Direct detection for Dark Matter in G2HDM and Z' search with Dark Matter at the LHC

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(1)Direct detection for Dark Matter in G2HDM and Z ′ search with Dark Matter at the LHC. Yu-Xiang Liu Supervisor: Professor Chuan-Ren Chen Department of Physics National Taiwan Normal University. This dissertation is submitted for the degree of Doctor of Philosophy. August 2020.

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(3) All is well that ends well. This work is dedicated to those good peoples that crossed my world-line. Most of all, to my mother..

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(5) Declaration I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other University. This work was done wholly or mainly while in candidature for a Ph.D degree at this University. This dissertation is my own work and jointly with others. Yu-Xiang Liu August 2020.

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(7) Acknowledgements 首先我要感謝我的指導教授，陳傳仁教授多年來的耐心指導與支持，不管是在研究上或是在人生方面。謝謝阮志強教授的指導，我從您身上學到許多。 Many thanks to my working partner Van Que Tran and Chrisna Setyo Nugroho, thank you for your help and kindness, I am very happy that we were working together. And thanks to Dr. Raymundi Ramos and Dr. Yue-Lin Sming Tsai for their advices and help. 謝謝Jason熱心的幫助，跟你一起做研究不會感到特別有壓力，卻又學到許多。感謝我的同學們：先至、銘杰、又澤、容格、煜彬、昌宏，有你們在身邊讓學校的生活感覺比較不枯燥。感謝我的朋友們：怡伶、JC、紅心、黑面、思翰、力宏、小光...以及其他沒有在此提到，但非比較不重要的人，很幸運有你們的陪伴。謝謝身邊許多人，支持我度過許多難關，我很感激。唯一的遺憾就是沒能讓媽媽看到我畢業，但希望我有讓你感到驕傲。 As my favorite motto says "It is good to have an end to journey towards, but it is the journey that matters, in the end." Thank you for all your suppoert in my journey of life..

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(9) Abstract In this thesis, we will study dark matter phenomenology in few different aspects. First, we study the complex scalar dark matter phenomenology in Gauged Two Higgs Doublet Model (G2HDM). The model has additional gauge SU (2)H × U (1)X that will leave an accidental Z2 symmetry after symmetry braking, which protects the stability of dark matter. We choose the scalar dark matter for our study, which can further categorized into doublet-like, triplet-like, and Goldstone-like dark matter by their composition of mixing elements. We test these dark matter by current constraints from PLANCK and XENON1T experiments. In the second part we study the dark matter search by means of the popular Higgs pair search in collider with an effective model. We use jet substructure technique for better resolution to find our signal from tremendous backgrounds at the LHC, and project the discovery rate for the High-Luminosity LHC. Keywords: Complex Scalar Dark Matter, Gauged Two Higgs Doublet Model, Jet Substructure, Pair Higgs Production Thesis Supervisors: Professor Chuan-Ren Chen.

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(11) Table of contents List of figures. xiii. List of tables. xvii. 1 Prelude. 1. 2 Brief Review of the Standard Model 2.1 SM Particles . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Gauge Sector . . . . . . . . . . . . . . . . . . 2.1.2 Fermion Sector . . . . . . . . . . . . . . . . . 2.2 The SM Higgs mechanism . . . . . . . . . . . . . . . 2.2.1 Masses of Gauge Boson and Coupling to Higgs 2.2.2 Masses of Fermion and Coupling to Higgs . . 3 Brife Review of Dark Matter 3.1 Weakly Interacting Massive Particle 3.2 WIMP DM Searches . . . . . . . . 3.2.1 Direct Detection . . . . . . 3.2.2 Indirect Detection . . . . . . 3.2.3 Collider Search . . . . . . . 4 Dark Matter in G2HDM 4.1 The G2HDM Model . . . . . . 4.1.1 Introduction . . . . . . . 4.1.2 Particles Content . . . . 4.1.3 The SSB of the Potential 4.1.4 Scalar Sector . . . . . . 4.1.5 Gauge Sector . . . . . . 4.1.6 Fermion Sector . . . . .. . . . . . . .. . . . . . . .. . . . . . .. . . . . . .. . . . . . .. and Relic Abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . .. . . . . .. . . . . . . .. . . . . . .. . . . . .. . . . . . . .. . . . . . .. . . . . .. . . . . . . .. . . . . . .. . . . . .. . . . . . . .. . . . . . .. . . . . .. . . . . . . .. . . . . . .. . . . . .. . . . . . . .. . . . . . .. 5 5 6 7 8 11 14. . . . . .. 19 20 22 23 25 25. . . . . . . .. 27 27 27 28 29 31 33 35.

(12) 4.2. 4.3. 4.1.7 Hidden symmetry . . . . . . . . . . . 4.1.8 DM Candidate in G2HDM . . . . . . Numerical Analysis . . . . . . . . . . . . . . 4.2.1 Constraints and Range of Parameters 4.2.2 Relic Density . . . . . . . . . . . . . 4.2.3 Direct Detection . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . 4.3.1 Doublet-like DM . . . . . . . . . . . 4.3.2 Triplet-like DM . . . . . . . . . . . . 4.3.3 Goldstone-like DM . . . . . . . . . . 4.3.4 Summary . . . . . . . . . . . . . . .. 5 Z ′ Search with Dark 5.1 Benchmark model 5.2 Collider Study . . 5.3 Summary . . . .. Matter . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 35 36 37 37 39 40 42 43 45 46 48. in LHC 51 . . . . . . . . . . . . . . . . . . . . . . . . . 52 . . . . . . . . . . . . . . . . . . . . . . . . . 53 . . . . . . . . . . . . . . . . . . . . . . . . . 60. 6 Summary. 63. References. 65.

(13) List of figures 3.1. The mass range in this figure are only approximate. Figure is taken from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Evolution of DM number density per comoving volume through the thermal freeze-out, where the solid line show the equilibrium value of comoving number density in different time, and dashed line show the actual relic density for the different value of σA v. Figure is taken from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 There are serveral different ways for searching DM, categorize by different type of interaction. Figure is taken from [3]. . . . . . . . . . . . . . . . 3.4 The current direct detection experimental limits of spin-independent WIMP-nucleon cross section. Figure is taken from [4]. . . . . . . . . . 4.1 4.2. 4.3. 20. 22 23 24. The relation between the ratio v∆ /vΦ and parameter fGp for all the scanned DM parameter space after SGSC. . . . . . . . . . . . . . . . . 37 The dominant Feynman diagrams of direct detection with Z gauge boson and Higgs boson as mediator. The diagram with heavy fermions mediator can be ignored for its insignificant contribution. . . . . . . . . 41 Doublet-like DM SGSC allowed regions projected on (mD , ΩD h2 ) (left) and (mD , σnSI ) (right) planes. The gray area in the left panel has no coannihilation or resonance. The gray area in the right panel is excluded by PLANCK data at 2σ. . . . . . . . . . . . . . . . . . . . . . . . . . 43.

(14) 4.4. Triplet-like DM SGSC allowed regions projected on (mD , ΩD h2 ) (left) and (mD , σnSI ) (right) planes. The gray area in the left panel has no coannihilation or resonance. The gray area in the right panel is excluded by PLANCK data at 2σ. In the right panel, the lower red solid line is the published XENON1T limit with isospin conservation, while the upper green solid line is the same limit with ISV with fn /fp = −0.5. Some of orange filled squares are above the published XENON1T limit due to ISV cancellation at nucleus level. . . . . . . . . . . . . . . . . . 4.5 Goldstone-like DM SGSC allowed regions projected on (mD , ΩD h2 ) (left) and (mD , σnSI ) (right) planes. The gray area in the left panel has no coannihilation or resonance. The gray area on the right is excluded by PLANCK data at 2σ. In the right panel, the lower red solid line is the published XENON1T limit with isospin conservation, while the upper green solid line is the same limit with ISV of fn /fp = −1.5. The small region of orange filled squares above the published XENON1T limit present ISV cancellation at nucleus level. . . . . . . . . . . . . . 4.6 A summary plot for the scalar potential parameter space allowed by the SGSC constraints (green region) and SGSC+RD+DD constraints (red scatter points) for the triplet-like DM. The numbers written in the first block of each column are the 1D allowed range of the parameter denoted in horizontal axis after the SGSC+RD+DD cut. . . . . . . . 4.7 A summary plot for the VEVs, MΦ∆ , gX and gH parameter space allowed by the SGSC constraints (green region) and SGSC+RD+DD constraints (red points) for the triplet-like DM. . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6. Higgs pair production accompanies ETmiss induced by a pair of heavy gauge boson Z ′ decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross section of pp→Z ′ Z ′ as mZ ′ varied from 500 GeV to 2 TeV at the 14 TeV LHC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The correlation of transverse momentum of Higgs and ∆R between its two daughter b-quark. The green line represent the relation 5.7. . . . . The symmetric distribution of two Higgs pT are indicate they are both daughter of heavy particles decay. . . . . . . . . . . . . . . . . . . . . . The ETmiss of signals for various masses of Z ′ boson (left) and signal versus various backgrounds. (right) . . . . . . . . . . . . . . . . . . . . The reconstructed invariant mass of Higgs from γγ (left) and fatjet (right) before the (C4) cut. . . . . . . . . . . . . . . . . . . . . . . . .. 45. 47. 49. 50 53 55 56 57 58 58.

(15) 5.7. The discovery potential of pp → Z ′ Z ′ → hhχχ at the 14 TeV LHC. . . 60.

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(17) List of tables 1.1. Table of acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.1. The quantum number of fermion fields under SM gauge, i denotes the generation of fermion. Notice that right-handed neutrinos are absences in SM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. Matter contents and their quantum number assignments in G2HDM. Note that H here is written explicitly as an SU (2)H doublet and, thus, the T stands for SU (2)H transposition. For doublets of a single SU (2) T stands for transposition under that same SU (2). . . . . . . . . . . . 28 4.2 The Z2 assignments in G2HDM model. . . . . . . . . . . . . . . . . . . 36 4.3 Parameter ranges used in the scans mentioned in the text. MX is fixed at 2 TeV in this work and MY is set to be zero throughout the scan. . 39 4.1. 5.1. 5.2. Select cuts and resulting cross sections for the signal and backgrounds of 14 TeV LHC. Here, we set mZ ′ = 500 GeV, mχ = 100 GeV, and ghZ ′ χ = gqq′ Z ′ = 0.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Select cuts and resulting cross sections for the signals of 14 TeV LHC. Here, we set mχ = 100 GeV, and gZ ′ χh = guu′ Z ′ = 0.6. . . . . . . . . . 62.

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(19) Chapter 1 Prelude The Standard Model (SM) is the remarkable achievement of decades of work by thousands of physicists, it describes three fundamental interactions for the elementary particles, except for gravitational interaction. Beside the prediction of Z and W bosons and many other discoveries, it matches almost all the experimental results. Since the force of gravity is so weak, so we only consider it when discusses in macroscopic scale, otherwise, the SM works very well. The Higgs boson discovered in 2012 by ATLAS [5] and CMS [6] at Large Hadron Collider (LHC) fits the frame work of SM, it is responsible to the masses of most particles of SM through Higgs mechanism [7–9], but we still cannot exclude the possibility of other particle beyond SM. Although SM can explain lots of experiment resultsts with great accuracy, but it is inadequate to describe our universe, left with many physics phenomena that cannot explain by SM: • In the current knowledge, we know the universe composite by 70% of dark energy, and 30% of matter. However only 5% are the atomic matter that we knew, and other 25% is still unknown, which we call it Dark Matter(DM). • The observed neutrino oscillation indicates neutrino must have mass, which is contradicted with SM. • The significant difference of quantity between matter and anti-matter in our observation • There are over 20 free parameters in SM, which is unacceptable for a Ultimate theory..

(20) 2. CHAPTER 1. PRELUDE • etc.. To answer all the problems, it will lead us to have more understanding with our universe, hence we need to have Beyond Standard Model (BSM). With many literatures for BSM, most of them can address only few parts of all the problems, in other words, we don’t have an "Ulimate theory" that can solve all the problems. In this thesis, I will show two different approaches to the DM studies. One is using an effective model to demonstrate one of the method to search DM in collider. In this method, we won’t need to constrcut a full physical model, but rather create a part of Lagrangian with the interactions that we are interesting in, and the couplings can be free parameters. And then, we take certain processes that derive from the Lagrangian for the collider simulation. With the results of simulation, we can constraints those parameters by the current experiment data. Another way to discusses the DM is constructing a full model that usually an extension of SM. Then performs the detail analysis to determine whether this model can compatible with current experiments. We will using natural units (ℏ = c = 1) throughout this thesis. Some of the acronyms used in this thesis are listed in Table 1.1..

(21) 3. Acronym 2HDM ATLAS BSM CKM CMB CMS CP DD DM dSphs EM EW EWPT G2HDM HP ID ISC ISV LHC LHT MET PLANCK PU QCD QED RD SGSC SM SSB VEV VS WIMP XENON1T. Description Two Higgs Doublet Model A Toroidal LHC ApparatuS Beyond the Standard Model Cabibbo-Kobayashi-Maskawa Cosmic Microwave Background Compact Muon Solenoid Charge Conjugation and Parity Direct Detection Dark Matter dwarf Spheroids Electromagnetism Electroweak Electroweak Precision Test Gauged Two Higgs Doublet Model Higgs Physics Indirect Detection Isospin Conserved Isospin Violation Large Hadron Collider Little Higgs Model with T-parity Missing Transverse Energy https://www.cosmos.esa.int/web/planck/home Perturbative Unitarity Quantum Chromodynamics Quantum Electrodynamics Relic Density Scalar and Gauge Sector Constraints Standard Model Spontaneous Symmetry Breaking Vacuum Expectation Value Vacuum Stability Weakly Interacting Massive Particle http://www.xenon1t.org Table 1.1 Table of acronyms..

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(23) Chapter 2 Brief Review of the Standard Model We will review some basic features of particles and interactions in SM, and the Higgs mechanism to give masses for particles.. 2.1. SM Particles. The particles have been separating from two sectors: gauge sector and fermion sector. Fermions are partilcle with spin- 12 . There are two types of fermions which have been categorized into quarks and leptons by there quantum number under SM gauge group, each of them has three generations (or families/flavor). For quarks each generation has up-type and down-type, up-quark (u) and down-quark (d) for first generation, charm-quark (c) and strange (s) for second generation, top-quark (t) and bottom-quark (b) for third generation. Up-type quark has 23 of electric charge and down-type quark has −1 of electric charge. Similarly, leptons have electron (e) and electron neutrino (νe ) 3 for first generation, muon (µ) and muon neutrino (νµ ) for second generation, tau (τ ) and tau neutrino (ντ ) for third generation. e µ and τ have +1 for electric charge, and neutrinos are chargeless (also massless in SM). Since fermions have half-integer spin, each of them has anti-particles∗ , which has the same mass, spin and lifetime, but with different electric charge and baryon/lepton number. There are three different types of boson to mediate three different interactions, gluon(g) for strong interaction, Z/W ± for weak interaction and photon (γ) for electromagnetic interaction. ∗. Except for Majorana fermion, but it’s not included in SM.

(24) 6. CHAPTER 2. BRIEF REVIEW OF THE STANDARD MODEL. The SM gauge structure is SU (3)c × SU (2)L × U (1)Y , which corresponds to strong interaction (subscript c stands for color), and electroweak interaction (subscript L referred to it couple to left-hand fermions, and subscript Y referred to hypercharge). Quarks comes in three different colors (red, blue and greed), and gluon comes in eight different states.† Confinement restrict that all the particles has to be color singlet, thus we cannot observe individual quark or gluon. Last, SM has one scalar boson that give the mass to particles through Higgs mechanism. We will discuss more about gauge bosons, fermions and Higgs in the following section.. 2.1.1. Gauge Sector. We start with Lagrangian of gauge boson kinetic terms: 1 1 a µν 1 a µν LGauge = − Bµν B µν − Wµν Wa − Gµν Ga 4 4 4. (2.1). repeated indices are summed. Each individual term represents different gauge group. The field strength terms are given below: For the U (1)Y interaction, the field strength is as same as in electromagnetism: Bµν = ∂µ Bν − ∂ν Bµ. (2.2). The next two groups are non-Abelian gauge group which has extra terms. The field strengh for SU (2)L is: a Wµν = ∂µ Wνa − ∂ν Wµa + gεabc Wµb Wνc. (2.3). where g is weak interaction coupling, and εabc is structure constant of SU (2) which a,b,c take values from 1 to 3. For SU (3)c , the form is similar to SU (2): Gaµν = ∂µ Gaν − ∂ν Gaµ + gs f abc Gbµ Gcν. (2.4). with different coupling gs and structure constant is f abc where a,b, and c take values from 1 to 8. †. Under SU (3)c symmetry group, we have night different states: a color octet, and a color singlet, but if gluon can have color singlet state, we should observe it as free particle, which contradicts to the experiment results..

(25) 7. 2.1. SM PARTICLES. To preserve the gauge invariance, we need to promote the derivative to covariant derivative. With this change, we get terms of gauge bosons interaction to fermions and scalar through the covariant derivative: Dµ = ∂µ − ig ′ Bµ Y − igWµa T a − igs Gaµ ta. (2.5). where g ′ is the coupling of hypercharge interaction. T a is the SU (2) generator, when taking the doublet representation it is just Pauli matrices divide by two: σ a /2, and ta is the SU (3) generator, which are Gell-Mann matrices. The Lagrangian in eq.(2.1) is invariant under infinitesimal transformation of U (1)Y , SU (2)L and SU (3)c :. U (1)Y :. ϕ′ = exp(iλY (x)Y )ϕ. SU (2)L :. ϕ′ = exp(iλaL (x)T a )ϕ. SU (3)c :. ϕ′ = exp(iλac (x)ta )ϕ. 1 ∂µ ϵY (x) g′ 1 Wµa ′ = Wµa + ∂µ ϵaL (x) + εabc Wµb ϵcL (x) g 1 Gaµ ′ = Gaµ + ∂µ ϵac (x) + f abc Gbµ ϵcc (x) gs Bµ′ = Bµ +. (2.6). where ϕ can be fermion of scalar field. However, the gauge invariance forbids massive gauge bosons, for example: 1 1 1 1 2 mB Bµ B µ ̸= m2B (Bµ + ′ ∂µ ϵY (x))(B µ + ′ ∂ µ ϵY (x)) 2 2 g g. (2.7). Similar argument holds in SU (2)L and SU (3)c ‡ .. 2.1.2. Fermion Sector. Table (2.1) are the SM fermion content. The SM fermions are chiral fermion which transformation differently under SU (3)c × SU (2)L × U (1)Y . Acting the projection operators: 1 1 PR = (1 + γ 5 ), PL = (1 − γ 5 ) (2.8) 2 2 on an unpolarized Dirac spinor to get left-handed and right-handed chiral fermion states: PR ψ ≡ ψR , PL ψ ≡ ψL (2.9) ‡. Gluons are keep massless since Higgs didn’t carried SU (3)c charge.

(26) 8. CHAPTER 2. BRIEF REVIEW OF THE STANDARD MODEL uL dL. QLi = U (1)Y SU (2)L SU (3)c. 1/6 2 3. !. uRi i. dRi. LLi. 2/3 −1/3 1 1 3 3. νL = eL −1/2 2 1. !. eRi i. −1 1 1. Table 2.1 The quantum number of fermion fields under SM gauge, i denotes the generation of fermion. Notice that right-handed neutrinos are absences in SM.. For the adjoint spinor, we can use the Hermitian property of γ 5 , and the anticommutation relation {γ µ , γ 5 }: ¯ R = ψ † γ 0 PR = ψ † PL γ 0 = (PL ψ)† γ 0 = ψ¯L ψP. (2.10). ¯ R = ψ¯L . We can use the properties of projection operators PR + PL = 1 Similarly ψP and PR 2 = PR , PL 2 = PL to rewrite the Dirac Lagrangian in terms of chiral fermion fields: ¯ µ γ µ ψ − mψψ ¯ L =ψ∂ ¯ µ γ µ (P 2 + P 2 )ψ − mψ(P ¯ 2 + P 2 )ψ = ψ∂ R L R L (2.11) µ µ ¯ L ∂µ γ PR ψ + ψP ¯ R ∂µ γ PL ψ − mψ¯L ψR − mψ¯R ψL = ψP = ψ¯R ∂µ γ µ ψR + ψ¯L ∂µ γ µ ψL − mψ¯L ψR − mψ¯R ψL The kinetic part seperated into one term with only ψL and one with only ψR . After promoting the derivative to covariant derivative, both terms will be gauge invariant for all the fermions list in Table (2.1). Since ψR is SU (2)L singlet, it does not carried SU (2)L charge, hence the transformation under SU (2)L × U (1)Y for fermion with different chirality is different: a. ψL′ = eiλL (x)T. a +iλ. Y. (x)Y. ψL. ψR′ = eiλY (x)Y ψR. (2.12). Therefore, the mass term of fermions is not gauge invariant, which implies that all fermions of SM are massless.. 2.2. The SM Higgs mechanism. To retain the gauge invariance of Lagrangian but also have massive particle at same time, we introduce a SU (2)L -doublet scalar field with a very specific potential that.

(27) 9. 2.2. THE SM HIGGS MECHANISM. will make the vacuum not invariant under SU (2)L × U (1)Y which we call spontaneous symmetry breaking (SSB). The SM Higgs field can be written as: . . . . ϕ1 + iϕ2  ϕ+ 1 ϕ= 0 = √  2 ϕ3 + iϕ4 ϕ. (2.13). Where ϕ1 , ϕ2 , ϕ3 and ϕ4 are real scalar fields and the √12 is the normalization constant that keeps the kinetic term of ϕ with correct normalization L ⊃ 12 ∂µ ϕi ∂ µ ϕi . Next, we assign ϕ to have hypercharge Y = 12 (We will elaborate this choice later), and it chosen to be color singlet since it will not participated in strong interaction. Now we write the Lagrangian related to ϕ field: Lϕ = (Dµ ϕ)† (Dµ ϕ) − V (ϕ) + LY ukawa. (2.14). The first term contains kinetic term of ϕ and interaction with gauge bosons through covariant derivative. The second term is the potential of ϕ, and the third term is the Yukawa couplings between ϕ and fermions. We will construct the potential to have SSB. The general form of potential we can have by ϕ is: V (ϕ) = −µ2 ϕ† ϕ + λ(ϕ† ϕ)2 (2.15) Where the µ2 is positive, so we can have nonzero vacuum value which will break the SU (2)L × U (1)Y symmetry, but leave U (1)EM symmetry§ . By minimizing the potential we will get: µ2 (2.16) ϕ† ϕ = 2λ Then we expand ϕ† ϕ in terms of its component in eq.(2.13): 1 ϕ† ϕ = (ϕ21 + ϕ22 + ϕ23 + ϕ24 ) 2. (2.17). In the convention we choose only ϕ3 component get the non-zero vacuum expectation value (VEV): s µ2 ⟨ϕ3 ⟩ ≡ v = , ⟨ϕ1 ⟩ = ⟨ϕ2 ⟩ = ⟨ϕ4 ⟩ = 0 (2.18) λ §. Which will leaving photon massless.

(28) 10. CHAPTER 2. BRIEF REVIEW OF THE STANDARD MODEL. Define a new field around the vacuum: ϕ3 = h + v. (2.19). Where ⟨h⟩ = 0. Now we have: . . . . ϕ1 + iϕ2  ϕ+ 1 ϕ= 0 = √  2 v + h + iϕ4 ϕ. (2.20). However, we can expand the field in another form: . . 3 X 0  1 ϕ = √ exp(i ξ a (x)σ a )  2 v+h a=0. (2.21). For σ a are the Pauli matrices with a = 1, 2, 3, and σ 0 = I2×2 . The gauge transformation of ϕ are: i For U (1)Y ϕ′ = e 2 λY (x) ϕ (2.22) i a a For SU (2)L ϕ′ = e 2 λ (x)σ ϕ We can choose a special value for λ’s, λY (x) = −2ξ(x) and λa (x) = 2ξ a , then the field will become:   1  0  (2.23) ϕ= √ 2 v+h It has only h field left. By choosing this special gauge (unitarity gauge) we can reduce the degree of freedom from four to one¶ , which means when we break the symmetry to have a physical field h and a non-zero VEV. Now the potential can be rewriten by using eq.(2.23): λ V = constant + λv 2 h2 + λvh3 + h4 4. (2.24). Recall the scalar mass term is in the form of 12 m2ψ ψ 2 , thus we can see that only h field √ got the mass mh = 2λv 2 . ¶ Where the SU (2)× U (1)Y are reduce to U (1)EM , and the remaining degree of freedom indicate the Photon is massless..

(29) 11. 2.2. THE SM HIGGS MECHANISM. 2.2.1. Masses of Gauge Boson and Coupling to Higgs. Moving on to the kinetic part in eq.(2.14). Recall the covariant derivative in eq.(2.5) and gs = 0 for Higgs since it participated in strong interaction, we can write: . . − 2i g(Wµ1 − iWµ2 )(v + h)  1 Dµ ϕ = √  2 ∂µ h + 2i (gWµ3 − g ′ Bµ )(v + h). (2.25). then we can expand the kinetic term as: 1 1 (Dµ ϕ)† (Dµ ϕ) = (∂µ h)(∂ µ h) + g 2 (v + h)2 (Wµ1 − iWµ2 )(W 1µ + iW 2µ ) 2 8 1 2 + (v + h) (−g ′ Bµ + gWµ3 )2 8. (2.26). The first term is just kinetic term of h. For the second term, we can rewrite as combination of W 1 ± W 2 : Wµ+. Wµ1 − iWµ2 √ , = 2. Wµ−. Wµ1 + iWµ2 √ = 2. (2.27). The the second term become: 1 2 g (v + h)2 Wµ+ W −µ 4 g2 g 2 v 2 + −µ g 2 v Wµ W + hWµ+ W −µ + hhWµ+ W −µ = 4 2 4. (2.28). The mass term of charged boson is in the form of m2 W + W − , thus the W boson mass is: gv MW ± = (2.29) 2 The rest of terms in eq.2.28 are representing the interaction between W and Higgs, by reading the parameters we can write down the couplings:‖ hWµ+ Wν− : hhWµ+ Wν− ‖. :. g 2 v µν M2 g = igMW g µν = 2i W g µν 2 v 2 g 2 µν MW 2! · i g = 2i 2 g µν 4 v i. (2.30). A coupling is obtained by pick up the parameter of that interaction in Lagrangian, then multiplying by i, and n! for numbers of identical particle participate in the interaction..

(30) 12. CHAPTER 2. BRIEF REVIEW OF THE STANDARD MODEL. Now we move on to the third term of the Lagrangian. (−g ′ Bµ + gWµ3 )2 = Wµ3 , Bµ. . . .  . g 2 −gg ′  Wµ3   Bµ −gg ′ g ′2. (2.31). We can easily check that the determinant of the matrix is zero, that’s means it must have one zero eigenvalue, hence one of the combination of Wµ3 and Bµ is massless∗∗ . With little calculation: eigenvalue,. eigenvector,  . λ = 0,. √. 1. g 2 +g ′ 2. g′ g.  ,. . λ = (g 2 + g ′ 2 ), √. 1 g 2 +g ′ 2. mixing-state √. 1 g 2 +g ′ 2. (g ′ Wµ3 + gBµ ). g  , √ 21 ′ 2 (gWµ3 − g ′ Bµ ) g +g −g ′. . We can rewrite the parameters as trigonometric functions √ √. g′. g 2 +g ′ 2. (2.32). . g g 2 +g ′ 2. = cosθW = cW ,. = sinθW = sW , where θW is the weak mixing angle or Weinberg angle. Now. we redefine the states:. cW Wµ3 + sW Bµ =Aµ. Photon. cW Wµ3 − sW Bµ =Zµ. Z-boson. (2.33). Now the third term of eq.(2.26) can be written in terms of Zµ : (g 2 + g ′ 2 ) (g 2 + g ′ 2 )v 2 (g 2 + g ′ 2 ) (g 2 + g ′ 2 ) (v + h)2 Zµ Z µ = Zµ Z µ + hZµ Z µ + hhZµ Z µ 8 8 4 8 (2.34) 1 2 µ The mass term of neutral gauge boson is in the form of 2 m Zµ Z , so the mass of Zµ is: q. MZ =. g2 + g′2v 2. (2.35). Similar to Wµ , we can write down the couplings between Z and Higgs: hZµ Zν : hhZµ Zν : ∗∗. M2 (g 2 + g ′ 2 )v µν g = 2i Z g µν 4 v 2 ′2 M2 (g + g ) µν 2! · 2! · i g = 2i 2Z g µν 8 v. 2! · i. Because it didn’t couple to Higgs field. (2.36).

(31) 13. 2.2. THE SM HIGGS MECHANISM. Although g and g ′ are free parameters, the SM makes no absolute predictions for MW and MZ , but by measuring MW and g, we can calculate v ≃ 246 GeV, then the couplings of hWµ+ Wν− and hhWµ+ Wν− are determined. We can determine g ′ , the couplings of hZµ Zν , and hhZµ Zν by further measuring MZ . Now we rewrite the covariant derivative in terms of physical fields we just discuss. 1 1 Wµ1 T 1 + Wµ2 T 2 = (Wµ1 − iWµ2 )(σ 1 + iσ 2 ) + (Wµ1 + iWµ2 )(σ 1 − iσ 2 ) 4 4 1 1 = √ Wµ+ σ + + √ Wµ− σ − 2 2 1 = (W + T + + W − T − ) 2 √ Where σ 1 + iσ 2 = 2σ + , σ 1 − iσ 2 = 2σ − and T ± = 2σ ± . We also have: Bµ =cW Aµ − sW Zµ. (2.37). (2.38). Wµ3 =sW Aµ + cW Zµ The covariant derivative from eq.(2.5) becomes:. g Dµ = ∂µ −igs Gaµ ta −i (Wµ+ T + +Wµ− T − )−iZµ (gcW T 3 −g ′ sW Y )−iAµ (gsW T 3 +g ′ cW Y ) 2 (2.39) We know photon coupled to charged particles, so we examine the photon coupling: gg ′ (gsW T 3 + g ′ cW Y ) = q (T 3 + Y ) ≡ eQ 2 g2 + g′. (2.40). In convention we define electric charge e and electric charge operator Q: gg ′ e= q = gsW = g ′ cW , 2 2 ′ g +g. Q = T3 + Y. (2.41). We can easily check charge operator for Table (2.1), then we will get the correct charge number for fermions. Also the ϕ0 has zero charge. Now we substitute g and g ′ in covariant derivative by e, sW , cW , and Q: Dµ = ∂µ − igs Gaµ ta − i. e e (Wµ+ T + + Wµ− T − ) − i Zµ (c2W T 3 − g ′ s2W Y ) − ieAµ Q 2sW s W cW (2.42).

(32) 14. 2.2.2. CHAPTER 2. BRIEF REVIEW OF THE STANDARD MODEL. Masses of Fermion and Coupling to Higgs. In section 2.1.2 we discuss the mass term of fermion in the form of −mψ¯L ψR and mψ¯R ψL are not gauge invariant. With introduction of the Higgs boson, we can construct combinations of fermion and Higgs field, so that the Lagrangian is invariant under gauge transformation. We start with Lepton, since the right-handed neutrino is absences in SM, the only combination will be: ¯ L ϕeR LeYukawa = − y e e¯R ϕ† LL + y e∗ L h. i. (2.43). The second term are just h.c. (Hermition conjugate) of the first term, which makes the Lagrangian Hermition. Since fermion field is dimension 3/2 and Higgs field is dimension 1, then y e must be a dimensionless constant. In general, y e could be complex constant, but we can just absorbe the phase by rephrasing the fermion field. In the unitarity gaue, we can rewrite the Lagrangian: yev ye ye eR eL + e¯L eR )] = − √ e¯e − √ h¯ ee LeYukawa = − √ [(v + h)(¯ 2 2 2. (2.44). The first term is mass of the electron: yev me = √ 2. (2.45). and the second term is interaction of electron with Higgs, which the coupling is Yukawa coupling of e: ye me h¯ ee : λe = √ = (2.46) v 2 We can see the coupling is proportional to the mass of electron. Similar deduction can applies to mu and tau lepton. The Yukawa couplings are free parameter, so the masses of fermion is not predicted in the SM. We only discussed one generation of leptons, but we know in the SM we have three generations of leptons, the SU (2)L doublet/singlet are transforming in the same way for different generation, so in general we can write down the Lagrangian: LlYukawa =.  3 X 3 X  − yl. . ¯. ij LLi ϕlRj. i=1 j=1.  + h.c.. (2.47).

(33) 15. 2.2. THE SM HIGGS MECHANISM. Now the couplings yije are complex numbers that form a 3 × 3 matrix, contains totally 9 complex couplings. Inserting the Higgs field in unitarity gauge: LlYukawa = −.  ! 3 3 X X v+h  √ yijl ¯lLi lRj  + h.c.. 2. i=1 j=1. (2.48). =(v + h)ℓ¯L Y l ℓR + h.c. Where ℓL = (e1 , e2 , e3 )TL , ℓR = (e1 , e2 , e3 )TR and Y l are 3 × 3 mass matrices of lepton √ with Yijl = 1/ 2yijl . In the above representation we write the leptons in generation space, and we want to find the leptons in mass eigenstate, which means we will diagonalize the mass matrix. We know a matrix can be diagonalize to a real diagonal matrix through the biunitary transformation: †. VRl Y l VLl =. Ml , v. v with Mlij = mli δij = √ yil δij 2. (i, j = e, µ, τ ). (2.49). VLl and VRl are two appropriate 3 × 3 unitary matrices†† . On the other hand, the lepton array in mass eigenstate have become: . . eL   ′ l†  ℓL = VL ℓL = µL  , τL. . . eR   ′ l†  ℓR = VR ℓR = µR   τR. (2.50). Now we can rewrite the Lagrangian in eq.(2.48) in mass eigenstate: l. M ℓR + h.c. LlYukawa =(v + h)ℓ¯′ L v =−. X i=e,µ,τ. mli l¯i′ li′ −. X i=e,µ,τ. yl √i l¯′ i li′ h 2. (2.51). Where ℓ′ = ℓL + ℓR . Since yel , yµl and yτ are free parameters in the SM, so the masses of the charged lepton can only be obtained through experiment. Moving on to the quark sector. As we can see in the lepton case, the Higgs doublet is only interacting with the lower component in the fermion doublet, so we introduce the conjugate doublet which transforms in the same way as the doublet under SU (2) ††. recall that for a unitary matrix U −1 = U †.

(34) 16. CHAPTER 2. BRIEF REVIEW OF THE STANDARD MODEL. transformation:. . . ϕ0∗ ϕ˜ = iσ 2 ϕ∗ =  −  −ϕ. (2.52). With this doublet, we can write down another Lagrangian term: LqYukawa. =.  3 X 3 X  − y u u¯Ri ϕ˜† QLj ij. . +. ∗ yijd d¯Ri ϕ† QLj . + h.c.. (2.53). i=1 j=1. Now the couplings yiju and yijd are complex numbers that form two 3 × 3 matrices, containing totally 18 complex couplings. Let’s discuss the quark masses by inserting the √ ϕ field with (0, v/ 2)T and rewrite the Lagrangian in the form of matrix in generation basis: . LqYukawa. . . . u1 d1   v d v u ¯ ¯ ¯    ⊃ −(¯ u1 , u¯2 , u¯3 )R √ yij u2  − (d1 , d2 , d3 )R √ yij d2   2 2 u3 L d3 L. (2.54). then we define the quark mass matrices as: v Yiju = √ yiju , 2. v Yijd = √ yijd 2. (2.55). Follow the procedure we had discussed in lepton sector, we diagnolize the mass matrices: Mu UR Y UL = , v Md , DR † Y d DL = v †. u. v with Muij = mui δij = √ yiu δij (i, j = u, c, t) 2 v with Mdij = mdi δij = √ yid δij (i, j = d, s, b) 2. (2.56). Where UL,R and DL,R are proper unitarity matrices to diagonal mass matrices  . u′L,R. =. † U l L,R uL,R. u    = , c t L,R.  . d. d′L,R. =. † Dl L,R ℓL,R. =.   s  . b. (2.57). L,R. When we expand the kinetic Lagrangian of fermion, we will have a term relate to charged current: µ. JL+ ∝ u¯L γ µ dL = u¯′L UL γ µ DL† d′L = u¯′L γ µ V † d′L. (2.58).

(35) 2.2. THE SM HIGGS MECHANISM. 17. Where UL† DL ≡ V is the Cabibbo–Kobayashi–Maskawa(CKM) matrix[10][11], which is unitary matrix. In general, CKM is not identity matrix, hence there are mixing between different generation of up-type quark and down-type quark, then we have generation-changing charge current. For the right-handed quark, since they are SU (2)L singlet, means the up-type and down-type right-handed quark are not tie together by charge current, thus the UR and DR have no physical consequence in the SM. On the other hand, the fermion bilinears enter the neutral current interactions (γ and Z mediator) can be writing down as: µ JL0 ∝ u¯L γ µ uL = u¯′L UL γ µ UL† u′L = u¯′L γ µ u′L. (2.59). The neutral current are automatically diagonal in generation basis, so the photon and Z boson couplings are the same in each generation, same argument can apply to downtype quark. As we see the SM do not have flavor changing neutral currents(FCNC) at tree level in SM, and the FCNC induced in 1-loop by W boson are far too small to detect in current technology.[12].

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(37) Chapter 3 Brife Review of Dark Matter Many observations indicate the existence of DM. Historically the first evidence was observed by Fritz Zwicky in 1937[13], he measured mass of the Coma cluster, and found it’s not compatible with the galaxies velocity within the cluster, the mass of visible matter∗ cannot sustain the orbits of galaxies. However, his work was overlooked for many years. Next breakthrough was provided by Vera Rubin et al.[14], they measured the rotation curve of 21 Sc galaxies, and found the rotation velocity grow and approximate to a constant as the radius grow. In contrast, the velocity at radius r related the visible matter is described by the Newtonian gravity v = (GM/r)1/2 was propotional to r−1/2 , which indicates it must have more mass than we can observe, and those matters do not emit or absorb light that we can measure. They only interact through gravity. Later, we have more experiment providing the evidence of exsistence of DM from scale as small as dwarf galaxies to as large as Cosmic Microwave Background (CMB), the detail will not within the scope of this thesis. There are few basic properties that we know about DM: • From "not" observe DM interact with light, hence in most of models, the DM are electric neutral. • DM is "almost" collisionless • DM obey the gravitational interaction like atomic matter. • DM must be stable, otherwise it will decay in the early universe. There are many different types of DM candidate with different mass and different strength of interactions as shown in Fig.3.1, but we will only focus on the Weakly ∗. Although at that time he did not know there has a halo of hot gas in the Coma cluster..

(38) 20. CHAPTER 3. BRIFE REVIEW OF DARK MATTER. Fig. 3.1 The mass range in this figure are only approximate. Figure is taken from [1]. Interacting Massive Particle (WIMP) scenario, which the DM mass are in GeV to TeV range with typical cross sections at weak scale.. 3.1. Weakly Interacting Massive Particle and Relic Abundance. Consider the DM χ interacts with SM particle X through a process: ¯ χχ ¯ ↔ XX. (3.1). In the early universe, when the temperature was high enough, particles were in thermal equilibrium, which means the process of creation and annihilation of particles were equally efficient. The temperature of universe is decreasing as it is exanding. When the temperature was lower than mχ , the creation of particles were supressed. However, the annihilation continued, so the number of density was decreased as the temperature dropped. At some point, the expansion rate of the universe are high enough so that annihilation processes were ineffective, thus the relic abundance of DM are almost remain the constant till today. The period are often addressed as DM freeze-out or decoupling. Follow the Kolb and Turner’s method[15] in deduction of relic density (RD). The evolution of number density n of DM can be described by Boltzmann equation: dn = −3Hn − ⟨σA v⟩ (n2 − n2EQ ) dt. (3.2). We assume the DM can only create and annihilate like in eq.3.1, and nχ = nχ¯ = n. Here, H = a/a ˙ is Hubble parameter that describe the expansion rate of the universe, a is scale factor of the universe and ⟨σA v⟩ is the thermally averge DM annihilation.

(39) 3.1. WEAKLY INTERACTING MASSIVE PARTICLE AND RELIC ABUNDANCE21 cross section times the relative DM velocity. The Big Bang Nucleosynthesis (BBN) happened when the temperature T ≃ O(1) MeV, the heavy particle like DM were created before BBN when it is radiation dominated period[16]. There are three terms on the right-hand side of eq.3.2, each stand for the expansion of the universe, annihilation of DM and creantion of DM. The Boltzmann equation can further represent in another form: . x dY ΓA  Y =− YEQ dx H YEQ. !2. . − 1. (3.3). where Y ≡ n/s, x ≡ mχ /T , s is the entropy density, and ΓA = nEQ ⟨σv⟩ is the equilibrium annihilation rate. Noted the conservation of entropy per comoving volume S = sa3 = constant, thus the number of particles per comoving volume Y ∝ na3 become constant when dY /dx = 0. Such case will lead to ΓA ≪H. The numerical solution of eq.3.3 were showed in Fig.3.2. As the cross section increase, DM can stay in equilibrium longer, which will decoupling in later time with smaller RD. As we discussed before, the universe is at high temperature in the early stage, thus the interaction rate between particles is faster than the expansion of the universe. As the temperature decreases ΓA is decreasing faster than H, and when the annihilation rate drops below the expansion rate of universe, DM will decouple from thermal bath: ΓA (Tf o ) = ⟨σA v⟩T =Tf o nEQ (Tf o )≃H(Tf o ). (3.4). We can get freeze-out temperature Tf o by solving numerically the Boltzmann equation: xf o.   √ 15 mχ 5 m M (a + 6b/x ) χ Pl fo  ≈ ln  √ 3 = 1/2 1/2 Tf o 8 2π g ∗ xf o. (3.5). where a, b are the parameters in the non-relativistic expansion ⟨σA v⟩ = a+b ⟨v 2 ⟩+O(v 4 ), and g∗ is the number of external degrees of freedom at freeze-out(in the SM, g∗ ≈ 120 for T ≈1 TeV and g∗ ≈ 65 for T ≈1 GeV). If the mχ is at GeV to TeV scale, and cross section at weak-scale, freeze-out happened at xf o ≈ 20 − 30, resulting DM density today as: xf o Ωχ h ≈ 0.1 20 2. . . 80 g∗. !. 3 × 10−26 cm3 /s a + 3b/xf o. !. (3.6).

(40) 22. CHAPTER 3. BRIFE REVIEW OF DARK MATTER. Fig. 3.2 Evolution of DM number density per comoving volume through the thermal freeze-out, where the solid line show the equilibrium value of comoving number density in different time, and dashed line show the actual relic density for the different value of σA v. Figure is taken from [2]. Thus we can see that for cross section on the order of 10−26 cm3 /s, will give Ωh2 ≈ 0.1. The current measurment by PLANCK gave Ωh2 = 0.120 ± 0.001[17]. The WIMP scenario can have RD in the correct order, which sometimes so-called "WIMP miracle".. 3.2. WIMP DM Searches. Many BSM provided different kinds of WIMP candidate. With the observation of galaxies rotation curve, CMB, gravitational lensing, etc., we knew the existence of WIMP and their RD today, or constraint the self-interaction strength from collision of.

(41) 23. 3.2. WIMP DM SEARCHES. Fig. 3.3 There are serveral different ways for searching DM, categorize by different type of interaction. Figure is taken from [3]. galxies. However for particles physics properties of WIMP such as mass, and interaction cross sections with SM particles, we need other experiments. For the search of WIMP, the most commend methods are direct detection (DD), Indirect detection (ID), and collider search, each has its advantage and its downside. Fig.3.3. 3.2.1. Direct Detection. As we discussed in the beginning of this chapter, the rotation curve of galaxies indicated the existence of DM, so as our Milky Way. Milky Way was immersed in a clump of DM that so-called Dark Halo, thus we expected many DM passing through Earth at any time. The idea of DD is to set up a bulk of target, waiting WIMP to scatter off the nucleus of the target material and depositing energy that can be measured by detector. Since the interaction rate of SM to DM particles were much smaller than SM particles to SM particles, thus such experiment is set up in underground facility to shield off the interference by cosmic rays. The DD experiment can further categorized into spin-dependent(SD) and spinindependent(SI) interaction. If the WIMP are axial-vector particle, the interactions of WIMP with baryonic matters will depend on the spin of the target material. Whereas if the WIMP are scalar or vector particle, the interaction is independent of the spin of the target material. The SI cross section is given by: σSI =. 4µA 2 [fp Z + fn (A − Z)]2 π. (3.7).

(42) 24. CHAPTER 3. BRIFE REVIEW OF DARK MATTER. Fig. 3.4 The current direct detection experimental limits of spin-independent WIMPnucleon cross section. Figure is taken from [4]. where N stands for a nucleus with mass number A and proton number Z, µA stands for WIMP-nucleus reduced mass, and (fp , fn ) are (proton, neutron) spin-independent effective couplings respectively. Thus, the exclusion limit for cross section depends on the target material; currently, the most stringent limit for mass roughly above 5 GeV were provided by XENON1T with liquid xenon target [18], and mass below 5 GeV to sub- GeV were limited by CRESST-III [19] with scintillating CaWO4 crystals. The limits with other experiments are showed in Fig.3.4. The shaded region in the figure at the bottom are the neutrino floor[20], which are the neutrino from several sources that will perfectly simulate WIMP signal, and need other method in the region below this limit [21]. It is worth mentioning that in some of models, the WIMP interactions with proton and neutron can be different that is so-called Isospin-Violating dark matter[22]. In the case of complex scalar dark matter which we will discuss in the later chapter, the cross section of χ¯ and χ were different in general [23]..

(43) 3.2. WIMP DM SEARCHES. 3.2.2. 25. Indirect Detection. Although the DM annihilation are suppressed after the thermal freeze-out, but it can still happen in the current universe. We are looking into the higher DM density regions like center of galaxies to increase the probability, and hopefully the final product will propagate to Earth to be captured by a satellite or telescope. In contrast, the DD probe the scattering of DM to SM particle, ID search for the DM annihilation that can relate to the cross section at freeze-out, and also the decay of DM. To detect the DM signal we need to exclude the astrophysical background, which has large uncertainties depending on the propagation models. The experiment typically looking into the γ-rays, electrons, positrons, anti-protons and neutrino final states.. 3.2.3. Collider Search. Beside scattering with SM particles, annihilation, another way to probe DM properties is creating DM in colliders. However, the DM would escape the detector without any interactions. Thus, if we only produced DM in the final state, it will not trigger the detector and be recorded as a event. On the other hand, we can look for the DM production associated with visible SM trace. When two protons collides† in the direction of beam pipe of accelerator, the total transverse momentum is approximated to zero. And by the conservation of momentum, the transverse momentum of final state should also add up to zero. Since we cannot detect the DM, the events of DM production will have non-zero total transverse momentum or so-called missing transverse energy(MET or noted as ETmiss ) in the final state. One of most analyzed process is mono-jet[24] search, where the event produce a pair of DM with a gluon or a quark jet from initial state radiation that balance the MET cause by the DM pair. However, there has downside for the collider DM search: the long-lived particle that decay ouside the detector will also record as MET, which we cannot distinguish from DM. The large background from irrelevant events, also the capability of collider will limited the upper bound of DM candidate.. †. ex: Large Hadron Collider (LHC).

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(45) Chapter 4 Dark Matter in G2HDM In this chapter, we will demonstrate a model dependent search for DM. For this purpose, we will first present an interesting model called Gauged Two Higgs Doublet Model(G2HDM)[25], it provides DM candidate and rich phenomenology to our discussion.. 4.1 4.1.1. The G2HDM Model Introduction. As we discussed in the earlier chapter, the SM is inadequate to explain our universe, hence people proposed various models to extend the particle contents of SM. One of most investigated model category for BSM is simply extends the scalar sector of the SM, the most familiar one is Two Higgs Doublet Model(2HDM) and it’s variants[26, 27]. To reduce the complexity in 2HDM, the intuitive choice is imposing certain symmetry, the most popular one is acting Z2 symmetry on the second Higgs doublet. For example, the Inert Higgs Double Model(IHDM)[28]. There are some interesting features of IHDM. One is the lightest neutral component of the second Higgs doublet can be the DM candidate, which the stability is protected by the Z2 symmetry. Another is the Z2 can be used to avoid FCNC at tree-level. However, this Z2 symmetry was imposed by hand without any justification, thus motivated the construction of Gauged Two Higgs Doublet Model(G2HDM). In G2HDM the two Higgs doublet are put together into a doublet H = (H1 , H2 )T of the new nonabelian SU (2)H gauge group. In this construction, the stability of neutral component of H2 is protected by the continuous symmetry SU (2)H , hence it can be a possible DM candidate. With the new gauge, we introduced a SU (2)H doublet and a SU (2)H.

(46) 28. CHAPTER 4. DARK MATTER IN G2HDM. triplet, the VEV of both scalar are generating messes to the new gauge bosons. The VEV of triplet also induced the SSB of SM, while the new scalar H2 do not develop any VEV and the neutral component of H2 could be DM candidate, whose stability is protected by the SU (2)H symmetry and Lorentz invariance. In order to simplify the scalar potential, we are imposing a U (1)X gauge group. To write down the gauge invariant Yukawa couplings under the new gauge group SU (2)H ×U (1)X , we are adding heavy SU (2)L singlet Dirac fermions, where the right-handed components will pair up with SM right-handed fermions to form SU (2)H doublets, while the left-handed components are singlet of SU (2)H , and both are singlet under SU (2)L gauge group.. 4.1.2. Particles Content. The gauge group of G2HDM are extended the SM gauge group to SU (3)c ×SU (2)L ×U (1)Y ×SU (2)H ×U (1)X . Matter Fields. SU (3)C × SU (2)L × SU (2)H × U (1)Y × U (1)X. T. H = (H1 , H2 ) ∆H ΦH. (1, 2, 2, 1/2, 1) (1, 1, 3, 0, 0) (1, 1, 2, 0, 1). QL = (uL , dL )T. (3, 2, 1, 1/6, 0). . UR = uR , uH R. T. (3, 1, 2, 2/3, 1). . T. (3, 1, 2, −1/3, −1) (1, 2, 1, −1/2, 0). NR = νR , νRH. . T. (1, 1, 2, 0, 1). . T. DR = dH R , dR LL = (νL , eL )T. ER = eH (1, 1, 2, −1, −1) R , eR H νL (1, 1, 1, 0, 0) H eL (1, 1, 1, −1, 0) H uL (3, 1, 1, 2/3, 0) dH (3, 1, 1, −1/3, 0) L Table 4.1 Matter contents and their quantum number assignments in G2HDM. Note that H here is written explicitly as an SU (2)H doublet and, thus, the T stands for SU (2)H transposition. For doublets of a single SU (2) T stands for transposition under that same SU (2). The SM Higgs is still within SU (2)L doublet H1 . Another SU (2)L doublet H2 contained a scalar DM candidate. Furthermore, H1 paired up with H2 to form a SU (2)H doublet H = (H1 , H2 )T , but only the H1 will get the VEV. The new fermions were introduced to have anomaly cancellation. Then we introduced a SU (2)H scalar.

(47) 29. 4.1. THE G2HDM MODEL. doublet ΦH which will give masses to the new fermions. Then we arranged the fermions in a way that the SM SU (2)L left-handed fermion doublets are singlet under SU (2)H , while the SM right-handed singlets are paired with right-handed new fermions to form SU (2)H doublet, and the remaining left-handed new fermions are singlet under SU (2)L ×SU (2)H . There are also a SU (2)H triplet ∆H . Both of ∆H and ΦH will give masses to the gauge bosons and both of them are singlet under the SM gauge group. The matter content is summarized in Table.4.1. 4.1.3. The SSB of the Potential. The full Higgs potential can be written into four parts, three self-interaction terms for each scalar and one mixing term. VT = V (H) + V (ΦH ) + V (∆H ) + Vmix (H, ∆H , ΦH ). (4.1). where the self-interaction of H1 and H2 is given by . . 2. . . . V (H) = µ2H H1† H1 + H2† H2 +λH H1† H1 + H2† H2 +λ′H −H1† H1 H2† H2 + H1† H2 H2† H1 (4.2) The self-interaction of ΦH is . V (ΦH ) =µ2Φ Φ†H ΦH + λΦ Φ†H ΦH. 2. (4.3). =µ2Φ (Φ∗1 Φ1 + Φ∗2 Φ2 ) + λΦ (Φ∗1 Φ1 + Φ∗2 Φ2 )2 where ΦH = (Φ1 , Φ2 )T . The self-interaction of the SU (2)H scalar triplet is . . . . V (∆H ) = − µ2∆ Tr ∆2H + λ∆ Tr ∆2H =−. µ2∆. . 2. 1 2 1 2 ∆3 + ∆p ∆m + λ∆ ∆ + ∆p ∆m 2 2 3 . . (4.4). 2. and all the possible interaction between H, Φ and ∆ . . . Vmix (H, ∆H , ΦH ) = + MH∆ H † ∆H H − MΦ∆ Φ†H ∆H ΦH . . . . + λHΦ H † H. . . . . . . Φ†H ΦH + λ′HΦ H † ΦH . . Φ†H H . . + λH∆ H † H Tr ∆2H + λΦ∆ Φ†H ΦH Tr ∆2H. . (4.5).

(48) 30. CHAPTER 4. DARK MATTER IN G2HDM where the ∆H filed can be written in the matrix form. √  ∆ /2 ∆ / 2 3 p √ ∆H =  = ∆†H ∆m / 2 −∆3 /2 . with ∆m = (∆p )∗. and. (∆3 )∗ = ∆3 (4.6). Note the full potential VT in eq.(4.1) is invariant under H1 → H1 , H2 → −H2 , Φ1 → −Φ1 , Φ2 → Φ2 , ∆3 → ∆3 and ∆p,m → −∆p,m as it has Z2 symmetry, but it is not adding by hand in here, but is automatically reproduce by SU (2)L × U (1)Y × SU (2)H × U (1)X , and we will deduce this discrete symmetry can extend to the whole Lagrangian of G2HDM in the later section. The different components in the SU (2)H doublet and triplet have opposite parity. As we mentioned before, the U (1)X is imposed to kill the unwanted terms, thus treating it as global symmetry is sufficient for this purpose. The gauge symmetry of G2HDM will be broken spontaneously by the VEVs of √ √ ⟨H1 ⟩ = (0, v/ 2)T , ⟨Φ2 ⟩ = vΦ / 2, and ⟨∆3 ⟩ = −v∆ . To calculate these VEVs, we first shift the fields around its VEV . . . . . . . −v∆ +δ3 GpH G+  H +     , ∆H =  2 , Φ = H1 = v+h , H = 0 0 H 2 G G vΦ√ +ϕ2 √ + i√ √1 ∆m H20 + i √H2 2 2 2 2. . √1 ∆p 2  v∆ −δ3 2. (4.7) To keep the accidental Z2 symmetry, only those Z2 -even fields get VEV. By putting VEVs back into VT , the total potential becomes VT (v, v∆ , vΦ ) =. 1 4 2 λH v 4 + λΦ vΦ4 + λ∆ v∆ + 2 µ2H v 2 + µ2Φ vΦ2 − µ2∆ v∆ 4 (4.8) 2 2 − MH∆ v 2 + MΦ∆ vΦ2 v∆ + λHΦ v 2 vΦ2 + λH∆ v 2 v∆ + λΦ∆ vΦ2 v∆. The value of VEVs can obtained by minimizing the potential in eq.(4.8) . . 2 2λH v 2 + 2µ2H − MH∆ v∆ + λHΦ vΦ2 + λH∆ v∆ =0. . . 2 =0 2λΦ vΦ2 + 2µ2Φ − MΦ∆ v∆ + λHΦ v 2 + λΦ∆ v∆. . (4.9). . 3 4λ∆ v∆ − 4µ2∆ v∆ − MH∆ v 2 − MΦ∆ vΦ2 + 2v∆ λH∆ v 2 + λΦ∆ vΦ2 = 0. The analytical solution of VEVs can express in the form of function of the scalar potential parameters, which are calculated in Ref.[25]. In general v∆ has three different roots, naturally we pick by minimum energy requirement. Once v∆ is decided, and subsequently we can solve v and vΦ , this indicated after SU (2)H is broken by v∆ , it triggered the symmetry breaking of SU (2)L × U (1)Y × U (1)Y ..

(49) 31. 4.1. THE G2HDM MODEL. The coefficients of quadratic terms for H1,2 in the full potential 4.1 can be read as 1 µ2H − MH∆ · v∆ + 2 1 µ2H + MH∆ · v∆ + 2. 1 2 λH∆ · v∆ + 2 1 2 λH∆ · v∆ + 2. 1 λHΦ · vΦ2 2 1 (λHΦ + λ′HΦ ) vΦ2 2. (4.10). respectively. The positive or negative of the total value of each coefficient will determine the VEV of H1,2 . By proper choice of parameters in eq.(4.10), we can obtain ⟨H1 ⟩ = ̸ 0 and ⟨H2 ⟩ = 0 which will cause the spontaneous breaking of SU (2)L × U (1)L symmetry. Similar argument can be applied to Φ field, we find the coefficients of the quadratic terms for Φ1,2 are 1 µ2Φ + MΦ∆ · v∆ + 2 1 µ2Φ − MΦ∆ · v∆ + 2. 1 2 λΦ∆ · v∆ + 2 1 2 λΦ∆ · v∆ + 2. 1 (λHΦ + λ′HΦ ) v 2 2 1 λHΦ v 2 2. (4.11). With proper choice of parameters, we obtained ⟨Φ1 ⟩ = 0 and ⟨Φ2 ⟩ = ̸ 0 Note the scalar potential VT is Hermitian, implying all the coefficients have to be real value, hence there are no CP-Violation in scalar sector of G2HDM.. 4.1.4. Scalar Sector. In terms of the fields in eq.(4.7), we can express the mass term as a 10 × 10 mass matrix, which block diagonalized into 3 sub-matrix. The first 3 × 3 block is collective of Higgs-like Z2 -even Scalars, which can be written in terms of the S = {h, ϕ2 , δ3 }T basis 2λH v 2  M20 =  λHΦ vvΦ  v (MH∆ − 2λH∆ v∆ ) 2 . vΦ 2. λHΦ vvΦ 2λΦ vΦ2 (MΦ∆ − 2λΦ∆ v∆ ). v (MH∆ − 2λH∆ v∆ ) 2  vΦ  (MΦ∆ − 2λΦ∆ v∆ )  2 3 (8λ∆ v∆ + MH∆ v 2 + MΦ∆ vΦ2 ). . 1 4v∆. (4.12) As we discussed in Ch.2, the mass eigenstates can be obtained by diagonalizing the mass matrix through the biunitary transformation since the elements in matrix are real numbers, the rotation matrix O is orthogonal matrix (OH )T · M2H · OH = Diag(m2h1 , m2h2 , m2h3 ) .. (4.13).

(50) 32. CHAPTER 4. DARK MATTER IN G2HDM. The physical Higgs fields hi (i = 1, 2, 3) is mixture of S, where hi = OijH Sj . We identified the lightest eigenvalue mh1 is the SM 125GeV Higgs, and mh3 > mh2 for other two heavy Higgs. The remaining Z2 -even are the massless Goldstone bosons G± , G0 and G0H , which do not mix with other scalar fields, and will absorbed by the longitudinal components of the gauge fields as in the SM case. Next we consider the Z2 -odd fields. Since H2 couples to all three multiplets H1 , ΦH and ∆H , after SSB it acquires mass from all three VEVs 1 1 m2H ± = MH∆ v∆ − λ′H v 2 + λ′HΦ vΦ2 2 2. (4.14). The second 3 × 3 complex scalar∗ block in terms of the basis of G = {GpH , H20∗ , ∆p } is 1 ′ MΦ∆ v∆ + 12 λ′HΦ v 2 λ vvΦ 2 HΦ  2 1 ′  MD =  λ vvΦ MH∆ v∆ + 12 λ′HΦ vΦ2 2 HΦ 1 − 21 MΦ∆ vΦ MH∆ v 2. . − 12 MΦ∆ vΦ  1  MH∆ v  2 (MH∆ v 2 + MΦ∆ vΦ2 ) . 1 4v∆. (4.15). It is easy to check the determine of this matrix is zero, implies it has at least one massless ˜ p with m ˜ p = 0. We eigenstate, which we identified as unphysical Goldstone boson G G ˜ respect to mD < m ˜ . The assign the remaining two physical states as D and ∆ ∆ analytical expression can be solved as m2D,∆ e =. −B ∓. √ B 2 − 4AC , 2A. . . (4.16). where the A, B, C parameters are A =8v∆ . . . . 2 2 B = − 2 MH∆ v 2 + 4v∆ + MΦ∆ 4v∆ + vΦ2 + 2λ′HΦ v∆ v 2 + vΦ2. . 2 C = v 2 + vΦ2 + 4v∆. . . . MH∆ λ′HΦ v 2 + 2MΦ∆ v∆ + λ′HΦ MΦ∆ vΦ2. . (4.17). . As the result, the possible DM in scalar sector can only be D, since the other Z2 -odd choice H ± has electric charge, which is intolerable for DM candidate. For this reason, we required mH ± > mD in our discussion. ∗. By our definition in eq.(4.7), these three fields are complex.

(51) 33. 4.1. THE G2HDM MODEL. 4.1.5. Gauge Sector. The gauge boson can be read from the kinetic terms of scalars h. i. L ⊃ Tr Dµ′ ∆†H D′µ ∆H + Dµ′ Φ† D′µ Φ + Dµ′ H † D′µ H. (4.18). For their different quantum number assignment, the covariant derivative reads as h. Dµ′ ∆H = ∂µ ∆H − igH Wµ′ , ∆H. i. (4.19) !. Dµ′ Φ. gH = ∂µ − i √ Wµ′p T p + Wµ′m T m − igH Wµ′3 T 3 − igX Xµ · Φ 2. (4.20). and !. Dµ′ H. gH = Dµ · 1 − i √ Wµ′p T p + Wµ′m T m − igH Wµ′3 T 3 − igX Xµ · H 2. (4.21). Where Dµ is covariant derivative of SU (2)L × U (1)Y , and √ Wµ′(p,m) = (Wµ′1 ∓ iWµ′2 )/ 2,. with T(p,m) =. 1 1 σ ± iσ 2 2. (4.22). which T a = σ a /2 (σ a is Pauli matrices). Note gH , gX are couplings of SU (2)H , U (1)X separately. The new guage boson W (p,m) are not SU (2)L generator means they are electric charge neutral and will not mix with SM W (p,m) bosons. Note that W ′± in left-right models have electric charge, to mark the difference, we use (p, m) instead of "±" in G2HDM. The mass of SM charged gauge boson are only come from VEV of SM Higgs v, not affect by the new gauge symmetry. 1 MW ± = gv 2. (4.23). On the other hand, the new gauge boson W ′a and X of SU (2)H and U (1)X acquired masses from the VEVs of H1 , Φ2 and ∆3 1 2 2 2 m2W ′( p,m) = gH v + vΦ2 + 4v∆ 4. (4.24). The remaining gauge bosons X, W ′3 , W 3 and B acquire masses from VEVs of H1 and Φ2 but not ∆3 2 2 1 2 v 2gX Xµ + gH Wµ′3 − gWµ3 + g ′ Bµ + vΦ2 −2gX Xµ + gH Wµ′3 8 . . (4.25).

(52) 34. CHAPTER 4. DARK MATTER IN G2HDM. However, there are only two Goldstone boson in H1 and Φ2 to absorb, hence only two out of X, W ′3 , W 3 , B can aquire masses; but the massless dark photon might not be phenomenologically desirable. This motivated us to introducing a Stueckelberg mass term[29, 30] to avoid this problem LStu =. 1 (∂µ a + MX Xµ + MY Bµ )2 2. (4.26). where the MX and MY are the Stueckelberg masses for the gauge fields Xµ and Bµ of U (1)X and U (1)Y respectively, and a represent the axion field. With the new mass terms, the mass matrix in the basis of V ′ = {B, W 3 , W ′3 , X} can be written as . M21 =.       . g ′2 v 2 4. ′. + MY2 ′. − g gv 4. − g gv 4. g2 v2 4 ggH v 2 − 4 2 − ggX2 v. 2. g ′ gH v 2 4 g ′ gX v 2 2. 2. + MX MY. g ′ gH v 2 4 ggH v 2 − 4 2 v 2 +v 2 gH ( Φ) 4 2 gH gX (v 2 −vΦ ) 2. g ′ gX v 2 2. + MX MY 2 − ggX2 v 2 gH gX (v 2 −vΦ ) 2. 2 gX. (v + vΦ2 ) + MX2 2.        . (4.27). The determinant of this matrix is zero, means we have at least one massless eigenstate that can be identified as photon. In general, the remaining eigenstates are massive which written as Z, Z ′ and Z ′′ , where mZ < mZ ′ < mZ ′′ . The mass matrix M1 can be diagonalized by a 4 orthogonal matrix. As mentioned in [31], the parameter MY has to be zero to keep the neutrinos chargeless, also MY will give correction to SM charge. Therefore, we consider MY = 0 in our study. Now the top-left 2 × 2 elements of M1 can be block diagonalized by Weinberg angle by applying a rotation M2Z = (OW )T · M21 (MY = 0) · OW , where . OW. and. 0  0  . M2Z =  . 0 . cW −sW  s cW  W =  0 0  0 0. 0 MZ2 SM − gH2 v MZ SM. 0 −gX vMZ SM. 0 0 1 0. 0  0   0  1 . (4.28). 0 gH v − 2 MZ SM 2 v 2 +v 2 gH ( Φ). 0 −gX vMZ SM 2 gX gH (v 2 −vΦ ). 4 2 gX gH (v 2 −vΦ ) 2. 2 2 (v 2 + vΦ2 ) + MX2 gX.        . (4.29). √ where MZ SM = g 2 + g ′2 · v/2 is exactly the SM gauge boson Z SM mass in eq.(2.35), and we confirm the first element is indeed massless. For the convenience, we will use Zi.

(53) 35. 4.1. THE G2HDM MODEL. with i = 1, 2, 3 to represent Z, Z ′ and Z ′′ in the following. Note the basis in eq.(4.29) is {A, Z SM , W ′3 , X}. We can find a rotation matrix OZ that diagonalize eq.(4.29). Thus, the final basis is     ZSM Z1      W ′  = O Z · Z  (4.30)  3   2 X Z3 G We can see ZSM is combination of Zi ’s. Note that O4×4 (MY = 0) = OW OZ. 4.1.6. Fermion Sector. As mentioned in sec.4.1, the SM QL is SU (2)L doublet and SU (2)H singlet, which can T H T pair with SU (2)L singlet and SU (2)H doublet: UR = (uR , uH R ) or DR = (dR , dR ) LYuk. . . ¯ L (DR · H) + yu Q ¯ L UR · H + H.c. ⊃yd Q ≈. H ˜ ¯ L dH ¯ ˜ =yd Q R H2 − dR H1 − yu QL uR H1 + uR H2 + H.c.. . . . . (4.31). ˜2 − H ˜ 1 )T and H ˜ 1,2 = iσ2 H ∗ . After SSB, only H1 acquired where we defined H ≡ (H 1,2 VEV, thus the new heavy fermion remains massless. To give masses to heavy fermion, we introduce a SU (2)H doublet and SU (2)L singlet ΦH = (Φ1 , Φ2 )T , then we can have the Yukawa term of new fermions ≈. H ˜ LYuk ⊃ − yd′ dL (DR · ΦH ) + yu′ uH L UR · ΦH + H.c.. . H. . . . . . ′ H ∗ H ∗ = − yd′ dL dH R Φ2 − dR Φ1 − yu uL uR Φ1 + uR Φ2 + H.c.. (4.32). √ H With the VEV of ⟨Φ2 ⟩ = vΦ / 2, the new fermions uH L,R and dL,R acquired masses √ √ yu′ vΦ / 2 and yd′ vΦ / 2, respectively. In addition, the triplet scalar v∆ did not contribute to the fermion mass. The lepton sector are in the same set up. Note the neutrino νL,R and their SU (2)H √ √ H H partner νL,R acquired Dirac mass MDν = yν v/ 2 and MDν = yν ′ vΦ / 2 The Yukawa couplings are invariant under H1 → H1 , H2 → −H2 , Φ1 → −Φ1 , Φ2 → Φ2 , f SM → f SM and f H → −f H for all fermions f = u, d, ν, e. which is the same Z2 symmetry we found in scalar sector.. 4.1.7. Hidden symmetry. Due to the choice of vacuum alignment where only the Z2 -even fields H1 , Φ2 and ∆3 get the VEV, we have a residue Z2 symmetry after SSB of the SU (2)L × U (1)Y ×.

(54) 36. CHAPTER 4. DARK MATTER IN G2HDM. SU (2)H × U (1)X gauge symmetry. It is also discussed in [25] that there is no gauge invariant higher dimensional operator that can lead to the decay of DM candidate in G2HDM. Since the DM candidate we will discuss in this thesis are the scalar DM, ∗ which are the combination of Z2 -odd fields GpH , H20 and ∆p , thus the stability of DM is protected by this Z2 symmetry. This is similar to the remaining U (1)EW after the SSB of SU (2)L × U (1)Y , which now is the broken of SU (2)H × U (1), we will refer this residue Z2 as the hidden parity (h-parity) in the Follow. This h-parity can extended to the fermion and gauge sectors, where we will find the new fermions and W ′(p,m) are all Z2 -odd, while the SM particle remain Z2 -even as it should be. But the new Higgs fields h2 and h3 as well as Z2 and Z3 are also Z2 -even. We summerized the h-parity of particles in Table,4.2 SM Z2 Even h1 , h2 , h3 W ± , Z1 , Z2 , Z3 fL,R H ˜ H± Z2 Odd D, ∆, W ′(p,m) fL,R Table 4.2 The Z2 assignments in G2HDM model.. 4.1.8. DM Candidate in G2HDM. In general, all the electric neutral Z2 -odd particles in this model can be DM candidate e.g. new heavy neutrino ν H , W ′(p,m) abd D. We will only focus on scalar DM in this ∗ thesis. As we mentioned before, the scalar DM are the combination of GpH , H20 and ∆p . Use the same rotation matrix that diagonalized M2D in eq.(4.15) to transform the basis G to the basis of physical fields ˜ T = OD T · G {G˜p , D, ∆}. →. ∗. D p D D D = O12 GH + O22 H20 + O32 ∆p. (4.33). D where Oi,j represents the elements of OD . The value of OD depend on the value of parameters in eq.(4.15). Eq.(4.33) shows the DM is combination of Goldstone boson, and element of SU (2)H doublet and triplet separately. Thus, we define three diffent types of DM, depend on their dominant component.. • 1. Doublet-like DM: fH2 > 2/3 • 2. Triplet-like DM: f∆p > 2/3 • 3. Goldston-like DM: fGp > 2/3 D 2 D 2 D 2 Where we define fGp = (O12 ) , fH2 = (O22 ) and f∆p = (O32 ) . By the unitarity of rotation matrix, we have fGp + fH2 + f∆p = 1. The reason for using the value 2/3.

(55) 4.2. NUMERICAL ANALYSIS. 37. fGp > 2/3. Fig. 4.1 The relation between the ratio v∆ /vΦ and parameter fGp for all the scanned DM parameter space after SGSC. to specify the dominant component is that we want to have large coverage of the parameter space for our scan. If we increase this value, for example 0.95, it is difficult to find points in the parameter space of our scan. In the mass matrix 4.15, the v∆ /vΦ ratio control the composition of Goldstone boson in DM. However, when the ratio close to 1, the Z ′ has large mixing with ZSM , which is disfavor by electroweak precision test(EWPT). Also, to get the value of fGp larger than ∼ 0.8 will resulting tachyonic DM mass. Thus, the value 2/3 is fair choice. ∗ In general, GpH , H20 and ∆p are complex fields, thus the scalar DM candidate are also complex scalar.. 4.2. Numerical Analysis. Now we want to test the G2HDM for experiment of DM searches, if we found points in parameter space that satisfied DM constraints, it will indicates the G2HDM can be a BSM candidate for DM. In this thesis we will illustrate our scalar DM candidate in the search of relic density and direct detection, both of them are relative strict constraints.. 4.2.1. Constraints and Range of Parameters. The constraints of scalar sector parameters were studied in [32], where they examined the vacuum stability(VS) of the scalar potential, the perturbative unitarity(PU) for.

(56) 38. CHAPTER 4. DARK MATTER IN G2HDM. scattering amplitudes, and the Higgs physics (HP) of Higgs diphoton and τ + τ − decay in LHC. The parameters for our DM study are all constrained by VS+PU+HP. By the gauge sector study in ref.[31] for the Z ′ constraints, we set our scan of vΦ start from 20TeV, and set the upper limit to 100GeV by considering the energy scale of future collider. In addition, we required the SM-like Z1 is the lightest among Zi , and the mass of Z1 within its 3σ of measured value 91.1876 ± 0.0021GeV. We will mark the scalar and gauge sector constraints as SGSC in the follows. Since we choose Z2 -odd scalars D as our DM in this thesis, there will be several constraints arise from mass spectrum. The mass of DM have to be lighter than all the other Z2 -odd, mH ± > mD , mW ′(p,m) > mD , and mf H > mD . We will implement these conditions in our scan. By eq.(4.24), the mass condition of mW ′(p,m) , it imposing the lower bound of gH 2mD gH min = q (4.34) 2 v 2 + vΦ2 + v∆ We want to eliminate fermion coannihilation as much as possible to simplify our analysis. However, the Yukawa couplings are proportional to fermions. Meanwhile, we want the new Yukawa couplings to be reasonably small in order to minimize their effects on perturbative unitarity and renormalization group running. Thus, we set the mf H = 1.2mD , and as suggest by the LHC SUSY colored particles search in [33], we set the lower limit of heavy fermions to 1.5TeV, thus we have mf H = max [1.5TeV, 1.2mD ]. (4.35). And as showed in sec.4.1.6, we found the new fermions acquired mass solely from Φ2 . By the condition in eq.(4.35), the constraints on Yukawa couplings are 1.5TeV 1.2mD √ , √ = max vΦ 2 vΦ 2 ". yf H. #. (4.36). Since mD only up to few TeV in our scan, and vΦ > 20TeV, we are safe for small Yukawa couplings. Before going to calculate the DM properties for relic density and direct detection, we randomly scan all the parameters in the range list in Table.4.3 for the model to test for varies of theoretical and phenomenological constraints, and passed those points satisfied constraints to DM search. The input parameters in scalar sector are λH , λΦ , λ∆ , λHΦ , λH∆ , λΦ∆ , λ′HΦ , λ′H , MH∆ , MΦ∆ , v∆ , and vΦ . We scanned vΦ in the range between 20TeV to 100TeV, except for Goldstone-like DM, as we discussed in.