Dark Matter and Collider Physics

Beyond the Standard Model: Dark Matter and Collider Physics

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Russell Colburn, B.S. Graduate Program in Physics

The Ohio State University 2017

Dissertation Committee: Linda Carpenter, Advisor

Chris Hill

Annika Peter

Stuart Raby c Copyright by

Russell Colburn

2017 Abstract

In this work we analyze dark matter annihilation in dwarf galaxies as well as scenarios of new physics relevant to current and future runs at the LHC. Our work with dark matter focuses on the implications for dark matter models from observations from the Fermi-LAT telescope while our work on LHC physics offers an explanation for the 750 GeV diphoton excess as well as a new possible decay channels of a scalar color octet. Using gamma-ray observations of dwarf spheroidal galaxies from the pass-7 Fermi-LAT experiment, we discuss the implications on dark matter models where dark matter annihilates to multiple final state configurations: where dark matter annihilates to third generation fermions and Standard Model vector bosons. We present limits in various slices of model parameter space both as limits on dark matter mass and in the context of effective operators. We visualize our bounds for models with multiple final state annihilations by projecting parameter space constraints onto triangles, a technique familiar from collider physics; and we compare our bounds to collider limits on equivalent models. We extend this analysis to the pass-8 Fermi-LAT observations of dwarfs by using a more robust “stacked” analysis using a joint-likelihood analysis to combine multiple dwarf galaxies. We once again look at dark matter annihilations to multiple final-state fermions and interpret these results in the context of the most popular simplified models, including those with s- and t-channel dark matter annihilation through scalar and vector mediators. Additionally, we compare our simplified model results to those of effective field theory contact interactions in the high-mass limit. We propose that the sbino, the scalar partner of a Dirac bino can explain the 750 GeV diphoton excess observed by the ATLAS and CMS collaborations. We analyze the minimal completion of the effective operator model in which the sbino couples to pairs of gauge bosons through loops of heavy sfermions, with the sfermion-bino coupling originating from scalar potential D-terms. We find that the sbino model may be fit the 750 GeV excess by considering gluon fusion processes with decay to diphotons. Finally, we explore the phenomenology of scalar fields in the adjoint representation of SM gauge groups at the LHC. We write a general set of dimension 5 effective operators in which SM adjoint scalars couple to pairs of standard model bosons. Using these effective

ii operators, we explore new possible decay channels of a scalar color octet into a gluon and a Z boson, another gluon, or a photon. We recast several analyses from Run I of the LHC to ﬁnd constraints on an a scalar octet decaying into these channels, and we project the discovery potential of color octets in our gluon+photon channel for the 14 TeV run of LHC.

iii Acknowledgments

There are many people to whom I owe sincere gratitude for their contributions to my work and education. I would first like to thank my advisor, Linda Carpenter, for her patience, mentorship, and guidance throughout my graduate career. I would also like to thank to Stuart Raby for providing opportunities to enrich my knowledge of particle physics by leading multiple reading courses in his spare time as well as teaching many courses on the subject. During the course of my research, I had the privilege of collaborating with Jesi Goodman and Tim Linden on various projects. Over the years, Jesi has also provided advice and knowledge making her akin to a second advisor in my mind. I have been lucky to have Chris Svoboda, Brian Daintan, Sushant More, Khalida Hen- dricks, and Humberto Gilmer as my office mates over the years. They have provided much assistance over the years from discussing physics to helping finding bugs in codes to helping maintain sanity. I would like to thank members of the OSU high energy and nuclear physics groups who have been more than willing to discuss physics and willing to assist with various issues over the years: Archana Anandakrishnan, Chuck Bryant, Zijie Poh, Shaun Hampton, Hong Zhang, Bowen Shi, Alex Dyhdalo, Dennis Bazow, Sarah Wesolowski, Heiko Hergert, Sebastion Kronig, Gojko Vujanovic, Mauricio Martinez Guerrero, Kenny Ng, Shirley Li, Hudson Smith, Abhishek Mohapatra, and Evan Johnson. Finally I would like to thank my family and friends who have provided support and encouragement over the course of my entire educational career.

iv Vita

October 29, 1988 ...... Born—Danville, IL

May, 2012 ...... B.S., Michigan State University, East Lansing, MI

Publications

Indirect Detection Constraints on s- and t-channel Simpliﬁed Models of Dark Matter. Linda M. Carpenter, Russell Colburn, Jessica Goodman, and Tim Linden. Phys. Rev. D94 (2016) 5, 055027, arXiv:1606.04138.

Supersoft SUSY Models and the 750 GeV Diphoton Excess, Beyond Eﬀective Operators. Linda M. Carpenter, Russell Colburn, Jessica Goodman. Phys. Rev. D94 (2016) 1, 015016, arXiv:1512.06107.

Searching for Standard Model Adjoint Scalars with Diboson Resonance Signatures. Linda M. Carpenter, Russell Colburn, JHEP 1512 (2015) 151, arXiv:1509.07869.

Indirect Detection Constraints on the Model Space of Dark Matter Eﬀective Theories. Linda M. Carpenter, Russell Colburn, Jessica Goodman. Phys. Rev. D92 (2015) 9, 095011, arXiv:1506.08841.

Fields of Study

Major Field: Physics

v Table of Contents

Page Abstract...... ii Acknowledgments...... iv Vita...... v List of Figures ...... viii List of Tables ...... xii

1 Introduction ...... 1 1.1 The Standard Model...... 2 1.2 Problems with the Standard Model...... 6 1.2.1 Dark Matter ...... 6 1.2.2 Hierarchy Problem...... 10 1.3 SUSY ...... 10 1.3.1 Dirac Gaugino SUSY Breaking...... 14 1.4 Eﬀective Field Theories ...... 15 1.4.1 Example: Fermi Theory of Weak Interactions...... 17 1.4.2 EFTs in the Search for Dark Matter...... 18 1.5 Synopsis...... 19

Chapters

2 Indirect Detection Constraints on the Model Space of Dark Matter Eﬀective Theories ...... 21 2.1 Introduction...... 21 2.2 Indirect Detection from Dwarf Spheroidal Galaxies...... 22 2.3 Models with Independent Annihilation Channels ...... 24 2.3.1 Fixed Annihilation Rate...... 25 2.3.2 Constraints for 4 Independent Channels...... 32 2.4 Constraints in Models with Interfering Channels ...... 34 2.4.1 Limits of the EFT and Collider Constraints...... 40 2.5 Conclusions...... 42

3 Indirect Detection Constraints on s- and t-Channel Simpliﬁed Models of Dark Matter ...... 44 3.1 Introduction...... 44 3.2 Indirect Detection from Dwarf Spheroidal Galaxies...... 46

vi 3.2.1 γ-ray Analyses of the Population of Dwarf Spheroidal Galaxies . . . 47 3.3 Generic DM Mass Bounds...... 50 3.4 EFT completions...... 53 3.5 Simpliﬁed Models...... 55 3.5.1 Vector mediator ...... 56 3.5.2 T-channel mediator ...... 61 3.6 Conclusions...... 66

4 Supersoft SUSY Models and the 750 GeV Diphoton Excess, Beyond Eﬀective Operators ...... 68 4.1 Introduction...... 68 4.2 Operators in Dirac Gauginos Models...... 70 4.3 UV completions...... 71 4.4 Tree Level Singlet Decays and Collider Bounds...... 73 4.5 Production and Loop Level Decays...... 75 4.6 Stability and Electroweak Constraints ...... 81 4.7 Conclusions...... 83

5 Searching for Standard Model Adjoint Scalars with Diboson Resonance Signatures ...... 86 5.1 Introduction...... 86 5.2 Eﬀective Operators...... 87 5.3 High Energy Models...... 89 5.4 Production and Decay of Scalar Adjoints ...... 91 5.5 Current Limits ...... 94 5.5.1 The Jet + γ Channel ...... 95 5.5.2 Dijet Channel...... 96 5.5.3 The Monojet Channel...... 97 5.5.4 Heavy Boson plus Jet Channel ...... 99 5.5.5 Combined Results ...... 102 5.6 Projection for LHC 14...... 105 5.7 Biadjoint Operators ...... 107 5.8 Conclusions...... 108

6 Conclusion ...... 110

Bibliography ...... 112

vii List of Figures

Figure Page

1.1 The comoving number density, Y , and the resulting relic density of a 100 GeV, P-wave annihilating DM particle as a function of temperature T (bottom) or time t (top). The solid line is for an annihilation cross section which yields the correct DM relic abundance and the bands have cross sections which differ by a factor of 10 (yellow), 100 (green), and 1000 (blue) from that value. The dashed contour traces the number density for a particle which remains in thermal equilibrium [1]...... 7 1.2 Cartoon of different search techniques for DM, where the flow of time in the diagram is indicated by the arrow...... 8 1.3 Feynman diagram of corrections to the Higgs mass from a fermion loop. . . 10 1.4 List of effective operators between the SM and DM for DM as a Dirac fermion, Majorana fermion, complex scalar, and real scalar up to dimension 7 [2]. . . 19

2.1 Combined 95% upper bounds on photon energy flux scaled by J-factor from 6 of largest J-factor dwarf spheroidal galaxies...... 24 2.2 Gamma ray spectrum from DM annihilation into various channels for a 100 GeV DM fermion...... 25 2.3 DM annihilation into pairs of fermions...... 27 2.4 Lower bound on DM mass in GeV for annihilations into b quarks, τ’s, and invisible particles for σv tot = σv Th in the upper plot and σv tot = 10 σv Th h i h i h i h i in lower plot...... 28 ∗ ∗ 2.5 Lower bound on operator coefficients Λ (upper) and Λ (lower) for σv tot = b τ h i σv Th...... 30 h i ∗ ∗ 2.6 Lower bound on operator coefficients Λ (upper) and Λ (lower) for σv tot = b τ h i 10 σv Th ...... 31 h i 2.7 Axis labeling for tetrahedron presentation of DM mass constraints when a fourth annihilation channel is allowed. The base corresponds to the top left of Fig. 2.8...... 32 2.8 The contours correspond to lower bounds (in GeV) on DM mass from annihilations into pairs of fermions (tt, bb, ττ, νν). The total annihilation rate is set to 33 times the thermal relic for all three plots. The left image corresponds to Rt = 0, the right to Rt = .3 and the bottom to Rt = .7...... 33

viii 2.9 DM annihilation into gauge bosons...... 34 2.10 Limits on effective operator coefficients for 10 times the thermal annihilation 3 rate given by Fermi dwarf bounds. The axes show the value of ki/Λ . Bounds on operators are shown for four DM masses, mχ = 10, 75, 150 and, 200 GeV (top left, top right, bottom left and bottom right respectively). Red points indicate that the operator coefficient is ruled out, blue points are unconstrained...... 36 2.11 The colored contours correspond to upper bounds on mχ (in GeV) from annihilation into all final states allowed from the operators in Eq. 2.4 with 3 ki κi/Λ . The amount to the gg final state is fixed in each of these plots ≡ as a percentage of the total rate, 0% (upper) and 70% (lower). The solid, dashed, and dotted lines correspond to a combination of κ1 and κ2 which turn off the final states Zγ, ZZ, and γγ respectively...... 38 2.12 These contours correspond to upper bounds on σv tot in units of the thermal h i relic, σv Th, from annihilation into all final states allowed from the operators h i 3 in Eq. 2.4 with ki κi/Λ . The amount to the gg final state is fixed in each ≡ of these plots as a percentage of the total rate, 0% (upper) and 70% (lower). The solid, dashed, and dotted lines correspond to a combination of k1 and k2 which turn off the final states Zγ, ZZ, and γγ respectively...... 39 2.13 Constraints from Fig. 2.12 reconfigured for the effective cut-off vs DM mass plane. The regions below the solid lines are excluded. The dashed lines correspond to curves of constant σv tot ...... 41 h i 2.14 Constraints from Fig. 2.12 reconfigured for the effective cut-off vs DM mass plane...... 42

3.1 Gamma ray spectrum from DM annihilation into various channels for a 100 GeV DM particle...... 47 3.2 A comparison of the annihilation cross section upper limits obtained using this work and the spectrum calculated by PPPC compared to the Fermi-LAT collaboration analysis [3] for the case of “standard” DM annihilations into b¯b (top) and τ +τ − (bottom) final states as a function of the DM mass and assuming a 95% confidence upper limit set at ∆LG( ) = 2.71/2...... 49 L 3.3 Stages in the analysis of the joint-likelihood limits on the annihilation of particle DM where the color corresponds to the difference in the log-likelihood value from the minimum (best fit) value. First, the limit is shown in an intermediate state, as a joint-likelihood analysis calculated independently in each spectral bin,with J-factors for each dwarf that are alllowed to float independently in each bin (top left). Second, a DM mass and annihilation pathway is fixed (100 GeV DM annihilating to a b¯b final state) and the J- factor for each dwarf is forced to be consistent over each spectral bin (bottom left). Finally, the combined limits are calculated by summing the LG( ) scan L in each energy bin to produce a limit on the DM cross section (right). . . . 50 3.4 The lower bound on the DM mass (GeV) for annihilations into b¯b, τ +τ −, and invisible particles for σv tot = σv Th. In the upper figure, the photon h i h i spectrum is generated with PPPC, while in the lower figure, the photon spectrum is generated with DarkSUSY...... 52

ix 3.5 The lower bound on the effective coupling (GeV) of DM to b-quarks for + − annihilations into b¯b, τ τ , and invisible particles for σv tot = σv Th. In h i h i the upper figure, the photon spectrum is generated with PPPC. In the lower figure, the photon spectrum is generated with DarkSUSY...... 54 3.6 The lower bound on the effective coupling (GeV) of DM to τ-leptons for + − annihilations into b¯b, τ τ , and invisible particles for σv tot = σv Th. In h i h i the upper figure, the photon spectrum is generated with PPPC. In the lower figure, the photon spectrum is generated with DarkSUSY...... 55 3.7 Diagrams for DM annihilation in simplified models with a vector mediator (upper left), scalar mediator (upper right), and a t-channel mediator (lower). 56 3.8 Upper bound on the allowed DM-mediator coupling gf as a function of mediator mass for DM annihilation to pure b’s (upper), pure τ’s (lower) for gχ = 1 in the vector mediator simplified model...... 58 3.9 Upper bound on the allowed DM-mediator coupling to gb as a function of mediator mass for mixed case of 30% τ’s and 70% b’s (upper) and 70% τ’s and 30% b’s (lower) for gχ = 1 in the vector mediator simplified model. . . . 59 3.10 Upper bounds on DM-mediator coupling as a function of mediator mass for a vector mediator model compared with assumptions from effective operator µ D5 (¯χγ χfγµf) with gχ = 1 and mχ = 150 GeV (upper) and 950 GeV (lower). 60 3.11 Upper bound on the allowed DM coupling to final states bb (upper) and ττ (lower) in the t-channel mediator simplified model...... 62 3.12 Upper bound on the allowed DM coupling vs. mediator mass for a t-channel simplified model where DM annihilates to 70% τ’s and 30% b’s. The upper plot has the bounds on the parameters related to the b interaction with DM and the lower plot is the same for τ interactions...... 63 3.13 Upper bound on the allowed DM coupling vs. mediator mass for a t-channel simplified model where DM annihilates to 30% τ’s and 70% b’s. The upper plot has the bounds on the parameters related to the b interaction with DM and the lower plot is the same for τ interactions...... 64 3.14 Upper bound on the allowed DM coupling to final states of down-type quarks (upper) and leptons (lower) in the t-channel mediator simplified model. . . 65

4.1 One loop diagrams contributing to singlet coupling to pairs of gluons and electroweak gauge bosons...... 72 4.2 SM singlet coupling to pairs of the lightest Higgs...... 74 4.3 gg S rate in pb for squark mass vs Dirac mass plane in p-p collisions at 8 → TeV (upper) and 13 TeV (lower)...... 77 4.4 Partial width (GeV) of S to gluon gluon in mD-ms plane...... 78 4.5 Ratio of diphoton to digluon partial widths in the squark mass vs. slepton mass plane...... 79 4.6 Scalar potential as a function of slepton vev and S vev...... 84

5.1 Decay through mediators to diboson ﬁnal states...... 91 5.2 Production cross section of a single scalar octet vs. mS in pp collisions at √s = 8 TeV with Λ2 = 10 TeV. These cross sections were generated with MadGraph 5...... 93

x 5.3 The rate σ BRgγ of the scalar octet resonance at 8 TeV where Λ2 = 10 × × TeV andΛ1 = 30 TeV. This rate includes the efficiency with which events passes the ATLAS selection criteria...... 95 5.4 Efficiency of our simulated events passing the selection criteria in the search channels considered for √s = 8 TeV analyses at the LHC. The upper two plots are the signal efficiencies for the ATLAS searches for photon plus jet (upper) and monojets (middle) corresponding to the decay modes S gγ → and S gZ(Z νν) respectively. The lower plot is the kinematic efficiency → → of the CMS search for hadronically decaying heavy bosons for the decay mode S gZ(Z qq)...... 100 → → 5.5 Bounds on the Λ1 Λ2 for 4 values of the octer mass mS. Exclusions follow − from 4 LHC 8 TeV analyses. The upper plot follows from searches in the pp X gγ final state. The lower plot follows from searches in the → → pp X jj final state...... 102 → → 5.6 Bounds on the Λ1 Λ2 for 4 values of the octer mass mS. Exclusions follow − from 4 LHC 8 TeV analyses. The upper plot follows from searches in the pp j + Emiss final state. The lower plot follows from searches in the → T pp j + V,V jj final state...... 103 → → 5.7 Combined 95% confidence level bound on EFT scale from all the channels considered for the choice Λ2 = .25Λ1. The black dashed line corresponds to where the validity of the EFT framework breaks down mS > 2Λi...... 104 5.8 a.)In the upper plot is the efficiency of signal events for S gγ which pass → the selection criteria at 14 TeV center of mass energy. b.)In the lower plot is the projected sensitivity for the 14 TeV run in the photon plus jet resonance channel ...... 106 5.9 Projected sensitivity of color octet searches in the gγ channel for the 14 TeV LHC run. Shown are reached for various valued of the octet mass in the Λ2 Λ1 plane. Solid lines indicate the 2 sigma reach while dotted lines − indicate 5 sigma reach...... 107

xi List of Tables

Table Page

1.1 SM particle content and quantum numbers...... 2 1.2 MSSM particle content and quantum numbers...... 12

4.1 Parameters for Higgs breaking minimum...... 82

5.1 Signal regions for ATLAS monojet search...... 98

xii Chapter 1 Introduction

The main goal of particle physics is to explain nature at the most fundamental level. The Standard Model of Particle Physics (SM), which uses elementary particles which interact through the strong, weak, and electromagnetic forces via particle exchange is the current paradigm of particle physics [4–8]. The SM has been an extremely successful theory in explaining experimental results and all current experimental observations remain consistent with the SM. While no experimental inconsistencies with the SM have been observed, there are both observational and theoretical problems that are beyond the scope of the SM. One of the major problems with the SM is that the SM does not contain a dark matter (DM) candidate which is needed to explain astrophysical and cosmological observations. Another major problem with the SM is the Hierarchy Problem which is a concern over why the Higgs 2 mass is mh (10 GeV) when quantum corrections to its mass would naively make the ∼ O Higgs mass at the Planck scale mh (Mpl). Some other problems in physics unsolved ∼ O by the SM are the origin of the matter-antimatter asymmetry in the Universe, neutrino masses, hierarchy of fermion masses, and the strong CP problem. A large portion of current research into particle physics attempts to address these problems with a variety of Beyond the Standard Model (BSM) models which attempt to explain one or more of problems with the SM. In many BSM scenarios, we expect new physics to show up at the TeV scale, which may allow us to probe this new physics in current and future experiments. Each BSM scenario will have its own phenomenology, but diﬀerent BSM scenarios can predict similar particle content or a similar collider signal. It is therefore important to consider a wide class of BSM scenarios and their unique phenomenologies when examining experimental observations. Below I will review our current understanding of particle physics by reviewing the SM and reviewing some of the problems with the SM. Then I will review supersymmetry as a BSM scenario and some of the features that can be contained within it as this will be relevant for Chapter 4. Finally, this chapter will ﬁnish with an overview of the use of

1 eﬀective ﬁeld theories as a tool for discussing phenomenology.

1.1 The Standard Model

The SM is a quantum ﬁeld theory which characterizes the strong, weak, and electromagnetic forces between quarks and leptons. The SM is constructed with the gauge group SU(3) × SU(2) U(1)Y , under which the quarks, leptons, and Higgs boson are charged. There × are three generations, or families, of spin-1/2 fermions in the SM with the same quantum numbers, but diﬀerent masses: three generations of up-type quarks, down-type quarks, charged leptons, and neutrinos. In addition to the fermions, there are the gauge boson force carriers associated with each of the SM gauge groups: gluons for the strong force

(SU(3)), and the W bosons and B bosons for the electroweak sector (SU(2) U(1)Y ). The × spin-0 Higgs boson breaks the electroweak sector down to electromagnetism. The quantum numbers for the SM ﬁeld content are listed in the Table 1.1 below where electric charge is deﬁned as Q = T3 + Y/2.

SU(3) SU(2) U(1)Y u q = d 3 2 1/3 u¯ 3¯ 1 -4/3

d¯ 3¯ 1 2/3

ν l = e 1 2 -1 e¯ 1 1 2

G 8 1 0

W 1 3 0

B 1 1 0

Φ 1 2 1

Table 1.1: SM particle content and quantum numbers.

The SM Lagrangian can be broken down into multiple sectors:

= gauge + fermion + Higgs + Yukawa. (1.1) L L L L L Starting with the gauge sector of the SM which contains the gauge kinetic terms for the

2 vector bosons, we have,

1 µν,a a 1 µν,i i 1 µν gauge = G G W W B Bµν, (1.2) L −4 µν − 4 µν − 4 where,

a a a abc b c G = ∂µG ∂νG + g3f G G , µν ν − µ µ ν i i i ijk j k W = ∂µW ∂νW + g W W , (1.3) µν ν − µ µ ν Bµν = ∂µBν ∂νBµ, − where f abc are the structure constants of SU(3) and ijk are the structure constants for SU(2). Since SU(2) and SU(3) are non-abelian groups, we see in Eqs. 1.2 and 1.3 that we will obtain 3- and 4-point interactions between the W bosons and between the gluons. The kinetic terms for the fermions in the SM can be written as,

† µ † µ † µ † µ † µ fermion = q iσ¯ Dµqf +u ¯ iσ¯ Dµu¯f + d¯ iσ¯ Dµd¯f + l iσ¯ Dµlf +e ¯ iσ¯ Dµe¯f , (1.4) L f f f f f

where the f index is the family index (f = 1, 2, 3) and Dµ is the covariant derivative which is given by,

a a i i 0 Y Dµ = ∂µ ig3 G igT W ig Bµ, (1.5) − T µ − µ − 2 where the ’s are the SU(3) generators, T ’s are the SU(2) generators, and Y is the hyper- T charge operator. The covarient derivatives contain both the kinetic terms for the fermions as well as to their interactions with the gauge bosons which are controlled solely by their quantum numbers. The Higgs part of the SM Lagrangian consists of both the Higgs kinetic term and the Higgs potential, † µ Higgs = (DµΦ) D Φ V (Φ), (1.6) L − where the Higgs potential has the form,

V (Φ) = µ2 Φ 2 + λ Φ 4. (1.7) | | | |

The Higgs kinetic term contains the kinetic terms for the Higgs doublet in addition to interactions between the gauge bosons and the Higgs. This Higgs potential is responsible

3 for electroweak symmetry breaking and mass generation within the SM [9–14]. The Φ ﬁeld is a complex SU(2) doublet and can be written as, 1 φ1 + iφ2 Φ = . (1.8) √2 φ3 + iφ4

When we take the vacuum expectation value (vev) of the Higgs ﬁeld, we can write it as,

1 0 0 Φ 0 = , (1.9) h | | i √2 v

where we have used SU(2) U(1)Y transformations to rotate away 0 φi 0 (i = 1, 2, 4). × h | | i Once we insert the vev into the potential it becomes, 1 1 V (v) = µ2v2 + λv4 (1.10) 2 4 which needs to be minimized with respect to v, and there are two cases to consider, µ2 0 ≥ and µ2 < 0. For the case where µ2 0, a stable minimum exists at φ = 0, otherwise ≥ qh i2 φ = 0 is an unstable minimum and the stable minima occurs at v = −µ . In both cases h i λ for a stable vacuum to exist, λ needs to be positive.

The second case where the vev takes on a non-zero value will result in the SU(2) U(1)Y × symmetry being broken to U(1)EM . We will redefine the Higgs field to represent the physical Higgs field using the Unitary Gauge in the following manner:

1 0 Φ = . (1.11) √2 v + h

The Higgs kinetic term will now contain three massive gauge bosons given by:

2 † µ 2 +µ − mZ µ (DµΦ) D Φ m W W + Z Zµ, (1.12) ⊂ W µ 2 where we deﬁne the following linear combinations of ﬁelds,

± W1 iW2 W = ∓ ,Z = sin(θW )B + cos(θW )W3, (1.13) √2 − and p gv g2 + g02v mW = , mZ = , (1.14) 2 2

4 and θW is the weak angle deﬁned by

g0 sin(θW ) = p . (1.15) g2 + g02

Of the four EW gauge bosons, there will remain one massless gauge ﬁeld deﬁned as

A = cos(θW )B + sin(θW )W3. (1.16)

This ﬁeld is recognized as the photon, which is the gauge ﬁeld associated with the unbroken

U(1)EM after electroweak symmetry breaking (EWSB). After EWSB, the Higgs potential is,

µ2 λ V (Φ) = − µ2h2 + λvh3 + h4, (1.17) 4λ − 4

p 2 which contains the tree-level Higgs mass given by mh = 2µ = √2λv. − The Yukawa sector of the SM governs the coupling of the Higgs to fermions and is given by, ij ij ij Yukawa = y d¯iΦ˜qj + y u¯iΦqj + y e¯iΦ˜lj + h.c., (1.18) L d u l

where Φ˜ = iσ2Φ∗ and the indices i, j = (1, 2, 3) are family indices. After the Higgs picks up a vev, the Yukawa terms will become masses for the fermions in the theory with the mass yij v matrices for the fermions, mij = √f , which needs to be diagonalized. As there is noν ¯ in f 2 the theory, the lepton sector can be rotated freely to be diagonal. Another consequence of the lack of aν ¯ in the theory is that neutrinos will remain massless within the SM. In the quark sector however there is a misalignment between the weak eigenstates and the mass eigenstates. This mismatch leads to the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix in the quark interactions with the charged W-bosons,

† ij µ + fermion u V σ¯ W dj + h.c., (1.19) L ⊂ i CKM µ

† where VCKM = Vu Vd and Vu,Vd are the unitary matrices to rotate the u, d fields from their weak eigenstates to their mass eigenstates. The CKM term in the SM Lagrangian allows for weak interactions to change quark flavors with off-diagonal terms in the CKM matrix. The 3 3 CKM matrix contains 3 mixing angles and a CP-violating phase which have been × measured through many weak-interaction processes. The Yukawa couplings in the SM also have a large range of values which range over 5 orders of magnitude resulting in a large

5 mass hierarchy amongst the fermions from .5 MeV for the electron to 173 GeV for the top quark.

1.2 Problems with the Standard Model

While the SM has been very successful experimentally in predicting cross sections, decay widths, electroweak observables, ﬂavour physics, the running of masses and couplings, etc., it is not without problems both theoretical and observational. Observational challenges of the SM include explaining the nature of DM, the origin of neutrino masses, and the matter-antimatter asymmetry. In addition to observational challenges, there are theoretical concerns with the SM, such as the Hierarchy Problem, the strong CP problem, the hierarchy of fermion masses, and the number of parameters in the theory. These theoretical and observational issues with the SM are at the heart of BSM physics as we try to understand and solve these problems. Below I will discuss the Hierarchy Problem and DM in more depth.

1.2.1 Dark Matter

One of the most convincing observational problems with the SM are astrophysical observations which imply the existence of DM. SM matter makes up approximately 5% of the energy density in the Universe, while DM makes up roughly 25% of the energy density, with the last 70% made up of dark energy. The fact that the Universe is made up of 5 times more invisible and unknown matter than known SM matter is a large problem with the SM. Some of the observations which imply that there needs to be DM in the Universe come from galactic rotation curves [15, 16] where we see the rotation speed of stars in galaxies rotating faster than the amount of visible matter in galaxy would suggest, implying that there is extra non-visible mass in galaxies. Other observations include gravitational masses of galaxy clusters [17, 18] from virial velocities and gravitational lensing which also imply some form of additional, invisible matter in galaxy clusters. In both the cases mentioned above, the amount of DM to explain these measurements is greater than the amount of visible matter within the galaxies/galaxy clusters of interest. There are also many other observations which are consistent with DM existing, including CMB measurements [19], cosmic structure [20], the Bullet Cluster [21], and others. While we have not yet detected a DM particle we have been able to understand some basic properties which should be exhib- ited by whatever composes DM. For example, we understand that it needs to be stable on the time scale of the Universe so that it is still around today. We know roughly how much 2 DM there should be since we have measured the relic abundance Ωch = 0.1199 0.0027 [19] ± in CMB measurements. We also know that DM candidates should be neutrally charged and interact very weakly at best with SM particles. In addition we know that during structure

6 formation in the early Universe, DM needs to be cold, meaning that DM particles moved slowly compared to the speed of light. Since the nature of DM is unknown, there are a wide range of models for DM such as weakly interacting massive particles (WIMPs), axions, sterile neutrinos, gravitinos, and others. These DM candidates often show up in BSM models as a side eﬀect of solving other problems in the SM such as the Hierarchy Problem or the Strong CP problem. The masses of allowed DM candidates range from sub-eV (i.e. axions) to the TeV scale or higher (i.e. WIMPs) producing a very large class of phenomena.

Figure 1.1: The comoving number density, Y , and the resulting relic density of a 100 GeV, P-wave annihilating DM particle as a function of temperature T (bottom) or time t (top). The solid line is for an annihilation cross section which yields the correct DM relic abundance and the bands have cross sections which diﬀer by a factor of 10 (yellow), 100 (green), and 1000 (blue) from that value. The dashed contour traces the number density for a particle which remains in thermal equilibrium [1].

The most studied classes of DM candidates are those of WIMPs because they possess many attractive properties. A WIMP particle will have a mass on the order the weak scale mDM 10 GeV-1 TeV and have weak scale couplings to the SM. Many BSM scenarios ∼ naturally produce WIMP DM such as the neutralino in SUSY or the lightest KK state in extra dimensional models with KK-parity. Another attractive feature of WIMPs is that they 7 are relevant for many types of DM searches which are discussed below. On cosmological grounds, WIMPs are able to produce the correct thermal relic rate in the early Universe naturally in what is called the “WIMP miracle” which is sketched in the next paragraph. In the early Universe, the Universe is hot and dense with all particles in thermal equilibrium, but as the Universe expands it will begin to cool, eventually getting to temperatures below the DM mass. When the temperature is below the DM mass, the number density of DM particles will become Boltzmann suppressed, however the Universe is also expanding at the same time which will cause the DM particles to become more dilute until they can no longer find other DM particles with which to annihilate. Once this occurs, the number of DM particles will become fixed in what is known as “freeze out” which will fix the DM relic density. For s-wave annihilation, relic density is given by,

r −10 −2 mχ 100 10 GeV Ωh2 = 1.69 , (1.20) × 20Tf g∗ σv h i where, mχ is DM mass, Tf is temperature at freezeout, σv is DM annihilation cross h i section, and g∗ is number of massless degrees of freedom at freeze out. For DM masses in the WIMP mass range an annihilation rate of σv 10−26cm3/s will get the correct relic h i ∼ abundance. This can be seen in Fig. 1.1, where if the DM has the correct annihilation cross section the DM density will asymptote to the correct relic density as the Universe cools and expands.

SM SM

Collider Indirect

DM DM

Direct Figure 1.2: Cartoon of diﬀerent search techniques for DM, where the ﬂow of time in the diagram is indicated by the arrow.

8 There are three general strategies in DM searches: direct, indirect, and collider searches as seen in Fig. 1.2. The strategies are able to probe different regions of DM parameter space and in many cases can cross check against other methods if a possible DM signal is found in an experiment. The goal in direct detection experiments is to look for nuclear recoil from a DM particle passing through the experiment and interacting with nuclei. These experiments are typically located deep underground and use many different techniques to achieve detection and manage the backgrounds from ambient radiation. The threshold energies required for these experiments to measure a signal are typically a few keV which means that low mass ( (GeV)) DM typically do not have enough energy to trigger an event. In the recent ∼ O past, there have been a large number of these types of experiments including LUX [22], XENON [23], ADMX [24], and CDMS [25], among many others, with more of this class of experiment coming online in the next few years. Indirect detection experiments, on the other hand, look for DM annihilation or decay products in the Universe which have propagated to the Earth. These experiments typically consist of a ground or space telescope which can look for an excess of SM particles over astrophysical backgrounds. Charged particles such as electrons, positrons, and protons will be deflected in astrophysical magnetic fields which makes it difficult to get a handle on which part of the sky they are coming from; therefore, experiments typically look for an excess over astrophysical modeling. Uncharged particles such as photons or neutrinos will not have this problem and therefore can point back to their source, allowing experiments to look in regions of the sky where a large DM density is expected such as the galactic center or dwarf spheroidal galaxies (dSphs). Example experiments of indirect detection are AMS-02 [26], Fermi-LAT [27, 28], VERITAS [29], IceCube [30], or Super-Kamiokande [31]. Collider searches look for the production of DM at particle colliders such as the LHC. DM should interact very weakly with the SM, which means that in a particle detector, there would be no energy deposits as it traverses the detector: to the detector, the DM particle is invisible. Since DM particles leave no energy in the detector, DM searches with a collider are looking for an event topology with a large amount of missing transverse momentum, alternatively called missing energy, (MET or E/ T ). The requirement of looking for MET in an event means that the DM particle would need to be made in association with one or more SM particles for the detector to see and trigger the event off of. These types of experiments work well for DM masses at or below (TeV) scale since DM particles must be produced. O Most models require two or more DM particles to be produced in association with very energetic SM particles to trigger off of. There are numerous searches by the ATLAS and CMS detectors which look for signals with large MET [32–35].

9 1.2.2 Hierarchy Problem

Figure 1.3: Feynman diagram of corrections to the Higgs mass from a fermion loop.

One of the most concerning theoretical problem with the SM is the Hierarchy Problem, which is essentially a question of why the Higgs boson is so light. If the Higgs boson is a fundamental scalar it will receive large quantum corrections to its mass since there is nothing analogous to chiral symmetry for fermions or gauge invariance for vector bosons to protect the mass. In the SM the Higgs boson will receive corrections to its mass from fermion loops (Fig. 1.3) which will be quadratically divergent at one loop order:

λ2 2 !! 2 f 2 2 ΛUV δmh = 2 ΛUV + 2mf log 2 , (1.21) −8π mf where ΛUV is the ultraviolet cut-off for the theory. This cut-off, where we expect new physics to appear can be very large; for instance the only known physics at high energies is 19 gravity which is characterized by the Planck scale, Mpl 10 GeV. If gravity was the only ≈ 15 scale above the electroweak scale, it would imply corrections to the Higgs mass δmh 10 ∼ GeV, while the physical Higgs mass is mh 100 GeV. ∼ To avoid excessive fine tuning of the bare parameters of the theory to achieve a light Higgs mass, some new physics should come into play at a much lower energy scale than the Planck scale, which will contribute to radiative corrections to the Higgs mass. There are many classes of BSM models which attempt to solve the Hierarchy Problem by adding new physics at an intermediate scale such as supersymmetry (SUSY) [36], extra dimensional models [37, 38], composite Higgs models [39], or cosmological relaxation [40]. If the scale of new physics is of (TeV), we could potentially see hints of new physics at the LHC or O other experiments.

1.3 SUSY

SUSY extends the Poincare group to include a fermionic operator, Q, which take bosons into fermions and fermions into bosons,

Q boson = fermion >, Q fermion = boson . (1.22) | i | | i | i 10 There does not need to be one operator, Q, in SUSY, we can in principle have more in what is referred to as extended SUSY and is characterized by N which is the number of SUSY operators, though we will keep our discussion to N = 1. The particle states in a supersymmetric theory can be organized into irreducible representations of the SUSY algebra called supermultiplets which contain an equal number of fermionic and bosonic degrees of freedom. The SM gauge groups are invariant under a SUSY transformation which means that all the ﬁelds within a supermultiplet have the same SM quantum numbers. The minimal supersymmetric extension of the SM is the Minimal Supersymmetric Stan- dard Model (MSSM) which takes the SM plus an additional Higgs doublet and adds a superpartner with opposite spin statistics. The extra Higgs doublet is necessary for anomaly cancellation and to give mass to up-type quarks as the superpotential must be a function of superﬁelds and not their conjugates. The MSSM contains both chiral supermultiplets and vector supermultiplets. A chiral supermultiplet will take on the form,

Φ = φ + √2θψ + θθF, (1.23)

where φ is a complex scalar, ψ is a Weyl fermion, F is the auxiliary ﬁeld, and θ is a Grassmann spinor. Vector superﬁelds can be expressed in the Wess-Zumino gauge as,

µ † 1 VWZ = θ¯σ¯ θAµ + θ¯θθλ¯ + θθθλ¯ + θθθ¯θD,¯ (1.24) 2

where Aµ is a spin-1 boson, λ is a spin-1/2 fermion, and D is an auxiliary field. Within the MSSM, the SM fermions and Higgs doublets are components of chiral supermultiplets, while the gauge fields are components of the vector supermultiplets. Under a SUSY transformation, the F-term of chiral multiplets and the D-term in vector multiplets are invariant up to a total derivative. To construct a Lagrangian which is invariant under SUSY transformations we want to select out the F- and D-terms which can be accomplished by integrating over the Grassmann variables θ and θ¯. For SM fermions, the corresponding scalar superpartner is referred to as a sfermion and will be denoted by a tilde (˜u, d,˜ e˜, etc.). The superpartners to the gauge bosons are fermions called gauginos. Since there are two Higgs doublets in the theory, they are referred to as Hu and Hd which both acquire a vev to give mass to the up-type quarks and the down-type quarks/leptons respectively. The ratio of the two Higgs vevs defines the quantity tan β vu/vd. The fermionic superpartners to the Higgs doublets are called ≡ Higgsinos.

11 SU(3) SU(2) U(1)Y u u˜ q = d q˜ = d˜ 3 2 1/3 u¯ u¯˜ 3¯ 1 -4/3

d¯ d¯˜ 3¯ 1 2/3 ν ˜ ν˜ l = e l = e˜ 1 2 -1 e¯ e¯˜ 1 1 2

G g˜ 8 1 0

W w˜ 1 3 0

B ˜b 1 1 0

Hu h˜u 1 2 1

Hd h˜d 1 2 -1

Table 1.2: MSSM particle content and quantum numbers.

Since the MSSM has an extra Higgs doublet, there will be extra physical scalars. When EW symmetry is broken down to electromagnetism 3 of the 8 degrees of freedom will be absorbed by the W and Z bosons which leaves us 5 physical scalars: two CP-even states h and H, one CP-odd state A, and two charged states H±. The h state is typically referred to as the light Higgs which would be identiﬁed with the 125 GeV scalar discovered at the LHC [41, 42]. SUSY solves the Hierarchy Problem by extending the chiral symmetry enjoyed by fermions to bosons since they will be related by a SUSY transformation. If one were to compute the radiative corrections to the Higgs mass in SUSY in addition to top quark loops, there will also be top squarks or stops in the loop,

2 λ˜ λ ∆m2 = t Λ2 t Λ2 + log terms, (1.25) h 8π2 UV − 8π2 UV

2 where λt˜ is the coupling between stops and the Higgs. With SUSY, λt˜ = λt and resolves the quadratic divergence at the lowest loop order and in fact SUSY solves this at all orders. The MSSM Lagrangian contains the Kahler potential, gauge kinetic function, and the superpotential. The Kahler potential contains the kinetic and gauge interactions terms of

12 the matter ﬁelds and is written as,

Z a a 2 2 ∗i 2gaT V j = d θd θ¯Φ e Φj, (1.26) L i

a a where Φ are superfields, ga are gauge couplings, T are group generators, V are vector superfields, the index a is over SM gauge groups, and the i, j indices of the group representation of the Φ fields. The gauge kinetic function contains the field strength terms for the gauge bosons and kinetic terms for gauginos and is written as, Z 2 α = d θW Wα + c.c., (1.27) L

2 where Wα = D¯ DαV is a chiral superﬁeld strength and D¯ and D are chiral covariant derivatives:

∂ µ α ∂ µ α Dα = i(σ θ¯)α∂µ,D = + i(θ¯σ¯ ) ∂µ, ∂θα − −∂θ α (1.28) ∂ ∂ D¯ α˙ i σµθ α˙ ∂ , D¯ i θσµ ∂ . = (¯ ) µ α˙ = α˙ + ( )α˙ µ ∂θ¯α˙ − −∂θ¯

The superpotential contains the Yukawa interaction terms and Higgs self-interactions and is given by,

Z 2 = d θ + h.c. = yuk + scalar, (1.29) L W L L where in the MSSM,

ij ij ij = y U¯iQjHu + y D¯iQjHd + y E¯iLjHd + µHuHd, (1.30) W u d l where the E¯, U¯, D¯, Hd, Hu, Q, and L are all chiral superﬁelds and i, j are family indices. Doing the θ integrals to recover Yukawa interactions will yield the following formula,

2 1 ∂ yuk = W ψiψj + h.c., (1.31) L −2 ∂Φi∂Φj θ,θ¯=0

where Φi is the chiral superfield and ψi is a component fermion field. Similarly, we can also obtain the scalar potential describing interactions amongst the component scalar fields with, 2 X ∂ X ∂ scalar = W Fi = W (1.32) L ∂Φ ¯ ∂Φ ¯ i i θ,θ=0 i i θ,θ=0

13 where we have used the equations of motion for the F fields. Within the MSSM there will be mass matrices which will need to be diagnolized for the squarks, sleptons, and electroweakinos which come from a mismatch between gauge and mass eigenstates. Since U(1)EM is unbroken the electroweak gauginos will mix with 0 like charged gauginos. There are four neutralinos denoted byχ ˜i which are made up of the neutral electroweakinos: bino, the neutral wino, and neutral Higgsinos. Similarly the ± charged winos and charged Higgsinos mix to give two charginosχ ˜i . 3(B−L)+2s R-parity is defined by Rp = ( 1) , where B is baryon number, L is lepton − number, and s is the spin of the component field. This assignment makes all the SM particles have Rp = +1 and all superpartners have Rp = 1. Requiring R-parity will restrict lepton − and baryon number violating operators (i.e. DQL¯ ) in the superpotential which would lead to unobserved effects like proton decay. One of the consequences of having R-parity in the theory is that it will make the lightest supersymmetric particle (LSP) stable, which in turn makes it a natural DM candidate. Since we have not discovered superpartners with the same mass as SM particles, we know that SUSY must be broken at low energies. There are many methods to go about breaking SUSY such as gauge mediation, anomaly mediation, or gravity mediation, though no mechanism is favored at the moment. One can incorporate SUSY breaking without knowing the precise method by incorporating soft SUSY breaking terms into the Lagrangian which explicitly break SUSY. The soft SUSY breaking terms in the MSSM are,

X 1 X 2 ∗ u d ˜ e soft = Miλiλi m f˜ f˜ A u¯˜iHuq˜j A d¯iHdq˜j A e¯˜iHd˜lj L 2 − f˜ − ij − ij − ij i f (1.33) 2 ∗ 2 ∗ m H Hu m H Hd + BµHuHd, − Hu u − Hd d where, Mi (i = 1, 2, 3) are gaugino masses, sum over f is over squarks and sleptons, mf are soft masses for squarks and sleptons, Au ,Ad ,Ae are trilinear scalar couplings, m2 , m2 ij ij ij hu hd are up- and down-type Higgs mass-squared terms, and Bµ is mass-squared term between

Hu and Hd. Another allowed term is a Dirac mass term for the gaugino mDλiψi, however this requires a fermion which is in the adjoint representation of the gauge group.

1.3.1 Dirac Gaugino SUSY Breaking

A possible extension to an R-symmetric MSSM is to allow the gauge sector of the theory to function as an N = 2 supersymmetry by adding in a Dirac mass term for the gauginos [43]. The matter ﬁelds in the theory would continue to work as an N = 1 supersymmetry which is required for a theory with chiral fermions such as the SM. In order to get Dirac gauginos we will require extra chiral superﬁelds to be added to

14 the MSSM which are adjoints under the SM gauge groups: an octet under SU(3), a vector

under SU(2), and a complete SM singlet for U(1)Y . The fermionic component of these chiral superfields will then mix with the MSSM gauginos and generate a mass term in the presence of D-term SUSY breaking in a hidden U(1)0. The superpotential term which will do this is the following: Z W 0αW aAâ D = d2θ α λaψa , (1.33) L Λ ⊂ Λ A where W 0 is a hidden sector U(1)’ field strength superfield, W a are the SM field strength â a superfields for the ath gauge group, A is the chiral, adjoint superfield, and ψA is the fermionic component field of Aâ. SUSY is broken in this scenario by the hidden sector 0 U(1) acquiring a D-term vev, which then generates a Dirac mass for the gaugino mD = D/Λ. Within the MSSM, the gauginos will only have a Majorana mass term, mλλ. The superpotential in Eq. 1.3.1 can be generated by integrating out heavy messengers which are charged under SM gauge groups and the hidden U(1)0 which have couplings to A. A consequence of this scenario is that contributions to the sfermion masses will be “supersoft”, meaning that they will receive finite corrections to their mass. Since the contributions are finite, there is no UV dependence. This will provide a particle spectrum with lighter scalars and heavier gauginos which gives rise to a rich phenomenology.

1.4 Eﬀective Field Theories

A powerful set of tools that has been used often in the history of particle physics are effective field theories (EFTs). An EFT is a low energy approximation of a more fundamental theory of nature which captures the relevant degrees of freedom at the lower energy scale and ignores higher energy degrees of freedom; for example we do not need to deal with the dynamics of QCD to understand the hydrogen atom. One advantage of using EFTs are that they allow one to accurately describe phenomena while not having to worry about substructure or dynamics at much shorter length scales, which drastically simplifies calculations. Another use for EFTs comes in cases where the high energy theory is currently unknown and an EFT will allow us to describe phenomena without knowing the details of the high energy theory. Since an EFT is a low energy or infrared (IR) theory, there will come a point when an EFT will start to break down and the high energy dynamics will become important. This means that for any EFT, there is some energy scale (sometimes referred to as the cutoff scale), Λ, associated with the EFT at which the details of the high energy theory become important and the EFT will begin to break down. An example of this is an EFT coming from “integrating out” a heavy particle in some high energy theory: the scale Λ will be the mass of this heavy particle, M. For low energies, E M, the heavy particle is 15 not a relevant degree of freedom in the theory since the heavy particle cannot be created on-shell and using an EFT will give accurate predictions. However, at energies E M, the ∼ heavy particle will be a relevant degree of freedom, that can produce phenomena such as resonances which are not captured within the EFT framework. In an EFT, we can write an effective Lagrangian in the form,

X ci = i, (1.33) L Λd−4 O i

where i are operators with mass dimension d which describe the light degrees of freedom O and all the dependence on the heavy degrees of freedom are contained in the scale, Λ, and coefficients, ci. The operators i can be organized by their dimensionality, di, and O classed into three categories: relevant (d < 4), marginal (d = 4), and irrelevant (d > 4). Irrelevant operators will be non-renormalizable and suppressed at low energies by powers of E/Λ. Relevant operators on the other hand become more important at lower energies, for example mass terms become negligible at very high energies but are very important at energies comparable to the mass. Marginal operators are important at all scales and will have dimensionless coefficients. If we have a high energy theory that we wish to describe with an EFT we need to understand the relevant degrees of freedom that we wish to describe. Once we have identified what will be relevant for the IR theory, the only remnant of the high energy degrees of freedom should be contained in the couplings in the EFT and the symmetries in the EFT. The EFT will in principle contain an infinite number of local non-renormalizable operators to capture the UV theory however only the operators with the lowest dimensionality are important. If we keep only the lower dimensionality operators, the neglected operators will only provide corrections to the EFT so we can compute to a given accuracy (E/Λ)di−4. ∝ When using EFTs to describe physics, one must always be aware of when the energies of a system start to approach the cutoff scale of the theory. In addition, since the EFT is only accurate to within corrections of (E/Λ)di−4, we must be careful to introduce higher dimensional operators if greater accuracy is required. There is also the case where an experiment may be able to probe scales beyond the scope of an EFT, so one has to be careful interpreting results in the context of an EFT. One approach to get around this is to assume a simplified model UV completion, where the UV theory contains minimal new degrees of freedom from the EFT. An example of this is a theory where there is a single new degree of freedom to describe an interaction, with all the rest of the UV theory at an even higher energy scale. Below, I will illustrate the use of EFTs by describing the Fermi Theory of Weak Inter- actions and EFTs in the search for DM.

16 1.4.1 Example: Fermi Theory of Weak Interactions

An example case of an EFT is the Fermi Theory of Weak Interactions which was developed in the 1930’s to describe beta decay processes. The beta decay process had a relatively long lifetime compared to other nuclear processes, and discovery of parity violation led to an EFT with a V-A interaction. The development of this V-A theory was long before the idea that beta decay was mediated by the W-boson charged current interactions, so this is a case where the EFT at the MeV scale was developed without knowing the full UV theory that appeared at the 80 GeV scale corresponding to the mass of the W-boson. ∼ The V-A theory contains as the relevant light degrees of freedom only the light quarks and leptons. The eﬀective potential takes on the form

GF † µ eff = JµJ + h.c., (1.33) L − √2

where Jµ is the V-A current:

X ij X Jµ = u¯iγµ(1 γ5)V dj + νγ¯ µ(1 γ5)l. (1.33) − CKM − ij l

These eﬀective operators are dimension 6 and thus are irrelevant operators with the coeﬃ-

cient having mass dimension [GF ] = 2. Since the eﬀective operators are irrelevant, they − will be suppressed at low energies. For instance, we can compute the decay width in the l ν l0ν l ν l0ν G2 m5 process l ¯l0 to be Γ( l ¯l0 ) F l . The energy of the system is the mass of → → ∝ 4 4 the decaying lepton and we see that Γ is suppressed by E /Λ . The extra ml is present to give the correct dimensionality for the decay width. This result will have corrections of 2 2 order ml /Λ which result from higher dimensional operators within the EFT framework, which we would have to include if we required more accuracy. In creating this EFT, we remove the W-boson from the SM which takes the non-local structure of W-boson interactions and replaces these interactions with local four-fermion interactions in the V-A theory. The local nature of the interactions is evident if one looks at the W-boson propagator in cases with a small momentum transfer,

qµqν ηµν + m2 ηµν − W q2 m2 . (1.33) q2 m2 W m2 − W −−−−−−→ W

We see that in this limit the propagator has been reduced to a contact interaction which 2 2 will have corrections of order q /mW . Matching conditions between the low-energy EFT coeﬃcient (GF ) and SM parameters (g,mW ) will be given by,

17 2 GF g = 2 . (1.33) √2 8mW

We see that this coeﬃcient contains the remnants of the high energy degree of freedom, the W-boson, in the low energy theory.

1.4.2 EFTs in the Search for Dark Matter

As we have not yet discovered a DM particle, the exact properties that DM possesses are unknown. For instance, we do not know if DM is a boson or a fermion, what SM particles DM interacts with, or how strongly. Various DM experiments can set upper bounds on the interaction rates from non-detection which preclude most tree-level couplings between a DM particle and SM particles. One method to attack this problem in recent years is to introduce EFTs as a way to describe how DM may interact with the SM and place bounds upon the effective operators [2, 44–46]. These EFTs can come from a wide variety of UV completions such as extra dimensional models or SUSY models, but since the EFT will be insensitive to the exact UV theory it allows a model independent approach to categorizing DM interactions. The general strategy when using an EFT with an unknown UV theory is to identify any relevant symmetries of the problem and write down all the allowed operators under those symmetries. In the case of DM, the symmetries of interest for the EFT are Lorentz invariance, the SM gauge symmetries, and some symmetry to make the DM particle stable such as a Z2 symmetry. Since we do not know the spin of DM particles, we write down all allowed effective operators for scalar, fermionic, vector, or higher spins. Fig. 1.4 has a list of example EFTs to describe DM interactions with the SM for fermionic or scalar DM which are relevant for LHC searches. In all of these scenarios there is the assumption that there is some particle which is much heavier than SM and DM particles, mSM , mDM mmed, which mediates the interaction between the SM and DM which has been integrated out of the theory. One of the key advantages of using EFTs to characterize DM interactions is that it provides a consistent framework to compare DM models across the different classes of DM experiments. In the cases of direct and indirect searches for DM the relevant energy scales are that of the DM mass or below so the EFT framework should provide an accurate way to describe interactions as long as the mediators are heavy. However, in collider searches for DM the energy scales in a collision can be well above the DM mass and can potentially probe features of the underlying UV theory which makes interpreting bounds from EFTs at colliders problematic.

18 Figure 1.4: List of eﬀective operators between the SM and DM for DM as a Dirac fermion, Majorana fermion, complex scalar, and real scalar up to dimension 7 [2].

A way to overcome this issue is to use a simplified model to characterize DM-SM interactions. A simplifed model postulates that there is some simple mediator section between DM and the SM with all other features of the UV theory at an energy much higher than the mediator mass. Common simplified models add an extra scalar or vector mediator which then communicates between the DM and the SM through s- or t-channel interactions. These simplified models then map onto EFTs in the high mediator mass limit, but allow for use in a broader range of energy ranges than an EFT.

1.5 Synopsis

In this chapter, we have reviewed the SM which describes the fundamental particles and interactions that we observe in nature. We observed that while the SM has been a resound- ing success in explaining phenomena, there are still shortcomings which are not explained within the context of the SM. Among the problems are the Hierarchy Problem of Higgs mass and the nature of DM, which were discussed brieﬂy. There are many approaches to

19 solving the problems within the SM, such as SUSY which can solve the Hierarchy Problem and the DM problem. Lastly, we have gone into some detail on using EFTs as a tool for describing phenomena in the search for new physics. The various problems within the SM provide hints that the solution to them may involve new TeV scale physics which we hope to observe in a broad class of experiments from colliders to astrophysical measurements. This thesis is organized as follows: In Chapter 2, we will discuss using data from the Fermi-LAT pass-7 dwarf spheroidal data to constrain DM models using an EFT framework where the annihilation to multiple channels is present. Chapter 3 will extend the work in Chapter 2 to using Fermi-LAT pass-8 data using a stacking technique for the dwarf galaxies to constrain annihilation into multiple channels and place bounds on simpliﬁed models as well as EFTs describing DM annihilation. In Chapter 4, we will discuss using a Dirac gaugino model which was motivated by a diphoton excess to explore signatures at the LHC, vacuum stability, and electroweak constraints of the model. Chapter 5 explores the use of EFTs to search for a color-octet scalars which have previously unexplored experimental signatures at the LHC. The work in this thesis highlights the importance of using EFTs in the search for signals and signatures of BSM physics. With the cost of conducting high energy physics, it is important that “no stone be left unturned” in regards to considering ways in which to look for new physics models. Since the LHC has seen no signatures of new physics, limits on new physics relevant to a broad range of BSM models has also been an important and useful result. The results from work in this thesis include providing additional collider signals in the search for exotic particles within the LHC which are important for understanding new physics, providing methods to constrain DM annihilation with a range of ﬁnal states, and exploring model space of Dirac gaugino models.

20 Chapter 2 Indirect Detection Constraints on the Model Space of Dark Matter Effective Theories

This Chapter is based on work published in Phys. Rev. D92 (2015) no.9, 095011

2.1 Introduction

In this era constraints on dark matter (DM) models are being synthesized from multiple experiments. There has been much recent work in collider physics, focusing on DM models, which include both UV complete theories and Effective Field Theory (EFT) scenarios. The same models studied in collider physics imply detectable signatures from DM annihilation in space. Due to gauge invariance or other theoretical considerations, many of these models, both EFTs and simplified models, predict couplings between DM and multiple species of SM particles. Thus, DM may be produced in many correlated final state channels at colliders, and may have multiple final state annihilation channels in space, which would contribute to total detectable photon or positron flux for satellite experiments. In this chapter we explore the indirect detection bounds from Fermi-LAT dwarf spheroidal galaxies [28] on models where DM annihilates into multiple final state channels. These bounds are among the tightest constraints on DM models. We study several EFT models with dimensions 6 and 7 effective operators. We choose operators which lead to unsuppressed DM annihilation rates in indirect detection processes, and which are being simultaneously studied in DM production processes at LHC. The dimension 6 operators we study are those which couple DM to third generation fermions pairs. The dimension 7 operators we consider are vector boson portals where there DM couples to multiple pairs of SM gauge bosons. The use of effective operators allows a great degree of model independence for DM

21 studies, while capturing some of the important kinematic features of DM processes [2, 44, 47]. For some models which are completed by loops, EFT based calculations have so far provided the best means for study at colliders. The limits of the effective operator paradigm are becoming more clear for collider analyses. In particular, UV completions of models with low scale messenger portals are less probeable by colliders, and models with low scale effective operator cut-offs may not be sensible at collider energies [48–51]. However, we expect that EFT analyses are reliable at the scale of indirect detection where the center of mass energy of the annihilation process is the same order as the DM particle mass itself. We expect that the EFT treatment is valid down to much lower scales, perhaps for mediator sectors in range of 10’s of GeV, as opposed to colliders where mediators of some hundred GeV to just under a TeV may not be appropriate. We will visualize the bounds we set in two slices of the total parameter space, the plane of fixed DM annihilation rate, and the plane of effective operator coefficients where the total annihilation rate varies. For the regions of fixed annihilation rates, we will use the constraints to produce a 2-D visualization of the bounds on a triangle, a technique familiar from collider physics [52]. We also compare bounds set with the dwarf limits to those set by collider constraints, and discuss the validity limits of EFTs for both cases. The format of this paper is as follows, in Section I we will discuss dwarf constraints on photon flux from DM annihilations. In Section II we will analyze constraints on models with non-interfering final state annihilations and present triangular visualizations of parameter space. In Section III we will analyze a popular set of vector boson portal models with interfering final state channels. Section IV presents results along with collider constraints and discusses EFT validity. Section V concludes.

2.2 Indirect Detection from Dwarf Spheroidal Galaxies

Dwarf spheroidal galaxies provide some of the tightest constraints on photon flux originating from DM annihilation as they are believed to contain a substantial DM component [53, 54]. This combined with their low astrophysical background makes them a good laboratory to search for DM. As no significant excess in the photon spectrum has been observed from dwarf data, we use upper bounds on photon flux obtained from Fermi-LAT data [28] to place constraints on DM mass and couplings in the scenarios discussed in the introduction. The photon flux (photons cm−2 s−1) at the earth expected from annihilation of DM in the area of interest is given by

Z Emax 1 X σv f dNγ Φγ = h i2 dEγJ. (2.0) 4π 2m dEγ f χ Emin f

Where the J-factor (GeV2cm−5) is the line of sight integral of the DM density ρ, integrated

22 over a solid angle, ∆Ω Z Z J = ρ2(r)dldΩ0. (2.0) ∆Ω l.o.s The additional terms in Eq. 2.2 are dependent on the particle properties of the DM. Here, dNf /dE is the differential photon energy spectrum per annihilation to final state f, σv f h i is the thermally averaged DM annihilation cross section and mχ is the DM mass. The Fermi-LAT collaboration has presented DM constraints from observation of 25 dwarf spheroidal galaxies [28]. Of these 25 Milky Way satellites, 15 were used to place constraints on annihilation cross sections of DM particles with masses between 2 GeV and 10 TeV into various SM channels. These annihilation constraints were derived assuming that the DM annihilates primarily to a single SM channel. This is not realistic when considering most UV completions of DM models, even ones that are quite simple. Here we discuss obtaining bounds on DM mass and annihilation cross sections assuming annihilation into multiple channels. The bin by bin integrated γ-ray energy flux upper limits at 95% CL for each of the 25 dwarf galaxies was presented in [28]. The J-factor for dwarfs is determined through dynamical modeling of their velocity dispersion profiles; we use J-factors presented in [28]. For a conservative limit we perform a very simple combination of six dwarfs with the highest J-factors. The limit in each bin is obtained from the dwarf which gives the strongest 95% CL upper limit on γ-ray flux scaled by the J-factor. The combined limits are shown in Fig. 2.1. We note that the Fermi-LAT collaboration has obtained its own combined dwarf limits on photon flux from DM annihilation. However, these limits assumed DM annihilation into one and only one visible channel at a time. Since our goal in this chapter is to analyze models that may have multiple final state annihilations we have forgone use of Fermi’s combination, though in future analyses, stronger bounds than ours may be obtained in each bin by ”stacking” the total signal from all dwarves and subtracting a summed astrophysical background.

We calculate the expected σv f for each kinematically accessible annihilation channel, h i and obtain the differential gamma ray spectra for annihilation channels from the Mathe- matica code PPPC 4 DM ID [55, 56]. As an example, we show the spectrum for several relevant SM pair annihilation channels for a 100 GeV mass DM particle in Fig. 2.2 below. Here, the dimensionless parameter x is the gamma ray energy scaled by the DM mass. Analysis of many models requires calculation of differential gamma ray spectra from particles produced in non-symmetric pairs, for example, in the analysis of models of section IV we have calculated the differential spectrum of the Zγ final state. Limits are obtained by binned comparison of total expected flux from all DM annihilations to our combined upper bounds.

23 95% CL Bin Upper Bounds __

5´10-28 ______

L -28 1 2´10 - __ s 3 __ cm __ 1 -28 - 1´10 __ __ GeV H

J -29 5´10 ______Flux ______-29 __

Energy 2´10 ______1´10-29

1000 5000 1´104 5´104 1´105

EΓ MeV

Figure 2.1: Combined 95% upper bounds on photon energy ﬂux scaled by J-factor from 6 of largest J-factor dwarf spheroidal galaxies. H L

2.3 Models with Independent Annihilation Channels

We will now discuss the method of constraining the parameter space of models with multiple independent annihilation channels. We will first assume an effective Lagrangian which is the sum of several independent operators, each of which couples DM to one and only one pair of SM particles. Therefore the Lagrangian has the form κ f = Σi i = Σi n χχXiXi. (2.0) L O Λi Here the χ is the DM, and X is some SM particle with particle index i. Any specific operator n will be gauge and Lorentz invariant and will have coefficient κ/Λi , where the effective cut-off Λ appears with the appropriate power to make the operator dimension 4.

The full parameter space of the EFT consists of the DM mass mχ and the i operator n coefficients κ/Λi . Each point in this parameter space specifies a total DM annihilation rate, the specific ratios of the i final state annihilation channels, and the resultant γ-ray flux. There are various slices of parameter space which can be studied. The first one we will consider is slices of the parameter space on which the total DM annihilation rate is held constant. Below we will show that along planes of parameter space with fixed annihilation rate, we will rule out masses and effective cut-offs below certain scales.

24 Γ spectrum, mDM=100 GeV dN dlogx ΜΜ 100 ΤΤ bb 10 ΓΓ

1 WW ZΓ Γ

0.1 ZΓ Z H L H L

-4 x 10 0.001 0.01 0.1 1

Figure 2.2: Gamma ray spectrum from DM annihilation into various channels for a 100 GeV DM fermion.

2.3.1 Fixed Annihilation Rate

The total annihilation rate, σv tot, is simply a linear sum of the thermally averaged anni- h i hilation cross sections σv O to particle Xi due to operator i, h i i O

σv tot = N σv Th = σv O + σv O + (2.0) h i h i h i 1 h i 2 ··· We will ﬁrst ﬁx the desired total annihilation rate. This rate may be anything we like, for simplicity we will consider it some numerical factor times the thermal annihilation rate

N σv Th. This constraint drops us 1 dimension in parameter space; certain coefficient h i n values κ/Λi will satisfy the constraint for any specific DM mass. Having fixed the total annihilations rate, we may then determine the limits of the operator coefficients which saturate the Fermi-LAT photon-flux bounds for any given DM mass. A natural choice for the total annihilation rate is the thermal rate. However, in the spirit of model independence, we will show Fermi-bounds on models with various annihilation rates which will correspond to models with various non-thermal histories. The complete theory will have to account for this history, as well as the presence (and absence) of specific operator coefficients. With annihilation rates greater than the thermal rate one can still maintain

the correct amount of DM through eﬀects such as Sommerfeld enhancement for T Tf or coannihilation at freeze-out. We note that models where the total visible annihilation rate is below the thermal rate can easily be saved from the prospect of overclosure by the addition of invisible DM annihilation channels which restore the total annihilation rate to

25 thermal. We discuss such models in the next subsection. The rate constraint conﬁnes us to a hyper-surface in the full parameter space, however we may greatly simplify the form of this constraint by dividing out by the total rate σv tot h i . We deﬁne the fractional annihilation rate as in [57] to be Ri = σv i/ σv tot, giving a h i h i constraint which is linear in the partial rates Ri.

R1 + R2 + R3 + = 1. (2.0) ··· The natural visualization for this parameter space of models with 3,4,5 annihilation channels etc. are a triangle, tetrahedron, 4-simplex, etc. As an example and for clarity of discussion we will analyze a simple model where DM may annihilate to multiple independent channels. We choose an EFT model where DM couples to the third generation of fermions via dimension 6 operators;

κt κb κτ κν f = 2 χΓχtΓt + 2 χΓχbΓb + 2 χΓχτΓτ + 2 χΓχνΓν. (2.0) L Λt Λb Λτ Λν Here Γ specifies the whether the fermionic currents are scalar, pseudo-scalar, vector etc. Under the conventions in [2] they correspond to operators D1, D2, etc. Annihilation rates will depend greatly on the Lorentz structure of the operator with operators leading to velocity and/or helicity suppression of the rate [58]. The strongest constraints from photon- flux measurements will apply to models with operators corresponding to unsuppressed DM annihilation rates. We have here included an annihilation into an invisible channel, specifically neutrinos. In setting constraints for fractional annihilation rates, we may consider this channel a stand-in for annihilation into any kinematically accessible invisible channel. The option of an invisible channel allows a total visible annihilation rate below the thermal rate while still avoiding overclosure of the universe. Recent work, for example [59], has shown that these annihilations may be significant in some scenarios. The relevant annihilation diagram are shown in Fig. 2.3.

Let us ﬁrst assume that the DM is light and that annihilation into top pairs in kinematically forbidden. Here, the total annihilation rate is the sum of the thermally averaged annihilation cross section into b’s, τ’s, and the invisible channel,

σv tot = σv b + σv τ + σv ν. (2.0) h i h i h i h i The crucial point here is that each operator in Eq. 2.3.1 corresponds to only one channel, 22 22 22 thus the constraint above has the form a κb/Λ + b κτ /Λ + c κν/Λ (where a, b, ∝ b τ ν and c depend on DM mass and ﬁnal state particle mass) with no terms of the form κiκj for i = j. Therefore, a triangle seems the natural choice for visualization of constraints where 6 the sides of the triangle correspond to the Rb,Rτ ,Rν axis.

26 1 χ¯ f¯

χ f

Figure 2.3: DM annihilation into pairs of fermions.

We see from Eq. 2.2 that once the annihilation rates σv f are fixed, upper bound on h i γ-ray flux corresponds to a lower bound on DM mass. To determine this mass bound we set the total annihilation cross section to a multiple of the thermal relic, σv tot = N σv thm, χ¯ γ χ¯ γ χ¯ Z χ¯ h i Wh+ iχ¯ g then for a given point in (Rb,Rτ ,Rν) space find the lowest mass which gives the maximum allowed γ-ray flux determined by Fig. 2.1. The mass limits depend on the total annihilation rate which we have fixed; as we increase the total annihilation rate, the mass bounds become stronger. The results are given in the Fig. 2.4 for two total annihilation rates, the thermal rate χ γ χ χ χ χ g and 10 times thermal rate. The blankZ spaces in the triangleZ correspond to a lowerW − bound of 0 GeV or no lower bound. We see that the strongest bounds come from annihilation into b’s while the weakest bounds come from annihilation into invisible particles as expected.

27 Figure 2.4: Lower bound on DM mass in GeV for annihilations into b quarks, τ’s, and invisible particles for σv tot = σv Th in the upper plot and σv tot = 10 σv Th in lower h i h i h i h i plot.

Here the points that form the triangle each correspond to a unique ratio of the partial annihilation rates Ri. At each vertex of the triangle, a single Ri = 1, that is, one channel saturates that total annihilation rate. Along the edges of the triangle only two annihilation channels contribute to the total annihilation rate, with the third partial rate set to 0. The 28 form of Eqn. 2.3.1 was satisfactory to derive a lower DM mass bound for a fixed total annihilation rate. We will now pick a specific Lorentz structure for our operators and calculate bounds on the effective operator coefficients. As an example, we will choose to constrain the vector current operators D5, whose annihilation rate is neither velocity nor chirality suppressed. κf µ f = 2 χγ χfγµf (2.0) L Λf Each mass that satisfies the rate constraint will correspond to specific values of the effective operator coefficient. With total annihilation rate fixed, points of lower mass will require lower effective cut-offs to satisfy the bound. We also plot on the triangle the lowest effective cut-off (largest operator coefficient) that satisfies the bound of total flux. Below ∗ we show the lower limits operator coefficients Λi = Λi/√κi which saturate the bounds on photon flux.

29 ∗ ∗ Figure 2.5: Lower bound on operator coeﬃcients Λ (upper) and Λ (lower) for σv tot = b τ h i σv Th. h i

30 ∗ ∗ Figure 2.6: Lower bound on operator coeﬃcients Λ (upper) and Λ (lower) for σv tot = b τ h i 10 σv Th h i

For the thermal total annihilation rate, we see that DM annihilating into mostly b’s or τ 0s, must be above 5-15 GeV in mass while the effective operator cut-off scales may not be smaller than a few hundred GeV. We point out that in our exclusion region, the effective operator paradigm appears to be justified as the effective cut-offs are well above the center

31 of mass energy for the annihilation process.

2.3.2 Constraints for 4 Independent Channels

Next we consider the case of allowing more than three annihilation channels. We now allow the DM mass to be larger than the top mass opening up an additional annihilation channel.

One can visualize the mass constraints on a tetrahedron where the “vertical” axis is Rt, see Fig. 2.7. Each slice of the tetrahedron would correspond to a ﬂat triangle with constraint equations given by

σv tot σv t = σv b + σv τ + σv ν h i − h i h i h i h i (1 Rt) = Rb + Rτ + Rν. (2.0) −

ττ νν bb

Figure 2.7: Axis labeling for tetrahedron presentation of DM mass constraints when a fourth annihilation channel is allowed. The base corresponds to the top left of Fig. 2.8.

In particular, the base of the tetrahedron would correspond to Rt = 0 and the constraints shown in the top left of Fig. 2.8. Below we show lower bounds on DM mass for triangle slices where the fractional annihilation rate to tops is Rt = 0, 0.3, and 0.7. The blank areas in the triangles of Fig. 2.8 correspond to the region of parameter space where the kinematic bound is stronger then that obtained from indirect searches.

32 Figure 2.8: The contours correspond to lower bounds (in GeV) on DM mass from annihilations into pairs of fermions (tt, bb, ττ, νν). The total annihilation rate is set to 33 times the thermal relic for all three plots. The left image corresponds to Rt = 0, the right to Rt = .3 and the bottom to Rt = .7.

33 For models with each operator corresponding to independent annihilation channels, this procedure may be iterated for any number of final state annihilation channels. In the next section we will discuss bounds on the parameter space of models where there is not a one to one correspondence between operators and annihilation channels. We first discuss the bounds on parameter space visualized on surfaces of fixed total annihilation rate. We will then discuss the bounds on surfaces of fixed operator coefficients.

2.4 Constraints in Models with Interfering Channels

Many examples of models exist in which a single parameter or operator coefficient in the Lagrangian demands DM annihilation to multiple final state channels. Further there may be multiple operators or parameters in the Lagrangian which contribute to annihilations into the same channel. We will now analyze a model in which both of the previous statements are true. We consider a popular set of effective operator models for collider and indirect studies (see, for example [60–65]), the vector boson portal models . We will consider a set of dimension 7 operators where a pseudo-scalar DM current couples to the SM field strength tensors. 1 κ1 5 µν χ¯ κ2 5 if¯ µν κ3 5 a µν = χγ¯ χBµνB + χγ¯ χW W + χγ¯ χG G (2.0) L Λ3 Λ3 µν i Λ3 µν a These operators are not velocity suppressed and they may naturally arise if the DM couple to the SM through loops of heavy messengers which carry SM charge [62]. Unlike the previous model discussed, gauge invariance ensures that two of the above operators contribute to multiple annihilation channels. Theseχ two operatorsf also have interfering contributions to 3 3 the same annihilation channels. The coefficients κ1/Λ and κ2/Λ control the coupling of DM to four pairs of electroweak gauge boson final states: ZZ, Zγ, γγ, and W +W −, see the relevant diagrams in Fig. 2.9. The thermally averaged annihilation cross section to

χ¯ γ χ¯ γ χ¯ Z χ¯ W + χ¯ g

χ γ χ χ χ χ g Z Z W −

Figure 2.9: DM annihilation into gauge bosons.

34 diboson ﬁnal state is given in Eq. 2.4 below.

2 s 2 κ2 mW 4 2 2 4 σvrel WW = 6 1 2 (16mχ 16mW mχ + 6mW ) h i 4πΛ − mχ − 2 2 2 s 2 (κ1sw + κ2cw) mZ 4 2 2 4 σvrel ZZ = 6 1 2 (16mχ 16mZ mχ + 6mZ ) h i 8πΛ − mχ − 2 2 2 swcw(κ2 κ1) 2 2 3 σvrel Zγ = 2− 6 (4mχ mZ ) h i 16πmχΛ − 2 2 2 4(κ1cw + κ2sw) 4 σvrel γγ = m h i 2πΛ6 χ 2 16κ3 4 σvrel gg = m h i πΛ6 χ We now explore the bounds on the parameter space of this model, the space consisting of the DM mass mχ and the three operator coefficients. For any given point in parameter space, the total photon flux due to annihilations results from the sum the annihilations into 3 each of the five diboson channels. We see that the operator with coefficient κ3/Λ controls the annihilation to gluons and only gluons, and therefore factorizes in the total annihilation rate as per the discussion before. However the total rate to other diboson pairs depends on 3 two operator coefficients. In this case, the total annihilation rate has the form (ki = κi/Λ )

2 2 2 ak + bk + c12k1k2 + ck = σv tot (2.0) 1 2 3 h i The coefficients a, b, c12, and c are functions of known SM parameters as can be seen from Eq. 2.4. The relationship between these coefficients is such that the quadratic surface described by Eq. 2.4 is an ellipsoid in (k1, k2, k3)-space. We may again study the system by fixing the total annihilation rate as in the previous section. Unlike the case in the previous section, the rate constraint is not linear; however for each specific DM mass, the couplings which saturate the annihilation rate are still constrained to lie on a 2-dimensional surface. This surface is a triangular section of an ellipsoid. The radii of the ellipsoid vary as the total annihilation rate is changed. We can again present dwarf constrains on the space of operator coefficients using a simple visualization, now on an ellipsoidal surface rather than a triangle. In Fig 9, we show the ellipsoidal surfaces which satisfy the constraint that the total DM annihilation rate is 10 times the thermal rate.

We consider four different DM masses for which this constraint is satisfied, mχ = 10, 75, 150 and, 200 GeV. For each DM mass, certain combinations of the operator coefficients will satisfy the rate constraint. As the DM mass is decreased, lower cut-off scales Λ will obey the annihilation rate constraint, just as in the previous section. For each point on the ellipsoid, we may check if the combined photon flux from diboson annihilations violates

35 or satisfies the dwarf constraints. Just as before, we find that parameter regions of low mass (hence low effective cut-off scale) are ruled out. In our scan of DM masses, we found that there is no allowed parameter space for DM masses much below 10 GeV which also satisfies the constraint σv tot = 10 σv T h. h i h i

1 _ 3 k_3 -3 135 3 Λ3 (GeV) 1_ 3 1_ 3 36.8 149 1_ 3 1_ 3 40.5 k_3 171 -3 3 1_ Λ3 (GeV) 3 1_ 46 4 3 . 215 1 _ 3 58.5 0 0

0 0 0 1_ 3 215 1_ 3 1_ 1_ 171 1_ 3 3 3 46.4 58.5 171 1_ 3 k_2 k_1 135 -3 -3 3 (GeV) 3 (GeV) 1 Λ3 1_ Λ3 k_2 _ 3 k_1 3 -3 1_ -3 36.8 3 ( ) 3 149 3 (GeV) Λ3 GeV 118 Λ3 1_ 1_ 3 1_ 3 1_ 3 3 32.2 46.4 108 135

1_ 3 292

1 k_3 _ 3 -3 1 321 3 (GeV) _ 3 Λ3 232 1 _ 3 1_ 368 3 k_3 -3 255 3 (GeV) Λ3 1_ 1_ 3 3 464 292 0 1 0 _ 3 0 368 1 0 _ 3 1_ 0 368 3 464 1_ 3 464 1_ 1_ 3 3 1 1_ 3 292 368 _ 3 368 368 1 k_2 _ 3 k_1 1_ 1_ -3 1_ -3 3 3 3 (GeV) 321 3 3 (GeV) k_2 255 k_1 Λ3 321 Λ3 -3 292 -3 3 3 ( ) 3 (GeV) Λ3 GeV 1_ Λ 1_ 3 1_ 3 1_ 3 3 292 292 255 232

Figure 2.10: Limits on effective operator coefficients for 10 times the thermal annihilation 3 rate given by Fermi dwarf bounds. The axes show the value of ki/Λ . Bounds on operators are shown for four DM masses, mχ = 10, 75, 150 and, 200 GeV (top left, top right, bottom left and bottom right respectively). Red points indicate that the operator coefficient is ruled out, blue points are unconstrained.

In the Fig. 2.10 plots, the red points are ruled out by the Fermi-LAT data, while the blue points are unconstrained. As we scan through parameter space increasing the DM mass diﬀerent annihilation channels open up which contribute to the total photon ﬂux.

The ﬁrst plot gives constraints for a 10 GeV DM particle. At such low masses mχ < mZ /2, only annihilations to gluons and photons are kinematically accessible. The photon energy

Eγ is equal to the DM mass mχ and the operator coeﬃcient values that yield a high diphoton annihilation rate and are ruled out. For DM masses above half of the Z mass, the Zγ annihilation channel open up and starts to increase in magnitude but we see the the

36 characteristic photon energy of the Zγ annihilation is less than that of the photons in the q 2 2 γγ annihilation channel, here Eγ = mχ 1 4m /m . Therefore the electroweak operator − χ Z coefficients in this mass region are not too strongly constrained. For higher DM masses, mχ > mW/Z , the WW and ZZ annihilation channels also open up as we see in the bottom two plots where mχ is 150 and 200 GeV respectively. We have so far proceeded by fixing the total annihilation rate. We will now present the same constraints in a semi-orthogonal plane of parameter space where the total DM annihilation rate is allowed to vary. In Figs. 2.11 and 2.12, we show the Fermi-LAT bounds in plane of operator coefficients. For simplicity in the figures we made the redefinition, 3 3 3 ki κi/Λ . We now allow the coefficients k1 κ1/Λ and k2 κi/Λ to vary freely. ≡ ≡ ≡ Here, k3 is determined at each point by fixing the fractional annihilation rate to gluons Rgg. We may then scan through increasing masses and determine which points in parameter 3 space are ruled out. Note, we are not holding k3 κ3/Λ constant; as mχ is varied this ≡ parameter must be varied in order for Rgg to remain at a fixed value. Each point in the plot corresponds to a unique value of the three operator coefficients so each specifies a particular total annihilation rate for DM. As in the previous section, the parameter space exclusions are derived from the upper bounds on γ ray flux from dwarfs. For fixed operator coefficient 4 values, the annihilation cross section goes as mχ, so the total photon flux for each point 2 increases goes like mχ. We may thus put an upper bound on DM masses for fixed values of operator coefficient. We plot upper limits on DM masses in Fig. 2.11. The top left, top right, and bottom plots in Figs. 2.11 and 2.12 correspond to 0%, 30%, and 70% annihilation rate into digluons respectively. The solid, dashed, and dotted lines represent the regions where RZγ, RZZ , and Rγγ go to zero respectively. The vanishing of each of these channels coincide with a specific choice of the couplings in Eq. 2.4 given by

κ1 = κ2 RZγ = 0 2 ⇒ sw κ2 = κ1 2 RZZ = 0 − cw ⇒ 2 cw κ2 = κ1 2 Rγγ = 0 (2.-1) − sw ⇒ As one would expect, the γγ channel tends to be the most powerful in constraining annihilation rates when allowed. As we can see in Fig. 2.11, much of the parameter space will be ruled out by the γγ channel since the limits are stringent enough that other channels do not turn on before photon lines rule them out.

37 Figure 2.11: The colored contours correspond to upper bounds on mχ (in GeV) from 3 annihilation into all final states allowed from the operators in Eq. 2.4 with ki κi/Λ . ≡ The amount to the gg final state is fixed in each of these plots as a percentage of the total rate, 0% (upper) and 70% (lower). The solid, dashed, and dotted lines correspond to a combination of κ1 and κ2 which turn off the final states Zγ, ZZ, and γγ respectively.

38 Figure 2.12: These contours correspond to upper bounds on σv tot in units of the thermal h i relic, σv Th, from annihilation into all final states allowed from the operators in Eq. 2.4 h i 3 with ki κi/Λ . The amount to the gg final state is fixed in each of these plots as a ≡ percentage of the total rate, 0% (upper) and 70% (lower). The solid, dashed, and dotted lines correspond to a combination of k1 and k2 which turn off the final states Zγ, ZZ, and γγ respectively.

39 Notice that when the operator coefficients are very small, flux bounds allow for much larger DM masses. If we look at the region where the γγ final state is off in parameter space, we still expect annihilations into the final states with W W/ZZ and Zγ which allow for strong constraints to be placed. In Fig. 2.12 we plot the upper bound on the total DM annihilation rate in the plane of effective operator coefficients.

2.4.1 Limits of the EFT and Collider Constraints

We will now say a few words about the maximum range of validity of the EFT for DM annihilation processes. The most natural completion of a model with a vector boson portal involves loops of heavy messenger particles which are charged under the SM gauge groups such as those found such as in [62]. For the EFT to be valid, the momentum transfer of the process should be less than twice the messenger mass. Since the square root of the c.o.m energy of the process is twice the DM mass, we get mχ < Mmess. We assume that the 2 3 loops roughly scale like fαigχ/Mmess where f is some numerical factor, Mmess a messenger mass, and gχ the coupling of the messenger to the DM. Therefore, our effective operator 3 2 3 coefficient ki/Λ corresponds to fαigχ/Mmess. We can then estimate the messenger masses in terms of our effective cut-off for the maximum perturbative value of the hidden sector 2−1/3 2−1/3 coupling gχ. We find Mmess Λ fαi16π . For EFT validity mχ < Λ fαi16π . → For order 1 values of f, the validity limits for the U(1) operator for example are roughly mχ < Λ; in regions of parameter space where the DM mass larger than the effective cut-off, derived bounds are not reliable. We now wish to synthesize our parameter space constraints with EFT-validity limits and with constraints from colliders. We will first reinterpret our results given in Fig. 2.12 in the plane of DM mass vs operator suppression scale for several specific combinations of operator coefficients κ1 κ2 and κ3. First, we will assume κ3 = 0, that is there are no annihilation of 3 3 DM into gluons. We now have a three dimensional parameter space of mχ, k1/Λ , k2/Λ . Our exclusions in this parameter space from Fermi-LAT data are shown in Fig. 2.13 for certain simple ratios of couplings κ1 and κ2. The excluded regions of parameter space lay under the colored lines. The dot and dashed lines in Fig. 2.13 correspond to curves of constant total annihilation rate. We recall that for the operators in Eq. 2.4 σv m4 /Λ6 and thus upper bounds on h i ∝ χ annihilation cross section or DM mass correspond to lower bounds on the suppression scale Λ. Below the solid purple line is the region of breakdown of EFT validity for the indirect annihilation process for the U(1) operator. Below the solid brown line is the region of breakdown of EFT validity for the indirect annihilation process for the SU(2) operator. Vector boson portal models like those in this section have been widely studied in the context of collider production. At the LHC, operators of this type will lead to the production of DM particle pairs in association with a single vector boson, pp χχ + V where V is a → 40 κ1=0,κ 2=1

GeV Λ( ) κ1=1,κ 2=0 1000 κ1=1,κ 2=1(Zγ channel off)

800 2 2 κ1=1,κ 2=-sw /cw κ1 (γγ channel off)

max EFT validityκ 1=max,κ 2=0 600 max EFT validityκ 1=0,κ 2=max

400 0.1<σv> <σv> 200 5<σv>

0 mχ(GeV) 0 200 400 600 800 10<σv> 100<σv>

Figure 2.13: Constraints from Fig. 2.12 reconfigured for the effective cut-off vs DM mass plane. The regions below the solid lines are excluded. The dashed lines correspond to curves of constant σv tot h i

gluon, photon, W or Z. No monoboson signal is currently in excess and therefore collider

constraints place lower bounds on the effective operator cut-offs Λi for each given DM mass. Studies on monoboson collider signatures for the pseudo-scalar DM current may be found, for example, in references [66] and [65]. In the plot below we consider the effects of the SU(2) operator only for a simple comparison of collider and indirect detection bounds. We show our exclusion contour from combined dwarf data, along with the ATLAS constrain from the mono-W channel (given by [66]) for the pseudo-scalar current operator. We have also plotted contours of constant total DM annihilation rate. We see that following lines of constant annihilation rate we arrive at a minimum DM mass bound. Following lines of constant operator coefficient we arrive at a maximum DM mass bound. The Fermi-LAT dwarf bounds are more constraining than collider analyses for DM masses above a few hundred GeV. It may be true that these bounds are more strict than collider bounds for regions of small Λ as well. The validity of the EFT prescription

for the collider bound requires that the momentum transfer √s < 2Mmess. The general procedure for collider bounds is to truncate the number of events, eliminating those where √s is larger than 2 times the maximum messenger mass [50]. Previous collider analyses for models with D11 type gluon operators estimate that the region of maximum EFT validity limits interpretation for cut-off scales below 350 GeV. For these low effective cut-off scales, event rates are severely truncated [67]. In our analysis for the SU(2) and U(1) operators, cut-off scales of around 500 GeV, imply messenger masses at or under roughly 1 TeV. We generated a sample of mono-W events for the process pp W χχ at 8 TeV c.o.m. energy → 41 Limits onΛ withκ 1=0,κ 2=1 Λ(GeV) 1000 Collider mono-W channel Combined dwarf limit 800 Max EFT validityκ 1=0,κ 2=max

600 0.1<σv> <σv> 400 5<σv>

200 10<σv> 100<σv>

0 mχ(GeV) 0 200 400 600 800

Figure 2.14: Constraints from Fig. 2.12 reconfigured for the effective cut-off vs DM mass plane

using in MADGRAPH [68]. Barring any other cuts, we found that less than 10 percent of the events pass the above truncation requirement for an eﬀective cut-oﬀ of 500 GeV and DM masses of 100 GeV. We do not know how far under 500 GeV the collider limits hold for these operators and this is a matter for further investigation.

2.5 Conclusions

We have given indirect detection constraints derived from Fermi-LATs dwarf spheroidal galaxy data on variety of models with multiple final state annihilations of DM. These models included DM portals from third generation fermions, SM vector bosons, and popular effective field theory models formally analyzed in collider physics. We have presented constraints in model space on surfaces of fixed total annihilation rate, and in planes of freely varying effective operator coefficients. In planes of fixed annihilation rate our bounds take the form of lower mass limits for DM mass/effective operator cut-off. Our mass bounds are in the 10-100s of GeV range, and we have presented these results using a visualization on triangles and ellipsoidal sections. We have also compared our constraints to those obtained by collider production for vector boson portal models. In general we find that dwarf constraints overtake collider constraints for DM masses greater than a few hundred GeV, and in regions of low effective cut-off where there is theory break down for collider applications. While existing results indicate that EFT validity fails under a few hundred GeV in collider studies, the exact range of maximum validity in vector boson portal models is yet to be determined. This is a course for future work. Dwarf constraints used are among the most limiting constraints for DM models, and therefore present the best possibility of ruling out regions of DM parameter space. We

42 reiterate that a more sophisticated combination of dwarf data, which doesn’t assume annihilation into only one channel would provide much tighter constraints on our parameter space, and this is another avenue for further work. We also give another caveat here, we have implicitly assumed the most uncontroversial DM profiles for dwarf galaxies. Assump- tions of different DM profiles will lead to wiggle room in our constraints and require further study.

43 Chapter 3 Indirect Detection Constraints on s- and t-Channel Simplified Models of Dark Matter

This Chapter is based on work published in Phys. Rev. D94 (2016) no.5, 055027

3.1 Introduction

Multiple astrophysical observations provide extremely strong evidence indicating the existence of a dark matter (DM) particle. However, these observations give little indication of how (or whether) DM may interact non-gravitationally with the Standard Model (SM), and vast regions of parameter space exist that may successfully explain interactions between DM and the SM. These interactions are currently probed by three classes of experiments: those that directly probe the scattering of DM with SM nuclei, those studying the production of DM in colliders, and those indirectly studying the annihilation of DM into stable SM final states. Any given model of DM may produce signals in more than one class of experiment, and observations and correlations between these experiments will reveal significant insight into the parameter space of viable DM models. A significant fraction of UV complete DM models demand that DM annihilate into multiple sets of SM particles. Fixed relations between various coupling constants in the theory may follow from symmetry considerations, such as gauge invariance or supersymmetric relations. In the context of indirect detection signatures for DM, we may expect to see a γ-ray spectrum formed from the composition of several annihilation final states - a scenario which typically results in a relatively smooth γ-ray spectrum. In this chapter, we will focus our analysis on a set of models where DM annihilates to SM fermions in the final state. These models have become popular to explain spectral features in the Galactic Center [69] and are popular portals in collider searches for DM [70]. To make our constraints generic,

44 we do not present exclusions assuming a 100% annihilation rate into a single SM channel, but consider composite DM annihilation spectra where the total annihilation rate among all fermionic channels is held constant. We compare the resulting DM annihilation spectrum and intensity against the stacked population of dwarf spheroidal galaxies [28], utilizing a joint-likelihood analysis to determine the minimum allowable DM mass that can annihilate at the thermal annihilation cross section while remaining consistent with the gamma-ray data. One well studied method of comparing or translating bounds from different search strategies is through the use of effective field theories (EFTs) [2, 44, 47]. In EFT models, one assumes that the only accessible new degree of freedom is the DM itself and that any particles mediating the DM-SM interaction can be consistently integrated out, leaving behind a set of operators representing effective interactions. This offers a model independent method of analyzing and comparing results from various DM searches while still capturing most of the kinematic features of the process being explored. Various EFT scenarios have been used to probe collider scenarios with DM couplings to light quarks and gluons [67], Higgs bosons [60, 71], and electroweak gauge bosons [60, 61, 63, 65, 66]. However, the limitations of EFTs are becoming apparent. The predictive power of EFTs becomes questionable once the effective cutoff and the mass of the mediating particles becomes smaller than the center of mass energy of the process [48–51]. Fortunately, in the case of indirect detection, the center of mass energy is generally much lower than in the case of collider production of DM. We therefore expect the EFT analysis to be valid so long as the effective cutoff scale exceeds twice the DM mass. In cases where the EFT model fails, one can turn to simplified models where a new sector is added to mediate interactions between the SM and DM. The particle content of the least complex simplified models consists of adding in a single mediating particle that communicates between DM and the SM at tree level. We evaluate the constraints provided by the Fermi-LAT dwarf analysis on a group of simplified models including s-channel annihilation through a heavy vector boson as well as t-channel annihilation through a scaler mediator. In each case, we show that the simplified models match the EFT predictions for DM annihilations in the heavy mediator limit. This provides a basic test indicating where the EFT breaks down in DM annihilation processes. The format of this paper is as follows. In Section II we will employ current observations of the γ-ray flux from the population of dwarf spheroidal galaxies and describe the joint- likelihood algorithm used to constrain the DM annihilation parameter space. In Section III, we produce generic lower limits on the DM mass in models where the DM is allowed to annihilate to multiple SM fermion final states. These mass bounds are then interpreted as effective coupling bounds in Section IV, assuming an effective model. In Section V we will present parameter space exclusions on models mediated by both vector and scalar particles,

45 and compare these limits against EFT models with contact interactions. In Section VI we will discuss the implications of our analysis and conclude.

3.2 Indirect Detection from Dwarf Spheroidal Galaxies

The dwarf spheroidal galaxies (dSphs) observed in the Milky Way provide some of the tightest constraints on the γ-ray flux from DM annihilation due to a combination of their: (i) high DM densities, (ii) relative proximity, and (iii) insignificant astrophysical γ-ray backgrounds [53, 54]. Observations by both the Fermi-LAT collaboration and several sets of external authors [3, 28, 72–77] have examined the population of dSphs and have found no convincing (5σ) evidence for a γ-ray excess coincident with the population of these sources. Using this information, these groups have set strong limits on the annihilation rate of DM as a function of its mass and final interaction states. The photon flux (photons cm−2 s−1) at Earth expected from DM annihilations in the region of interest is given by:

Z Emax 1 X σv f dNγ Φγ = h i2 dEγJ, (3.0) 4π 2m dEγ f χ Emin f where the J-factor (GeV2cm−5) is the line of sight integral of the DM density ρ, integrated over a solid angle: ∆Ω, Z Z J = ρ2(r)dldΩ0. (3.0) ∆Ω l.o.s The additional terms in Eq. 3.2 are dependent on the particle properties of the DM. Here, dNf /dE is the differential photon energy spectrum per annihilation to final state f, σv f is h i the thermally averaged DM annihilation cross section and mχ is the DM mass. Throughout this chapter, we calculate the expected σv f for each kinematically accessible annihilation h i channel, and obtain the differential gamma ray spectra for annihilation channels, both from the Mathematica code PPPC 4 DM ID [55, 56] and from DarkSUSY [78]. These codes generate the prompt spectrum to photons from various 2-body final states and subsequent decays and are normalized per one annihilation. For a given 2-body final state, the exact nature of the underlying interaction between DM and the SM is unnecessary as the spectra will be determined by kinematics and SM interactions, though this fails to hold for higher- body decays as the underlying interaction will influence the kinematics. In addition to the prompt spectrum of photons, there are also contributions to the photon spectrum from inverse Compton scattering and synchrotron radiation from propagating charged particles.

In particular for high mass DM (mχ > 100 GeV) with electrons in final state, photons from inverse Compton scattering can be∼ significant effects within the Fermi-LAT energy range. Since the effects of inverse Compton are ignored, the limits presented here should be considered as conservative for high mass DM annihilations into leptonic final states. In 46 Figure 3.1 we show the resulting γ-ray spectrum for several choices of relevant SM pair annihilation channels assuming a 100 GeV DM particle. Here, the dimensionless parameter

x = Eγ/mχ is the gamma ray energy scaled by the DM mass.

Γ spectrum, mDM=100 GeV dN dlogx ΜΜ 100 ΤΤ bb 10 ΓΓ

1 WW ZΓ Γ

0.1 ZΓ Z H L H L

-4 x 10 0.001 0.01 0.1 1

Figure 3.1: Gamma ray spectrum from DM annihilation into various channels for a 100 GeV DM particle.

3.2.1 γ-ray Analyses of the Population of Dwarf Spheroidal Galaxies

Most notably for this analysis, the Fermi-LAT collaboration has recently produced limits utilizing 6 years of Fermi-LAT data processed through the updated Pass 8 event reconstruc- tion algorithm [3]. In addition to performing an analysis of 25 individual dSphs, this analysis (like several before it) produces a joint-likelihood (or “stacked”) analysis of 15 dSphs with high confidence J-factors and background models. The analysis proceeds in the following way. The Fermi-LAT data is divided into 24 energy bins. For each energy bin and at the location of each dSph, a source with an extended γ-ray emission profile fit to observations is placed and the log-likelihood fit, LG( ), of this source is computed as a function of the L source normalization in the energy bin. The ∆LG( ) of this fit is computed by comparing L this model against a background model (with no dSph) to the γ-ray data. The Fermi-LAT team has publicly released these likelihood profiles for each dwarf 1. Noting that the WIMP paradigm mandates that the DM cross section and spectrum be equivalent in all dSphs, the joint-likelihood analysis for a given DM model is computed as follows. For each dwarf the flux in each energy bin is computed by multiplying the expected

1https://www-glast.stanford.edu/pub data/1048/ 47 DM flux by the astrophysical J-factor in that dwarf galaxy, and then the corresponding ∆LG( ) is calculated, compared to the null hypothesis that DM does not annihilate. Since L the J-factors of individual dwarf galaxies are highly uncertain, the astrophysical J-factor is allowed to shift from its measured value (Jmeas) to its best fit value (Jbf ), incurring a 2 2 log-likelihood cost of ∆LG( ) = (Jbf -Jmeas) / (2σ ). The 95% upper limits computed by L J the Fermi-LAT collaboration are set when the total ∆LG( ) for all dwarf galaxies exceeds L the model with no DM annihilation by 2.71/2. At present the Fermi-LAT collaboration has produced constraints on the DM annihilation cross section for DM models that annihilate directly into a few choice final states (e+e−, µ+µ−, τ +τ −, uu¯, b¯b,W+W−). Since many additional final states (or combinations of these final states) are well motivated, we utilize the Fermi-LAT likelihood profiles for each dwarf galaxy to produce a joint-likelihood analysis which can be analyzed for any γ-ray spectral shape. Specifically, we utilize the individual likelihood profiles calculated by the Fermi-LAT team for each individual dSph, and then produce an independent likelihood-minimization algorithm to compute ∆LG( ) for a joint-likelihood analysis of the entire dwarf galaxy L population. We utilize the best-fit J-factors and uncertainties calculated by [79]. We note that this J-factor estimation is identical to that employed by the Fermi-LAT team, allowing for a direct comparison between our results and those of [3]. As a check of our analysis, we compare our upper bounds with those of [3] assuming DM annihilation dominated by a single final state channel. As seen in Figure 3.2, we obtain results consistent with those obtained by the Fermi-LAT collaboration in the cases of annihilations directly to b¯b and τ +τ −. While we find a small (<10%) offset in the b¯b channel, we note that this is smaller than the deviation stemming from the usage of different particle physics models to characterize the γ-ray flux from the annihilation to a given DM final state. In Figure 3.3 we demonstrate how the joint-likelihood algorithm is employed to set limits on the annihilation cross section for a given DM mass and annihilation final state. For illustrative purposes, we first (top left) show an intermediate state of the analysis (not directly computed in the analysis routine), where no DM spectral information is added to the fitting algorithm, and the J-factor of each dwarf galaxy is allowed to float independently in each energy bin. This provides an upper limit on the particle physics flux (E2 dN σv ) dE h i from DM annihilation in each energy bin. In this case, the limits are relatively weak, as we are only integrating information from a small energy slice of the Fermi-LAT data. Second (bottom left), we integrate a particle physics model (in this case 100 GeV DM particles annihilating to b¯b final states) and restrict the J-factors in each energy bin to be equivalent. The addition of this particle physics model effectively reweights each energy bin based on its contribution to the total combined limit. For example, energy bins exceeding 50 GeV become unimportant as the 100 GeV DM particle produces only a negligible γ-ray flux above this energy. Some bins show small (non statistically significant excesses) which

48 signiﬁcantly weakens their contribution to the cross section limit. Finally (right), we show the combined limit on this model, obtained by summing the joint-likelihood analysis in each energy bin shown from Stage 2. In this case, we ﬁnd a total upper limit (∆LG( ) = 2.71/2) L of approximately σv = 2 10−26 cm3s−1. h i ×

χχ → �� <σ�> (��-�)

2.× 10 -25

1.5× 10 -25 Current Analysis Fermi-LAT 1.× 10 -25

5.× 10 -26

�χ(��) 200 400 600 800 χχ → ττ <σ�> (��-�)

7.× 10 -25

6.× 10 -25

5.× 10 -25 Current Analysis

4.× 10 -25 Fermi-LAT

3.× 10 -25

2.× 10 -25

1.× 10 -25 �χ(��) 200 400 600 800

Figure 3.2: A comparison of the annihilation cross section upper limits obtained using this work and the spectrum calculated by PPPC compared to the Fermi-LAT collaboration analysis [3] for the case of “standard” DM annihilations into b¯b (top) and τ +τ − (bottom) ﬁnal states as a function of the DM mass and assuming a 95% conﬁdence upper limit set at ∆LG( ) = 2.71/2. L

49 ) 1 −

s 25 No Spectral Fitting

3 10− 4.0

26 10− Combined (GeV cm 27 26 > 10− 8x10 v − 3.0

<σ 28 10−

) 26 1 5x10− 29 − s dN/dE 10− ) 3

2 0.5 5 50 500 L E Energy (GeV) 26

(cm 2.0 3x10− >

24 Fit to DM Spectrum ∆LG( 10− v 26 )

1 2x10− χχ->bb, χ=100 GeV <σ

− 25

s 3x10− 3 1.0 25

(cm 10− 26

> 10−

v 26 3x10− <σ 10 26 0 − 0.5 5 50 500 Energy (GeV)

Figure 3.3: Stages in the analysis of the joint-likelihood limits on the annihilation of particle DM where the color corresponds to the difference in the log-likelihood value from the minimum (best fit) value. First, the limit is shown in an intermediate state, as a joint-likelihood analysis calculated independently in each spectral bin,with J-factors for each dwarf that are alllowed to float independently in each bin (top left). Second, a DM mass and annihilation pathway is fixed (100 GeV DM annihilating to a b¯b final state) and the J-factor for each dwarf is forced to be consistent over each spectral bin (bottom left). Finally, the combined limits are calculated by summing the LG( ) scan in each energy bin to produce a limit on L the DM cross section (right).

3.3 Generic DM Mass Bounds

−2 Noting that the DM annihilation rate scales as mχ , we see from Eq. 3.2 that an upper bound on the gamma-ray flux from DM annihilation can be translated into a lower bound on DM mass, under the assumption of a fixed total DM annihilation rate. Here we present lower mass bounds on DM annihilating into multiple final state channels. This analysis will proceed following the method presented in Ref. [80]. We will assume that DM annihilates into i final state channels, with partial annihilation rates denoted σv i. Specifically, we h i express the total DM annihilation cross section as a simple sum of partial rates:

σv tot = N σv Th = σv 1 + σv 2 + . (3.0) h i h i h i h i ··· We have expressed the total DM annihilation rate as some numerical factor times the thermal annihilation cross section, N σv Th. The ﬁrst step in this analysis will be to ﬁx h i the total DM annihilation rate to some desired number. While the total annihilation rate

50 may take any value, the least controversial choice is the thermal rate (N = 1), though we note that annihilation rates above or below this level may still be allowable in certain particle physics models, or within certain non-thermal evolution histories for the universe. One caveat to this assumption is that the total annihilation cross section to visible channels may be sub-thermal, with an additional annihilation component to invisible ﬁnal states that “bleeds” oﬀ part of the total annihilation rate.

Deﬁning the partial rates into each channel as in [80], Ri = σv O / σv tot, we may h i i h i divide out by the total rate to get a single constraint

1 = ΣRi = R1 + R2 + R3... (3.0)

We consider the parameter space of this general analysis to consist simply of the i specific partial DM annihilation rates into each channel. By fixing the value of the total annihilation rate we now have i independent partial annihilation rates and one constraint equation, Eq. 3.3, reducing our parameter space to i 1 parameters. For three, four, or more annihilation − channels we we may then visualize the parameter space as a triangle, tetrahedron, and N-plex as described in [52, 80]. We will now consider a generic scenario where we allow DM to annihilate into multiple final state fermions. Here we will limit ourselves to three final state annihilation channels, bb, ττ, and neutrinos. The annihilation into final state neutrinos may be thought of as a generic stand-in for any invisible annihilation channel, which would allow the total DM annihilation rate into visible channels to fall below the thermal annihilation cross section. The parameter space is three dimensional. However, once the total annihilation rate is set, we obtain a parameter space that may be represented in two dimensions. Specifically, we can visualize the parameter space by constructing a triangle as in Fig 3.4. The points on this triangle specify various admixtures of the partial rates that satisfy the constraint equation above. The vertices of the triangle represent an annihilation rate which is saturated by a single channel. Each outer edge of the triangle corresponds to an annihilation rate which is saturated by only two channels. Each point on the triangle specifies a different admixture of partial annihilation rates into the τ +τ −, b¯b and invisible channels. Thus, each point specifies a unique γ-ray spectrum. Having fixed the total DM annihilation rate, we use our our joint-likelihood analysis from the previous sections to place lower limits on the DM mass, as shown in Figure 3.4. The color contours plotted on the triangle correspond to lower mass bounds on DM at each particular point. It is clear that the strongest bounds stem from models in which DM annihilates mostly into b-quarks while the weakest bounds result when the invisible channel takes up a significant fraction of the total DM annihilation cross section. In fact, there is no lower bound when the model allows for greater than 90% annihilation into invisibles. We note that in the region where annihilation proceeds only into b’s and τ’s, the lower bound on

51 the DM masses ranges from 90-150 GeV.

Figure 3.4: The lower bound on the DM mass (GeV) for annihilations into b¯b, τ +τ −, and invisible particles for σv tot = σv Th. In the upper ﬁgure, the photon spectrum is h i h i generated with PPPC, while in the lower ﬁgure, the photon spectrum is generated with DarkSUSY.

It should be stressed that the constraints presented in Fig. 3.4 are extremely general.

52 We assume only that the the total DM annihilation cross section is set at the thermal rate, and that the DM annihilation cross section is dominated by annihilations into b¯b, τ +τ −, and invisible particles. One may then translate these DM mass bounds, in a less generic way, into various bounds on parameter space of DM models.

3.4 EFT completions

In order to remain agnostic about the high energy details of the DM model, we may describe the interaction between DM and SM fermions as a set of eﬀective operators. This allows for the translation of our lower bounds on the DM mass to bounds on the eﬀective coupling of DM to SM particles. Here, we choose to consider DM annihilating to b¯b, τ +τ − and invisible particles as above. We write a set of dimension 6 operators which couples DM to SM fermions:

1 ¯ 1 1 f = 2 χ¯ΓχbΓb + 2 χ¯Γχτ¯Γτ + 2 χ¯Γχν¯Γν, (3.0) L Λb Λτ Λν where Λ is the effective cut-off scale for each operator and Γ specifies if the fermionic currents are scalar, pseudo-scalar, vector, axial vector and so on. Once the nature of the current is specified, the above model has 4 parameters. There are three effective cut-off scales that specify the DM coupling to each type of fermion, and the fourth parameter is the DM mass mχ. Each operator in the above equation will contribute to one and only one final state annihilation channel. The total annihilation rate is factorized as a simple sum of the partial rates, which depends on the effective operator couplings with a form proportional 22 22 22 to: a 1/Λb + b 1/Λτ + c 1/Λν . Here we choose to analyze a model with a vector current,

κf µ f = 2 χγ χfγµf. (3.0) L Λf With a Lorentz structure specified, we can now translate our generic bounds from the last section into bounds on the effective operator couplings, depicting these limits on an identical triangle morphology. Again, we fix the total DM annihilation rate to the thermal rate. At each point on the triangle, we utilize the lower bounds on the DM mass calculated above, which in each case corresponds to a specific value for the effective opperator coefficients. In

Figs. 3.5 and 3.6 we plot lower bounds on the Λb and Λτ eﬀective operator cut-oﬀs using γ-ray spectra generated by both PPPC [55] and DarkSUSY [78].

53 Figure 3.5: The lower bound on the effective coupling (GeV) of DM to b-quarks for anni- + − hilations into b¯b, τ τ , and invisible particles for σv tot = σv Th. In the upper figure, h i h i the photon spectrum is generated with PPPC. In the lower figure, the photon spectrum is generated with DarkSUSY.

Importantly, we note that at each point in the parameter space the lower limit on the

operator cutoff is significantly larger than our threshold of twice the DM mass (2mχ). We thus expect that the use of effective operators in this analysis is justified since the effective cutoff significantly exceeds the center of mass energy of the annihilation process.

54 Figure 3.6: The lower bound on the effective coupling (GeV) of DM to τ-leptons for an- + − nihilations into b¯b, τ τ , and invisible particles for σv tot = σv Th. In the upper figure, h i h i the photon spectrum is generated with PPPC. In the lower figure, the photon spectrum is generated with DarkSUSY.

3.5 Simpliﬁed Models

We now discuss UV completions for a simple scenario where DM annihilates to SM fermions. At tree level, the simplest means for DM to talk to the SM is via two basic types of

55 interactions: an s-channel vector mediator, or a t-channel scalar mediator (see Fig. 3.7) [51, 81]. We show Feynmann diagrams for these simple scenarios below. In our discussion, we will focus on s-channel vector mediator and t-channel scalar mediator models where DM annihilates to fermionic final states. The case of an s-channel scalar mediator is interesting but more complicated, as the s-channel mediator must mix with the Higgs in order to remain consistent with electroweak symmetry. In this case, the annihilation to final state fermions maybe suppressed in favor of annihilation into Higgses and the couplings to SM fermions will no longer be variable. The annihilation rate for simplified models is completely determined by the model parameters (i.e. the mediator couplings and masses), thus a bound on the annihilation rate directly gives constraints on the model parameters.

χ f χ f

Z' S

- - χ f χ f

χ f

~ Q - χ f

Figure 3.7: Diagrams for DM annihilation in simpliﬁed models with a vector mediator (upper left), scalar mediator (upper right), and a t-channel mediator (lower).

3.5.1 Vector mediator

A vector mediated model can be produced in extensions of the SM gauge groups by an extra U(1) that is spontaneously broken in the complete UV theory to obtain a mass for the mediator. We will assume that a UV completion of this model exists that is gauge invariant under new symmetries, anomaly free, and able to generate mass for the mediator. The Lagrangian describing interactions between fermionic DM (χ), the vector mediator

(Vµ), and the SM fermions (fi) is [51]:

µ V A 5 X µ V A 5 = Vµχγ¯ (g g γ )χ + Vµfγ¯ (g g γ )f, (3.0) L χ − χ f − f f

56 V A where the vector (gf ) and axial-vector couplings (gf ) are assumed to be ﬂavour diagonal. Using the narrow width approximation, the thermally averaged annihilation cross section to fermionic ﬁnal states takes on the form:

f 2 2 !1/2 Nc mχ mf σv (χχ¯ V ff¯) = 1 h i → → 2π[(M 2 4m2 )2 + Γ2 M 2 ] − m2 V − χ V V χ  2 " m2 ! m2 !# m2 2 !  V 2 V 2 f A 2 f A 2 A 2 f 4mχ  gχ gf 2 + 2 + 2 gf 1 2 + gχ gf 2 1 2 , × | | | | mχ | | − mχ | | | | mχ − MV 

f f where Nc = 3 for quarks and Nc = 1 for leptons, MV is the mass of the vector mediator,

and the mf is the mass of the annihilation products. The total width of the vector mediator is:

2 1/2 2 MV X i 4mi V 2 A 2 mi V 2 A 2 ΓV = N 1 g + g + 2 g 4 g , (3.0) 12π c − M 2 | i | | i | M 2 | i | − | i | i V V where the sum is taken over all kinematically allowed decay modes. The term in the annihilation rate proportional to gA 2 gV 2 is p-wave suppressed and has been ignored | χ | | f | above. In order to ensure the validity of the narrow width approximation, one must require 2 Γv/MV < 0.25 which corresponds to couplings gi < 1. ∼ ∼ A In this analysis we will consider the case of pure vector couplings, gi = 0, to correspond with the EFT analysis in Section IV. The parameter space of the simplified model consists V of the DM mass mχ, the mediator mass MV , and the i mediator-fermion couplings gi . Using our joint-likelihood upper limits computed in Sections II and III, we place bounds on the parameter space of this model. In the treatment of this simplified model, unlike the EFT treatment above, we will allow the total DM annihilation rate to float. We will instead fix the DM mass and also specify the ratios of the partial rates; for example we may consider 100 percent annihilation into b¯b, or 33.3 percent annihilation rates into b¯b, τ +τ −, and invisible channels respectively. For each fixed DM mass and ratio of partial rates, we determine the maximal allowed value of the total annihilation rate that saturates the γ-ray flux bounds. This then allows us to compute the values of the vector mediator mass and vector couplings which correspond to the total annihilation rate bound. In Fig. 3.8 we plot exclusions in the plane of the couplings and mediator mass, for various values of DM mass given the specified admixture of partial annihilation rates. We plot exclusions for partial rates consisting of: 100% b-quark - 0% τ, 0% b-quark - 100%

57 τ, 30% b-quark - 70% τ, and 70% b-quark - 30% τ. The region above the curves are excluded. Unsurprisingly we see that parameter space is excluded when the couplings are large, and hence the partial annihilation rate is large. We note another kinematic feature,

the parameter space is most restricted for mediator masses MV 2mχ where the mediating ∼ vector particle is produced on resonance. We see that as the vector mediator is made much heavier than the DM mass, the total annihilation cross section drops and the parameter space is less constrained.

�� (χχ → ��) 0.6

0.5

0.4 mχ=50 GeV,<σv>=0.53<σv> Therm

mχ=150 GeV,<σv>=1.2<σv> Therm

� 0.3 � mχ=250 GeV,<σv>=1.7<σv> Therm

mχ=550 GeV,<σv>=3.7<σv> Therm 0.2 mχ=950 GeV,<σv>=6.7<σv> Therm

0.1

0.0 0 500 1000 1500 2000

��(��) �� (χχ → ττ) 0.6

0.5

0.4 mχ=50 GeV,<σv>=0.5<σv> Therm

mχ=150 GeV,<σv>=1.6<σv> Therm

τ 0.3 � mχ=250 GeV,<σv>=3.2<σv> Therm

mχ=550 GeV,<σv>=9.4<σv> Therm 0.2 mχ=950 GeV,<σv>=24.<σv> Therm

0.1

0.0 0 500 1000 1500 2000

��(��)

Figure 3.8: Upper bound on the allowed DM-mediator coupling gf as a function of mediator mass for DM annihilation to pure b’s (upper), pure τ’s (lower) for gχ = 1 in the vector mediator simpliﬁed model.

58 �� % �� % ττ 0.6

0.5

0.4 mχ=50 GeV,<σv>=0.56<σv> Therm

mχ=150 GeV,<σv>=1.4<σv> Therm

� 0.3 � mχ=250 GeV,<σv>=2.2<σv> Therm

mχ=550 GeV,<σv>=4.7<σv> Therm 0.2 mχ=950 GeV,<σv>=8.4<σv> Therm

0.1

0.0 0 500 1000 1500 2000

��(��) �� % �� % ττ 0.6

0.5

0.4 mχ=50 GeV,<σv>=0.56<σv> Therm

mχ=150 GeV,<σv>=1.6<σv> Therm

� 0.3 � mχ=250 GeV,<σv>=2.8<σv> Therm

mχ=550 GeV,<σv>=6.7<σv> Therm 0.2 mχ=950 GeV,<σv>=13.<σv> Therm

0.1

0.0 0 500 1000 1500 2000

��(��)

Figure 3.9: Upper bound on the allowed DM-mediator coupling to gb as a function of mediator mass for mixed case of 30% τ’s and 70% b’s (upper) and 70% τ’s and 30% b’s (lower) for gχ = 1 in the vector mediator simpliﬁed model.

In Fig.3.10 we compare the exclusion limits from a vector mediator simplified model and µ effective operator model with the vector current operator (¯χγ χfγµf). We have chosen two benchmark masses for the DM particle, one heavy (950 GeV) and one light (150 GeV) and additionally assume 100% annihilation into b quarks. The vertical line indicates where the effective cut-off of the EFT is less than 2mχ. The mapping from effective coupling to the UV parameters goes as: Λ mV /√gχgf . In general, we find that the effective theory over ∼ constrains the model. For light DM masses the EFT converges with the simplified model

59 exclusion for cutoﬀs above approximately 600 GeV. For heavier DM masses the EFT does not closely match the simpliﬁed model, and begins to converge only at extremely large coupling values and mediator masses. It is clear from Fig.3.10 (left), that EFT models with lower DM mass more accurately represent bounds on a more complete model.

�� (��)� � χ= �� (χχ → ��)

1.4

1.2

1.0

0.8

� Simplified Model � EFT 0.6

0.4

0.2

0.0 500 1000 1500 2000

��(��)

�� (��)� � χ= �� (χχ → ��)

3.0

2.5

2.0

� Simplified Model � 1.5 EFT

1.0

0.5

0.0 1000 2000 3000 4000 5000

��(��)

Figure 3.10: Upper bounds on DM-mediator coupling as a function of mediator mass for a µ vector mediator model compared with assumptions from eﬀective operator D5 (¯χγ χfγµf) with gχ = 1 and mχ = 150 GeV (upper) and 950 GeV (lower).

60 3.5.2 T-channel mediator

In t-channel completions of eﬀective operator models, DM exchanges a massive mediator in a t-channel scattering process to produce two SM particles. We take the DM particle to be a fermion, and the mediator to be a scalar. In this case, the mediator must carry SM quantum numbers; the annihilation to quarks will require a colored scalar mediator, while the annihilation to leptons will require a mediator with electroweak quantum numbers. Supersymmetry is the most well known theoretical structure in which to embed the model, but here we will not consider the details of a complete supersymmetric model. The interaction between DM and the right handed SM fermions that we will consider have the form:

X ∗ = giφ χP¯ Rfi + h.c., (3.0) L i i where the sum is over fermions, PR is a right projection operator, and φi are the mediators. One can also consider coupling to left-handed fields in a similar fashion. Unlike the vector simplified model above which required only a single vector boson as mediator, here we will require a different mediator for each flavor of Fermion. This will ensure that we do not run aground of flavor violating constraints. This feature mimics supersymmetric models. The annihilation cross section corresponding to a t-channel mediator as described in Eq. 3.5.2 and in the massless final-state limit is [51]:

N f g4m2 ¯ c i χ σv (¯χχ fifi) = 2 2 2 , (3.0) h i → 32π(Mi + mχ) where Mi are the the mediator masses, and Nc a color factor. We note that in the case of light mediators there will be non-trivial bounds on the mediator mass from pair production at colliders. The exact collider bounds will depend on the details of the mediator decay.

If Mi > mf + mχ, then the mediator will have a non trivial decay width to a SM fermion plus missing energy. If this condition is not satisfied the mediator must have additional SM couplings in order to avoid being absolutely stable. The parameter space consists of the DM mass, the i couplings gi and the i mediator masses Mi. Once these are specified one can calculate all of the specific partial annihilation rates. As before, the total annihilation rate factorizes into a simple sum of the partial annihilation rates, since the annihilation channels do not interfere; the existence of any one mediator contributes only to a single annihilation channel. We proceed in a similar fashion to the discussion of the vector mediator simplified model. We will fix the DM mass, and will specify the partial annihilation rate ratios into the final state fermion channels. We may then determine the values of gi and Mi that saturate the total photon-flux bounds. 61 �-�� (χχ → ��) 2.0

1.5

mχ=50 GeV,<σv>=0.53<σv> Therm

mχ=150 GeV,<σv>=1.2<σv> Therm

� 1.0 � mχ=250 GeV,<σv>=1.7<σv> Therm

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mχ=950 GeV,<σv>=6.7<σv> Therm 0.5

0.0 0 200 400 600 800 1000

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mχ=550 GeV,<σv>=9.4<σv> Therm

mχ=950 GeV,<σv>=24.<σv> Therm 0.5

0.0 0 200 400 600 800 1000

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Figure 3.11: Upper bound on the allowed DM coupling to ﬁnal states bb (upper) and ττ (lower) in the t-channel mediator simpliﬁed model.

62 �-�� % �� % ττ

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mχ=950 GeV,<σv>=13.<σv> Therm 0.4

0.2

0.0 100 200 300 400 500

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Figure 3.12: Upper bound on the allowed DM coupling vs. mediator mass for a t-channel simpliﬁed model where DM annihilates to 70% τ’s and 30% b’s. The upper plot has the bounds on the parameters related to the b interaction with DM and the lower plot is the same for τ interactions.

63 �-�� % �� % ττ

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0.6 mχ=550 GeV,<σv>=4.7<σv> Therm

mχ=950 GeV,<σv>=8.4<σv> Therm 0.4

0.2

0.0 100 200 300 400 500

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Figure 3.13: Upper bound on the allowed DM coupling vs. mediator mass for a t-channel simpliﬁed model where DM annihilates to 30% τ’s and 70% b’s. The upper plot has the bounds on the parameters related to the b interaction with DM and the lower plot is the same for τ interactions.

64 �-�� (χχ →�-�� )

1.4

1.2

1.0 mχ=50 GeV,<σv>=0.15<σv> Therm

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0.6 mχ=550 GeV,<σv>=1.3<σv> Therm

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mχ=950 GeV,<σv>=17.<σv> Therm 0.4

0.2

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Figure 3.14: Upper bound on the allowed DM coupling to ﬁnal states of down-type quarks (upper) and leptons (lower) in the t-channel mediator simpliﬁed model.

In Figs. 3.11- 3.14 we plot bounds on the DM model parameter space. For each curve we have fixed the DM mass, and the bounds are plotted for five different DM masses,

mχ = (50, 150, 250, 550, 950) GeV. We also consider various scenarios for the admixture of partial annihilation rates. In Fig. 3.11 we consider the scenarios were DM annihilated 100% into b quarks and 100% τ’s. In Figs. 3.12 and 3.13 we consider annihilations to 30% b’s - 70% τ’s and 70% b’s - 30% τ’s. In Fig. 3.14 we consider a more ﬂavor democratic scenario

65 where we consider annihilation into 30% down-type quarks - 70%charged leptons and 70% down-type quarks - 30% charged leptons respectively. We plot exclusions in the coupling vs mediator mass plane. The region above the curves is excluded. As in the previous simpliﬁed model analysis, we see that regions with large couplings have large annihilation cross section and are thus excluded. We also note that when the DM mass is light, the DM annihilation cross section scales like m2 /M 4, therefore ∼ χ i the total photon ﬂux drops sharply with the fourth power of the mediator. Therefore we see that when DM masses are light, our limits considerably weaken as the mediator mass increases.

3.6 Conclusions

By utilizing data from the Fermi-LAT Collaboration’s joint-likelihood analysis of the dwarf spheroidal galaxies of the Milky Way [3], we have produced stringent, and generic, limits on the DM annihilation cross section in models where DM annihilates to multiple final state channels. Specifically, we have produced new joint-likelihood analyses of 15 dSphs with relatively well-measured J-factors and calculated the lower limits on the DM mass for models which annihilate at the thermal cross section to arbitrary admixtures of b¯b, τ +τ − and invisible final states. We note that the tools presented here could be generalized to produce limits on the DM annihilation rate assuming any DM mass and γ-ray spectrum. We have analyzed the resulting limits in two ways. First, we have presented the constraints as lower limits on the cut-off scales in an EFT approach. We find that this approach is justified as the lower-limits on the cut-off scales typically significantly exceeds the center of mass energy for the DM annihilation event. We have additionally explored simplified models as completions of the EFT scenario. Specifically, we have considered a vector mediated s-channel completion as well as a scaler-mediated t-channel completion. We have presented constraints for each model in the parameter space of the mediator mass vs. the mediator-SM coupling, and have considered flavor restricted scenarios where DM couples only to b-quarks and τ’s, as well as a flavor democratic annihilation scenario. In order to test the validity of the effective operator paradigm we compared the results from our EFT scenario against those derived for the vector mediated simplified models. We find that for

mediator masses above a few times 2mχ the results matched fairly well. However the EFT severely over-constrained the cut-oﬀs for models close to the resonance production limit. There are signiﬁcant extensions which could be followed along these lines. We note that the tools presented here could be generalized to produce limits on the DM annihilation rate assuming any DM mass and γ-ray spectrum. Of particular interest would be models where DM annihilates into electroweak gauge bosons and photons. These models are interesting because they generally require a loop level process in order to couple DM to the SM. Another

66 very fruitful avenue would be to interpret our stacked constraints in the supersymmetric paradigm of the PMSSM.

67 Chapter 4 Supersoft SUSY Models and the 750 GeV Diphoton Excess, Beyond Effective Operators

This Chapter is based on work published in Phys. Rev. D94 (2016) no.1, 015016

4.1 Introduction

Both the CMS and ATLAS collaborations have reported an excess in the diphoton resonance channel in the first stages of the LHC’s 13 TeV run. This excess seems to correspond to a new boson with mass of approximately 750 GeV. ATLAS reports a diphoton resonance with mass 747 GeV at 3.6σ local significance while CMS reports a diphoton resonance of mass 760 GeV and a local significance of 2.6σ [82, 83]. If the excess persists, it would be a smoking gun for a new sector beyond the SM. Many beyond the SM (BSM) scenarios might accommodate such a resonance including models with exotic axion-like states, models with strong couplings, extra dimensions, heavy Higgses and more [84–104]. The production process of the new state may be variable, the diphoton resonance may be produced in association with other states, or alone [82]. Thus, production may be through gluon or quark fusion to a single new resonance, vector boson fusion, or production in association with other states. The simplest assumption for a BSM particle candidate which decays to two photons is a scalar field that couples to SM gauge bosons through a dimension 5 operator. This would indicate the existence of some heavy “messenger” particles with SM charges that couple both to SM gauge bosons and to the singlet field responsible for the possible excess. This mechanism is, in principle, exactly the same as the SM Higgs coupling to photon and gluon pairs. One guess, then, for the identity of the possible 750 GeV state is simply a heavy Higgs as would appear in a Type II two Higgs doublet model like the Minimal Supersym-

68 metric Standard Model (MSSM). This possibility would be quite exciting as supersymmetry (SUSY) is the leading candidate for BSM physics. However, it has been pointed out that a heavy Higgs in the minimal MSSM scenario fails to reproduce the observed rate of the excess. The MSSM would have to be extended by adding multiple sets of vector like chiral superfields to enhance the signal [86]. There are, however, alternate SUSY scenarios which would fit the excess. We propose that a well studied supersymmetric scenario contains all of the necessary pieces to fit the excess, that of an R-symmetric MSSM. In these scenarios gauginos are Dirac particles, rather than Majorana [43, 105]. The Dirac gauginos gets its mass by “marrying” the fermionic component of chiral superfield which is an adjoint under SM gauge groups. The superpartners of these new fermion fields are complex scalar particles in the adjoint representation of their respective gauge symmetry. Thus, there exists scalars and pseudoscalars which are a color octet, a SU(2) triplet, and SM singlet fields. The bino superpartner offers a scalar and pseudoscalar SM singlet candidate, and in this chapter we explore the possibility that the 750 GeV resonance is the real scalar part of the SM singlet superfield whose fermionic component marries the bino. In keeping with R-symmetric parlance we can refer to this as the sbino. The sbino fields may couple to SM gauge bosons in a variety of ways. First we may simply consider a set of general effective operators which couple scalar fields to SM field strength tensors. These operators are consistent with all symmetries of the theory, and extremely similar to operators which produce the Dirac gaugino masses themselves. We may complete these operators through loops in which the scalar fields couple to a heavy fermion or scalars which are charged under the SM gauge groups. However, these heavy fields need not be added to the model to explain the excess. In the case of fermion these may simply be the messenger fields already necessary to create Dirac gaugino masses. Even simpler, and the case we will study in this chapter, the messenger fields may be the superpartners of the fermions themselves which have Kahler potential couplings to the new chiral superfields. This means that Dirac gaugino models already contain the necessary fields, operators, and couplings to produce the diboson coupling needed to explain the LHC excess. We will explore a simplified version of a Dirac gaugino model calculating the loop level couplings of the sbino to pairs of gluons and photons. We calculate the gluon fusion production cross section and calculate relative decay rates into gluons, photons, and light Higgs fields of the sbino. We find that we can match observed resonant diphoton production rate at 13 TeV with a 750 GeV sbino, while obeying constraints from alternate decay channels, Higgs sector mixing and vacuum stability. This paper is organized as follows, in Section II we review the formalism of models with Dirac gauginos and introduce couplings of the sbino to pairs of dibosons as an effective operator. In section III we discuss the UV generation of the effective operator through

69 Kahler potential couplings between the sbino and the SM sparticles. In section IV we compute tree level decay widths for the sbino and discuss current collider constraints from alternate decay channels of the 750 GeV ressonance. In section V we compute the sbino production cross section through gluon fusion and decay rate to photons. In section VI we discuss constraints on the parameter space of Dirac gaugino models from the Higgs potential and vacuum stability. In section VII we conclude.

4.2 Operators in Dirac Gauginos Models

We consider a class of SUSY models in which gaugino mass terms are Dirac as opposed to Majorana [43, 105]. In such models, the gauginos get mass by “marrying” chiral fields which are adjoints under SM gauge groups. The Dirac mass is generated by the superpotential operator Z 0 α 0 2 WαW A D W = d θ λψA (4.0) Λ ⊃ Λ where W is the SM gauge field strength tensor, A is the chiral adjoint field, and W 0 is the field strength tensor of a hidden U(1) gauge group. The hidden sector U(1) is broken at a high scale and gets a nonzero D-term vev. After the symmetry breaking this operator, known as the supersoft operator, contains a Dirac mass for the gaugino with mass mD D/Λ. ≡ This SUSY breaking can be embedded in the framework of gauge mediation [106]. One may add to the theory a set of heavy messenger particles that are charged under the SM and the new U(1) gauge group. These messengers couple the SUSY breaking sector to the gauginos and SM adjoint fields, thus communicating the SUSY breaking to the visible sector. The new chiral multiplets, A, contain real and imaginary scalar degrees of freedom. One might assume that the real and imaginary fields have large mass of order mD. However, the masses of the real and imaginary parts of the multiplets may be quite split. These masses depend greatly on the details of the messenger sector of the model [107–109]. Since most models have large negative mass contributions for one combination of the fields, it is quite natural to assume that one of the adjoints is much lighter than the other, and may be thus accessible to colliders in the TeV range. Consistent with all symmetries of the theory, we may also write a set of operators extremely similar to those that yield the Dirac gaugino masses in Eq. 4.2. These operators couple the scalar adjoint fields to the square of SM field strength tensors. We will denote the new chiral fields as follows; O denotes the SU(3) adjoint, T the SU(2) adjoint, and S the SM singlet. We then have a complete set of gauge invariant operators

Z α α α WY αW S W2αW T W3αW O W = d2θ Y + 2 + 3 (4.0) Λ1 Λ2 Λ3

70 where WY is the hypercharge field strength, Wi is the appropriate SU(2) or SU(3) field strength tensor, and Λi is the appropriate cut-off scale which may be different for each operator. Integrating over the fermionic coordinates these operators become terms in the Lagrangian, 1 µν 1 µν 1 µν SrB Bµν + TrW Wµν + OrG Gµν (4.0) L ⊃ Λ1 Λ2 Λ3 These operators couple the real part (as indicated by the subscript) of the scalar states to the square of the SM field strength tensors. Similarly, we may expect to generate operators which couple the imaginary part of the scalar states to the SM field strength tensor and its dual. 1 µν 1 µν 1 µν = SiB B˜µν + TiW W˜ µν + OiG G˜µν (4.0) L Λ1 Λ2 Λ3 The full set of such operators were discussed in reference [110]. Interestingly, we may also write in the Lagrangian a gauge invariant term which couples pairs of gluons to the SU(2) adjoint, with Higgs fields inserted to ensure gauge invariance. This operator provides another interesting prospect for scalar states accessible by gluon fusion.

4.3 UV completions

We expect that the operators coupling the SM ﬁeld strength tensors to the chiral adjoints may be completed by considering integrating out loops of heavy “messengers”. One possibility is simply to consider loops of the heavy messengers which generate the Dirac gaugino masses themselves. These ﬁelds carry SM quantum numbers, the simplest messenger sector are those in which the messengers are fundamentals and antifundamentals under SU(5), see for example [108, 109]. Though messengers may couple to the chiral adjoint through large Yukawa like couplings, we expect that the messenger mass scale is very large, thereby suppressing the operator. Another possibility is to consider loops containing the scalar superpartners of the fermions. Dirac gaugino models have a unique Kahler potential coupling between the new chiral states and the SM sfermions generated by integrating out SM D-terms. The scalar potential involving the singlet scalar contains the terms

∗ † 1 V mD(S + S )DY + qgY DY f˜f˜ + DY DY (4.0) ∼ 2 where f˜ are the SM sfermions, q are the hypercharges of each sfermion, and DY is the U(1) hypercharge D-term. Once the SM D-term is integrated out, one ﬁnds a trilinear coupling between the real component of the scalar and each pair of sfermions with non-zero hypercharge,

∗ † V qgY mD(S + S )f˜f˜ . (4.0) ∼ 71 There is also a coupling to the Higgs fields which we further discuss below. We see that the real part of the scalar field will couple to pairs of SM gauge bosons through loops of sfermions. The couplings will be proportional to the hypercharge of each field and the square of the Dirac mass. The relevant diagrams can be seen in Fig. 4.1. It should be noted that couplings from superpotential terms induce additional loop contributions to the decay of the singlet. For example, in the µ/SSM [111] or the MRSSM [112] one would expect additional diagrams with charged Higgsinos running in the loops. However, in what follows we have chosen not to fully specify the Higgs potential and we expect that these additional contributions will not qualitatively change our results as the charged Higgsinos are expected to be heavy.

∼ ∼ q q ∼ SR q SR

∼ q q∼

∼ γ, Z, W f, H γ, Z, W ∼ f, H ∼ SR f, H SR

∼ γ ∼ f, H , Z, W f, H γ, Z, W

Figure 4.1: One loop diagrams contributing to singlet coupling to pairs of gluons and electroweak gauge bosons.

To compute each contribution to the diphoton diagram, we must add up all fields which have hypercharge. These fields include three generation of squarks and sleptons, QL, uR, dR and LL, eR plus the Higgs doublets, Hu and Hd. However, only the three generations of squarks contribute to the digluon channel. Thus we expect to complete our operators by calculating loops of the roughly TeV scale scalar “messengers”. We may control the ratio of the digluon channel to that of the of electroweak gauge boson channel by varying the mass scale of the squarks with respect to the sleptons. Making the squarks much heavier than the sleptons will reduce the digluon coupling while maintaining or enhancing the coupling to pairs of electroweak gauge bosons. In the limit of mass degenerate left and right handed states, the coupling of the singlet to dibosons will vanish due to a diagramatic cancelation. This is equivalent to the cancelation which occurs in the coupling of the sgluon (color 72 octet adjoint) to pairs of gluons as noted in reference [113, 114]. Thus, the strength of the effective operators coupling the sbino to gauge bosons may be dialed by changing the mass differences between left and right handed states. We also note that by varying the masses of left handed to right handed particles in general, we may control the ratio of the coupling of S to the U(1) and SU(2) gauge bosons. In this way we may set the scales of our operators in Eq. 4.2. In general models of Dirac gauginos have quite a flexible spectrum. In the simplest models, the mass ratio of the gauginos and sparticles differ by the square root of a loop factor. However the mass splittings between gauginos and scalars can in principle be arbitrary. In many completions of Dirac gaugino models, the physics which sets the scalar adjoint masses may also effect the SUSY spectrum; gauginos may have some mix of Dirac and Majorana masses, models may have extra R-symmetric gauge mediated contributions, for example see [43, 107]. In addition, SUSY models with Dirac gauginos are generally less constrained by LHC searches than other SUSY scenarios [115]. The sparticle spectrum and decay chains are quite different in Supersoft scenarios than in other SUSY models, and mass constraints on squarks and sleptons can be weaked considerably.

4.4 Tree Level Singlet Decays and Collider Bounds

From arguments above, we also note that the singlet state S couples to the Higgs ﬁelds. In particular we the real part of the singlet couples to both Hu and Hd through the hypercharge D-term. 1 † † V = gY mDSR(huhu hdhd ) (4.0) 2 − We thus expect that the singlet will have tree level decays into the light Higgs boson. For

TeV scale values of mD, the tree level width may only be suppressed at low values of tan

β. The trilinear coupling is proportional to gY mD TeV, and the width is by ∼ 2 q (gY mD) 2 2 2 2 2 Γh = (cos α sin α) 1 4mh/mS. (4.0) 16πmS − − where mS is the mass of the singlet, mD is the Dirac bino mass and α is the Higgs mixing angle. The tree level singlet decay to pairs of the heavy Higgses, will of course be suppressed if the mass of the heavy states is more than half of the singlet mass. The 750 GeV resonance may be consistent with a narrow width, or an intermediate size width of order 10’s of GeV. However, as we discuss below we expect that resonances with a larger width dominated by diHiggs decays to be incompatible with current collider constraints. Smaller resonance widths into light Higgses may be ﬁt with judicious choices of parameters, for example a sbino Dirac mass of 1.5 TeV and tanβ 1.1 yields a tree level width of 66 MeV. ∼ Other tree level decays may follow from additional Higgs sector operators. The physics

73 h0 SR

h0 Figure 4.2: SM singlet coupling to pairs of the lightest Higgs.

of various Higgs sector operators in Dirac gaugino models has been studied, for example in references [108, 111, 112, 116–118]. Some examples of models with additional operators are Dirac gaugino models that include the µ-less MSSM [111] in which the scalar is given a tree level coupling to Hu and Hd and the MRSSM [112] which introduces two additional doublets, the R-Higgses. These models introduce tree level coupling of the singlet to Higgsino (or R-Higgsino) pairs. If the Higgsinos are light enough, this opens another avenue for a decay of the singlet which would provide a sizable width into invisible or highly mass degenerate states. We will now briefly discuss collider limits on a scalar resonance from decay channels other than diphotons. Many models predict a large enhancement of the diHiggs production rate through decay of a heavy resonance. Current limits on diHiggs resonant production for a state of mass 750 GeV are tightest in the 4b channel [119–121]. ATLAS has placed limits on the total production cross section times branching fraction in this channel at 42 fb at 8 TeV. An extrapolation, then, for the limit on total rate into this channel at 13 TeV is approximately 200 fb [88]. If the singlet has a total production cross section large enough to explain the diphoton excess, its partial decay width to diHiggses, will be limited by this constraint. Other modes that will be associated with a particle decaying to diphotons include ZZ, WW, and Zγ. That is, if the singlet decays into two photons, it will also decay into other pairs of electroweak gauge bosons with fixed ratios as dictated by gauge invariance. The coefficients for the effective operators coupling the singlet to diboson pairs is derived directly from the effective superpotential terms in Eq. 4.2 and, after accounting for SU(2) U(1) ×

74 breaking, are given by

2 2 cw sw gγγ = + (4.1) Λ1 Λ2 1 1 gZγ = cwsw( ) Λ2 − Λ1 2 2 sw cw gZZ = + Λ1 Λ2 1 gWW = Λ2 where Λ is the effective operator cut-off scale. For our scenario we expect the ZZ and γZ resonant signals at a smaller rate of production than diphotons. However these signals will become observable in the coming data sets if the diphoton resonance persists. Finally, the singlet scalar field can mix with the Higgs. This can greatly enhance the singlet decay rate to ZZ and WW if the mixing parameter is large enough. Scaling up limits from searches for ZZ resonances at 8 TeV, the limits at 13 TeV are 89 fb and 40 fb with the production modes of gluon fusion and vector boson fusion respectively. Scaling up limits from searches for WW resonances give limits on the allowed 13 TeV production rate of 174 fb and 70 fb from gluon fusion and vector boson fusion production modes respectively. The ZZ and WW constraints will limit the allowable Higgs-singlet mixing. In this chapter we will aim for a Higgs-singlet mixing of 1 percent or less. In the gluon fusion production mode, resonant dijet limits from the 8 TeV run will also be important, though due to the much higher background the limits are much more relaxed. The limits on the allowed dijet production at 13 TeV is 104 fb after scaling up the 8 TeV results [88]. ∼ 4.5 Production and Loop Level Decays

We now discuss loop level couplings of the singlet ﬁeld to pairs of electroweak gauge bosons. As stated above the loop level coupling of the singlet to gluons is mediated by squark loops. The value of the eﬀective coupling of the singlet S to pairs of gluons is given by

2 gY gs mD 2 2 2 2 Nc(ΣqQL C(0, 0, ms, mQL , mQL , mQL ) + ΣqQR C(0, 0, ms, mQR , mQR , mQR )). 16π mS (4.-2)

Here C is the dimensionless Passerino-Veltman form factor, mQL and mQR are the masses of the left and right handed species of squarks, and the sums are taken over each species of left or right handed squark. The hypercharge of each species is given by qQi and we see that within each generation of squarks there will be cancelations due to sign differences in hypercharge. If all squarks are to contribute to the loop we must sum over 3 generations of up and down type left and right handed squarks. The value of the effective coupling of the singlet S to the U(1) gauge boson is given by 75 3 gY mD 3 2 3 2 2 2 (ΣNcqLC(0, 0, ms, mL, mL, mL) + ΣNcqRC(0, 0, ms, mR, mR, mR)), (4.-2) 16π mS where the sums are taken over every state with hypercharge, not just over squarks. Here qi denotes the hypercharge of the field in the loop. Many more particles contribute to this coupling than to the effective coupling of the singlet to gluons. In principle the value of these two effective couplings depends on the hypercharges and masses of the light states that contribute to the loop. We may choose various masses for the left and right handed, and up and down type squarks and sleptons, and a complete spectrum will be given by the parameters of the high energy theory. Here however we may simplify the theory by considering some particles to be arbitrarily high in mass. There are several options for this ‘simplified model’ of Supersoft SUSY. One choice might be to consider all squarks except the lightest, presumably the right handed stop, to be very massive. For the sake of simplicity we will consider a slightly different simplified model. We will send the soft masses of all of the left handed states very high, effectively decoupling them from the theory. This will serve as an existence proof that we may find points in parameter space which fit the excess. We will assume that the only sfermions to have small soft masses will be the right-handed squarks and right-handed sleptons. We will also set the masses of all 3 generations of up and down right-handed squarks to be equal. We will will consider that masses of all three generations of sleptons are equal. In this way the couplings of the singlet and SU(3) and U(1) gauge boson are of comparable order. With the singlet coupling to gluons above we may now calculate gluon fusion production rates for the singlet. In order to calculate the production cross section of the process gg S → in proton-proton collisions, we have implemented a our model into Feynrules [122] inputting right handed up type squarks with D-term coupling to the sbino singlet field. This model was renormalized using the the NLOCT [123] package and imported into Madgraph5@NLO [124] to calculate cross sections. The singlet mass was set to 750 GeV and cross sections were calculated. We have scaled our results by a k-factor of 2 for both 8 and 13 TeV production, as is consistent, for example, with loop induced heavy Higgs production at similar masses [125]. Below, in Fig. 4.3, we show contour plots of the total gluon fusion in the process pp S in the mq˜ vs. mD plane at 13 and 8 TeV. →

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0.68 5000

400 500 600 700 800 900 1000

�� (��)

Figure 4.3: gg S rate in pb for squark mass vs Dirac mass plane in p-p collisions at 8 → TeV (upper) and 13 TeV (lower).

Again, the total production cross section will depend on how many light squarks are allowed to run in the loop, and what their masses and hypercharges are. The total production

rate via gluon fusion increases with the square of mD. The cross section also increases dramatically as the squarks are made lighter.

77 We may also compute the partial decay width, Γgg, of the singlet state into digluon

pairs. This depends on the chosen value of mD as well as the value of the squark masses.

In Fig. 4.4 below we show a contour plot of the this width in the mD mq˜ plane. −

Γ��(��) 10 000

8000

0.06

6000 0.04 (��) � �

4000 0.02

0 2000

400 500 600 700 800 900 1000

�� (��)

Figure 4.4: Partial width (GeV) of S to gluon gluon in mD-ms plane.

The ratio of the S width to photons and to gluons depends on the loop form factors, but also on the admixture of ﬁelds contributing to the photon loop vs the gluon loop. In our simple model with only right handed squarks and sleptons contributing to loops, we can express the ratio of partial widths of the singlet into photons over partial width into gluons as

4 4 cwgY Γγγ/Γgg = 2 4 Nc g3 × !2 Σq3 C(0, 0, m2, m , m , m ) + ΣN q3 C(0, 0, m2, m , m , m ) lR s eR eR eR c QR s QR QR QR 2 . ΣqQR C(0, 0, ms, mQR , mQR , mQR )

The form factors are quite sensitive to the ratio of the squark and slepton masses and the ratio is independent of the Dirac mass. We see that we may vary the ratio of partial widths

78 to gluons and photons by varying the squark and slepton masses. In particular, in the regime that the squarks are heavier than the sleptons, we ﬁnd that the partial width to photons may be made appreciable. Below we have created a contour plot of the ratio of decay widths of the singlet to gluons and photons over the squark, slepton mass plane Fig. 4.5.

�� Γ γγ/Γ�� 400

395

0.03

390

0.02 (��) ��

� 385

10-2

380 0

375 400 500 600 700 800 900 1000

�� (��)

Figure 4.5: Ratio of diphoton to digluon partial widths in the squark mass vs. slepton mass plane.

There are various points in the overall parameter space that give a total diphoton rate in the 5-10 fb range at the 13 TeV run of LHC. However, we may describe the general

requirements for viable points the parameter space. Allowed points generally require mD to be in the multi TeV range, with squarks masses in the 400-1000 GeV range. These values maximize the overall production cross section. By keeping the sleptons near the on-shell threshold mass, the partial decay rate to photons is maximized. The simplest possible region in parameter space is one in which the tree-level diHiggs decay is suppressed by choosing low tanβ, with the singlet decay dominated by gluons and photons. As the tree-level decay width to light Higgses is made larger, it begins to dominate the singlet branching fraction. Therefore, maintaining a high enough partial width to photons requires that the overall

79 production rate by scaled up-which begins to impinge on the diHiggs resonance limits. The viable region of parameter space in the minimal model, will therefore be one in which the overall singlet decay width is small, with limited allowed tree-level decays to diHiggses.

An example point has values (mD, mq˜, me˜) of (3.5 TeV, 420 GeV, 375 GeV) with a diphoton rate σγγ of 5.7 fb. Here we have chosen to suppress the partial width into Higgses, by working at tanβ of 1. The total decay width of the singlet is purely to gauge bosons. At this point in parameter space the partial width to gluons, Γgg, is 31.6 MeV. The branching fraction into photons at this point is roughly 0.95%. There is a very small mixing between the S and the light Higgs at this point-as we show in the following section. The rate for diboson channels is thus σWW and σZZ are 4.7 fb and 2.3 fb. At this point in parameter space, the total production cross section for the scalar is 603 fb at 13 TeV, which is within bounds for dijet production. This point relies on quite light squarks in order to pump up the gluon fusion production rate. These, however, are very mass degenerate with the NLSP which we have, in this case, chosen to be a right handed stau in a gauge mediated scenario. In our scenario, and Dirac gaugino scenarios in general, the gauginos have large Dirac masses and are heavier than the sfermions. The squarks decay chain is thus highly nonstandard, the squark decays through an off-shell gaugino to a stau which then decays to tau and gravitino. Collider bounds on stops and light flavor squarks in compressed MSSM scenarios still allow for squark masses down to 400 GeV[126][127]. It is therefore quite probable that this non-standard decay scenario is unconstrained. An alternate point might be one where the singlet has a more appreciable partial width into light Higgses, however, this will require a large Dirac mass as discussed above. For example, with parameters (mD, mq˜, me˜) of value (15 TeV, 750 GeV, 375 GeV) we may have a partial width into Higgses of 40 MeV, and a Higgs-singlet mixing of 1 percent. This produces a diphoton rate of 4.8 fb. The total production cross section of the singlet is 361 fb with a digluon rate of 106 fb. The total rate in the diHiggs channel is 168 fb, nearly saturating the bound. Due to Higgs mixing the production rates in the WW and ZZ channels are 54 and 26 fb respectively. Our points in parameter space are in accordance with 8 TeV constraints in alternate singlet channels, and follow from the most minimal model involving Dirac gauginos. This serves as an existence proof that Dirac gaugino models may fit the 750 GeV excess. However, in the next section we will explore the behavior of the scalar potential for these models where we will see additional caveats placed on the parameter space. We will find that points like our first one above are in accordance with vacuum stability, while points like the second-with large Dirac mass are meta-stable.

80 4.6 Stability and Electroweak Constraints

We now briefly examine the compatibility of our minimal scenario with Higgs sector constraints and briefly discuss vacuum stability. Dirac gaugino models allow for various Higgs sector operators in addition to new trilinear couplings involving the SM adjoints. In order to specify a scalar potential we must choose a set of Higgs operators. Here we choose to complete our Higgs sector in the framework of the µ-less MSSM, where the µ term is en- hanced by a tree-level coupling between the Higgs and adjoint fields. This framework has several advantages, additional quartic couplings appear in the Higgs potential, and loop effects involving the adjoint fields substantially raise the tree-level Higgs mass. The viability of the Higgs sector of such models was studied in [116]and [108]. We note we require a heavy Higgsino in this scenario. In the superpotential, we will also include a supersymmetric masses for our adjoints as well as usual Yukawa couplings for sleptons. The additional 2 superpotential terms involving the singlet are thus, W = λSHuSHd +MSS . The soft terms in the potential, arising once SUSY is broken, include soft masses and b terms for the Higgs fields, a soft mass and b term for the S field as well as a linear term for the S field. In addition, we include quartic terms for the S field and sleptons,

2 2 2 2 2 2 2 2 2 2 2 Vsoft = m hu + m hd + m S + m e˜R + m ˜l + bhuhd + bSS + h.c + tSS u| | d| | S| | e| | l | | ∗ 2 ˜˜∗ ∗ 2 Vq = λSS(SS ) + λl(ll +e ˜Re˜R) . (4.-2)

1 2 D-terms also give a contribution to the scalar potential, VD = 2 ΣDi . For example, the U(1) D-term is

0 1 2 1 2 1 2 2 ∗ DY = g ( hu hd ˜l + e˜R ) + mD(S + S ). (4.-2) 2| | − 2| | − 2| | | | We may now analyze the potential. With parameters picked below we may enforce the conditions to ﬁnd the Higgs breaking minimum.

81 tan(β) 1

λS 0.4

λSS 0.7

λl 1.2

mDS 3.5 TeV MS 120 GeV − b 5 109 GeV2 × 4 2 bS 5 10 GeV − × 8 3 tS 1.21 10 GeV × Table 4.1: Parameters for Higgs breaking minimum.

The above parameters produce a real scalar sbino of the correct mass to fit the excess, while maintaining a small mixing parameter between the Higgs and singlet fields of 1.7 × 10−3. We have set the slepton soft masses such that the mass eigenstates of right and left handed slepton masses are 376 GeV and 5040 GeV respectively. This point has parameters consistent with our first point in the previous section which predicted the correct diphoton excess. However, we must check for the existence of alternate vacua. In particular we may look in the direction in which the Higgs vev is zero where the potential is given by

1 2 02 ∗ 2 1 02 ∗ 2 1 02 2 ∗ ∗ 2 2 2 V = (g + g )(˜l˜l ) + (g )(˜eRe˜ ) (g 2y )(˜eRe˜ )(˜l˜l ) + λl( ˜l + e˜R ) 8 2 R − 2 − e R | | | | 0 ∗ 2 g mD 2 ∗ 0 2 ∗ 2 2 2 2 + λSS(SS ) ˜l (S + S ) + g mD e˜R (S + S ) + m e˜R + m ˜l − 2 | | | | e| | l | | 2 2 2 2 ∗ 2 1 2 + (m + 4M ) S + m (S + S ) + (bSS + h.c.) + (tSS + h.c.). S S | | D 2 In analyzing the potential, we find an extremum with right and left handed slepton vevs equal to zero and S at non-zero vev. The Higgs vacuum has a lower energy than this extremum. However, for certain values of parameters, we find additional minima in which either the left or right handed slepton has large vevs. These charge breaking minima may have lower energy than the Higgs vacuum, rendering the Higgs vacuum meta-stable. The meta-stability arises from the large D-term coupling between the singlet and the sleptons. The effect of such trilinear terms on vacuum stability is well known, for example see [128]. We see from the minimization conditions we may, for example, set the vev of the left handed slepton, ˜l to zero. Then the vev of fielde ˜R is given by

∂V 2 02 2 2 =e ˜R(me + g e˜R + 2√2mDS + 2λle˜R) = 0. (4.-2) ∂e˜R 82 This is a cubic equation with one zero root. The third term in parentheses follows from the large trilinear coupling between the singlet and the sleptons, proportional to mD. We ﬁnd for certain values of parameters the remaining roots are imaginary, thus there is a single real solution for the vev ofe ˜R corresponding to the extremum described above. There is a 2 critical point for the remaining roots when the value ofe ˜R passes through zero. Then for other values of parameters, in particular large mD compared to the slepton soft mass, there are three real solutions for the slepton vev corresponding to a minimum or saddle point at zero, and symmetric charge breaking vacua at large slepton vev. The necessary condition for avoidance of additional charge breaking vevs is

m2 m6 2√2t 2b + 4m2 + 2m2 + 8M 2 e 2λ e 0. (4.-2) S S D S S 0 SS 03 3 − − 2√2g mD − 16√2g mD ≤ For large Dirac mass and insufficient slepton soft mass, charge breaking vacua will develop. A similar condition holds if one considers the left handed slepton direction. In general this feature of the minimal model limits the size of mD. Our point given above has no charge breaking vacua. In Figure 4.6 we show a contour plot of the potential value in the S vse ˜R plane given our parameters. We find a single U(1) preserving extremum with higher energy than our Higgs potential. However points with much larger values of Dirac mass will have only metastable Higgs breaking vacua. As the Dirac mass is made larger with all other parameters fixed, charge breaking vacua develop and move out to large values ofe ˜R and S vevs along the elongated ‘horns’ seen in Figure 4.6. Our very minimal Dirac gaugino model may fit the diphoton excess while maintaining consistency with bounds from alternate decay channels. However, maintaining stability of the scalar potential requires points in parameter space with low values of the Dirac mass. The lifetime of the possible metastable vacua remains to be calculated, and it is topic for study whether the thin wall approximation is appropriate in these cases. Since the original appearance of this work, several extensions of the Dirac gaugino scenario have been proposed [129][130]. In particular [129] attempt to lower the singlet Dirac mass by including additional fields and trilinear couplings to enhance the loop coupling of the S field to gauge bosons while avoiding additional charge breaking vacua. Other extension of the Dirac gaugino scenario may yield less constrained parameter spaces for fitting the excess.

4.7 Conclusions

We have proposed that the LHC diphoton excess may be explained minimally in a model with Dirac gauginos. The signal follows from production and decay of the sbino, the real component of the scalar partner of the ﬁeld which gives Dirac mass to the bino. This particle is a scalar SM singlet. In the most minimal case, the sbino couples to pairs of dibosons

83 �� =� -200

10 -300 1× 10

-400 -1× 10 10 (��) � � -2× 10 10 -500

-3× 10 10

-600

-400 -200 0 200 400

�� (��)

Figure 4.6: Scalar potential as a function of slepton vev and S vev.

through loops of squarks and sleptons. The sbino may exhibit tree level decay width to pairs of Higgs bosons, and possibly also to Higgsinos. As a simple viability proof we have calculated the production cross section and decays of the sbino in a very simple model in which we consider only the loop contributions of up type right handed squarks and right handed sleptons. We ﬁnd the the sbino-gluon- gluon coupling, and hence the gluon fusion production cross section of the sbino, is highly sensitive to the squark masses running in the squark loop. We have produced production cross sections for the full one loop computation. The total branching fraction of the sbino into diphotons may be on the percent level. However, we may vary the maximize photon to gluon branching ratio by choosing slepton masses lighter than those for squarks. We ﬁnd that in general, we do not run aground of the 8 TeV dijet search. However, a constraining feature in our model is a large rate in the resonant diHiggs channel. We have shown this is within the bounds of LHC exclusion limits at 8 TeV if we limit ourselves to low tanβ. Further study of the Higgs sectors of the model would be fruitful and it is possible that with further studies of parameter space, more viable points will be found. For example, by opening up decays to light Higgsinos, which would appear to searches as very mass degenerate particles, the sbino branching fraction may be dumped into an invisible channel. This may allow a broadening of the total sbino decay width and thus

84 partial width into light Higgses without impinging on additional collider constraints. This possibility would entail models with larger Dirac masses therefore, a study of the lifetime of the metastable vacuum would be necessary. It is a topic of further study to determine how minimal Dirac gaugino scenarios might be extended. For example, the Dirac bino scenario mixed with other SUSY mediation scenarios, for example anomaly mediation as in [131]. Such models might lend themselves toward SUSY mass spectra and additional couplings that might further be able to increase the singlet production rate or decay into diphotons. As many have already pointed out, due to gauge invariance, if our state produced a diphoton signal, it must also be seen in the ZZ and Zγ and possibly WW channels. These production rates are only a factor of a few less than the diphoton rate, and thus must be observed soon if the resonance is to stay. As a ﬁnal note, we mention that Dirac gaugino models also contain an scalar SU(2) adjoint. Loops similar to those we discussed couple the neutral component of the SU(2) adjoint to gluons and photons. In the case of gluon coupling, the loops come with powers of Higgs insertions to preserve SU(2) quantum numbers. This state also couples to electroweak boson pairs through loops of sleptons and it may therefore be possible to explain the excess as a swino.

85 Chapter 5 Searching for Standard Model Adjoint Scalars with Diboson Resonance Signatures

This Chapter is based on work published in JHEP 1512 (2015) 151

5.1 Introduction

With Run II of the LHC now in progress, we are now able to probe well into the TeV range in the search for new particles. One discoverable class of heavy particles which appears in many Beyond the Standard Model (BSM) scenarios is that of scalar fields in adjoint representations of the Standard Model (SM) gauge groups. Scenarios with scalar adjoints are quite diverse and candidate fields include chiral states that mix with gauginos in R- symmetric extensions of the Minimal Supersymmetric Standard Model [43][105], triplet Higgs fields, various states in Minimal Flavor Violating models [132], and KK partners of gauge bosons in extra-dimensional models [38]. Proposed search channels for these states vary depending on the scenario, but include such signatures as jets plus missing energy, tt and dijet resonances, and searches for W/Z + jets [133][113][114][134][135]. In this chapter we will explore the consequences of couplings between scalar adjoint fields and pairs of SM gauge bosons. We will begin in the framework of effective operator analysis. As we have seen from recent studies in Higgs physics and dark Matter phenomenology, this is an extremely useful tool for systematically cataloging all possible interactions consistent with symmetries [136][47]. We first present the most general effective Lagrangian up to dimension 5 in which scalar fields in adjoint representations of the SM gauge groups couple to pairs of gauge bosons. These operators are allowed by all symmetries of the theory, we will therefore assume they are generically non-zero. Some dimension 5 operators involving scalar adjoints have been well studied, for example the dimension 5 coupling between a

86 scalar color octet and two gluons. Here, we will present several additional operators which induce new diboson couplings. We also discuss possible UV completions of the effective theory in several scenarios. These completions will involve integrating out a messenger sector of fields which are charged under multiple SM gauge groups. We will consider both supersymmetric and non-supersymmetric completions of the effective operators. We will see that for typical completions, for example simple messenger sectors consisting of vector-like fermions, gauge invariance demands the simultaneous existence of multiple dimension 5 operators. In general individual operators cannot be arbitrarily ignored. We then study the collider phenomenology induced by the operators in our effective Lagrangian. We choose to explore the phenomenology of a scalar color octet whose production and decay proceeds through loop-level couplings to pairs of SM bosons. In particular we focus on the effective couplings between the scalar octet and a gluon and photon/Z-boson. We argue that new proposed decay channels for the octet opens a window to constrain or discover various models of octet scalars. We recast several 8 TeV LHC analyses to find current constraints on effective operators coefficients. These analyses include searches for dijet resonances, searches for excited quarks, and the inclusive search in the mono-jet channel. We then present the discovery potential for the scalar octet in a jet plus photon channel for the 14 TeV run of LHC. This paper proceeds as follows, in section 2 we consider effective operators where a scalar SM adjoint may couple to gauge bosons. In section 3 we discuss possible UV completions of the effective Lagrangian. In Section 4 we focus on the production and decay modes of a scalar color octet at LHC given our set of effective operators. In section 5 we present the bounds on scalar color adjoints in our scenario coming from a recast of existing searches LHC Run 1. In section 6 we present a sensitivity search for scalar adjoints in S gγ → channel in the 14 TeV run of LHC. Section 7 concludes.

5.2 Eﬀective Operators

We will now explore the couplings of SM adjoint scalars to pairs of gauge bosons, beginning with operators of the lowest dimension. We consider two new spin 0 adjoint ﬁelds, color octet, S, with quantum numbers (8, 1) under SM gauge groups, and a weak triplet T with quantum numbers (1, 3) under SM gauge groups. At dimension 4, pairs of scalar adjoints have couplings to pairs of gauge bosons through their kinetic terms. However, couplings between a single adjoint and a pair of gauge bosons must come at higher dimension. We will now consider the complete set of gauge and Lorentz invariant operators of dimension 5 where a scalar adjoint couples to pairs of gauge bosons. For the scalar color octet the gauge and Lorentz invariant dimension 5 operators are

87 abc d µν 1 a,µν L = SaGb Gc,µν + SaG Bµν (5.0) Λ2 Λ1 µν where Λi is the scale of new physics at which the operator is generated. Here, G is the

SU(3) field strength tensor and Bµν is the U(1) field strength tensor. The first operator couples the color adjoint to a pair of gluons, and is familiar from previous studies of color adjoint phenomenology. In this operator, the color indices of the gluons and adjoint scalar are contracted symmetrically with dabc. The second operator is also gauge and Lorentz invariant. In this operator, the Lorentz indices are contracted between field strength tensors, while the color index is contracted between the gluon and the color octet. This is an effective coupling between the scalar octet, one gluon, and a photon/Z-boson. Similarly we may consider a similar operator governing the SU(2) adjoint T

1 µν L = TiWi Bµν (5.0) ΛTWB µν where W is the SU(2) field strength tensor, and Bµν is the U(1) field strength tensor. Lorentz indices are contracted between the field strength tensors, while the SU(2) indices are contracted between the adjoint T and the SU(2) field strength tensor. Unlike SU(3),

SU(2) does not have a factor dabc, therefore we may not write a coupling between the field T and two W µν’s as above. However, we may extend the list of possible gauge invariant operators if we allow the insertion of Higgs fields to soak up SU(2) indices. These operators will be of dimension 7, but once Higgs vevs are inserted, they become operators of effective dimension 5. These dimension 7 operators are

1 † µν 1 † µν 1 † µν L = 3 [H TH]B Bµν + [H TH]W Wµν + 3 [H TH]G Gµν ΛTBB ΛTWW 3 ΛT GG 1 a µν † i + 3 S Ga [H WµνH] ΛSGW In the first three terms we have contracted the SU(2) indices of the adjoint T with those of the Higgs fields to make an SU(2) singlet; this is indicated by brackets. In the last term, SU(2) indicies are contracted between the SU(2) field strength tensor and the Higgs doublets. The SU(2) structure of these operators is quite reminiscent of loop induced operators in two Higgs doublet scenarios which couple the Higgs fields to SM field strength tensors as, for example, in references [137][138]. The operators above are of dimension 7, however once the Higgs vev is inserted one get operators of effective dimension 5, which v are suppressed by two powers of Λ . In the appendix we make a more complete list of operators for more complicated models where the scalar color octet may have additional quantum numbers under the other SM gauge groups. We expect the operators above may be generated at loop level by various UV completions of the theory. We give some illustrative

88 examples in the next section.

5.3 High Energy Models

SM scalar adjoints appear in various BSM scenarios. Here we are considering models in which the dominant decay of these states is to pairs of SM gauge bosons. There are a variety of frameworks in which these operators may be completed. One scenario that naturally lends itself this type of phenomenology may be models of extended technicolor. Such models typically contain SM adjoint scalar which are bound states of techni quarks/leptons that are charged under the extended technicolor gauge group in addition to SM gauge groups. Often the scalar adjoints are the lightest states in the theory. Effective operators like those in equations 1 and 2 couple the techni-adjoint to pairs of SM gauge bosons, see for example [139][140]. It is a model building question as to which these couplings dominate over couplings between the adjoints and pairs of SM fermions. However the simplest realizations of such models are currently disfavoured by electroweak fit constraints and other theoretical challenges. A very simple and calculable realization of our effective operators is one where they are produced by integrating out heavy fields which have quantum numbers under several SM gauge groups. One of the simplest completions may be to include one or more generations of heavy vector-like quarks or leptons which carry hypercharge in addition to SU(2) or SU(3) quantum numbers. Sets of these vector-like ”messenger” fields may couple to our scalar adjoint fields through Yukawa-like terms. For example, a set of vector-like quarks which carry hypercharge have quantum numbers (3, 1)Y and may couple to a scalar octet to generate the Feynmann diagrams in Fig. 5.1. These diagrams correspond to the operators in

Eqn 1. The exact ratio of the operator coefficients Λ2 and Λ1 will depend on the hypercharge of the messengers. A general estimate for the ratio of effective couplings might be g3/g1, though this depends on the details of messenger sector. The effective coupling to U(1) gauge bosons is non-negligible and will provide additional production and decay channels for a scalar octet. This opens new windows for collider searches to illuminate new physics sectors. One large class of models which contain SM adjoint scalars are supersymmetric theories in which gauginos are Dirac as opposed to Majorana [43][105]. In Dirac gaugino models, gauginos get mass by mixing with new chiral fields which are adjoints under the SM gauge groups. In order to generate the Dirac Mass the relevant super-potential operator is

Z W 0 W αA Dλψ W = d2θ α = A (5.0) Λ Λ where A denotes the chiral adjoint field, W is a SM gauge field strength and W 0 is the field strength of a hidden U(1) gauge group which is broken at a high scale. This operator is

89 expected to be generated by integrating out heavy messenger ﬁelds which are charged under the SM as well as the hidden-sector U(1) gauge group, thus the UV model is a form of gauge mediation [106]. The hidden-sector U(1) ﬁeld gets a D-term vev, thus the operator becomes a Dirac mass for the SM gaugino, ’marrying’ it to the fermionic piece of the chiral adjoint.

Defining D/Λ mD we find the operator in Equation 4 is equivalent to a Dirac mass term ≡ mDλψA for the gaugino. The chiral multiplet A is complex, it thus contains both real and imaginary scalar fields which are adjoints under a SM gauge group. The masses of the real and imaginary parts of the scalar adjoint depend greatly on the details of the messenger sector of the model see for example, see for example [107],[108],[109]. Many models lead to tachyonic masses for one or the other adjoint, and fixes may produce a great range of adjoint masses. If the scalar adjoints are lighter than the gauginos and sparticles in the theory, then we expect their decays to follow through loops to sets of SM gauge bosons or fermions. We note that consistent with all symmetries in the theory we may also write the superpotential term

Z W Y W αA W = d2θ α (5.0) Λ where A is the a chiral SM adjoint field, W Y is the hypercharge field strength and W is appropriate SU(2) or SU(3) field strength tensor. This operator is nearly identical to the ’supersoft’ operator which generates Dirac gaugino masses. We may also write

Z dabcW aW bSc W = d2θ 3 3 (5.0) Λ0

Where W3 is the SU(3) field strength and S in the chiral SU(3) adjoint. Upon integration over θs, we see that these superpotential terms produce in the Lagrangian our operators from Eqns 1 and 2. These operators are presumably generated much the same way as the supersoft term is, by integrating heavy fields out of the theory as we discuss below. We note that the Lagrangian terms in Eqns 1 and involve the real part of the adjoint. We might also expect similar operators generated by messenger loops to couple imaginary part of A and the SM field strength tensor and dual. Many aspects of the phenomenology of scalar adjoints resultant from R symmetric SUSY models have been studied thoroughly [113][141]. It is known, for example, that an octet- gluon-gluon coupling may be generated by integrating out squarks from the R symmetric theory. Similar diagrams involving squarks and sleptons will generate the the operators in Eqn 5. Squarks and sleptons couple to the real part of the chiral adjoints through Kahler potential D-terms. Due to a cancelation of diagrams, it is known that operators following from sparticle loops are suppressed when left and right sparticle masses are degenerate. Another source of operators in Eqn 5 and 6 are loops of the heavy messenger fields.

90 These messengers carry SM quantum numbers, and in many completions are often various sets of fundamentals and anti-fundamentals under SU(5), see for example [108],[109]. The relative size of the operators in Eqns 5 and 6 will depend on the details of the messenger sector, and on the sparticle spectrum. Effective dimension 7 operators like those of Eqn 3 may be produced through loops of squarks and sleptons, as these sparticles are charged under multiple SM gauge groups and couple to the Higgs. Thus in ’supersoft’ models we find numerous effective operators which couple scalar adjoints to pairs of gauge bosons. As a phenomenological note, the real part of the supersoft adjoints may also decay to quarks/leptons through loops. These decays, however, do not always dominate adjoint branching fractions - there are various regions of parameter space where these decays are subdominant to decays into gauge bosons. This region of parameter space will be of special interest to the phenomenological discussions of the next section.

g g

S S

γ g

Figure 5.1: Decay through mediators to diboson ﬁnal states.

5.4 Production and Decay of Scalar Adjoints

We will now consider the collider phenomenology of our effective operators. We will focus on the phenomenology of colored states as they should be produced with large cross section at the LHC. As such, we will study the production and decay of a single scalar color octet. We will assume that, besides mass and kinetic terms, the octet has only dimension 5 couplings to SM gauge bosons resulting from the operators in Eqn 1. Our simple model has three parameters, the mass of the octet mS, and the two effective operator coefficients Λ1 and Λ2. In a UV complete model we would know the exact ratio between the operator coefficients, not knowing the details of the high energy model, however, we will consider these two parameters to be independent. The two operator coefficients control the couplings of the scalar color octet to three pairs of SM gauge bosons gg and gγ and gZ. We consider here only the single octet production process pp > S, which proceeds through gluon fusion. The total production cross section is proportional to the gluon-gluon decay width, Γgg, of the octet, and is given by,

91 16π2 Z 1 dx m2 σ pp S ε f x f S ( ) = Γgg m2 g( ) g( ) (5.0) → smS S x sx s where mS is the mass of the scalar, fg are the parton distribution functions (PDFs) for gluons, x is the momentum fraction of the initial gluons, and ε = 1/32 takes into account the interchange of summed and averaged over states in the decay rate and production cross section. The octet decay width into gluons Γgg is given by

3 40 mS Γgg = 2 . (5.0) 3 32πΛ2

3 We note that the decay width scales like mS leading to fairly wide resonances for larger masses of the octet.

The production cross section of the octet decreases with increasing parameter mS, it

also decreases with increasing scale of the effective cut-off Λ2. Higher order corrections to the production of color-octet scalars can have very large effects. A K-factor is used to scale up the tree level production cross section of the scalar octet. A typical K-factor for the process gg S at √s = 14 is between 2.4 and 3.6 for a TeV scale octet [142]. In the → analyses in this chapter we choose a K-factor of 2 for octet production at √s = 8 TeV and a K-factor of 2.5 for a sensitivity projection at √s = 14 TeV.

In Fig. 5.2 we plot the octet production cross section vs. mS for the process gg S in 8 → TeV proton-proton collisions. In the figure we have chosen the effective operator coefficient

Λ2 to be 10 TeV, but this total production cross section may easily be re-scaled for various values of the effective cut-off. We will now consider the decays of the scalar octet. In our model there are three possible diboson decay channels for the octet, S gg, gγ, gZ . That is, decays into two gluons, → a gluon photon pair, and a gluon Z pair. Using the effective Lagrangian in Eqn 1, we may calculate the branching fractions. In the limit that the octet is much heavier than the Z

boson, that is mZ << mS, the branching fractions are given by,

2 Γgγ cW BRgγ = = 10 2 2 Γgγ + Γgg + ΓgZ 1 + 3 Λ1/Λ2 2 ΓgZ sW BRgZ = = 10 2 2 Γgγ + Γgg + ΓgZ 1 + 3 Λ1/Λ2 Γgg 1 BRgg = = 3 2 2 Γgγ + Γgg + ΓgZ 1 + 10 Λ2/Λ1 where sW and cW are the sine and cosine of the weak angle and Λi are the effective operator coefficients. By varying the ratio of effective couplings Λ1/Λ2, we may change the branching fraction rates into the various diboson channels. For generic completions of the effective

92 Production Cross Section L2=10Tev, s=8 TeV

H , L

1 L pb H

S 0.01 ® pp Σ

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Figure 5.2: Production cross section of a single scalar octet vs. mS in pp collisions at H L √s = 8 TeV with Λ2 = 10 TeV. These cross sections were generated with MadGraph 5.

theory, for example including sets heavy vector-like quarks with electric charge in the range

1/3, we expect the ratio of the branching fractions Γgγ/Γgg to be in the percent range. This branching fraction is not insignificant, and together with the large octet production cross section it will mean that there is a substantial signal in the gγ resonance channel. As we will see below, this is an interesting and a relatively clean channel, it will therefore be important for scalar octet searches. Above we have discussed single production of scalar octets however pair production of color octet scalars can occur. The production is dominated by gluon fusion. Due to kinematics and p-wave suppression, the pair production cross section falls rapidly with increasing mS . Pair production of octets falls below the fb range for LHC energy of 14 TeV when mS < 1.5 TeV [134]. Possible signals for pair production are SS 4g, 3g+V, 2g+VV → where the∼ V is a γ or Z. Final state topologies will vary depending on the choice of and decay channel for V. In the most realistic models, the branching fraction S V g is in the → percent range, as mentioned above, therefore total production and decay into 2g + VV final states will be highly suppressed. However the final process pp S > V gS > gg with two → reconstructed octet resonances may be an interesting possible discovery channel for lighter octet masses. So far we have discussed signals for color octet scalars. However we note that all of our mentioned completions will also contain an SU(2) adjoint scalars which should couple to pairs gauge bosons due to equivalent new physics. In the case of supersymmetric completions

93 we may naturally evolve not only the dimension 5 operator in equation 2, but also the effective dimension 7 operators of equation 3, providing a very rich possible phenomenology. Though in this chapter we will not carry out a full collider simulation, we may comment on possible event topologies for the discovery of this state. Possible production channels for the SU(2) triplet scalar, T, involve vector boson fusion, in which the adjoint is produced along with a pair of forward jets qq jj + T ; and associated production where the adjoint → is produced along with an electroweak gauge boson through a quark fusion process, qq → V + T . For special models where the dimension 7 operators may be non-zero, gluon fusion may also be a possible production channel gg T where the cross section is proportional → to SU(2) breaking Higgs vevs. We expect the SU(2) adjoint to decay through the dimension 5 couplings to two electroweak vector bosons. This makes for quite interesting final state signals, as we may hope to reconstruct the heavy vector bosons in the decay from pairs of leptons, or, perhaps, as a hadronic ”fat” jets. We may also observe a significant number of events with relatively hard photons. This is an interesting topic for further collider studies.

5.5 Current Limits

We will now discuss the bounds on parameter space which result from various 8 TeV analyses in Run I of the LHC. In our model, production and decay of the octet produces many interesting final state topologies, therefore there are multiple search channels that can be relevant in the constraint or discovery of the scenario. Both ATLAS and CMS have performed searches for heavy resonances decaying into various final states, and one standard search for new colored resonances is in the dijet resonance channel, pp X jj [143][144]. → → Here a heavy state, X, is produced and then decays hadronically. These searches constrain the parameter space of our color adjoint operators as the scalar octet generically has significant branching fraction into the gluon-gluon final state. In addition ATLAS and CMS have performed searches for resonances which decay to a photon and jet final state [145] [146]. This search is relevant to our model since the scalar has a decay mode S gγ. The → monojet search pp j + Emiss is an important inclusive search channel for constraining → T new physics [67][147]. This analysis will be relevant to our decay channel S gZ, where → the Z decays to missing energy (Z νν). Finally, CMS has a search for a heavy resonance → decaying to a jet and a hadronically tagged vector boson pp X j + V [148]. This → → analysis will be relevant to our decay channel S gZ, where the Z decays hadronically → (Z jj). Below, we will explore the bounds on our parameter space from each of these → studies.

94 5.5.1 The Jet + γ Channel

We ﬁrst discuss constraints from searches for γ + j resonances on the process gg S γg. → → The benchmark models used in these analyses include production of non-thermal quantum black holes (QBH) and excited quarks (q∗), however the search constrains our color octet model as well. The ATLAS search for photon plus jets used the following search criteria:

At least one isolated photon with pT > 125 GeV •

At least one hard jet with pT > 125 GeV. • Photon required to have angular separation ∆R(γ, j) > 1.0 between leading photon • and all other jets with pT > 30 GeV.

Photon and jet required to be in the central region of the detector with ηγ < 1.37 • | | and ηj < 2.8. | | Pseudo-rapidity separation between jet and photon of ∆η(γ, j) < 1.6. • | |

Highest pT γ and jet candidates used to compute mγj, which is binned. •

Σ´Ε´BRgΓ L1 L2=3, L2=10 TeV, s=8 TeV 1

H , L 0.1 L pb

H 0.01 Γ g BR ´ Ε

´ 0.001 Σ

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Figure 5.3: The rate σ BRgγ of the scalar octet resonance at 8 TeV where Λ2 = 10 × × H L TeV andΛ1 = 30 TeV. This rate includes the eﬃciency with which events passes the ATLAS selection criteria.

95 The SM background for this search is dominated by γ+j production and QCD dijet production. The SM production of γ +j lacks a resonant process and is generated via Compton scattering of quarks and gluons, and through quark-antiquark annihilations. Electroweak processes account for about 1% of background. The analysis performed by ATLAS observed no excess of events over SM backgrounds, and reported limits on production and decay of their benchmark models to γj final states. We recast the ATLAS search at √s = 8 TeV with 20.3 fb−1 of data in order to rule on the parameter space of our color octet model. We implemented the effective Lagrangian using FeynRules [122], then using MADGRAPH 5 [149] we generated events for the process gg S γg in pp collisions at 8 TeV. The events were showered with Pythia [150] and run → → through the PGS detector simulator. We then implemented the ATLAS cut-based analyses on our data to determine the efficiency of the ATLAS search for our model. A plot of signal efficiency vs mS is given in Fig. 5.4. In general, we find signal efficiencies of order 50 percent for scalar masses between 1 and and 4 TeV, with the efficiency dropping rapidly at low invariant mass.

For each point in parameter space we specify the values of mS,Λ1,Λ2. We then calculate the total number of events which pass the selection criteria, Nevt = σ L BRgγ, in × × × each invariant mass bin. In Fig. 5.3 we plot the total rate σ BRgγ vs. scalar mass mS × × for the benchmark values Λ2 = 10TeV and Λ1 = 30 TeV. At this benchmark point, for octet masses below about 2 TeV, the total rate is in the few fb range, and therefore potentially detectable. We show the 95% confidence level (CL) exclusions in the Λ1-Λ2 plane for various values of the scalar mass in Figs. 5.5 and 5.6. We see that exclusions become weaker as the scalar mass increases, as is expected due to a rapidly decreasing production cross section. For a given mass, the production rate is set by the parameter Λ2, with the rate decreasing as Λ2 increases. The branching fraction into the gγ final state is set by the ratio of scales Λ1/Λ2. Thus the total signal rate decreases as this ratio of scales increases. We may exclude effective cut-offs in the 5-10’s of TeV range for mS of the order a few TeV. We note that the effective operator paradigm is not valid to arbitrarily low cut-off scales, but will be valid only when the √s < 2Mmess where Mmess are the mass of the UV messengers integrated out of the theory.

5.5.2 Dijet Channel

Dijet searches tightly constrain the existence of new colored resonances. The ATLAS search for dijet resonances was done at √s = 8 TeV with 20.3 fb−1 of data. The benchmark models that are considered by ATLAS include excited quarks, QBHs, heavy W 0 gauge ﬁelds decaying to quarks, and scalar octet particles decaying to a pair of gluons. The ATLAS dijet search criteria is as follows:

96 At least 2 anti-kt jets with rapidity y < 2.8. • | |

Two hardest jets (leading and subleading) are required to have pT > 50 GeV. •

An invariant mass cut of mjj > 250 GeV is placed on the two hardest jets • 1 Separation of rapidities ylead ysublead < 0.6. • 2 | − | SM backgrounds in this channel are dominated by QCD processes with a sub-percent mixture of additional SM processes. The background has a smoothly falling spectrum as the dijet invariant mass of the two highest pT jets (leading and sub-leading), mjj increases [151]. ATLAS analyzed these results to constrain a scalar octet benchmark model. Eﬃciency cuts for this model range from 61% to 63%. The ATLAS analysis showed no excess for dijet invariant masses up to 4.5 TeV and placed limits on production rates of their octet. The 95% CL upper-limit on σ A with 100% branching fraction to gg ﬁnal state was reported × by ATLAS. We recast these results to set bounds on the parameter space of model in which the scalar octet has multiple decay channels. By rescaling the ATLAS total rate for the process gg S gg we place bounds in the Λ1 Λ2 plane for various values of mS. We → → − show exclusions in Fig. 5.5.

For a given octet mass mS, the total production rate depends on the eﬀective coeﬃcient

Λ2, with the rate decreasing as Λ2 increases. The total branching fraction of the octet into gluons is then given by the ratio of scales Λ1/Λ2, with the digluon rate increasing as this ratio is increased. We thus expect this search to put the tightest limits on the scalar octet model when Λ1 is large compared to Λ2, that is, when the eﬀective couplings Sgγ and SgZ are suppressed.

5.5.3 The Monojet Channel

The mono-jet search channel is a standard inclusive analysis that applies to many BSM scenarios. In the model we are considering, the process gg S Zg will lead to a host → → of interesting signal topologies, among which is a monojet signature. This ﬁnal state is achieved when the Z decays invisibly. We will now recast the ATLAS √s = 8 TeV, 20.3 fb−1 monojet analysis to rule on our scenario. Monojet searches look for an event topology with a hard jet in the central detector region and several hundred GeV of missing energy. The ATLAS monojet search selection criteria is as follows:

miss Events required to have E > 150 GeV and at least one jet of pT > 30 GeV with • T η < 4.5. | |

Leading jet with pT > 120 GeV and η < 2.0. • | | 97 miss For a monojet like topology, the ratio pT /E > 0.5. • T Separation between leading jet and direction of missing transverse momentum • miss ∆φ(j, ET ) > 1.0.

Veto of events with electrons or muons with pT > 7 GeV or isolated tracks with • pT > 10 GeV and η < 2.5. | | Events passing the cuts were divided into 9 signal regions (SR) deﬁned by the minimum amount of Emiss in the SR. Limits where 95% CL limits were placed on σ A for the T × × diﬀerent SR as no excess of events was observed. The signal regions for the ATLAS analysis are shown in the table below.

Signal Regions SR1 SR2 SR3 SR4 SR5 SR6 SR7 SR8 SR9 miss Minimum ET (GeV) 150 200 250 300 350 400 500 600 700 95% CL for σ A (fb) 726 194 90 45 21 12 7.2 3.8 3.4 × × Table 5.1: Signal regions for ATLAS monojet search.

The main backgrounds for this search are dominated by Z( νν)+jets and W +jets → production. Smaller contributions come from Z/γ∗( l+l−)+jets, multijet, tt¯, and diboson → processes. The search found no excess of monojet events in the 8 TeV dataset. We utilize this search results to constrain parameter space. We generate events for the process pp S Zg ννg in MADGRAPH 5. The events are showered with Pythia → → → and then run through the PGS detector simulator. We then implemented the mono-jet cut based analysis to calculate the efficiency for events in the various signal regions. The calculated efficiency for scalar octets model in various signal regions is found in Fig. 5.4. The overall efficiency of the mono-jet search for this process is fairly high. We note that our events differ from many models searched for in the monojet channel. For example a cast of

DM models in which a gluon is emitted as initial state radiation favor softer gluons with pT peaked at low values. In our events however, the gluon in emitted as a hard decay product

in the ﬁnal state, thus will have a much harder pT which is more amenable to passing a

hard pT cut. We then calculate the total expected event rate and compare to the reported 95% CL

exclusion limits. The exclusions limit results in the Λ1-Λ2 plane can be seen in Fig. 5.6

for various values of the octet mass mS. The mono-jet search does signiﬁcantly worse at constraining the scalar octet model than the dijet or γj searches. This is in part because the

98 multiplicative eﬀects of the octet branching ratios into the gZ channel and the Z branching fraction into neutrinos.

As before, the overall production rate of octets decreases with increasing scale Λ2. The

branching fraction into the gZ ﬁnal state decreases as the ratio Λ1/Λ2 increases. Thus the shape of the exclusion due to this search is similar to that from the gγ search. We note that the ratio of gγ to gZ events is strictly related by gauge invariance given the form of our

operator with coeﬃcient Λ1, therefore we expect that any signal for scalar octets observed in the gγ would eventually produce a signal in the gZ channel as well.

5.5.4 Heavy Boson plus Jet Channel

The CMS collaboration has searched for heavy resonances which decay to a jet and a massive vector boson in the process pp X j + V . The search also applies to heavy → → resonances which decay to two heavy vector bosons. In these searches the vector bosons decay hadronically, V jj, and is reconstructed with a hadronic tag. Benchmark models → considered for the CMS analysis are excited quarks (q∗ qW , q∗ qZ), Randall-Sundrum 0 0 → → gravitons (GRS WW ), and heavy W bosons (W WZ). The search also applies to our → → scenario where a heavy scalar octet decays to a gluon and hadronically decaying Z boson, gg S Zg, Z jj. → → → To identify a jet as being Z/W-tagged, the CMS analysis used jet pruning techniques,

and require the pruned ”fat-jet” with to have mass between 70 GeV< mj < 100 GeV. The search criteria for the CMS analysis is as follows:

At least two jets with pjet > 30 GeV and η < 2.5. • T | |

At least one of the two highest pT jets is required to be vector-boson tagged ”fat”-jet. • The pseudorapidity separation between the leading jets ∆η(j, j) < 1.3. • | |

A cut on the invariant mass of the two highest pT jets of mjj > 890 GeV •

99 Efficiency ATLAS j+Γ 0.7

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Figure 5.4: Efficiency of our simulated eventsH passingL the selection criteria in the search channels considered for √s = 8 TeV analyses at the LHC. The upper two plots are the signal efficiencies for the ATLAS searches for photon plus jet (upper) and monojets (middle) corresponding to the decay modes S gγ and S gZ(Z νν) respectively. The lower → → → plot is the kinematic efficiency of the CMS search for hadronically decaying heavy bosons for the decay mode S gZ(Z qq). → →

100 The dominant source of background for this search stems from multijet production with tt¯, W +jets, and Z+jets contributing less than 2%. Data was binned by the invariant mass of the jet and tagged ”fat”-jet mjj. No significant excess in data was reported. We may thus use this search to constrain our parameter space. Using MADGRAPH 5 we simulated parton level events events for the process gg S → → Zg, Z jj. We estimated the search efficiency for these events using the CMS ”fat”jet- → tagging efficiency and cuts for kinematic efficiency. The tagging efficiency reported by CMS for a single Z-tagged fat-jet ranges from roughly 45% to 30%, with the Z-tag doing better at lower invariant mass. In our analysis we took the Z-tag jet efficiency to be 40%. In Fig.

5.4 we plot the search efficiency vs octet mass mS in the Z+j resonant final state. Using the production cross section, branching fraction and efficiency for our events, we may then calculate the total number of expected events in each invariant mass bin for each point in our parameter space. We then compare to 95% CL upper bounds in each mass bin from the CMS search to constrain the parameters. Exclusions are shown in the Λ1 Λ2 − plane for various scalar masses mS in Fig. 5.6. The shape of the exclusion line follows from previous discussions of the gγ and gZ final states.

101 5.5.5 Combined Results

EFT Bounds 8 TeV S®Γg 102 103 104 105

105 105

mS=1 TeV L 104 104 mS=2 TeV GeV H 2

L mS=3 TeV

mS=4 TeV 103 103

102 102 2 3 4 10 10 10 105

L1 GeV EFT Bounds 8 TeV S®gg H L 102 103 104 105 105 105

4 4 10 10 mS=1 TeV

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103 103 mS=4 TeV

102 102 2 3 4 10 10 10 105

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H L Figure 5.5: Bounds on the Λ1 Λ2 for 4 values of the octer mass mS. Exclusions follow − from 4 LHC 8 TeV analyses. The upper plot follows from searches in the pp X gγ → → ﬁnal state. The lower plot follows from searches in the pp X jj ﬁnal state. → →

102 EFT Bounds 8 TeV S®gZ Z® 102 103 104 H îîL

104 104

mS=1 TeV L

GeV m =2 TeV H S 2 L = 103 103 mS 3 TeV

102 102 102 103 104

L1 GeV EFT Bounds 8 TeV S®gZ Z®qq H L 102 103 104 H L

104 104

mS=2 TeV L

GeV m =3 TeV H S 2 L = 103 103 mS 4 TeV

102 102 102 103 104

L1 GeV

H L Figure 5.6: Bounds on the Λ1 Λ2 for 4 values of the octer mass mS. Exclusions follow − from 4 LHC 8 TeV analyses. The upper plot follows from searches in the pp j + Emiss → T ﬁnal state. The lower plot follows from searches in the pp j + V,V jj ﬁnal state. → →

103 Bounds on EFT scale with L1 L2=4 L2 GeV 5 ´ 104 H L

´ 4 1 10 Photon+jet 5000 Dijet Monojet

1000 Heavy boson+jet

500

MS GeV 0 1000 2000 3000 4000 H L Figure 5.7: Combined 95% conﬁdence level bound on EFT scale from all the channels considered for the choice Λ2 = .25Λ1. The black dashed line corresponds to where the validity of the EFT framework breaks down mS > 2Λi.

Combined results for all of the ﬁnal state channels under consideration are displayed in

Fig. 5.7. In the figure we have fixed the scale Λ2 to be 10 TeV, and the ratio Λ1/Λ2 to be 4. This benchmark point is representative of a class of a UV completions with vector-like messengers in the multi-TeV range with hypercharge of order unity. Effective cutoffs may be excluded between a few and 10s of TeV over the octet mass range. We see that for high mS regions the dijet search is the most exclusive, while the γ +j search is the most exclusive for the low mS regions. While the branching fraction of the octet into γ +j is in the percent range for this benchmark point, the background for this channel is quite low. Thus we find that in some of the parameter space of the theory, searches in the γ + j channel may be competitive with dijet searches. Due to a very large background, the dijet channel signal efficiency for very light octets is quite low. We might expect that in searches for scalar octets with masses in the 100s of GeV range, γ + j will generally be the most powerful. We will now discuss the limits of the effective operator paradigm. The effective operators which we have been studying are generated by integrating out some heavy messenger particles which couple to the scalar octet and to gauge bosons. We know that the effective operator treatment is invalid for scales of the effective cut-off that are below the center of mass energy of the process, that is where the messenger fields are very light. In this low cut-off regime we cannot use the effective terms to accurately calculate the octet production cross section [50]. However, we are assuming that if the octet prefers to decay via loop 104 level couplings, it must have no tree level decays to messenger fields. That is, the mass of the octet must be less than twice the messenger mass mS < 2Mmess and thus the messengers must be offshell in the process. Gladly, we note that whenever this offshell condition,

√s < 2Mmess, is satisfied, that is exactly the region where the EFT is valid as well. Since the effective operator coefficients are about equal to the UV messenger masses, may thus place a viability condition on the effective operator treatment that the coefficients should satisfy mS < 2Λ. In Fig. 5.7 we have drawn the line of effective operator viability, operator coefficients too light compared to the octet mass are unviable.

5.6 Projection for LHC 14

We will explore the reach of the 14 TeV run of LHC into our parameter space. Below we produce a sensitivity study for the process gg S gγ in pp collisions at √s = 14 TeV. → → This search is based on the ATLAS 8 TeV jγ resonance search. In this analysis will follow the same cuts as the 8 TeV ATLAS analysis above,

At least one isolated photon with pT > 125 GeV •

At least one hard jet with pT > 125 GeV. • Photon required to have angular separation ∆R(γ, j) > 1.0 between leading photon • and all other jets with pT > 30 GeV.

Photon and jet required to be in the central region of the detector with ηγ < 1.37 • | | and ηj < 2.8. | | Pseudo-rapidity separation between jet and photon of ∆η(γ, j) < 1.6. • | |

Highest pT γ and jet candidates used to compute mγj, which is binned. • Our model is implemented in Feynrules. For each point in our parameter space we generate signal events for our process using Madgraph5. We shower the events with Pythia and run them through the detector simulator PGS. We then run a cut based the analysis on the events to determine the search efficiency. In Fig. 5.8 we show a plot of signal efficiency vs octet mass mS. In general we find efficiencies similar to those of the 8 TeV analysis. We may then calculate the total number of events Nevt = σ L BRgγ in each invariant × × × mass bin.

105 Efficiency ATLAS j+Γ 14 TeV 0.7

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5 20 fb-1

4 100 fb-1

3 300 fb-1

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0 MS GeV 0 1000 2000 3000 4000 5000 H L Figure 5.8: a.)In the upper plot is the eﬃciency of signal events for S gγ which pass → the selection criteria at 14 TeV center of mass energy. b.)In the lower plot is the projected sensitivity for the 14 TeV run in the photon plus jet resonance channel .

As in the 8 TeV search, the SM background for this process is dominated by γ +j events. In order to estimate the background at 14 TeV we scale up the four parameter fit used to model the SM background in ATLAS’s 8 TeV analysis. We use a simple criteria for estimating sensitivity to new physics. The signal to root background ratio is denoted S/√B where S and B are the expected number of signal and background events respectively. For Poisson statistics, a search is sensitive to the existence of a new physics if S/√B 2 and has discovery potential if S/√B 5. ≥ ≥ 106 We plot the sensitivity for the 14 TeV process gg S gγ in Fig. 5.8. In the plot we → → fixed the benchmark effective operator coefficients Λ2 = 30 TeV and Λ1 = 90 TeV, and then plotted the expected S/√B ratio vs. the octet mass mS. We have shown the sensitivity for various values of the run luminosity. For this benckmark point, we find that a octets of −1 −1 mS . 1.8 TeV have 5 sigma discovery potential in 100 fb of luminosity. In 1 ab octets of mS . 3 TeV have discovery potential in this channel. In the Fig. 5.9 we shown the reach in the Λ1 Λ2 plane for various values of the octet mass. −

EFT Bounds 14 TeV, ab-1 103 104 105 106 106

105 105 mS=1 TeV L

GeV m =2 TeV H S 2 L mS=3 TeV 104 104

103 103 3 4 10 10 105

L1 GeV

Figure 5.9: Projected sensitivity ofH colorL octet searches in the gγ channel for the 14 TeV LHC run. Shown are reached for various valued of the octet mass in the Λ2 Λ1 plane. − Solid lines indicate the 2 sigma reach while dotted lines indicate 5 sigma reach.

5.7 Biadjoint Operators

For the Monohar Wise model, a color octet scalar is also an SU(2) doublet Su = (8, 2)1,Sd =

(8, 2)−1 [132]. We may write the most general terms eﬀective Lagrangian which couples these ﬁelds to pairs of SM gauge bosons,

107 1 abc † µν µν 1 abc µν µν 1 † a µν µν L = 2 d H SuaGb Gc + 2 d HSdaGb Gc + 2 H SuGa B Λgg1 Λgg2 Λgb1 1 µν µν 1 † µν µν 1 µν µν + 2 HSdGa B + 2 [H W Su]Ga + 2 [HW Sd]Ga Λgb2 Λgw1 Λgw2

Were Gµν is the SU(3) field strength tensor, W µν is the SU(2) field strength tensor and Bµν is the U(1) fields strength tensor. In the first 4 terms SU(2) indices of the scalar octets are contracted with those of the Higgs. In the last two operators SU(2) indices are contracted between the Higgs, scalar octet and W µν as indicated by brackets. Finally we may consider the effective Lagrangian of a biadjoint under SU(3) and SU(2)

1 j µν i 1 † j µν i 1 † j µν i L = SaGa Wµν + 3 [H SaH]Ga Bµν + 3 [H SaH]Ga Gµν (5.0) Λsgw Λsgb Λsgg

5.8 Conclusions

We have considered a set of effective dimension 5 operators through which SM adjoint scalars couple to pairs of SM gauge bosons. These operators may present themselves in a variety of models. We have considered supersymmetric and non-supersymmetric UV completions for these scenarios including the popular set of ’R-symmetric’ SUSY models, which demand the existence of scalar fields which are adjoints under the SM gauge groups. These operators open up new phenomenological possibilities in the search for scalar adjoint states. In this chapter we study the phenomenology of the production and decay of a single color octet scalar which couples to pairs of SM gauge bosons, gg, Zγ, Zg. We have calculated bounds on this scenario from several analyses from the 8 TeV run of LHC. We have placed lower limits on effective operator coefficients for a variety of octet masses using γ+jet, dijet, monojet and V+j search channels. We have also explored the discovery potential of the octet scalar in the gluon-photon final state for the 14 TeV run of LHC. These operators open the possibility for a variety of new production and decay channels for scalar octet searches. For example, the decays of the octet scalar in the Z g channel alone miss include such final state topologies as jet plus a hard dilepton pair, as well as jets plus ET , and jet+hadronic tagged Z topologies. Though the branching fraction into leptons is small, this may be an interesting low background channel to search in for new states. Similarly interesting decay channels exist for equivalent operators pertaining to SU(2) adjoint scalars. We also note that we have here considered only single scalar octet production through gluon fusion. There are, however, stranger production channels to be studied. For example, even if the Sgg coupling were somehow small, the Sgγ coupling would allow S production through the processes gg > g∗ > Sγ and even gg > g∗ > SZ. Models with such adjoints have a

108 variety of collider topologies which would be an interesting topic of further study. As discussed in the section on high energy models, the dimension 7 operators listed in Eqn 3. are produced at loop level by integrating out heavy squarks/sleptons in R-symmetric SUSY models. This set of operators presents more possibilities for phenomenological study if the operators are not too suppressed. For example once Higgs vevs are inserted, the operators allow production of the SU(2) adjoint though gluon fusion. The decay of this adjoint may have a striking leptonic signature. More complex scenarios which involve SM adjoints may have even more striking signatures. For example, in the Monohar-Wise model there exists scalar color octets which are fundamentals and anti-fundamentals of SU(2). Effective couplings of these adjoints to pairs of vector bosons may lead to very interesting decay chains. Finally we note that UV completions of our models predict correlations of signals across multiple channels. A combinations of final state searches will be the most powerful tool to rule on the parameter space of these models. This will also help to determine to the correct UV physics if a scalar adjoint signal is seen in a channel. For example, in an R-symmetric- MSSM scenario the detection of a scalar color octet in the dijet resonance channel would demand a specific rate in the j+photon channel as well. Measuring these rate would give valuable information about the messenger and squark sectors of the theory.

109 Chapter 6 Conclusion

Physics beyond the Standard Model is important for furthering our understanding of nature at the most fundamental level. Looking at BSM phenomenology is therefore extremely vital in being able to sort through the plethora of BSM models to identify what will explain experimental data. In Chapter 2, we discussed DM constraints from the Fermi-LAT telescope in cases where multiple annihilation channels were present using EFTs to characterize DM-SM interactions. Bounds on DM mass and EFT coefficients were placed using fixed total annihilation rates using a triangular visualization for third generation fermions and we were able to place bounds on DM masses in the range of 10-100’s of GeV. We also discussed the case where we have interfering channels which are present simultaneously; in particular we set limits on the case of DM annihilation into diboson final states where gauge invariance requires multiple diboson final states simultaneously. This has the striking result that in cases where DM annihilates to EW bosons there always will exist at photon line in either a γγ or Zγ final state which will place powerful bounds on DM mass. In comparing to the collider constraints for mono-W signals, we find that the dwarf constraints on DM EFTs are more powerful when the DM mass is above a few hundred GeV. The dwarf constraints also lack issues with EFT validity unlike collider constraints where the use of an EFT is questionable. Chapter 3 extended the work presented in Chapter 2 by using a more sophisticated technique of combining data from multiple dwarf spheroidal galaxies and extending the EFT framework by discussing simplified models. We combined the dwarf spheroidal galaxies in the Fermi-LAT dataset by using a joint-likelihood analysis which is able to provide much more powerful constraints in DM parameter space. We presented bounds on DM annihilation to multiple third generation fermions using a triangular visualization for DM mass and EFT coefficients. Using s- and t-channel simplified models as completions of EFTs we set bounds in the space of mediator mass vs. mediator-SM couplings. In addition we compared bounds for simplified models with bounds on appropriate effective operators where we showed that for high mediator mass the simplified model bounds map onto the

110 EFT bounds. Chapter 4 explored using a minimal Dirac gaugino SUSY model in order to explain a diphoton excess seen at the ATLAS and CMS experiments. We used the sbino, the scalar partner of the field which gives the bino a Dirac mass, as a candidate to describe this excess. The sbino will communicate with SM dibosons through loops of right-handed squarks and sleptons; allowing the sbino to be produced via gluon fusion and decay to pairs of photons. In this scenario, we found that if the diphoton excess was confirmed that ZZ and Zγ channels would be observed soon after as the production rates into these channels is only a few below the diphoton channel. The rate into diHiggs decay is found to be a major channel for detection of this signal and with the 8 TeV run was the most found to be the channel which constrained the model the most. We also considered conditions under which our minimal model will have a stable scalar potential without generating charge breaking minimas and we found that vacuum stability conditions favor small values of the Dirac mass. Finally, in Chapter 5 we discussed using effective operators of dimension 5 to look for scalar particles which are adjoints under SM gauge groups. These types of scalars can show up in many BSM scenarios such as in Dirac gaugino SUSY models discussed above. We studied the phenomenology of production of a single color-octet scalar at the LHC, where we use an EFT to couple the scalar to the SM diboson final states: gg, gγ, Zg. We recast several existing searches in the 8 TeV run of the LHC to place bounds on EFT coefficients in this scenario and found that the jet+photon final state to be of comparable sensitivity to a dijet final state. Projections for the discovery potential and expected limits from 14 TeV run of the LHC were also presented for the jet+photon final state. For other adjoint scalars we write down a complete list of effective operators and note that there are many interesting event topologies which are interesting channels for further studies.

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