ' 12πmZ0 ' 12π mZ0 feﬀ

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

0.045 m 100 GeV 3 < πd 1 , (7.34) 2 70 GeV m ∼ N Ad c 7.3. SEARCHES FOR THE DARK PHOTON 131

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,

+ − 1 2 2 Γ(Ad ` ` ) αc mA 30 eV (7.35) → ' 3 θ d ' for the benchmark model CiDM1. Hadronic decays are also allowed, but are subdominant 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 electromagnetic current is suppressed by a factor of relative to the coupling to the dark quarks µ µ dd + dem AdµJ + cθAdµJ . (7.36) L L ' d em In this limit, the branching fraction into SM leptons is negligible:

2c2g2 64g2c2 m4 ¯ θ θ πd −4 Br(Ad ``) 2 4 4.0 10 (7.37) → ' Ncgd ' Nc feff ' × 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 132 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

0 −+ there is only one light quark. Therefore, the lightest meson is ηd (0 ), in analogy 0 with QCD. Using the SM η as a prototype and the SM a0 meson as a typical hadronic 0 state, the mass of the ηd is estimated to be

√3 mη0 1.7Λd m 0 Λ . (7.38) ηd d ' √Nc ma0 ' √Nc

0 The ηd becomes light in the large Nc limit. For Nc & 10, the dark photon can decay 0 0 to the ηd. However, ηd 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 η0 A∗A∗ [214, 218]. The decay width d → d d 2 is suppressed by an additional factor of (Λd/fφ) relative to the φ decay mode of 0 [214, 218, 219] because the ηd has to mix with the φ to mediate the decay. The resulting decay width is

4 2 2 2 α m m 0 Λ 0 + − e ηd d 1 Γ(ηd e e ) 3 4 10 . (7.39) → ' (4π) fφ 10 years

0 The cosmological relic abundance of ηd is suﬃciently small to make up a small fraction of the matter density of the Universe.

0 −− Unlike the ηd, 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

m4 θ ωd , (7.40) ωd (m2 + m2 )2 ' ωd Ad the decay width is [218]

2αg2c2 m5 αc2 Λ5 1 Γ(ω e+e−) d θ ωd θ d (7.41) d 3 m4 3 f 4 20 m → ' Ad ' eﬀ ' 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 mA Λd and both the d 7.3. SEARCHES FOR THE DARK PHOTON 133

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 the Ad 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 (10), the number of dark hadrons is moderate. Because ∼ O 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 mA /Λd 1, there can be a large number of dark hadrons d 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 0 < −5 > Br(Z ) 10 . From Eq. 7.33, this corresponds to masses of the Ad 4 GeV. ∼ ∼ When the Ad decays with a mass mA Λd, it decays into a pair of dark quarks, d ¯ 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. 134 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

7.4 Discussion

This article introduced a composite inelastic dark matter model with dominant inelastic 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

mL < θ6P Θd 0.08. (7.42) ' Λd ∼

Discovering the elastic subcomponent will place a lower limit on mL 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,

vesc vmin cos γmax = − , (7.43) ve 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,

mN ER 2µ2 elastic vmin(ER) = , (7.44) q√ 1 mN ER + δm inelastic  2mN ER µ 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 mπ 70 GeV. This leads to an ∼ d ∼ 7.4. DISCUSSION 135

estimate of

Λd δm mπ = 150 MeV (7.45) ' d p > 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

2 1 FDM(q ) = 2 2 , (7.46) 1 + q rDM

−1 where rDM = 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 −1 factors can be estimated by extrapolating rDM Λd and behave similarly to having −1 → mA r . d ∼ 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., ¯ ¯ ΨH ΨH ΨLΨL 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 products 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 estimates 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 136 CHAPTER 7. PARITY VIOLATION IN CIDM MODELS

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 experiments 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 motion 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 scattering, which has an exponentially falling recoil energy spectrum [37]. A distinguishing feature of DAMA’s measured modulation spectrum is that it is suppressed at energies 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 (100 keV) more massive than the ground state [195, 147, 235, 138]. In O 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 comparison 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 factor 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 inelastic 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

2 2 4 Fdm(q ) = c0 + c1q + c2q + . (8.1) ···

If the ﬁrst term in this expansion vanishes, the elastic scattering rate is multiplied 2 by additional factors of q = 2mN ER and goes to zero as ER 0. Form factor- → suppressed scattering can arise from multipole, polarization, and charge-radius interactions 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 mN ER 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 8.1. DIRECT DETECTION PHENOMENOLOGY 139 L 0.03 keVee kg 0.02 cpd H

0.01 Amplitude

0.00 Modulation 0 2 4 6 8 Recoil Energy HkeVeeL

Figure 8.1: Comparison of the modulation amplitude for two scattering scenarios: completely inelastic scattering (solid) and inelastic scattering with a subdominant elastic charge-radius component (dashed). The points show the modulation amplitude measured 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 understanding the underlying structure of the theory. This article addresses how directional detection experiments, which can measure the direction of the nuclear recoil in addition 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 140 CHAPTER 8. DIRECTIONAL DETECTION

detector mass is

dR ρ dσ = 0 v , (8.2) dE m m dE R dm N R

3 where ρ0 = 0.3 GeV/cm 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 2m E nel 2m E nin dσ = c2 N R + c2 N R 0 , dE el Λ2 in Λ2 dE R R (8.3)

where cel,in are the dimensionless couplings for the elastic and inelastic interactions, Λ is the scale for new physics, and

2 dσ0 mN σN (fpZ + fn(A Z)) 2 = 2 2 2 − FH(ER) (8.4) dER 2v µ fp | |

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,

2 2 3j1( q r0) −s2|q|2 FH(ER) = | | e , (8.5) | | q r0 | |

2 2 1/3 where s = 1 fm, r0 = √r 5s , and r = 1.2A 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, 2 leading to a power series in q and vrel. Eq. 8.3 includes the leading non-vanishing term 8.1. DIRECT DETECTION PHENOMENOLOGY 141

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 2 the spectrum as long as Λ mN ER [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 dσ v = d3v f(~v + ~v ) v . (8.6) dE e dE R Zvmin R The minimum velocity is set by the kinematics of the scattering process:

mN ER 2µ2 elastic vmin(ER) = (8.7) q√ 1 mN ER + δm , inelastic  2mN ER µ 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 wherex ˆ points towards the galactic center,y ˆ points in the direction of the galactic rotation, andz ˆ 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) ≈ 142 CHAPTER 8. DIRECTIONAL DETECTION

The velocity of the Earth in the sun’s rest frame is given by

2π(t t0) 2π(t t0) ~v⊕ v⊕ ˆ1 cos − + ˆ2 sin − , (8.10) ≈ yr yr where t0 is the spring equinox ( March 21), v⊕ = 29.8 km/s is the orbital speed of ≈ the Earth [37], and

ˆ1 = (0.9931, 0.1170, 0.01032) (8.11) − ˆ2 = ( 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 distribution 2α 2α −(v/v0) −(vesc/v0) f(v) = N e e 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 8.1. DIRECT DETECTION PHENOMENOLOGY 143

χ2 function is

Nexp pred obs 2 Xi Xi χ (mdm, δm, σn, v0, vesc, α) = − , (8.13) σ i=1 i ! X pred where Nexp is the number of experiments included in the ﬁt, Xi is the predicted obs experimental result, Xi 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. 144 CHAPTER 8. DIRECTIONAL DETECTION

Figure 8.2: The 95% contours in the mdm δm parameter space corresponding to − cel/cin = 0 (blue), 0.4 (magenta), 0.6 (yellow) and 0.8 (green) for the scenario with nin = 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

2 2 2 χ (mdm, δm, σn, v0, vesc, α) = χmin + ∆χ (CL), (8.15) where ∆χ2(95%) = 12.6 for six degrees of freedom. The minimal χ2 found was 2 2 χ = 4.1 for cel/cin= 0 and thus every prediction was forced to have a χ 16.7 min ≤ corresponding to the 2σ band. The best-ﬁt point for the completely inelastic scenario is

(v0, vesc, α) = (279, 586, 0.80) (8.16) −39 2 (mdm, δm, σn) = (59 GeV, 121 keV, 5.9 10 cm ), ×

2 2 where σn = σN(µN /µn) 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 8.1. DIRECT DETECTION PHENOMENOLOGY 145

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) −39 2 (mdm, δm, σn) = (74 GeV, 98 keV, 1.2 10 cm ). ×

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 recoil 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 146 CHAPTER 8. DIRECTIONAL DETECTION

5.00

1.00 0.50 keV

Counts 0.10 0.05

0 20 40 60 80 Recoil Energy HkeVnrL

Figure 8.3: Estimated recoil spectrum at LUX with 1000 kg-days and 30% efficiency during 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-fit point 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 direction 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 0 ~v = vxˆ and scatters oﬀ a nucleus at rest. The recoiling nucleus has velocity ~vR and makes an angle θ with thex ˆ-axis. Energy and momentum conservation yield an expression for the recoil velocity,

2µv vR = cos θ, (8.18) mN which can be written in the frame-invariant form

vmin vˆ vˆR = 0. (8.19) · − v

The diﬀerential directional scattering rate per unit detector mass is

d2R ρ dσ = 0 v2 (8.20) dERd cos γ mdmmN dER 3 d vf(v)δ(~v vˆR ~ve vˆR vmin). × · − · − Z 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

∞ ˆ 3 f(w) = d v f(v)δ(~v vˆR w) = 2π dv vf(v), (8.21) · − Z Zw 148 CHAPTER 8. DIRECTIONAL DETECTION

0.020

0.010

0.005 kg

cpd 0.002 FFeDM Γ 0.001 dcos 4

dR 510

2104 iDM 1104 1.0 0.8 0.6 0.4 0.2 0.0 0.2 cos Γ

Figure 8.4: cos γ spectrum during the summer for the best-ﬁt parameters when cel = 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 to dominate over the inelastic scattering.

where w = ~ve vˆR + 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

2α ˆ 2 2 −(vesc/v0) f(w) = πN (w vesc)e ( − 2 2α 2α v0 1 w 1 vesc + Γ , Γ , Θ(vesc w). α α v0 − α v0 !) − h i h i 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 vˆR = 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 fˆ(w). In particular, there is a maximum value of cos γ above which the rate is zero:

vesc vmin cos γmax = − . (8.22) ve 8.2. DIRECTIONAL DETECTION 149

5 kg-yr

100 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. The shaded 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 a detector of CF3I. The dashed lines show the projected sensitivity for the DMTPC CF4 detector after 5 and 100 kg-yr [236]. These projected sensitivies should be even stronger for a CF3I detector, which would use a heavier nucleus to test the inelastic hypothesis.

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 depends 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 inelastic 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 150 CHAPTER 8. DIRECTIONAL DETECTION

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 unambiguously, even with next-generation experiments such as LUX and XENON100. In particular, directional detection experiments can detect elastic wide angle scatters that are kinematically forbidden in inelastic transitions. To evade current null experiments, 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 underlying 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 composite [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 several 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. Bibliography

[1] F. Zwicky, Ap. J 86, 217 (1937).

[2] E. Corbelli and P. Salucci, (1999), astro-ph/9909252.

[3] E. Kolb and M. Turner, The Early Universe (Westview Press, New York, 1990).

[4] S. Dodelson, Modern Cosmology (Academic Press, San Francisco, 2003).

[5] M. Rauch et al., Ap. J 489, 7 (1997).

[6] C. Pryke et al., Astrophys.J. 568, 46 (2002), astro-ph/0104490.

[7] C. B. Netterﬁeld et al., Ap. J 571, 604 (2002).

[8] G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rept. 267, 195 (1996), hep-ph/9506380.

[9] J. Terning, Modern Supersymmetry (Clarendon Press, Oxford, 2006).

[10] S. P. Martin, (1997), hep-ph/9709356.

[11] H. P. Nilles, Phys. Rept. 110, 1 (1984).

[12] T. Appelquist, H.-C. Cheng, and B. A. Dobrescu, Phys. Rev. D64, 035002 (2001), hep-ph/0012100.

[13] N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Phys. Lett. B513, 232 (2001), hep-ph/0105239.

152 BIBLIOGRAPHY 153

[14] LEP Working Group for Higgs boson searches, R. Barate et al., Phys. Lett. B565, 61 (2003), hep-ex/0306033.

[15] Particle Data Group, C. Amsler et al., Phys. Lett. B667, 1 (2008).

[16] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich, and G. Weiglein, Eur. Phys. J. C28, 133 (2003), hep-ph/0212020.

[17] R. Gaitskell, Ann.Rev.Nucl.Part.Sci. 54, 315 (2004).

[18] CDMS, D. S. Akerib et al., Phys. Rev. D73, 011102 (2006), astro-ph/0509269.

[19] CDMS, D. S. Akerib et al., Phys. Rev. Lett. 96, 011302 (2006), astro- ph/0509259.

[20] CDMS, Z. Ahmed et al., Phys. Rev. Lett. 102, 011301 (2009), 0802.3530.

[21] CDMS II collaboration, Z. Ahmed et al., Science 327, 1619 (2010).

[22] E. Armengaud et al., Phys. Lett. B687, 294 (2010), 0912.0805.

[23] G. Angloher et al., Astropart. Phys. 23, 325 (2005), astro-ph/0408006.

[24] G. Angloher et al., (2008), 0809.1829.

[25] J. Schmaler et al., (2009), 0912.3689.

[26] G. J. Alner et al., Astropart. Phys. 28, 287 (2007), astro-ph/0701858.

[27] V. N. Lebedenko et al., Phys. Rev. D80, 052010 (2009), 0812.1150.

[28] D. Y. Akimov et al., (2010), 1003.5626.

[29] XENON10, J. Angle et al., Phys. Rev. D80, 115005 (2009), 0910.3698.

[30] XENON10, J. Angle et al., Phys. Rev. D80, 115005 (2009), 0910.3698.

[31] XENON100, E. Aprile et al., (2010), 1005.0380.

[32] R. Bernabei et al., Riv. Nuovo Cim. 26N1, 1 (2003), astro-ph/0307403. 154 BIBLIOGRAPHY

[33] R. Bernabei et al., (2003), astro-ph/0311046.

[34] R. Bernabei et al., Int. J. Mod. Phys. D13, 2127 (2004), astro-ph/0501412.

[35] R. Bernabei et al., Eur. Phys. J. C56, 333 (2008), 0804.2741.

[36] A. Drukier, K. Freese, and D. Spergel, Phys.Rev. D33, 3495 (1986).

[37] J. D. Lewin and P. F. Smith, Astropart. Phys. 6, 87 (1996).

[38] M. C. Smith et al., Mon. Not. Roy. Astron. Soc. 379, 755 (2007), astro- ph/0611671.

[39] S. Chang, A. Pierce, and N. Weiner, Phys. Rev. D79, 115011 (2009), 0808.0196.

[40] D. S. M. Alves, S. R. Behbahani, P. Schuster, and J. G. Wacker, (2009), 0903.3945.

[41] D. S. M. Alves, S. R. Behbahani, P. Schuster, and J. G. Wacker, (2010), 1003.4729.

[42] B. Holdom, Phys. Lett. B166, 196 (1986).

[43] M. Pospelov and T. ter Veldhuis, Phys. Lett. B480, 181 (2000), hep- ph/0003010.

[44] S. Ahlen et al., Int. J. Mod. Phys. A25, 1 (2010), 0911.0323.

[45] D. N. Spergel, Phys. Rev. D37, 1353 (1988).

[46] S. Dimopoulos and H. Georgi, Nucl. Phys. B193, 150 (1981).

[47] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), hep-ph/9905221.

[48] N. Arkani-Hamed et al., JHEP 08, 021 (2002), hep-ph/0206020.

[49] UA2, J. Alitti et al., Phys. Lett. B235, 363 (1990).

[50] CDF, X. Portell, AIP Conf. Proc. 842, 640 (2006), hep-ex/0609017. BIBLIOGRAPHY 155

[51] D0, V. M. Abazov et al., Phys. Lett. B660, 449 (2008), 0712.3805.

[52] CDF, CDF Note 9093 (2007).

[53] CDF, A. Abulencia et al., Phys. Rev. Lett. 97, 171802 (2006), hep-ex/0605101.

[54] L. Randall and R. Sundrum, Nucl. Phys. B557, 79 (1999), hep-th/9810155.

[55] K. Choi, A. Falkowski, H. P. Nilles, and M. Olechowski, Nucl. Phys. B718, 113 (2005), hep-th/0503216.

[56] D. E. Kaplan and M. D. Schwartz, Phys. Rev. Lett. 101, 022002 (2008), 0804.2477.

[57] J. Alwall et al., JHEP 09, 028 (2007), 0706.2334.

[58] T. Sjostrand, S. Mrenna, and P. Z. Skands, JHEP 05, 026 (2006), hep- ph/0603175.

[59] J. Alwall et al., Eur. Phys. J. C53, 473 (2008), 0706.2569.

[60] S. Catani, Y. L. Dokshitzer, M. H. Seymour, and B. R. Webber, Nucl. Phys. B406, 187 (1993).

[61] J. Pumplin et al., JHEP 07, 012 (2002), hep-ph/0201195.

[62] W. Beenakker, R. Hopker, M. Spira, and P. M. Zerwas, Nucl. Phys. B492, 51 (1997), hep-ph/9610490.

[63] Conway, J. PGS: Pretty Good Simulator, http://www.physics.ucdavis.edu/ conway/.

[64] D0, V. M. Abazov et al., Phys. Rev. Lett. 101, 011601 (2008), 0803.2137.

[65] S. Mrenna, G. L. Kane, G. D. Kribs, and J. D. Wells, Phys. Rev. D53, 1168 (1996), hep-ph/9505245.

[66] H. Baer, C.-h. Chen, F. Paige, and X. Tata, Phys. Rev. D52, 2746 (1995), hep-ph/9503271. 156 BIBLIOGRAPHY

[67] I. Hinchliﬀe, F. E. Paige, M. D. Shapiro, J. Soderqvist, and W. Yao, Phys. Rev. D55, 5520 (1997), hep-ph/9610544.

[68] B. K. Gjelsten, D. J. Miller, 2, and P. Osland, JHEP 12, 003 (2004), hep- ph/0410303.

[69] B. K. Gjelsten, D. J. Miller, 2, and P. Osland, JHEP 06, 015 (2005), hep- ph/0501033.

[70] S. Dimopoulos, S. D. Thomas, and J. D. Wells, Nucl. Phys. B488, 39 (1997), hep-ph/9609434.

[71] G. F. Giudice and R. Rattazzi, Phys. Rept. 322, 419 (1999), hep-ph/9801271.

[72] K. Choi, A. Falkowski, H. P. Nilles, M. Olechowski, and S. Pokorski, JHEP 11, 076 (2004), hep-th/0411066.

[73] R. Kitano and Y. Nomura, Phys. Lett. B631, 58 (2005), hep-ph/0509039.

[74] T. Gherghetta, G. F. Giudice, and J. D. Wells, Nucl. Phys. B559, 27 (1999), hep-ph/9904378.

[75] ALEPH, R. Barate et al., Phys. Lett. B469, 303 (1999).

[76] ALEPH, A. Heister et al., Eur. Phys. J. C35, 457 (2004).

[77] OPAL, G. Abbiendi et al., Eur. Phys. J. C40, 287 (2005), hep-ex/0503051.

[78] CDF, T. Aaltonen et al., (2007), 0712.2534.

[79] CDF, T. Aaltonen et al., Phys. Rev. D78, 012002 (2008), 0712.1311.

[80] CDF, C. Henderson, (2008), 0805.0742.

[81] N. Arkani-Hamed et al., (2007), hep-ph/0703088.

[82] J. Alwall, M.-P. Le, M. Lisanti, and J. G. Wacker, Phys. Lett. B666, 34 (2008), 0803.0019. BIBLIOGRAPHY 157

[83] C. Balazs, M. S. Carena, A. Freitas, and C. E. M. Wagner, JHEP 06, 066 (2007), 0705.0431.

[84] T. Junk, Nucl. Instrum. Meth. A434, 435 (1999), hep-ex/9902006.

[85] A. L. Read, Modiﬁed frequentist analysis of search results (The CL(s) method), Workshop on Conﬁdence Limits, CERN (2000).

[86] K. Kawagoe and M. M. Nojiri, Phys. Rev. D74, 115011 (2006), hep- ph/0606104.

[87] CDF, (2009), 0903.4001.

[88] LEP-EWWG, http://www.cern.ch/LEPEWWG.

[89] G. L. Kane, T. T. Wang, B. D. Nelson, and L.-T. Wang, Phys. Rev. D71, 035006 (2005), hep-ph/0407001.

[90] P. Batra, A. Delgado, D. E. Kaplan, and T. M. P. Tait, JHEP 02, 043 (2004), hep-ph/0309149.

[91] P. Batra, A. Delgado, D. E. Kaplan, and T. M. P. Tait, JHEP 06, 032 (2004), hep-ph/0404251.

[92] R. Harnik, G. D. Kribs, D. T. Larson, and H. Murayama, Phys. Rev. D70, 015002 (2004), hep-ph/0311349.

[93] S. Chang, C. Kilic, and R. Mahbubani, Phys. Rev. D71, 015003 (2005), hep- ph/0405267.

[94] A. Birkedal, Z. Chacko, and M. K. Gaillard, JHEP 10, 036 (2004), hep- ph/0404197.

[95] N. Polonsky and S. Su, Phys. Lett. B508, 103 (2001), hep-ph/0010113.

[96] A. Maloney, A. Pierce, and J. G. Wacker, JHEP 06, 034 (2006), hep- ph/0409127. 158 BIBLIOGRAPHY

[97] K. S. Babu, I. Gogoladze, and C. Kolda, (2004), hep-ph/0410085.

[98] M. Dine, N. Seiberg, and S. Thomas, Phys. Rev. D76, 095004 (2007), 0707.0005.

[99] Y. Nomura and B. Tweedie, Phys. Rev. D72, 015006 (2005), hep-ph/0504246.

[100] A. Delgado and T. M. P. Tait, JHEP 07, 023 (2005), hep-ph/0504224.

[101] A. Brignole, J. A. Casas, J. R. Espinosa, and I. Navarro, Nucl. Phys. B666, 105 (2003), hep-ph/0301121.

[102] J. A. Casas, J. R. Espinosa, and I. Hidalgo, JHEP 01, 008 (2004), hep- ph/0310137.

[103] R. Dermisek and J. F. Gunion, Phys. Rev. Lett. 95, 041801 (2005), hep- ph/0502105.

[104] R. Dermisek and J. F. Gunion, Phys. Rev. D73, 111701 (2006), hep- ph/0510322.

[105] P. C. Schuster and N. Toro, (2005), hep-ph/0512189.

[106] R. Dermisek and J. F. Gunion, Phys. Rev. D76, 095006 (2007), 0705.4387.

[107] R. Dermisek and J. F. Gunion, Phys. Rev. D79, 055014 (2009), 0811.3537.

[108] S. Chang, P. J. Fox, and N. Weiner, JHEP 08, 068 (2006), hep-ph/0511250.

[109] S. Chang, P. J. Fox, and N. Weiner, Phys. Rev. Lett. 98, 111802 (2007), hep- ph/0608310.

[110] S. Chang, R. Dermisek, J. F. Gunion, and N. Weiner, Ann. Rev. Nucl. Part. Sci. 58, 75 (2008), 0801.4554.

[111] LEP Higgs Working for Higgs boson searches, (2001), hep-ex/0107032.

[112] OPAL, G. Abbiendi et al., Eur. Phys. J. C37, 49 (2004), hep-ex/0406057. BIBLIOGRAPHY 159

[113] DELPHI, J. Abdallah et al., Eur. Phys. J. C38, 1 (2004), hep-ex/0410017.

[114] ALEPH, S. Schael et al., Eur. Phys. J. C47, 547 (2006), hep-ex/0602042.

[115] OPAL, G. Abbiendi et al., Eur. Phys. J. C27, 311 (2003), hep-ex/0206022.

[116] OPAL, G. Abbiendi et al., Eur. Phys. J. C27, 483 (2003), hep-ex/0209068.

[117] R. Dermisek, (2008), 0807.2135.

[118] V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf, and G. Shaugh- nessy, Phys. Rev. D79, 015018 (2009), 0811.0393.

[119] CLEO, W. Love et al., Phys. Rev. Lett. 101, 151802 (2008), 0807.1427.

[120] G. Hiller, Phys. Rev. D70, 034018 (2004), hep-ph/0404220.

[121] B. A. Dobrescu and K. T. Matchev, JHEP 09, 031 (2000), hep-ph/0008192.

[122] R. Dermisek and J. F. Gunion, Phys. Rev. D75, 075019 (2007), hep- ph/0611142.

[123] P. W. Graham, A. Pierce, and J. G. Wacker, (2006), hep-ph/0605162.

[124] N. E. Adam et al., (2008), 0803.1154.

[125] J. R. Forshaw, J. F. Gunion, L. Hodgkinson, A. Papaefstathiou, and A. D. Pilkington, JHEP 04, 090 (2008), 0712.3510.

[126] M. Jezabek, Z. Was, S. Jadach, and J. H. Kuhn, Comput. Phys. Commun. 70, 69 (1992).

[127] A. Haas, private communication, 2008.

[128] CDF, D. E. Acosta et al., Phys. Rev. Lett. 88, 161802 (2002).

[129] P. Artoisenet, J. M. Campbell, J. P. Lansberg, F. Maltoni, and F. Tramontano, Phys. Rev. Lett. 101, 152001 (2008), 0806.3282. 160 BIBLIOGRAPHY

[130] CDF, F. Abe et al., Phys. Rev. Lett. 79, 578 (1997).

[131] CDF, F. Abe et al., Phys. Rev. Lett. 79, 572 (1997).

[132] R. Bernabei et al., Eur. Phys. J. C67, 39 (2010), 1002.1028.

[133] R. Schoenrich, J. Binney, and W. Dehnen, (2009), 0912.3693.

[134] B. Feldstein, A. L. Fitzpatrick, and E. Katz, JCAP 1001, 020 (2010), 0908.2991.

[135] S. Chang, A. Pierce, and N. Weiner, JCAP 1001, 006 (2010), 0908.3192.

[136] M. Lisanti and J. G. Wacker, (2009), 0911.4483.

[137] G. Duda, A. Kemper, and P. Gondolo, JCAP 0704, 012 (2007), hep- ph/0608035.

[138] K. Schmidt-Hoberg and M. W. Winkler, JCAP 0909, 010 (2009), 0907.3940.

[139] C. McCabe, (2010), 1005.0579.

[140] J. Kopp, T. Schwetz, and J. Zupan, JCAP 1002, 014 (2010), 0912.4264.

[141] J. Diemand, M. Kuhlen, and P. Madau, Astrophys. J. 657, 262 (2007), astro- ph/0611370.

[142] J. March-Russell, C. McCabe, and M. McCullough, JHEP 05, 071 (2009), 0812.1931.

[143] M. Fairbairn and T. Schwetz, JCAP 0901, 037 (2009), 0808.0704.

[144] M. Kuhlen et al., JCAP 1002, 030 (2010), 0912.2358.

[145] M. Lisanti and J. G. Wacker, (2009), 0911.1997.

[146] S. Yellin, Phys. Rev. D66, 032005 (2002), physics/0203002.

[147] S. Chang, G. D. Kribs, D. Tucker-Smith, and N. Weiner, Phys. Rev. D79, 043513 (2009), 0807.2250. BIBLIOGRAPHY 161

[148] W. Seidel, CRESST, Wonder Workshop, Italy, 3/22/2010.

[149] R. Bernabei et al., Phys. Lett. B389, 757 (1996).

[150] S. Pecourt et al., Astropart. Phys. 11, 457 (1999).

[151] D. R. Tovey et al., Phys. Lett. B433, 150 (1998).

[152] K. Fushimi et al., Phys. Rev. C47, 425 (1993).

[153] K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell, and M. Kamionkowski, Phys. Rev. D70, 083501 (2004), astro-ph/0406355.

[154] B. A. Dobrescu and I. Mocioiu, JHEP 11, 005 (2006), hep-ph/0605342.

[155] D. E. Kaplan, G. Z. Krnjaic, K. R. Rehermann, and C. M. Wells, JCAP 1005, 021 (2010), 0909.0753.

[156] G. D. Kribs, T. S. Roy, J. Terning, and K. M. Zurek, Phys. Rev. D81, 095001 (2010), 0909.2034.

[157] R. F. Lang and N. Weiner, (2010), 1003.3664.

[158] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B429, 263 (1998), hep-ph/9803315.

[159] L. Susskind, Phys. Rev. D20, 2619 (1979).

[160] C. J. Hogan, Rev. Mod. Phys. 72, 1149 (2000), astro-ph/9909295.

[161] S. Weinberg, Phys. Rev. Lett. 59, 2607 (1987).

[162] V. Agrawal, S. M. Barr, J. F. Donoghue, and D. Seckel, Phys. Rev. D57, 5480 (1998), hep-ph/9707380.

[163] N. Arkani-Hamed and S. Dimopoulos, JHEP 06, 073 (2005), hep-th/0405159.

[164] L. Susskind, (2003), hep-th/0302219. 162 BIBLIOGRAPHY

[165] N. Arkani-Hamed, S. Dimopoulos, and S. Kachru, (2005), hep-th/0501082.

[166] G. F. Giudice and A. Romanino, Nucl. Phys. B699, 65 (2004), hep-ph/0406088.

[167] J. D. Wells, (2003), hep-ph/0306127.

[168] R. Mahbubani and L. Senatore, Phys. Rev. D73, 043510 (2006), hep- ph/0510064.

[169] M. Cirelli, N. Fornengo, and A. Strumia, Nucl. Phys. B753, 178 (2006), hep- ph/0512090.

[170] M. Tegmark, A. Aguirre, M. Rees, and F. Wilczek, Phys. Rev. D73, 023505 (2006), astro-ph/0511774.

[171] S. Hellerman and J. Walcher, Phys. Rev. D72, 123520 (2005), hep-th/0508161.

[172] A. Pierce and J. Thaler, JHEP 08, 026 (2007), hep-ph/0703056.

[173] R. Barbieri, L. J. Hall, and V. S. Rychkov, Phys. Rev. D74, 015007 (2006), hep-ph/0603188.

[174] J. McDonald, Phys. Rev. D50, 3637 (1994), hep-ph/0702143.

[175] H. Davoudiasl, R. Kitano, T. Li, and H. Murayama, Phys. Lett. B609, 117 (2005), hep-ph/0405097.

[176] X. Calmet and J. F. Oliver, Europhys. Lett. 77, 51002 (2007), hep-ph/0606209.

[177] Particle Data Group, W. M. Yao, J. Phys. G 33, 1 (2006).

[178] M. Machacek and M. Vaughn, Nucl. Phys. B 222, 83 (1983).

[179] H. Murayama and A. Pierce, Phys. Rev. D65, 055009 (2002), hep-ph/0108104.

[180] S. L. Glashow, Triniﬁcation of all Elementary Particle Forces, Ed. K. Kang, H. Fried, and P. Frampton. Singapore: World Scientiﬁc (1984).

[181] B. Pendleton and G. G. Ross, Phys. Lett. B98, 291 (1981). BIBLIOGRAPHY 163

[182] K. Inoue, K. Kakuto, and Y. Nakano, Prog. Theor. Phys. 63, 234 (1980).

[183] K. Griest and D. Seckel, Phys. Rev. D 43, 3191 (1991).

[184] WMAP, D. N. Spergel et al., Astrophys. J. Suppl. 170, 377 (2007), astro- ph/0603449.

[185] M. E. Peskin and T. Takeuchi, Phys. Rev. D46, 381 (1992).

[186] D. Tucker-Smith and N. Weiner, Nucl. Phys. Proc. Suppl. 124, 197 (2003), astro-ph/0208403.

[187] L. Bergstrom, P. Ullio, and J. H. Buckley, Astropart. Phys. 9, 137 (1998), astro-ph/9712318.

[188] J. Gunion, H. Haber, G. Kane, and S. Dawson, The Higgs Hunter’s Guide (Addison-Wesley Publishing Company, California, 1990).

[189] A. Birkedal, A. Noble, M. Perelstein, and A. Spray, Phys. Rev. D74, 035002 (2006), hep-ph/0603077.

[190] J. Hisano, S. Matsumoto, M. M. Nojiri, and O. Saito, Phys. Rev. D71, 063528 (2005), hep-ph/0412403.

[191] G. Zaharijas and D. Hooper, Phys. Rev. D73, 103501 (2006), astro-ph/0603540.

[192] A. Djouadi, Phys. Rept. 457, 1 (2008), hep-ph/0503172.

[193] F. Maltoni and T. Stelzer, JHEP 02, 027 (2003), hep-ph/0208156.

[194] O. DeWolfe, A. Giryavets, S. Kachru, and W. Taylor, JHEP 02, 037 (2005), hep-th/0411061.

[195] D. Tucker-Smith and N. Weiner, Phys. Rev. D64, 043502 (2001), hep- ph/0101138.

[196] S. Nussinov, Phys. Lett. B165, 55 (1985). 164 BIBLIOGRAPHY

[197] R. S. Chivukula and T. P. Walker, Nucl. Phys. B329, 445 (1990).

[198] J. Bagnasco, M. Dine, and S. D. Thomas, Phys. Lett. B320, 99 (1994), hep- ph/9310290.

[199] M. Y. Khlopov, (2008), 0806.3581.

[200] C. Kouvaris, Phys. Rev. D78, 075024 (2008), 0807.3124.

[201] T. A. Ryttov and F. Sannino, Phys. Rev. D78, 115010 (2008), 0809.0713.

[202] R. N. Mohapatra, S. Nussinov, and V. L. Teplitz, Phys. Rev. D66, 063002 (2002), hep-ph/0111381.

[203] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner, Phys. Rev. D79, 015014 (2009), 0810.0713.

[204] A. Katz and R. Sundrum, JHEP 06, 003 (2009), 0902.3271.

[205] D. E. Morrissey, D. Poland, and K. M. Zurek, JHEP 07, 050 (2009), 0904.2567.

[206] M. Goodsell, J. Jaeckel, J. Redondo, and A. Ringwald, JHEP 11, 027 (2009), 0909.0515.

[207] C. Cheung, J. T. Ruderman, L.-T. Wang, and I. Yavin, Phys. Rev. D80, 035008 (2009), 0902.3246.

[208] R. H. Helm, Phys. Rev. 104, 1466 (1956).

[209] M. Pospelov, Phys. Rev. D80, 095002 (2009), 0811.1030.

[210] D. Feldman, Z. Liu, and P. Nath, Phys. Rev. D75, 115001 (2007), hep- ph/0702123.

[211] S. Cassel, D. M. Ghilencea, and G. G. Ross, Nucl. Phys. B827, 256 (2010), 0903.1118.

[212] Particle Data Group, C. Amsler et al., Phys. Lett. B667, 1 (2008). BIBLIOGRAPHY 165

[213] N. Arkani-Hamed and N. Weiner, JHEP 12, 104 (2008), 0810.0714.

[214] B. Batell, M. Pospelov, and A. Ritz, Phys. Rev. D79, 115008 (2009), 0903.0363.

[215] R. Essig, P. Schuster, and N. Toro, Phys. Rev. D80, 015003 (2009), 0903.3941.

[216] M. Reece and L.-T. Wang, JHEP 07, 051 (2009), 0904.1743.

[217] M. Baumgart, C. Cheung, J. T. Ruderman, L.-T. Wang, and I. Yavin, JHEP 04, 014 (2009), 0901.0283.

[218] P. Schuster, N. Toro, and I. Yavin, Phys. Rev. D81, 016002 (2010), 0910.1602.

[219] P. Schuster, N. Toro, N. Weiner, and I. Yavin, (2009), 0910.1839.

[220] BABAR, B. Aubert et al., (2009), 0902.2176.

[221] CLEO, W. Love et al., Phys. Rev. Lett. 101, 151802 (2008), 0807.1427.

[222] CLEO, P. Rubin et al., Phys. Rev. D75, 031104 (2007), hep-ex/0612051.

[223] BABAR, B. Aubert et al., (2009), 0908.2821.

[224] J. D. Bjorken, R. Essig, P. Schuster, and N. Toro, Phys. Rev. D80, 075018 (2009), 0906.0580.

[225] Dark Forces Workshop, SLAC National Accelerator Laboratory, 9/24/09- 9/26/09. http://www-conf.slac.stanford.edu/darkforces2009/.

[226] P. Gondolo, Phys. Rev. D66, 103513 (2002), hep-ph/0209110.

[227] D. P. Finkbeiner, T. Lin, and N. Weiner, Phys. Rev. D80, 115008 (2009), 0906.0002.

[228] G. Sciolla et al., J. Phys. Conf. Ser. 179, 012009 (2009), 0903.3895.

[229] K. Miuchi et al., Phys. Lett. B654, 58 (2007), 0708.2579.

[230] S. Burgos et al., Nucl. Instrum. Meth. A600, 417 (2009), 0807.3969. 166 BIBLIOGRAPHY

[231] D. Santos, O. Guillaudin, T. Lamy, F. Mayet, and E. Moulin, J. Phys. Conf. Ser. 65, 012012 (2007), astro-ph/0703310.

[232] M. J. Strassler and K. M. Zurek, Phys. Lett. B651, 374 (2007), hep-ph/0604261.

[233] M. J. Strassler and K. M. Zurek, Phys. Lett. B661, 263 (2008), hep-ph/0605193.

[234] T. Han, Z. Si, K. M. Zurek, and M. J. Strassler, JHEP 07, 008 (2008), 0712.2041.

[235] Y. Cui, D. E. Morrissey, D. Poland, and L. Randall, JHEP 05, 076 (2009), 0901.0557.

[236] G. Sciolla, Mod. Phys. Lett. A24, 1793 (2009), 0811.2764.

[237] W. Dehnen and J. Binney, Mon. Not. Roy. Astron. Soc. 298, 387 (1998), astro-ph/9710077.

[238] J. Binney and S. Tremaine, Galactic Dynamics (Princeton University Press, Princeton, 2008).

[239] G. Gelmini and P. Gondolo, Phys. Rev. D64, 023504 (2001), hep-ph/0012315.

[240] C. Savage, G. Gelmini, P. Gondolo, and K. Freese, JCAP 0904, 010 (2009), 0808.3607.

[241] L. W. Kastens, S. B. Cahn, A. Manzur, and D. N. McKinsey, Phys. Rev. C80, 045809 (2009), 0905.1766.

[242] P. Sorenson, private communication, 8/2009.

[243] I. Cholis, D. P. Finkbeiner, L. Goodenough, and N. Weiner, JCAP 0912, 007 (2009), 0810.5344.

[244] D. P. Finkbeiner, T. R. Slatyer, N. Weiner, and I. Yavin, JCAP 0909, 037 (2009), 0903.1037.

[245] PAMELA, O. Adriani et al., Nature 458, 607 (2009), 0810.4995. BIBLIOGRAPHY 167

[246] The Fermi LAT, A. A. Abdo et al., Phys. Rev. Lett. 102, 181101 (2009), 0905.0025.

[247] J. Chang et al., Nature 456, 362 (2008).

[248] H.E.S.S., F. Aharonian et al., Astron. Astrophys. 508, 561 (2009), 0905.0105.