complex scalar dark matter in gauged two higgs doublet

Complex Scalar Dark Matterin

Gauged Two Higgs Doublet Model

(G2HDM)

Chrisna Setyo Nugroho

Supervisor: Professor Chuan-Ren Chen

Professor Tzu-Chiang Yuan

Department of PhysicsNational Taiwan Normal University

This dissertation is submitted for the degree ofDoctor of Philosophy

July 2019

This work is presented to my parents!

Declaration

I hereby declare that except where specific reference is made to the work of others, thecontents of this dissertation are original and have not been submitted in whole or in part forconsideration for any other degree or qualification in this, or any other University. This workwas done wholly or mainly while in candidature for a Ph.D degree at this University. Thisdissertation is my own work and jointly with others.

Chrisna Setyo NugrohoJuly 2019

Acknowledgements

Alhamdulillah, thanks to the most Merciful and Benelovent i have been able to finish thiswork.

I offer my deepest gratitude to Prof. Tzu-Chiang Yuan for his patience and supportduring my entire graduate school life as a PhD student at National Taiwan Normal University(NTNU). He shared with me his insights of physics, taught me many life lessons as well asguided me to become a professional physicist.

My sincere thank is addressed to my co-supervisor, Prof. Chuan-Ren Chen for his helpand advices during my stay at NTNU. I also deeply thank Prof. Chiang-Hung VincentChang who has provided me a lot of support and help, especially during the early years ofmy graduate school life. I would like to thank Prof. Pham Quang Hung for his help andcollaborations.

Many thanks to my friends Dr. V. Q. Tran, Mr. Yu-Xiang Lin and Mr. Chia-Feng Changfor providing me a lot of ideas and help in research and daily life in Taiwan. I also offer myspecial thanks to Dr. Raymundo Ramos and Dr. Yue-Lin Sming Tsai for their importantadvices and help.

Finally, i thank my parents in Indonesia and my family in Taipei for their endless supportto me.

Abstract

In this thesis, we study the complex scalar dark matter phenomenology in Gauged Two HiggsDoublet Model (G2HDM). It is shown that a accidental Z2 symmetry arises naturally fromthe gauge invariance of SU(2)L ×U(1)Y ×SU(2)H ×U(1)X in the model and hence protectsthe stability of dark matter. The complex scalar dark matter in the model is categorized intoinert doublet-like, SU(2)H triplet-like and Goldstone boson-like. While the inert doublet-likedark matter is ruled out by XENON1T data, the SU(2)H triplet-like and Goldstone boson-likedark matter satisfy the relic density from PLANCK and all other experimental constraintsfrom XENON1T, Fermi-LAT and LHC. We discuss in detail the constraints on the parameterspace coming from the four pillars of dark matter phenomenology – relic density, direct andindirect detection, and collider searches.Keywords: Complex Scalar Dark Matter, Gauged Two Higgs Doublet ModelThesis Supervisors: Professor Chuan-Ren Chen and Professor Tzu-Chiang Yuan

Table of contents

List of figures xiii

List of tables xvii

1 Introduction 1

2 Short Review of the Standard Model 52.1 The Weak Interaction and the SM Particle Content . . . . . . . . . . . . . 52.2 The SM Gauge Interactions and Symmetry Breaking . . . . . . . . . . . . 8

2.2.1 The SM Field Contents . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 The SM Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 13

3 The G2HDM Model 173.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 The Particle Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Higgs Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 The SSB of the Potential and the Mass Spectrum . . . . . . . . . . . . . . 22

3.4.1 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . 223.4.2 Scalar Mass Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 233.4.3 Gauge Boson Mass Spectrum . . . . . . . . . . . . . . . . . . . . 253.4.4 Fermionic Mass Spectrum . . . . . . . . . . . . . . . . . . . . . . 27

3.5 The Accidental Z2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 283.6 Theoretical Constraints on the Scalar Sector . . . . . . . . . . . . . . . . . 29

4 WIMP Dark Matter and Its Constraints 334.1 Astrophysical Evidence of the Dark Matter . . . . . . . . . . . . . . . . . 33

4.1.1 Galactic Rotation Curves . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . 34

xii Table of contents

4.2 WIMP as Thermally Produced Dark Matter . . . . . . . . . . . . . . . . . 354.3 Dark Matter Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3.1 Dark Matter Direct Search . . . . . . . . . . . . . . . . . . . . . . 384.3.2 Dark Matter Indirect Search . . . . . . . . . . . . . . . . . . . . . 394.3.3 Dark Matter Collider Search . . . . . . . . . . . . . . . . . . . . . 40

5 Dark Matter in G2HDM: Constraints and Methodology 435.1 Dark Matter Properties in G2HDM and Experimental Constraints . . . . . 44

5.1.1 Relic Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1.2 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1.3 Indirect Detection: Gamma-ray from dSphs . . . . . . . . . . . . 505.1.4 Collider Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Dark Matter in G2HDM: Numerical Results 576.1 Inert Doublet-like DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 SU(2)H Triplet-like DM . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.3 SU(2)H Goldstone Boson-like DM . . . . . . . . . . . . . . . . . . . . . . 716.4 Constraining Parameter Space in G2HDM . . . . . . . . . . . . . . . . . . 75

7 Summary 79

Appendix A Relevant Couplings 81A.1 Dominant Couplings for Dark Matter . . . . . . . . . . . . . . . . . . . . 81

Appendix B Benchmark Points for Monojet 85

References 87

List of figures

2.1 The fundamental particles in the Standard Model. (Source: wikipedia.org) . . . . 7

3.1 A summary of the parameter space allowed by the theoretical and phenomeno-logical constraints. The red regions show the results from the theoreticalconstraints (VS+PU). The magenta regions are constrained by Higgs physicsas well as the theoretical constraints (HP+VS+PU). Figure is taken from [26]. 30

4.1 Rotation curves of spiral galaxies as observed by Rubin et al. [31]. At largeradial distance from the center, most of galaxies exhibit constant circularvelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Dark matter number density per comoving volume as function of its massover the temperature of the universe. Figure is taken from [33]. . . . . . . 37

4.3 The current (solid) and projected (dotted/dashed) bounds on the spin-independentWIMP DM-nucleon cross section. The orange band indicated the neutrinofloor which is a typical extraterrestrial background. This plot is adopted from[35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4 Indirect DM constraints for few final states: from IceCube [37], AMS02 [38],H.E.S.S. [39], PLANCK [40], CTA projected sensitivity [41], and Fermi-MAGIC collaborations [42]. The black dotted-dashed line denotes the typ-ical annihilation cross section ⟨σv⟩ = 3× 10−26cm3s−1. Figure adoptedfrom [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1 The Feynman diagrams of dark matter annihilation into W+W− pair. The hi

and Zi denote the three Higgs boson and three neutral gauge boson mediators.The t on the bottom left panel stands for t-channel diagram. . . . . . . . . 45

5.2 The Feynman diagrams of dark matter annihilation into h1h1 pair. The hi

denote the three Higgs boson mediators. The t (u) represents t (u)-channeldiagram respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

xiv List of figures

5.3 The Feynman diagrams of dark matter annihilation into ZZ pair. The hi

denote the three Higgs boson mediators. The t (u) represents t (u)-channeldiagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.4 The diagrams relevant for f f final states. The hi, Zi, and f Hi denote the

three Higgs boson, three neutral gauge boson as well as three heavy fermionmediators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.5 The dominant Feynman diagrams for Higgs bosons (left) and Z bosons (right)exchange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 The Feynman diagrams of leading contributions for monojet plus missingenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.7 The typical diagram used to study perturbative unitarity in scalar-scalarscattering. The Si, S j (Sk, Sl) denote the scalar particles in the initial (final)states respectively. These scalars can be one of the following pairs ( hh√

2, G0G0√

2,

H0∗2 H0

2 , H+H−, φ2φ2√2

, G0HG0

H√2

, GpHGm

H , δ3δ3√2

, ∆p∆m) as described in the text. . 54

6.1 Left: the doublet-like DM relic density as function of the DM mass. Right:the DM mass versus the DM-neutron elastic scattering cross section. Thegray scatter points agree with the SGSC constraints. The blue scatter pointsagree with PLANCK data within 2σ region. . . . . . . . . . . . . . . . . 58

6.2 Feynman diagrams of dark matter annihilation into ττ , cc and bb final states. 586.3 Relevant diagrams for h1h1 final state. For the doublet case, both D and

∆ exchange in t- and u-channels are important. For the triplet-like andGoldstone-like dark matter, only D exchange is relevant. . . . . . . . . . . 61

6.4 Relevant diagrams for ZLZL final state. The first three diagrams scale withthe center-of-mass energy. The sum of these three diagrams cancel the energydependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.5 Relevant diagrams for W+L W−

L final state. . . . . . . . . . . . . . . . . . . 626.6 Triplet-like SGSC allowed regions projected on the (mD, ΩDh2) (left) and

(mD, σSIn ) (right) planes. The gray area on the left has no coannihilation or

resonance. The gray area on the right is excluded by PLANCK data at 2σ .Some orange squares are above the XENON1T limit due to ISV cancellationat the nucleus level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

List of figures xv

6.7 The annihilation cross section at the present universe (left) and DM-neutronelastic scattering cross section (right) for f∆ > 2/3 case. In the left panel,the annihilation final state is classified to be three main types: W+W−

(blue), bb (green), and h1h1 (orange). However, the exclusion by ID ismarked by unfilled and light colors. In the right panel, the region allowedby SGSC+RD+ID+DD constraints is marked by filled dark blue squares.However, the region excluded by SGSC+RD+ID and SGSC+RD+DD ismarked in orange crosses and light blue squares. Projected sensitivities fromthe CTA experiment for the W+W− and bb final states are also shown. . . 70

6.8 Correlation between the ratio v∆/vΦ and the mixing parameter fGP afterapplying the constraints from the scalar and gauge sectors. . . . . . . . . . 72

6.9 Goldstone boson-like SGSC allowed regions projected on the (mD, ΩDh2)(left) and (mD, σSI

n ) (right) planes. The gray area on the left has no coanni-hilation or resonance. The gray area on the right is excluded by PLANCKdata at 2σ . The orange squares above the XENON1T limit present the ISVcancellation at the nucleus level. . . . . . . . . . . . . . . . . . . . . . . . 73

6.10 The present time total annihilation cross section by dominant annihilationchannel (left) and the DM-neutron elastic scattering cross section (right) forfGP > 2/3. Some blue filled squares above the XENON1T limit are due tothe ISV cancellation at the nucleus level. Projected sensitivities from theCTA experiment for the W+W− and bb final states are also shown. . . . . 75

6.11 A summary plot for the scalar parameter space allowed by the SGSC con-straints (green region) and SGSC+RD+DD constraints (red scatter points).The numbers written in the first block of each column are the 1D allowedrange of the parameter denoted in the horizontal axis after SGSC+RD+DDcut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.12 A summary plot table of the parameter space of the two VEVs vΦ and v∆,two mass scales MH∆ and MΦ∆, and two new gauge couplings gH and gX .The color scheme is the same as Fig. 6.11. . . . . . . . . . . . . . . . . . . 78

A.1 The DD∗W+W− coupling for p-channel interaction. . . . . . . . . . . . . 81A.2 The dominant DD∗Zi and DD∗hi couplings for the inert doublet-like dark

matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.3 The dominant DD∗Zi and DD∗hi couplings for the SU(2)H triplet-like dark

matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.4 The dominant DD∗Zi and DD∗hi couplings for the SU(2)H Goldstone boson-

like dark matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

List of tables

1.1 Table of acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Fermion and Higgs fields and the corresponding SM quantum numbers. Theindex i denotes the fermion generation. The numbers in the last column arethe hypercharge Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Matter contents and their quantum number assignments in G2HDM. . . . . 193.2 The Z2 assignments in G2HDM model. . . . . . . . . . . . . . . . . . . . 29

5.1 Parameter ranges used in the scans mentioned in the text. . . . . . . . . . . 55

B.1 10 benchmark points for the mono-jet of the SU(2)H triplet-like DM. . . . . 86

Chapter 1

Introduction

The Standard Model (SM) [1–3] provides a very good explanation about the fundamentalinteraction between particles observed in nature. Except for gravitational interaction, the SMhas been able so far to explain the strong force which is responsible to put the neutron andproton together inside the nucleus of an atom, the electromagnetic force which is involved inthe interaction between charged particles, and the weak force which explains the radiativebeta decay. In addition, it also predicts the existence of the new massive particles W andZ which are called the gauge bosons that act as mediators in weak interaction. Moreover,the masses of these gauge bosons as well as all other particles in nature, except photon(the electromagnetism mediator), gluons (the strong force mediators) and neutrinos, werebelieved to be originated from the Higgs mechanism [4–6]. The Higgs mechanism predictsthe existence of the new scalar boson that has been discovered in 2012 by both ATLAS [7]and CMS [8] experiments at the Large Hadron Collider (LHC) with a mass about 125 GeV.This discovery completes all the particles content in SM.

Despite of its amazing achievements in explaining three out of the four fundamentalparticle interactions in nature, the SM can not explain the following puzzles:

• The observed neutrino masses which can be inferred by the neutrino oscillation experi-ments. In SM, neutrinos are assumed to be massless.

• The Baryon Asymmetry of the Universe (BAU) that accounts for the dominance of thematter over anti-matter in the universe.

• The existence of the dark matter which accounts for about 80% of the total matter and30% of the total energy density of the universe.

• The observed dark energy which is the dominant form of the energy density budget inthe universe.

2 Introduction

• etc.

To explain these issues one needs to go Beyond the Standard Model (BSM). Thereare a lot of BSM models in the literature constructed to attack different open problemsin physics. For instance, there is the Minimal Superymmetric Standard Model (MSSM)based on supersymmetry (SUSY) which predicts that every bosonic degree in the SM hasits fermionic partner and vice versa. In this model, there exists the lightest supersymmetricparticle (LSP) that provides a viable dark matter candidate whose stability protected by theR symmetry. Two Higgs Doublet model (2HDM) is the next non-minimal version of theSM. Instead of having only one Higgs doublet, it has two Higgs doublets which renders thescalar potential more complicated than that in the SM. In the case that the additional Higgsdoublet does not acquire the vacuum expectation value (VEV), this becomes the Inert HiggsDoublet Model (IHDM) which provides viable dark matter candidate called the inert darkmatter. However, one needs to impose by hand the additional ad hoc Z2 symmetry in thescalar potential to make the dark matter stable.

This unpleasant Z2 discrete symmetry in IHDM has motivated Huang, Tsai and Yuan [9]to construct a so-called “Gauged Two Higgs Doublet Model” (G2HDM for short) whichembeds the two Higgs doublets in the popular 2HDM into a doublet of a non-abelian gaugegroup SU(2)H . The neutral component of the second doublet can be a dark matter candidatewhich is stable under the natural protection of this new gauge group rather than the ad hoc Z2

symmetry in the IHDM. This thesis explores the complex scalar dark matter phenomenologyin the G2HDM model. The content of this thesis is outlined as follows:

• Chapter 2 describes the brief summary and main features of the SM.

• Chapter 3 explores the construction of G2HDM model. Its particle content, scalarpotential, gauge and Yukawa interactions are discussed.

• Chapter 4 summarizes the WIMP dark matter and its various search strategies.

• Chapter 5 discusses the dark matter in G2HDM, experimental constraints of darkmatter and methodology used in our analysis.

• Chapter 6 summarizes the numerical results of our dark matter study and the allowedparameter space of G2HDM.

• Finally, we conclude this thesis in Chapter 7.

Some relevant Feynman rules involving the dark matter and 10 monojet benchmarkresults are listed in two appendices.

3

The metric and units used throughout in this thesis are gµν = (+1,−1,−1,−1) andh = c = 1 (natural units). We follow closely Peskin and Schroeder [10] for other conventions.

Some of the acronyms used in this thesis are listed in Table 1.1.

4 Introduction

Acronym Description2HDM Two Higgs Doublet ModelATLAS A Toroidal LHC ApparatuSBAU Baryon Asymmetry of the UniverseBR Branching RatioBSM Beyond the Standard ModelCKM Cabibbo-Kobayashi-MaskawaCMB Cosmic Microwave BackgroundCMS Compact Muon SolenoidCP Charge Conjugation and ParityCTA Cherenkov Telescope ArrayDD Direct DetectionDM Dark MatterdSphs dwarf SpheroidsEM ElectromagnetismEW ElectroweakEWPT Electroweak Precision TestFCNC Flavor Changing Neutral CurrentFermi-LAT Fermi Large Area TelescopeG2HDM Gauged Two Higgs Doublet ModelHP Higgs PhysicsID Indirect DetectionIHDM Inert Higgs Doublet ModelISC Isospin ConservedISV Isospin ViolationLHC Large Hadron ColliderMSSM Minimal Supersymmetric Standard ModelPandaX Particle and Astrophysical Xenon DetectorPLANCK https://www.cosmos.esa.int/web/planck/homePU Perturbative UnitarityQCD Quantum ChromodynamicsQED Quantum ElectrodynamicsRD Relic DensitySGSC Scalar and Gauge Sector ConstraintsSM Standard ModelSSB Spontaneous Symmetry BreakingSUSY SupersymmetryVEV Vacuum Expectation ValueVS Vacuum StabilityWIMP Weakly Interacting Massive ParticleXENON1T http://www.xenon1t.org

Table 1.1 Table of acronyms.

Chapter 2

Short Review of the Standard Model

The gauge theory of the electroweak interaction or the Standard Model (SM) will be summa-rized in this chapter. We will list some of the important properties in SM and their roles inparticle physics.

2.1 The Weak Interaction and the SM Particle Content

The Standard Model (SM) was motivated by the weak interaction observed in many decayprocesses starting from β decay, inverse β decay, muon decay, and many other fundamentalprocesses in nuclear physics. The first attempt to understand the nature of weak interactionin neutron β decay was done by Fermi by taking analogous current-current interaction inQuantum Electrodynamics (QED). The neutron β decay can be explained quite well atenergy scale around few GeV by using a particular form of the Fermi Theory in the pictureof current-current interaction Hamiltonian density as

HI =GF√

2J†

µJµ , (2.1)

where the current is written in terms of hadronic and leptonic currents of the form

Jµ = Jhadµ + Jlept

µ (2.2)

= pγµ(1− γ5)n+ νγµ(1− γ5)e , (2.3)

where p, n, ν , and e stand for fermion fields of the proton, neutron, neutrino, and electron.The constant GF that appears in the above equation is called the Fermi coupling constant and

6 Short Review of the Standard Model

its value has been determined from low energy experiment as [11]

GF = 1.1663787×10−5GeV−2. (2.4)

From the magnitude of Fermi constant above, the weak interaction indeed quite weak, andextremely short ranged. All particles that exist in nature (hadrons and leptons) were knownto interact weakly. Furthermore, their interactions are represented by SU(2)L multiplets ofweak isospin group. As a consequence, the currents in Eq. (2.3) encodes the weak isospinstructure within it, and the corresponding currents are called weak isospin currents. TheV −A (vector minus axial vector) structure of the weak currents stems from the fact thatthe parity is violated in weak processes. Moreover, this also implies that only left-handedcomponent of the fermion fields contribute in the weak interaction. The electric charge ofany particles involved in weak interaction can be realized by assigning additive quantumnumber known as weak hypercharge U(1)Y such that

Q = T3 +Y , (2.5)

where Q, T3 and Y correspond to the electric charge (in unit of e), the third component ofweak isospin generator, and the weak hypercharge of the particle. This assignment was resultfrom the experimental fact that both weak isospin and weak hypercharge quantum numberare violated in weak interaction but leave the electric charge intact.

The shortcoming of the four fermion interaction described above is that they are non-renormalizable. This can easily be seen from the existence of inverse square dimension ofthe Fermi coupling constant GF . At the same era, it was shown that the gauge theory canbe used to explain physical interaction and it is compatible with renormalizability. Thus,it is natural to incorporate the gauge theory to describe the weak interaction such that ittakes the form of V −A structure in the low energy limit. The gauge theory dictates theneed of the mediators in every interaction. This so called gauge boson acts as a messengerthat brings the information being exchanged during the interaction. Due to the short rangednature of the weak interaction, one expects that the gauge boson to be massive. This iswhere the Higgs mechanism enters the game. Utilizing the Higgs mechanism will causethe gauge symmetry to be spontaneously broken, but as an advantage, one obtains massivegauge bosons. This is exactly what we want, since the weak interaction violates both ofweak isospin and weak hypercharge quantum number, one can employ the Higgs mechanismto spontaneously break the SU(2)L and U(1)Y symmetry to get the corresponding massivegauge bosons while preserving the electromagnetic gauge symmetry. Diagrammatically, this

2.1 The Weak Interaction and the SM Particle Content 7

Fig. 2.1 The fundamental particles in the Standard Model. (Source: wikipedia.org)

can be written in the following

SSB : SU(2)L ×U(1)Y →U(1)EM , (2.6)

where SSB denotes spontaneously symmetry breaking. With this, one can construct thegauge theory of the electroweak interaction or known as the Standard Model (SM).

The Standard Model (SM) was developed to explain the weak and electromagneticinteraction of all particles observed in nature. All fundamental particles known so far arelisted in Fig. 2.1. One notices that instead of lepton and hadron that appear in the V −Acurrent, the SM matter fields consist of leptons and quarks which appear in three families.This is due to the fact that hadron is the bound state of quarks. Quarks and leptons are spin 1/2elementary particles that respect Fermi-Dirac statistics [12] widely known as fermions. Themain difference between those two species is that quarks are involved in strong interactionwhich is responsible to hold protons and neutrons inside the nucleus, while leptons are inertto it.

According to the weak isospin quantum number (flavor), there are 6 types of quarks andleptons, six flavors for quarks: up-quark (u), down-quark (d), charm-quark (c), strange-quark(s), top-quark (t) and bottom-quark (b), and six flavors for leptons: electron (e), electronneutrino (νe), muon (µ), muon neutrino (νµ ), tau (τ) and tau neutrino (ντ ). They are arrangedinto three groups with similar quantum number assignments even though their masses arenot the same. Each family consists of two quark and two lepton flavors. As an example,the second family has c,s quarks and νµ ,µ leptons. In unit of e > 0, up-type quarks (u,c, t)have similar electric charges of +2

3 , and down-type quarks (d,s,b) have electric charges−1

3 . For leptonic groups, e,µ,τ have identical electric charges of −1 and the neutrinos areelectrically neutral. Moreover, for each fermion listed on the Fig. 2.1, there is a correspondingantiparticle which has the same properties as the original fermion except its opposite electric


charge. The antiparticle of u quark is anti u quark (u), the antiparticle of electron (e−) is thepositron (e+) etc. It is known that leptons exist as free particles in nature while quarks alwaysappear in form of hadronic bound states due to the nature of strong interaction. Furthermore,hadron that is formed by odd number of quarks is identified as baryon, while meson consistsof quark and antiquark pair.

Moving forward to the messenger of the SM interaction, there are gauge bosons thatfollow the Bose-Einstein statistics [13]. The bosons are characterized by their integral spinquantum number. The gauge bosons are spin 1 particles that mediate every interactiondescribed by the gauge interaction. The weak interaction is mediated by three massive gaugebosons W+, W− and Z. The electromagnetic interaction occurs by exchanging massless spinone photon (γ). Furthermore, these two interactions belong to the unified framework of theelectroweak interaction. Finally, the strong interaction which is responsible for the nuclearforces requires massless colored gluons as their messenger.

Lastly, the SM requires a special boson to generate masses for all the fundamentalparticles listed above. This boson is called the Higgs boson which is a spin zero scalar particleresponsible for the spontaneously symmetry breaking in SM. This boson was discovered in2012 [7, 8] at the large hadron collider (LHC). It was shown that this particle has a massaround 125 GeV [14] while its other properties are still under careful investigation.

2.2 The SM Gauge Interactions and Symmetry Breaking

In this section we will discuss the SM field contents according to their quantum numberassignments under the corresponding gauge group, SM Lagrangian density which describeshow particles interact to each other, and finally explain the mass generation via spontaneouslysymmetry breaking (SSB) or Higgs mechanism.

2.2.1 The SM Field Contents

Gauge theory describes the interaction of all fundamental particles in nature. This theorycontrols the behaviour of each particle under the corresponding gauge transformation whichis internal transformation on the field. This transformation is also known as local phaserotation and it depends on some parameters similar to the Euler angles that parameterizedthree dimensional rotation in Euclidean space. The particle interaction can be written interms of the Lagrangian density analogous to those of classical mechanics formalism. Themain difference is that the Lagrangian density is the function of the fields that representthe elementary particles involving in the interaction. Gauge symmetry is reached when the

2.2 The SM Gauge Interactions and Symmetry Breaking 9

Lagrangian density is invariant under the gauge transformation. Taking into account thestrong interaction, the Standard Model gauge group can be expressed as

GSM = SU(3)C ×SU(2)L ×U(1)Y , (2.7)

where the SU(3)C represents strong interaction with color quantum number and SU(2)L ×U(1)Y stands for electroweak interaction. For the sake of simplicity, we only explain thematter fields in terms of one family of lepton that consists of the electron and its neutrino.This can be implemented to the quarks and other families in straightforward manner. Theelectron family is expressed in terms of multiplets of SU(2)L as

LL =

(νL

eL

), R = eR , (2.8)

where the left-handed and right-handed fermionic field is defined as

ψL,R =12(1∓ γ5)ψ. (2.9)

The left-handed multiplets of the Standard Model correspond to the SU(2)L doublet of weakinteraction, while the right-handed particles belongs to singlets. Each multiplet has the uniquequantum number assignment corresponds to its respective representation of the group. Incase of SU(2)L group, the left-handed quantum numbers are +1/2 for the electron neutrinoνe and −1/2 for the electron, while 0 is assigned for the right-handed part of the particles.Moreover, the hypercharge quantum number U(1)Y for the left-handed doublet leptonicfamily is −1/2 while the right-handed singlet carries a hypercharge −1. These assignmentsfollow from the condition that the electric charge of each particle satisfies the relation inEq. (2.5).

The quark sector can also be written in a similar way. The difference between leptonicfamilies and quark families is that the latter one always involve massive particles while theformer one is not. This is due to the fact that the neutrinos are massless in the StandardModel. Another feature that does not possessed by the leptonic families is that the quarkfamilies transform non-trivially under SU(3)C gauge group. As a consequence, the quarkparticles also carry SU(3)C quantum number. Finally, the Higgs boson is introduced as adoublet with respect to SU(2)L and has the hypercharge +1/2. In Table 2.1, we collect thefermionic (quarks and leptons) and Higgs fields as well as their quantum number assignmentsunder the full Standard Model gauge group.


Fields SU(3)C SU(2)L U(1)Y

LLi =

(νLeL

)i

1 2 −12

eRi 1 1 −1

QLi =

(uLdL

)i

3 2 +16

uRi 3 1 +23

dRi 3 1 −13

H =

(φ+

φ 0

)1 2 +1

2

Table 2.1 Fermion and Higgs fields and the corresponding SM quantum numbers. The indexi denotes the fermion generation. The numbers in the last column are the hypercharge Y .

2.2.2 The SM Lagrangian

Armed with the Standard Model gauge group, it is straightforward to write down the La-grangian density of the SM. First, let us write down the kinetic terms for the gauge fields

LGauge =−14

GaµνGµν

a − 14

FaµνFµν

a − 14

BµνBµν (2.10)

where Gaµν , Fa

µν and Bµν are the gauge field strengths defined as

Gaµν ≡ ∂µGa

ν −∂νGaµ +gs f abcGb

µGcν , (2.11)

Faµν ≡ ∂µW a

ν −∂νW aµ +gε

abcW bµW c

ν , (2.12)

Bµν ≡ ∂µBν −∂νBµ , (2.13)

where f abc and εabc denote the structure constants of SU(3)C and SU(2)L gauge groups. Theconstants gs and g stand for the coupling constants of gauge interaction associated with thegroups SU(3)C and SU(2)L respectively. The coupling constant of U(1)Y g′ does not appearyet as there is no self interacting gauge fields in Abelian group. However, as we will seelater on, it will appear in the interaction between the gauge field and the fermions. TheLagrangian density in Eq. (2.10) is invariant under infinitesimal gauge transformations ofSU(3)C, SU(2)L as well as U(1)Y given by


δGaµ(x) =

1gs

(Dµεs(x)

)a=

1gs

(∂µε

as (x)−gs f abcGb

µ(x)εcs (x)

), (2.14)

δW aµ (x) =

1g

(Dµε(x)

)a=

1g

(∂µε

a(x)−gεabcW b

µ (x)εc(x)

), (2.15)

δBµ(x) =1g′

∂µε(x) , (2.16)

where εas (x), εa(x), and ε(x) correspond to the local infinitesimal transformation parameters

of the SU(3)C, SU(2)L, and U(1)Y gauge groups respectively. The Lagrangian density ofthe quarks and leptons that satisfy gauge invariance can be realized via minimal couplings.This can be written in terms of their associated group transformations as

LMatter = iLLi /DLLLLi + ilRi /DlRlRi + iQLi /DQLQLi + iqRi /DqRqRi , (2.17)

where /D ≡ Dµγµ and Dµ is the covariant derivative. The index i runs over the family of thequarks and leptons. The four covariant derivatives arise due to the fact that the left-handedand right handed fermions belong to different representations of the gauge groups as can beeseen from Table 2.1. Their expressions are given by

Dµ

LLLL =(

∂µ + igσ i

2W i

µ + ig′YLL

2Bµ

)LL , (2.18)

Dµ

lRlR =(

∂µ + ig′YlR

2Bµ

)lR , (2.19)

Dµ

QLQL =(

∂µ + igsλ a

2Ga

µ + igσ i

2W i

µ + ig′YQL

2Bµ

)QL , (2.20)

Dµ

qRqR =(

∂µ + igsλ a

2Ga

µ + ig′YqR

2Bµ

)qR , (2.21)

with λ a (a = 1, · · · ,8) are the eight Gell-Mann matrices correspond to SU(3)C generatorsand σ i are the Pauli matrices of SU(2)L generators. The LL, lR, QL, and qR denote theleft-handed leptonic doublets, right-handed leptonic singlets, left-handed quark doublets andright-handed quark singlets respectively. This minimally coupled Lagrangian is invariantunder the infinitesimal local transformation given by


δLL(x) = −iσ i

2ε

i(x)LL(x)−i2

ε(x)LL(x) , (2.22)

δ lR(x) = − i2

ε(x)lR(x) , (2.23)

δQL(x) = −iλ a

2ε

as (x)QL(x)− i

σ i

2ε

i(x)QL(x)−i2

ε(x)QL(x) , (2.24)

δqR(x) = −iλ a

2ε

as (x)qR(x)−

i2

ε(x)qR(x) , (2.25)

The scalar Higgs Lagrangian density which gives the spontaneous symmetry breakingLH is given by

LHiggs =(

DHµH)†

Dµ

HH −V (H,H†) (2.26)

with V (H,H†) denotes the Higgs potential for the Higgs doublet H. We will write down theexplicit form of the minimal Higgs potential V (H,H†) in the next subsection. The covariantderivative of the Higgs with minimal coupling is given by

Dµ

HH =(

∂µ + igσ i

2W i

µ − ig′YH

2Bµ

)H , (2.27)

which is invariant under the infinitesimal transformation of the form

δH(x) = −iσ i

2ε

i(x)H(x)+i2

ε(x)H(x) . (2.28)

Finally, we can write down the Yukawa Lagrangian LYukawa density which describes theinteraction between matter fields and the Higgs field as

LYukawa =−QLiYi jd dR jH − QLiY i j

u uR jH − LLiY i je eRiH +h.c. (2.29)

where H = iσ2H∗. The i, j denote the family indices. Here we expand the qR explicitly sincethe Yukawa couplings between uR and dR are different. The full SM Lagrangian densitytherefore can be expressed as

LSM = LGauge +LMatter +LHiggs +LYukawa . (2.30)

This Lagrangian density describes all the interaction known in nature except the gravitationalone. Notice that all the particles that we have now are still massless.


2.2.3 The Higgs Mechanism

The minimal Higgs potential that (minimal in the sense that only one Higgs doublet isinvolved) invariant under the SM gauge transformation is given by

V (H,H†) =−m2H†H +λSM

4(H†H)2 , (2.31)

with λSM > 0 to ensure vacuum stability and such that the extremum of the potential locatedat

∂V∂ ⟨H⟩ = ⟨H†⟩

(−m2 +

λSM

2(H†H)

)= 0 , (2.32)

∂V∂ ⟨H†⟩ =

(−m2 +

λSM

2(H†H)

)⟨H⟩= 0 . (2.33)

This leads to two solutions of the extremum point as

⟨H⟩ = ⟨H†⟩= 0 , (2.34)

⟨H†H⟩ =2m2

λSM. (2.35)

The first solution corresponds to the local maximum while the second one is the true minimum

of the potential. The vacua are degenerate and form a circle of radius(

2m2

λSM

)1/2. The stability

of the theory requires us to take the second solution. We choose the minima of the form

⟨H⟩ =

(0v√2

), v =

2m√λSM

(2.36)

and furthermore expand the theory around the ground state (minima of the potential) byshifting the Higgs field as

H → H + ⟨H⟩= H +

(0v√2

). (2.37)

By doing this we fix the vacuum and therefore single out a particular direction in the groundstate. Thus, there is a special direction in the vacuum of our theory and this particular pointclearly breaks the corresponding symmetry that we have in the beginning. This mechanismrefers to the spontaneously breakdown of the symmetry (SSB) or the Higgs mechanism. Thisoccurs when the symmetry of the theory is broken by the ground state.


The mass spectrum of the Standard Model can be obtained by substituting the Higgsfield as in Eq. (2.37) and retain only the quadratic terms of the fields (taking into account themixing terms as well). This will give three massive gauge bosons as

W±µ =

(W 1µ ∓ iW 2

µ )√2

with mass mW± = gv2, (2.38)

Zµ = −sinθW Bµ + cosθWW 3µ with mass mZ =

√g2 +g′2

v2, (2.39)

where the weak mixing angle θW is related to the weak couplings as

sinθW =g′√

g2 +g′2. (2.40)

One can immediately see that cosθW =mW±mZ

. Note that after the SSB, our gauge fields havebeen redefined as

W±µ ∓ i

mW∂µφ

± →W±µ , (2.41)

Zµ − 1mZ

∂µ χ → Zµ , (2.42)

where χ is defined as χ ≡ i√2

(φ 0 −φ 0

). Thus, after the SSB, the massless gauge field

receives one additional degree of freedom and become massive. The gauge bosons are saidto have eaten the Goldstone bosons during the SSB and hence become massive. In additionto the massive gauge fields, there is one massless degree of freedom written as

Aµ = cosθW Bµ + sinθWW 3µ , (2.43)

which is what we called the photon.Furthermore, when the Higgs field generates non-zero vacuum expectation value (VEV),

the Yukawa interactions induce the mass matrices for quarks and charged leptons with thegeneric form M f = Y f v/

√2, where f denotes the charged leptons or quarks fields. On the

other hand, due to the fact that only the left-chirality of the neutrinos are present, the neutrinosare massless in this SM framework. As an additional remark, the general form of the chargedfermion mass matrices M f are given by 3×3 complex matrices which can be brought into


diagonal forms by using two unitary matrices via the following transformations

V uL Mu(V u

R )† = diag(mu,mc,mt) , (2.44)

V dL Md(V d

R )† = diag(md,ms,mb) , (2.45)

V eL Me(V e

R)† = diag(me,mµ ,mτ) , (2.46)

where V uL , V u

R , V dL , V d

R , V eL and V e

R are different unitary matrices operate in the associatedflavor vector space, i.e.

uL =V uL u′L , uR =V u

R u′R , dL =V dL d′

L , dR =V dR d′

R ,eL =V eL e′L , eR =V e

Re′R . (2.47)

The primed fields denote the physical fields with definite masses.Next, the interaction between the gauge fields and the matter fields are categorized into

three parts: the electromagnetic current, the neutral current, and the charged current. Theexpression of the first two currents are described by

Lem =− e(

Qu (uLγµuL + uRγ

µuR)+Qd(dLγ

µdL + dRγµdR)

+Qe (eLγµeL + eRγ

µeR))

Aµ ,(2.48)

and

LNC = − esinθW cosθW

[12

uLγµuL −

12

dLγµdL +

12

νLγµ

νL −12

eLγµeL

− sin2θW

(Qu (uLγ

µuL + uRγµuR)+Qd

(dLγ

µdL + dRγµdR)

+ Qe (eLγµeL + eRγ

µeR))]

Zµ . (2.49)

The electric charges of up-type quarks, down-type quarks and charged leptons are denoted asQu, Qd and Qe respectively. These interaction currents are unchanged under the transforma-tions given in Eq. (2.47).

Finally, under the transformation in Eq. (2.47), the charged current is modified into thefollowing form

LCC =− e√2sinθW

(u′Lγ

µ W+µ VCKMd′

L + νLγµ W+

µ (V eL )

†e′L +h.c.)

(2.50)

where VCKM =V uL (V

dL )

† is the usual Cabibbo–Kobayashi–Maskawa (CKM) mixing matrixin the quark sector [15, 16]. Due to the massless neutrinos, there is no corresponding mixingmatrix in the lepton sector. We note that the neutral current interaction of the neutrinos with


the Z boson is invariance under this field redefinition. Thus we can drop all the primes in thephysical fermion fields in the SM Lagrangian.

Chapter 3

The G2HDM Model

The Gauged Two Higgs Doublet Model (G2HDM) proposed by Huang et al. [9] will bediscussed in this chapter. The model was constructed to explain the observed dark matter relicabundance by embedding the two Higgs doublets into a non Abelian gauge group SU(2)H

in such a way that the DM stability is protected by the SU(2)H gauge group instead of theusual ad hoc Z2 discrete group. First, we will discuss the important motivation behind thecreation of the G2HDM model. We further discuss the particle contents in this model andtheir role. The potential consistent with the underlying symmetry of the gauge group aswell as its SSB breaking pattern will be briefly reviewed. As a bonus, the existence of theaccidental Z2 in the model will be elaborated. Lastly, we will summarize the correspondingtheoretical constraints coming from the vacuum stability, perturbative unitarity as well asHiggs phenomenology on the scalar sector.

3.1 Motivation

The discovery of the 125 GeV Higgs boson at the LHC was certainly a milestone in thehistory of particle physics. It completes, at least, the Standard Model particle contentsdiscussed in the chapter 2. However, the SM alone can not answer the remaining real openproblems in physics such as the origin of the neutrino mass, the observed dark energy thatdominates our universe, and the existence of the unknown matter, which constitutes almostone third of the energy density budget in our universe, dubbed as the dark matter problem.In order to explain these problems, many authors have tried to construct various modelsby enlarging the particle contents of the SM either by adding new fermions into the SM,extending the SM scalar sector as well as embedding the SM gauge group into higher groups.Among the many beyond the Standard Model (BSM) models available in the market, thegeneral two Higgs doublet model (2HDM) is a simple extension [17] by just adding one

18 The G2HDM Model

more Higgs doublet to the SM. One class of 2HDM model, the inert Higgs doublet model(IHDM) [18–21], advertised the lightest neutral component of the second Higgs doubletto be the DM candidate. The stability of the DM candidate in this model is achieved byassigning a discrete Z2 symmetry on the scalar potential of the model. However, accordingto the study of reference [22, 23], the discrete and continuous global symmetry is stronglydisfavored by the gravitational effects. In order to avoid these unpleasant features as well asproviding the DM candidate, a recent study in [9] has constructed a model called GaugedTwo Higgs Doublet model (G2HDM) in which the two Higgs doublets H1 and H2 are puttogether into a doublet H = (H1,H2)

T of a new non-abelian SU(2)H gauge group. Theneutral component of H2 is stable thanks to the SU(2)H gauge symmetry and hence it is aviable DM candidate. Other particles are incorporated in the G2HDM model including aSU(2)H doublet, a SU(2)H triplet, and heavy SU(2)L singlet Dirac fermions. Their roles willbe explained in the following session. In addition, The SM right-handed fermions are pairedwith new heavy right-handed fermions to form SU(2)H doublets. After the SSB, the VEV ofSU(2)H doublet induces the heavy fermions masses. In order to simplify the scalar potential,a U(1)X group is employed. Let us list some of the important properties of G2HDM:

• It is free of gauge and gravitational anomalies;

• It is renormalizable;

• The stability of inert DM candidate (H2) is protected by SU(2)H gauge symmetry;

• After SSB, the accidental Z2 symmetry survives such that all the SM particles belongto the Z2 even particles while some of the new scalars, W ′ and new heavy fermionsare odd. The lightest odd particle if neutral can be a DM candidate whose stability isprotected by the accidental Z2 symmetry;

• No flavour changing neutral currents at tree level for the SM sector;

• the VEV of the triplet induces SU(2)L symmetry breaking while that of ΦH generatemasses to the new fermions via SU(2)H-invariant Yukawa couplings;

• etc.

Some phenomenology of G2HDM at the LHC had been studied previously in [9, 24]for Higgs physics and in [25] for the search of the new gauge bosons. Recently, systematicstudies on theoretical and phenomenological constraints for both the Higgs and gauge sectorsin G2HDM have been presented in [26] and [27] respectively.

3.2 The Particle Contents 19

3.2 The Particle Contents

The full gauge group of G2HDM model is given by SU(3)C ×SU(2)L ×U(1)Y ×SU(2)H ×U(1)X . The 125 GeV Higgs boson is contained within the SU(2)L doublet H1. There is alsoanother SU(2)L doublet H2 which contains scalar dark matter candidate. These H1 and H2

doublets are grouped together into SU(2)H doublet H = (H1,H2)T . In addition, there is also

SU(2)H triplet ∆H and doublet ΦH which transform under trivial representation of the SMgauge group. Furthermore, all the scalars in this model carry a particular U(1)X quantumnumber. Moving to the fermion sector, the SM left-handed SU(2)L doublets transformtrivially under SU(2)H , while the SM right-handed SU(2)L singlets along with the newright-handed singlets are paired together to form SU(2)H doublets. Due to the anomalycancellation, the existence of new heavy left-handed fermions is a must. These new heavyfermions belong to the singlet representation of both SU(2)L and SU(2)H gauge groups. Thefull matter contents of the G2HDM model together with their respective quantum numberassignments are collected in Table 3.1.

Matter Fields SU(3)C SU(2)L SU(2)H U(1)Y U(1)X

QL = (uL dL)T 3 2 1 1/6 0

UR =(uR uH

R)T 3 1 2 2/3 1

DR =(dH

R dR)T 3 1 2 −1/3 −1

uHL 3 1 1 2/3 0

dHL 3 1 1 −1/3 0

LL = (νL eL)T 1 2 1 −1/2 0

NR =(νR νH

R)T 1 1 2 0 1

ER =(eH

R eR)T 1 1 2 −1 −1

νHL 1 1 1 0 0

eHL 1 1 1 −1 0

H = (H1 H2)T 1 2 2 1/2 1

∆H =

(∆3/2 ∆p/

√2

∆m/√

2 −∆3/2

)1 1 3 0 0

ΦH = (Φ1 Φ2)T 1 1 2 0 1

Table 3.1 Matter contents and their quantum number assignments in G2HDM.

20 The G2HDM Model

3.3 Higgs Potential

The most general Higgs potential which is invariant under the full G2HDM gauge group canbe expressed as

VT =V (H)+V (ΦH)+V (∆H)+Vmix (H,∆H ,ΦH) , (3.1)

where the first three terms on the right hand side describe the self interaction of the corre-sponding scalar fields while the last term denotes the mixing interaction among them. Theself interaction of the SU(2)L and SU(2)H scalar doublet H is given by

V (H) = µ2H(HαiHαi

)+λH

(HαiHαi

)2+

12

λ′Hεαβ ε

γδ(HαiHγi

)(Hβ jHδ j

),

= µ2H

(H†

1 H1 +H†2 H2

)+λH

(H†

1 H1 +H†2 H2

)2+λ

′H

(−H†

1 H1H†2 H2 +H†

1 H2H†2 H1

),

(3.2)

where (α , β , γ , δ ) and (i, j) stands for the SU(2)H and SU(2)L indices respectively. Theseindices run from 1 to 2. The upper and lower indices are related as Hαi = H∗

αi. Note thatV (H) in Eq. (3.2) consists of all the possible renormalizable terms involving H1 and H2 andit is automatically invariant under H1 → H1 and H2 →−H2. Unlike IHDM, this Z2 symmetryis not imposed by hand. The gauge invariance of SU(2)L ×SU(2)H implies this symmetryautomatically! The self interaction of ΦH can be written as

V (ΦH) = µ2ΦΦ

†HΦH +λΦ

(Φ

†HΦH

)2,

= µ2Φ (Φ∗

1Φ1 +Φ∗2Φ2)+λΦ (Φ∗

1Φ1 +Φ∗2Φ2)

2 , (3.3)

where ΦH = (Φ1 Φ2)T belongs to SU(2)H doublet and SU(2)L singlets. Note that V (ΦH)

in Eq. (3.3) is invariant under Φ1 →−Φ1 and Φ2 → Φ2. The self interaction of the SU(2)H

scalar triplet is given by

V (∆H) = −µ2∆Tr(∆

2H)+λ∆

(Tr(∆

2H))2

,

= −µ2∆

(12

∆23 +∆p∆m

)+λ∆

(12

∆23 +∆p∆m

)2

, (3.4)

where the triplet fields ∆H is written in the following matrix form

∆H =

(∆3/2 ∆p/

√2

∆m/√

2 −∆3/2

)= ∆

†H with ∆m = (∆p)

∗ and (∆3)∗ = ∆3 . (3.5)

3.3 Higgs Potential 21

Note that V (∆H) in Eq. (3.4) is invariant under ∆3 → ∆3 and ∆p,m →−∆p,m. Finally, themixing terms describing all possible interaction between the scalar fields can be expressed as

Vmix (H,∆H ,ΦH) = +MH∆

(H†

∆HH)−MΦ∆

(Φ

†H∆HΦH

)+λHΦ

(H†H

)(Φ

†HΦH

)+λ

′HΦ

(H†

ΦH

)(Φ

†HH)

+λH∆

(H†H

)Tr(∆

2H)+λΦ∆

(Φ

†HΦH

)Tr(∆

2H). (3.6)

Furthermore, Eq. (3.6) can also be expressed in terms of their fundamental fields H, ∆H andΦH as the following

Vmix (H,∆H ,ΦH) = +MH∆

(1√2

H†1 H2∆p +

12

H†1 H1∆3 +

1√2

H†2 H1∆m − 1

2H†

2 H2∆3

)−MΦ∆

(1√2

Φ∗1Φ2∆p +

12

Φ∗1Φ1∆3 +

1√2

Φ∗2Φ1∆m − 1

2Φ

∗2Φ2∆3

)+λHΦ

(H†

1 H1 +H†2 H2

)(Φ∗

1Φ1 +Φ∗2Φ2)

+λ′HΦ

(H†

1 H1Φ∗1Φ1 +H†

2 H2Φ∗2Φ2 +H†

1 H2Φ∗2Φ1 +H†

2 H1Φ∗1Φ2

)+λH∆

(H†

1 H1 +H†2 H2

)(12

∆23 +∆p∆m

)+λΦ∆ (Φ

∗1Φ1 +Φ

∗2Φ2)

(12

∆23 +∆p∆m

). (3.7)

Note that Vmix in Eq. (3.7) is invariant under H1 → H1, H2 →−H2, Φ1 →−Φ1, Φ2 → Φ2,∆3 → ∆3 and ∆p,m →−∆p,m. Thus gauge invariance implies the whole scalar potential isinvariant under this discrete Z2 symmetry automatically.

As an additional remark, one notes that the coefficient of µ2∆

in V (∆H) has a negative signas opposed to that of µ2

H and µ2Φ

in V (H) and V (ΦH) terms. Based on the above potential,the coefficients of the quadratic terms for H1 and H2 can be extracted as

µ2H − 1

2MH∆ · v∆ +

12

λH∆ · v2∆ +

12

λHΦ · v2Φ , (3.8)

µ2H +

12

MH∆ · v∆ +12

λH∆ · v2∆ +

12(λHΦ +λ

′HΦ) · v2

Φ , (3.9)

respectively. Since these coefficients consist of several terms, the total value of thesecoefficients can have either positive or negative values, even if one choose the positive valueof µ2

H . Thus, if one sets Eqs. (3.8) and (3.9) to be negative and positive respectively, one canthen obtain ⟨H1⟩ = 0 and ⟨H2⟩= 0 which will trigger the spontaneous breakdown of SU(2)L

symmetry.

22 The G2HDM Model

By expanding the potential, one can also obtain the coefficients of the quadratic terms forΦ1 and Φ2 which are given by

µ2Φ +

12

MΦ∆ · v∆ +12

λΦ∆ · v2∆ +

12(λHΦ +λ

′HΦ) · v2 , (3.10)

µ2Φ − 1

2MΦ∆ · v∆ +

12

λΦ∆ · v2∆ +

12

λHΦ · v2 , (3.11)

respectively. Analogous to the previous case of H1 and H2, with the positive value of µ2Φ

,we can obtain ⟨Φ1⟩ = 0 and ⟨Φ2⟩ = 0 by choosing some particular combinations on theparameter space.

In (3.4), if µ2∆> 0, SU(2)H is spontaneously broken by the VEV ⟨∆3⟩=−v∆ = 0 with

⟨∆p,m⟩= 0 by using an SU(2)H rotation. In fact, this induces the symmetry breaking of theother gauge symmetries in this model.

Note that the scalar potential in G2HDM is CP-conserving due to the fact that all termsin V (H), V (ΦH), V (∆H) and Vmix(H,∆H ,ΦH) are Hermitian, implying all the coefficientsare necessarily real. Therefore, there is no CP violation in the scalar sector of this model.

3.4 The SSB of the Potential and the Mass Spectrum

3.4.1 Spontaneous Symmetry Breaking

Similar to the case of the SM in chapter 2, the spontaneous symmetry breaking can berealized by expressing the scalar fields in terms of their fluctuation around the VEVs as

H1 =

(G+

v+h√2+ i G0√

2

), H2 =

(H+

H02

), ΦH =

(Gp

HvΦ+φ2√

2+ iG0

H√2

), ∆H =

(−v∆+δ32

1√2∆p

1√2∆m

v∆−δ32

).

(3.12)

where v, vΦ and v∆ are VEVs which can be expressed as a function of the scalar potentialparameters after minimizing the scalar potential. The set ΨG ≡ G0,G+,G0

H ,GpH are

Goldstone bosons to be absorbed later by the gauge fields. We note that only the Z2 evenfields are getting VEVs. Thus the accidental Z2 symmetry is not spontaneously broken whichmay lead to cosmological domain wall problem.

3.4 The SSB of the Potential and the Mass Spectrum 23

By putting the value of the VEVs, v, vΦ, v∆ into the potential VT in Eq. (3.1), the totalpotential becomes

VT (v,v∆,vΦ) =14[λHv4 +λΦv4

Φ +λ∆v4∆ +2

(µ

2Hv2 +µ

2Φv2

Φ −µ2∆v2

∆

)−(MH∆v2 +MΦ∆v2

Φ

)v∆ +λHΦv2v2

Φ +λH∆v2v2∆ +λΦ∆v2

Φv2∆

].

(3.13)

Next, after minimizing the potential in Eq.(3.13), we will have the following equations:(2λHv2 +2µ

2H −MH∆v∆ +λHΦv2

Φ +λH∆v2∆

)= 0 , (3.14)(

2λΦv2Φ +2µ

2Φ −MΦ∆v∆ +λHΦv2 +λΦ∆v2

∆

)= 0 , (3.15)

4λ∆v3∆ −4µ

2∆v∆ −MH∆v2 −MΦ∆v2

Φ +2v∆

(λH∆v2 +λΦ∆v2

Φ

)= 0 . (3.16)

By solving this set of coupled equations, one can get solutions for v, vΦ and v∆ in termsof other parameters in the potential. In addition, one can see the effects of triplet’s VEVv∆ in breaking the SU(2)L × U(1)Y and U(1)X gauge group after its SSB on the SU(2)H

symmetry.

3.4.2 Scalar Mass Spectrum

As in the SM case, the mass spectrum can be extracted after the SSB. In the scalar sector, weobtain the following three diagonal blocks in the mass matrix. The first 3×3 block in thebasis of S = h,φ2,δ3 is given by

M 2H =

2λHv2 λHΦvvΦv2 (MH∆ −2λH∆v∆)

λHΦvvΦ 2λΦv2Φ

vΦ

2 (MΦ∆ −2λΦ∆v∆)v2 (MH∆ −2λH∆v∆)

vΦ

2 (MΦ∆ −2λΦ∆v∆)1

4v∆

(8λ∆v3

∆+MH∆v2 +MΦ∆v2

Φ

) .

(3.17)

This symmetric matrix can be diagonalized by an orthogonal matrix OH ,

(OH)T ·M 2H ·OH = Diag(m2

h1,m2

h2,m2

h3) . (3.18)

The lightest eigenvalue mh1 is the mass of h1 which is identified as the 125 GeV Higgsboson observed at the LHC, while mh2 and mh3 are the masses of heavier Higgses h2 andh3 respectively. The physical Higgs hi (i = 1,2,3) is a mixture of the three components ofS: hi = OH

jiS j. Thus the SM-like Higgs boson in this model is a linear combination of the

24 The G2HDM Model

neutral components of the two SU(2) doublets H1 and ΦH and the real component of theSU(2)H triplet ∆H .

The second block of 3×3 matrix in the basis of D = GpH ,H

0∗2 ,∆p is given by

M 2D =

MΦ∆v∆ +12λ ′

HΦv2 1

2λ ′HΦ

vvΦ −12MΦ∆vΦ

12λ ′

HΦvvΦ MH∆v∆ +

12λ ′

HΦv2

Φ

12MH∆v

−12MΦ∆vΦ

12MH∆v 1

4v∆

(MH∆v2 +MΦ∆v2

Φ

) . (3.19)

This matrix can also be diagonalized by an orthogonal matrix OD

(OD)T ·M 2D ·OD = Diag(m2

Gp,m2D,m

2∆) . (3.20)

One eigenvalue of Eq. (3.19) is zero (i.e. mGp = 0) and identified as the unphysical Goldstoneboson Gp. The mD and m

∆(mD < m

∆) are masses of two physical fields D and ∆ respectively.

Their analytical expression is given by the following

M2D,∆

=−B∓

√B2 −4AC

2A, (3.21)

where the A,B,C parameters read

A = 8v∆ ,

B = −2(MH∆

(v2 +4v2

∆

)+MΦ∆

(4v2

∆ + v2Φ

)+2λ

′HΦv∆

(v2 + v2

Φ

)), (3.22)

C =(v2 + v2

Φ +4v2∆

)(MH∆

(λ′HΦv2 +2MΦ∆v∆

)+λ

′HΦMΦ∆v2

Φ

).

The physical field D can be a DM candidate in G2HDM. Furthermore, in a particularparameter space where the expression inside the square root of Eq.(3.21) is quite small,the ∆ and the D fields can be degenerate, and hence one must include the coannihilation incalculating the thermal relic of the D. There are other neutral fields νH

L , νHR or W ′(p,m) in the

model that can be qualified as DM candidate as well, depending on which one is the lightest.In this work, we will assume D, a complex scalar field, is the lightest among them.

The final mixing matrix is a 4×4 diagonal block with

m2H± = MH∆v∆ −

12

λ′Hv2 +

12

λ′HΦv2

Φ , (3.23)

m2G± = m2

G0 = m2G0

H= 0 , (3.24)

where mH± is mass of the physical charged Higgs H±, and mG±, mG0, mG0H

are masses ofthe four Goldstone boson fields G±, G0 and G0

H , respectively. Note that we have used the


minimization conditions Eqs. (3.14), (3.15) and (3.16) to simplify various matrix elements ofthe above mass matrices. The six Goldstone particles G±, G0, G0

H and Gp,m will be absorbedby the longitudinal components of the massive gauge bosons W±, Z, Z′ and W ′(p,m) afterSSB.

3.4.3 Gauge Boson Mass Spectrum

The gauge bosons mass can be extracted after the spontaneously symmetry breaking of theHiggs fields as discussed in the previous session. This can be read directly from the kineticterms of the scalars ∆H , Φ and H as

L ⊃ Tr[(

D′µ∆H

)† (D′µ

∆H)]

+(

D′µΦ

)† (D′µ

Φ)+(

D′µH)† (

D′µH), (3.25)

where the covariant derivatives of each fields are given by

D′µ∆H = ∂µ∆H − igH

[W ′

µ ,∆H

], (3.26)

D′µΦ =

(∂µ − i

gH√2

(W ′p

µ T p +W ′mµ T m

)− igHW ′3

µ T 3 − igX Xµ

)·Φ , (3.27)

and

D′µH =

(Dµ ·1− i

gH√2

(W ′p


)− igHW ′3

µ T 3 − igX Xµ

)·H , (3.28)

here Dµ is the SU(2)L covariant derivative which separately operates on H1 and H2, gH (gX)

is the SU(2)H (U(1)X) gauge coupling constant, and

W ′µ =

3

∑a=1

W ′aT a =1√2

(W ′p


)+W ′3

µ T 3, (3.29)

in which the SU(2)H generators is equal to the half of Pauli matrices T a = τa/2 acting onthe SU(2)H space, W ′(p,m)

µ = (W ′1µ ∓ iW ′2

µ )/√

2, and

T p =12(τ

1 + iτ2)=(0 10 0

), T m =

12(τ

1 − iτ2)=(0 01 0

), (3.30)

are the usual SU(2) fundamental representation ladder operators.

26 The G2HDM Model

The SM charged gauge boson W± obtained its mass entirely from v, so it is given by

MW± =12

gv , (3.31)

same as the SM.Due to their quantum number assignments, the SU(2)H gauge bosons W ′a and the U(1)X

gauge boson X receive their masses from ⟨∆3⟩, ⟨H1⟩ and ⟨Φ2⟩. The terms contributed fromthe doublets are similar with that of the standard model. Since ∆H transforms as a tripletunder SU(2)H , i.e., in the adjoint representation, the contribution to the W ′a masses arisefrom the term

L ⊃ g2HTr

([W ′µ ,∆H

]† [W ′µ ,∆H

]). (3.32)

Therefore, the W ′(p,m) receives a mass from ⟨∆3⟩, ⟨Φ2⟩ and ⟨H1⟩

m2W ′(p,m) =

14

g2H(v2 + v2

Φ +4v2∆

), (3.33)

while gauge bosons X and W ′3, together with the SM W 3 and U(1)Y gauge boson B, acquiretheir masses from ⟨Φ2⟩ and ⟨H1⟩ only but not from ⟨∆H⟩:

18

(v2(

2gX Xµ +gHW ′3µ −gW 3

µ +g′Bµ

)2+ v2

Φ

(−2gX Xµ +gHW ′3

µ

)2), (3.34)

where g and g′ are the SM SU(2)L and U(1)Y gauge couplings.Following [28–30], we add the following Stueckelberg mass term to avoid additional

massless gauge boson appearing in Eq.3.34 in the form of

LStu =+12(∂µa+MX Xµ +MY Bµ

)2, (3.35)

where MX and MY are the Stueckelberg masses for the gauge fields Xµ and Bµ of U(1)X andU(1)Y respectively, and a is the axion field, we have the following matrix in the basis ofV ′ =

B,W 3,W ′3,X

:

M 2G =

g′2v2

4 +M2Y −g′gv2

4g′gHv2

4g′gX v2

2 +MX MY

−g′gv2

4g2v2

4 −ggHv2

4 −ggX v2

2g′gHv2

4 −ggHv2

4g2

H(v2+v2Φ)

4gHgX(v2−v2

Φ)2

g′gX v2

2 +MX MY −ggX v2

2gHgX(v2−v2

Φ)2 g2

X(v2 + v2

Φ

)+M2

X

. (3.36)


As a result, this mass matrix has only one zero mode corresponding to the photon, and threemassive modes Z,Z′,Z′′. This mass matrix can be diagonalized by general 4×4 orthogonalmatrix. After making a rotation in the 1− 2 plane by the usual Weinberg angle θw, themass matrix M 2

G will further transform into a block diagonal matrix with the vanishing firstcolumn and first row. The nonzero 3-by-3 block matrix can be further diagonalized by anorthogonal matrix OG, ZSM

W ′3

X

= OG ·

ZZ′

Z′′

, (3.37)

where ZSM is the SM Z boson without the presence of the W ′3 and X bosons. For the rest of

this work, we will set MY equal to zero [27].

3.4.4 Fermionic Mass Spectrum

To complete the discussion of the mass spectrum in this session, let us write down themost general Yukawa couplings in this model and the corresponding fermionic mass terms.Starting from the quark sector, we set the quark SU(2)L doublet, QL, to be an SU(2)H singletand including additional SU(2)L singlets uH

R and dHR which together with the SM right-handed

quarks uR and dR, respectively, to form SU(2)H doublets consistent with the table 3.1, i.e.,UT

R = (uR uHR )2/3 and DT

R = (dHR dR)−1/3, where the subscript represents hypercharge, we

have 1

LYuk ⊃ ydQL (DR ·H)+ yuQL

(UR·

≈H)+H.c.,

= ydQL(dH

R H2 −dRH1)− yuQL

(uRH1 +uH

R H2)+H.c., (3.38)

where≈H≡ (H2 − H1)

T with H1,2 = iτ2H∗1,2. After the EW symmetry breaking ⟨H1⟩ = 0, u

and d obtain their masses but uHR and dH

R remain massless since H2 does not get a VEV.To give masses to the additional species, we employ the SU(2)H scalar doublet ΦH =

(Φ1 Φ2)T , which is singlet under SU(2)L, to write down the following Yukawa couplings for

the left-handed SU(2)L,H singlets uHL and dH

L and the SU(2)H doublets UR and DR,

LYuk ⊃ − y′ddHL (DR ·ΦH)+ y′uuH

L(UR · ΦH

)+H.c.,

= − y′ddHL(dH

R Φ2 −dRΦ1)− y′uuH

L(uRΦ

∗1 +uH

R Φ∗2)+H.c., (3.39)

1A ·B is defined as εi jAiB j where A and B are two 2-dimensional spinor representations of SU(2)H .

28 The G2HDM Model

where Φ has Y = 0, Y (uHL ) = Y (UR) = 2/3 and Y (dH

L ) = Y (DR) =−1/3 with ΦH = (Φ∗2 −

Φ∗1)

T . With ⟨Φ2⟩ = vΦ/√

2, uHR (uH

L ) and dHR (dH

L ) obtain masses y′uvΦ/√

2 and y′dvΦ/√

2,respectively. Note that there is no contribution from v∆ for the masses for both the SM andnew fermions.

The lepton sector is similar to the quark sector as

LYuk ⊃ yeLL (ER ·H)+ yν LL

(NR·

≈H)− y′eeH

L (ER ·ΦH)+ y′ννHL(NR · ΦH

)+H.c.,

= yeLL(eH

R H2 − eRH1)− yν LL

(νRH1 +ν

HR H2

)− y′eeH

L(eH

R Φ2 − eRΦ1)− y′νν

HL(νRΦ

∗1 +ν

HR Φ

∗2)+H.c., (3.40)

where ETR = (eH

R eR)−1, NTR = (νR νH

R )0 in which νR and νHR are the right-handed neutrinos

and their SU(2)H partner respectively, while eHL and νH

L are SU(2)L,H singlets with Y (eHL ) =

−1 and Y (νHL ) = 0 respectively. Notice that neutrinos are purely Dirac in this setup, i.e., νR

paired up with νL having Dirac mass MνD = yνv/

√2, while νH

R paired up with νHL having

Dirac mass MνH

D = y′νvΦ/√

2. As a result, the lepton number is conserved, implying vanishingneutrino-less double beta decay.

We note that the accidental Z2 symmetry in the scalar sector can be extended to thefermion sector as well. Indeed all the above Yukawa couplings are invariant under H1 → H1,H2 →−H2, f SM → f SM and f H →− f H for all fermion f = u,d,ν ,e.

3.5 The Accidental Z2 Symmetry

As mentioned in the previous session, the stability of the dark matter candidate in this modelis protected by the accidental discrete Z2 symmetry which is automatically implied by theSU(2)L ×U(1)Y × SU(2)H ×U(1)X gauge symmetry. However, as the gauge symmetryexperiences SSB, one may expect that this symmetry is no longer there. In other words,after the SSB, there is no symmetry that prevents the dark matter to decay. Interestingly, thisis not the case. Thanks to its special vacuum alignment where H2 fields do not acquire aVEV, the accidental Z2 symmetry remains intact after SSB. Thus besides the two well-knownaccidental global symmetries of baryon number and lepton number in the SM, there is also aaccidental discrete Z2 symmetry in G2HDM. All the field content arrange themselves intoa particular element of this Z2 group. All the SM fermions ( f SM

L,R ) belong to Z2 even, whilethe new heavy fermions ( f H

L,R) classified as Z2 odd. The SM Higgs boson (h1) as well as itsheavy partners (h2,h3) are members of Z2 even due to their mixings. On the other hand, thescalars that mix together in the second block of Eq.(3.19) (D, ∆) is a part of Z2 odd. The SMgauge boson (Z) along with its heavy partners (Z′,Z′′) align themselves into Z2 even. Lastly,

3.6 Theoretical Constraints on the Scalar Sector 29

although it was a part of the off-diagonal SU(2)H gauge boson, the W ′(p,m) is placed into Z2

odd cell. This is very surprising since both Z′ and W ′(p,m) have the same SU(2)H origin, theycan be separated into opposite sides of Z2 members. The summary of the Z2 assignments forall the particles in G2HDM is collected in the table 3.2 below.

Z2 Even h1, h2, h3 W±, Z, Z′, Z′′ f SML,R

Z2 Odd D, ∆, H± W ′(p,m) f HL,R

Table 3.2 The Z2 assignments in G2HDM model.

3.6 Theoretical Constraints on the Scalar Sector

This section summarizes the theoretical and Higgs phenomenological constraints on thescalar sector parameters space of G2HDM model discussed in [26].

This study focused on constraining the parameter space of the potential under severalphysical arguments and phenomenological results. The first physical consideration is theboundedness of the scalar potential. This relies on the fact that the potential must be boundedfrom below to ensure the vacuum stability (VS) of the theory. Second, one needs to examinewhether the theory satisfies the perturbative unitarity (PU) or not. This kind of check canbe realized by calculating the scattering amplitudes at very high energy. If the amplitudesare well behaved and do not go to infinity as the energy gets higher, then the unitarity ofthe theory is maintained. In the scalar sector of G2HDM model, this can be achieved byconsidering all possible 2 → 2 scattering amplitudes in the scalar sector and evaluate them inthe high energy limit. Thus, all scalars appear in this model need to be taken into accountfor these scattering amplitudes. Finally, the experimental constrains coming from the LHCHiggs data are needed to further limit the allowed parameter space of the model.

In [26], the surviving parameter space of the model have been evaluated under thetheoretical constraints from vacuum stability (VS), perturbative unitarity (PU) and the 125GeV Higgs physics (HP) data including the Higgs boson mass and signal strengths of Higgsboson decays into diphoton and τ+τ− from the LHC. The study showed that out of the eightλ−parameters, only two of them λH and λHΦ are essentially constrained by (VS+PU+HP).Other couplings like λ ′

H , λ ′HΦ

and λΦ∆ are less constrained. This study also concluded thatsome of the parameters such as MH∆, MΦ∆ and the VEVs are constrained only by HP but notby (VS+PU). In the numerical set up for the scanning in [26], the two parameters MH∆, MΦ∆

are varied in the range of [−1,1] TeV, v∆ ∈ [0.5,20] TeV, while v and vΦ were fixed at 246GeV and 10 TeV respectively.

30 The G2HDM Model

10−4

10−2

1

λΦ

10−4

10−2

1

λ∆

−20

0

20

λH

Φ

−20

0

20

λH

∆

−20

0

20

λΦ

∆

−20

0

20

λ′ H

Φ

10−4 10−2 1

−20

0

20

λ′ H

10−4 10−2 1 10−4 10−2 1 −20 0 20−20 0 20−20 0 20−20 0 20

10−4 10−2 1λΦ

10−4 10−2 1λ∆

−20 0 20λHΦ

−20 0 20λH∆

−20 0 20λΦ∆

−20 0 20

λ′HΦ−20 0 20

λ′H

10−4

10−2

1

10−4

10−2

1

10−4

10−2

1

−20

0

20

−20

0

20

−20

0

20

−20

0

20

10−4 10−2 1λH

10−4

10−2

1

λH

λH0.13,2.70

λΦ < 4.19

λ∆ < 5.03

λHΦ

-5.69,3.52

λH∆

-3.90,9.25

λΦ∆

-5.41,13.25

λ′HΦ

-0.38,17.11

−20 0 20

−20

0

20

λ′H-23.82,2.54

Fig. 3.1 A summary of the parameter space allowed by the theoretical and phenomenologicalconstraints. The red regions show the results from the theoretical constraints (VS+PU).The magenta regions are constrained by Higgs physics as well as the theoretical constraints(HP+VS+PU). Figure is taken from [26].

3.6 Theoretical Constraints on the Scalar Sector 31

We show a summary of allowed regions of parameter space in Fig. 3.1. The upper redtriangular block corresponds to (VS+PU) constraints, while the lower magenta triangularblock corresponds to the (VS+PU+HP) constraints. The diagonal panels indicate the allowedranges of the eight couplings λH,Φ,∆, λ ′

H , and λHΦ,H∆,Φ∆, λ ′HΦ

under the combined constraintsof (VS+PU+HP).

Chapter 4

WIMP Dark Matter and Its Constraints

The summary of the particle dark matter along with their experimental constraints will bediscussed in this chapter. We start with the astrophysical evidence of the dark matter whichprovides a strong motivation in studying this open problem. Next, we try to explain the darkmater problem from particle physics point of view by studying its thermal evolution in theuniverse. Finally, the corresponding experimental constraints coming from the dark mattersearches in the lab and sky will be given.

4.1 Astrophysical Evidence of the Dark Matter

There are many evidences that support the existence of the dark matter. Most of the evidencescome from astrophysical observations such as galactic rotation curves, gravitational lensing,structure formation, the observed cosmic microwave background (CMB), baryon acousticoscillation and the matter power spectrum, etc. In this section, we will discus the galacticrotation curves and gravitational lensing which provide strong evidence of the dark matter.

4.1.1 Galactic Rotation Curves

One of the strongest pieces of evidence for DM comes from studying the rotational velocityof stars which measure the circular speed of the star orbiting a particular galaxy. It is knownthat stars orbiting the galaxy is dictated by the gravitational potential. From Newtoniangravity, using the fact that the gravitational force is balanced by centrifugal force of the star,one can extract the star’s circular velocity, vc, as

vc(r) =

√GM

r(4.1)

34 WIMP Dark Matter and Its Constraints

1980ApJ...238..471R

Fig. 4.1 Rotation curves of spiral galaxies as observed by Rubin et al. [31]. At large radialdistance from the center, most of galaxies exhibit constant circular velocity.

where M is the mass enclosed by the galaxy, r is the distance between galaxy and the star,and G is the Newton’s gravitational constant. If the star located outside the galactic disk, Mwill be constant as there is no other massive object around. In this case vc ∝ r−1/2. However,the results of observations concluded that the circular velocity curve is constant or M(r) ∝ r.This gives us a hint that there is unknown matter apart from the visible matter and this unseenmatter distributes outside the galaxy disk. Figure 4.1 displays the 21 Sc rotation curves asobserved by Rubin et al. in [31]. This clearly shows that at large distance from the galacticcenter, the circular velocities are approaching constant values.

According to the observed rotation curves, we can conclude that the dark matter massdensity distribution is given by

ρ(r) ∝M(r)

r3 ∼ 1r2 . (4.2)

This conclusion was made by assuming that the dark matter is spherically symmetric dis-tributed around the galaxy center in contrast to the visible matter that mostly located at thegalactic disk.

4.1.2 Gravitational Lensing

It is well known from Einstein’s theory of General Relativity that light will be bent when itpropagates under the influence of gravitational potential. Thus, one expects the trajectory ofthe light will be deviated around massive objects. This is the main idea of the gravitational

4.2 WIMP as Thermally Produced Dark Matter 35

lensing. The deflection angle δφ of the light around an object with the mass M is given by

δφ ≈ 4GNMb

, (4.3)

where GN is the usual Newton’s constant and b is the impact parameter. It is clear thatwhen the mass M equals to zero (no massive object), light will follow straight line during itspropagation. This equation also tells us that when the deflected angle is known (to be exactobserved), we can further obtain the information about the mass of the corresponding object.

The incidental light from a source will pass through the massive object before it gets tothe observer’s "eyes". When the observer, the massive object as well as the light source arealigned in a straight line, the light is focused as if it passed through optical lenses. Therefore,the observer will see multiple images which originated from the same object and he/she willbe able to obtain its mass distribution. The observations of the galaxies in [32] exhibit thedensity profile ρ(r)≈ 1

r2 consistent with the one extracted from galactic rotation curves.

4.2 WIMP as Thermally Produced Dark Matter

The interaction of the dark matter so far was deduced from its gravitational interaction alone.Up to now, there is no other kind of interaction can be inferred from the dark matter. However,there are several notions in the literature that try to explain the existence of the dark matterbased on its interaction with the visible SM matter. One of the famous dark matter particlecandidate is the weakly interacting massive particle (WIMP). In this scenario, the dark matterwas assumed to be thermally produced in the early universe and interacts effectively withother particles. Moreover, the dark matter was in thermal equilibrium with the cosmic plasmabefore it decoupled when the temperature of the universe is below the dark matter mass.

In early universe, dark matter can be created thermally when the temperature of theuniverse is above the dark matter mass. The standard model (SM) particles in this era haveenough energy to produced the dark matter via the following annihilation process

f f ↔ χχ , (4.4)

where f stands for SM particle, and χ is the dark matter. The left-right arrows represent thatas long as dark matter and the SM particle are in thermal equilibrium and the reaction rateis very large compared to the Hubble expansion rate, the dark matter pair as well SM paircan be produced. However, as the expansion rate exceeds the reaction rate, the dark matterdecouples from the SM particles. The dark matter is said to experience the freeze-out. After


the freeze-out, the dark matter abundance is solely controlled by the Hubble expansion of theuniverse alone.

In order to study the abundance of the dark matter using WIMP scenario, one needsto solve the Boltzmann equation that express the number density evolution with respect oftime as a function of expansion of the universe and dark matter annihilation (creation) crosssection. This can be quantitatively written as

dnχ

dt+3Hnχ =−⟨σv⟩

[n2

χ − (neqχ )2], (4.5)

where nχ is the dark matter number density, H is the Hubble expansion rate that account forthe universe expansion, neq

χ is the dark matter number density in equilibrium, and ⟨σv⟩ is thethermally averaged cross section for dark matter annihilation into f f . The term freeze-outrefers to the moment that the reaction rate is of the same order as the Hubble expansion rate

⟨σv⟩neqχ ≈ H . (4.6)

The solution of of Boltzmann equation is plotted in figure 4.2. From this figure, onecan see that the dark matter number density is indeed depends on the thermally averagedcross section ⟨σv⟩. Dark matter with bigger cross section will decouple later which leads tosmaller relic abundance, while those with the small cross section decouples earlier and givesbigger abundance.

From the solution of the Boltzmann equation, one can then determine the dark matterfractional relic abundance which is defined as dark matter energy density over the criticaldensity of the universe. This can be written as [34]

Ωχ ≈ 10−26cm3/s⟨σv⟩ ≈ 0.1

(0.01

α

)2( m100GeV

)2, (4.7)

by assuming weakly interacting DM with the coupling α ∼ 0.01 and the DM mass mχ oforder 100 GeV, one gets the correct relic abundance of Ωχh2 = 0.12± 0.001 as observedby PLANCK collaboration. This observed relic abundance drawn from the weak scaleinteraction gives us a hint that solving the dark matter problem may have something to dowith the TeV scale physics currently probed by the LHC. This also goes under the jargon"WIMP miracle" even though it just one possibility of the combination between couplingover the mass of the dark matter.

4.3 Dark Matter Searches 37

Fig. 4.2 Dark matter number density per comoving volume as function of its mass over thetemperature of the universe. Figure is taken from [33].

4.3 Dark Matter Searches

The interaction of the dark matter particle with the SM particles in the early universe motivatesthe study of non-gravitational interaction of the dark matter in recent time. This opens upnew possibility to detect dark matter via its scattering with SM particles either elastically ornon-elastically. The dark matter can elastically be scattered with the SM particles. It can alsoannihilate into the SM particles in the center of galaxies where the density of dark matter ishigh. It can be also be produced in the collider experiment. In this session, we will reviewdark matter search directly, indirectly as well as via collider search.


Fig. 4.3 The current (solid) and projected (dotted/dashed) bounds on the spin-independentWIMP DM-nucleon cross section. The orange band indicated the neutrino floor which is atypical extraterrestrial background. This plot is adopted from [35].

4.3.1 Dark Matter Direct Search

From galactic rotation curves, we learn that the dark mater halos can be extended out to avery large distance away from the galactic center. As our solar system moves around thegalactic center, we expect that the dark matter "wind" pass through the earth. This darkmatter "wind" is expected to scatter off the nuclei target in the prepared experiment on theearth. Based on this scattering, one can study the dark matter property. This kind of methodis called dark matter direct detection. The recoil energy from dark matter-nucleon scatteringis typically lied between 1 to 100 keV range, depending on the mass of the dark matter andnucleus target. The scattering rate is typically of order 0.01 events day−1 kg−1. With sucha very low event rate, one needs to take into account the possible background in order toextract the DM detection signal. One of possible background coming from the surroundingradioactivity material. This can be avoided by shielding the nucleus target and neglect theevent that originated from electron scattering.

The dark matter nucleon interaction can be considered as non-relativistic scattering. Thisis because the typical velocity of the dark matter is about 300 km/s which is of order 10−3c.In this setup, the interaction can be categorized into spin independent and spin dependentinteraction. In the spin independent case, scattering cross section is proportional to the atomic


number A. This implies that by using the target with large nucleus number, one can probe thespin independent cross section more effectively due to the coherent contributions. The spindependent cross section on the other hand, do not increase as the atomic number becomeslarger. This kind of search usually is done by using the nucleus with unpaired protons orneutrons. The existing experimental direct search of the dark matter is presented in theFig. 4.3.

4.3.2 Dark Matter Indirect Search

Even though the dark matter annihilation is suppressed in the recent universe, it can stilloccurred in the place with high dark matter density. These include the galaxy centers, dwarfgalaxies and other astrophysical sources. Another possibility is that dark matter decays intothe SM particles. Both annihilation and decay products of the dark matter will become thecomponents of cosmic rays which propagate through the space. The typical propagation timeof the cosmic ray is usually much longer than the lifetime of the produced particles. Thus,right after produced, the SM particles will eventually decay into more stable particles whichare known as the secondary particles. These include electrons, positrons, nucleons, and thephotons (gamma ray). One of the interesting aspects of indirect detection business is thegamma ray observation. The gamma ray flux coming from the WIMP annihilation can bewritten as

dΦ

dΩdE=

σv8πm2

χ

× dNdE

×∫

l.o.sds×ρ

2 (−→r (s,Ω)) , (4.8)

where the left hand side is the differential gamma ray flux with respect to the solid angleand the energy. The first factor in the right hand side is the current WIMP annihilation crosssection. The second factor denotes the number of gamma rays produced per annihilation atgiven energy or the energy spectrum. The third factor account for the integration along theline of sight (l.o.s) which is oriented with an angle Ω away from the galactic plane. The −→rstands for the radial distance between the annihilation event and the galactic center whichis a function of both s and Ω. The integrand is the usual dark matter density distributionwhich is dependent on particular model. According to Eq. 4.8, we can learn that the indirectsearch of the dark matter provides several properties of the dark matter. This will probe howthe dark matter halo is distributed, the annihilation cross section of the dark matter todaywhich can be quite different from the early universe relevant for relic density and also thedark matter mass. Apart from the complicated statistical methods and uncertainties, the mainidea of indirect search experiment is the following [36]: once the experiment observed andmeasured the gamma ray flux coming from a particular source, one then evaluates the flux


10 50 100 500 100010-29

10-27

10-25

10-23

10-21

Fig. 4.4 Indirect DM constraints for few final states: from IceCube [37], AMS02 [38],H.E.S.S. [39], PLANCK [40], CTA projected sensitivity [41], and Fermi-MAGIC collab-orations [42]. The black dotted-dashed line denotes the typical annihilation cross section⟨σv⟩= 3×10−26cm3s−1. Figure adopted from [43].

with the relevant background. In case of null signal, one then proceeds to choose the desireddark matter density profile and also the particular annihilation products required to get theenergy spectrum. Finally, by using Eq. 4.8, one can extract the limit on σv

m2χ

. The summary ofrecent indirect dark matter search is presented in Fig. 4.4.

4.3.3 Dark Matter Collider Search

Unlike the two previous dark matter detection which are surrounded by a lot of backgroundand uncertainties, the collider search provide more friendly environment since the colliderparameters such as center mass energy and luminosity are precisely determined. In addition,for low dark matter mass, the collider search provide a complementary study with that ofdirect search since there is no limited energy threshold like direct detection case. Due toits zero electric charge as well as its stability, the suitable strategy for WIMP dark mattersearch is to look at the missing energy. One then proceeds to select the events with largemissing energy to overcome the SM background and further identifying the dark mattersignal. However, this missing energy can not solely be interpreted as the dark matter signal


since it only tells us about the existence of the neutral and stable particle which can evendecay outside the detector.

Another way to look for dark matter collider signal is by studying the invisible decayof the SM Higgs and Z boson. When the dark matter mass is below half of the Higgs mass(around 62.5 GeV), then one may expect that the Higgs will decay into dark matter pair. Thecurrent limit on the branching ratio (BR) of Higgs invisible decay [11] is < 24%. The sameargument also hold for invisible Z boson decay. When the dark matter mass is less than halfof Z boson mass, it should be taken into account. The current bound on the BR of invisible Zdecay [11] is < 20%.

Chapter 5

Dark Matter in G2HDM: Constraintsand Methodology

In this chapter we will discuss the dark matter phenomenology within the G2HDM model. Inparticular, we are interested in complex scalar dark matter that motivated the construction ofthis model. First, we discuss the properties of complex scalar dark matter in G2HDM model.We further demonstrate our strategy on imposing several dark matter phenomenologicalconstraints starting from the observed relic density, the direct search, indirect search, as wellas collider search. We summarize our parameter scan being used in this thesis in the table 5.1.

Complex scalar dark matter is a linear combination of the interaction eigenstates in thebasis of Gp

H , H0∗2 , and ∆p. After applying the proper rotation matrix OD to diagonalize the

mixing matrix in Eq. (3.19), the physical eigenstate of complex scalar dark matter can bewritten as

D = OD12Gp

H +OD22H0∗

2 +OD32∆p. (5.1)

One sees that complex scalar dark matter D can be characterized according to its component,whether it looks like inert doublet-like DM, the SU(2)H triplet-like DM, the Goldstoneboson-like DM, or the equal mixture between all of its components.

In this chapter, we will study complex scalar dark matter phenomenology according tothe following three cases:Case 1: the doublet-like DM1 by requiring fH∗

2> 2/3,

Case 2: the SU(2)H triplet-like DM if requiring f∆p > 2/3,Case 3: the SU(2)H Goldstone boson-like DM if requiring fGP > 2/3.

1Hereafter, when we refer to the doublet-like, triplet-like or Goldstone-like DM, it means the inert doublet-like, SU(2)H triplet-like or Goldstone boson-like DM, unless indicated otherwise.

44 Dark Matter in G2HDM: Constraints and Methodology

The rationale for choosing more than 2/3 to specify the dominant component of the darkmatter composition is due to the number of parameters in G2HDM model as listed in thetable 5.1. From this table, one see that we scan 14 parameters within the range allowed bythe theoretical and phenomenological consideration as will be elaborated later. In order toget big coverage of the parameter space that we scan, we learn that the coverage is optimizedwhen we require one dominant component to be greater than 2/3. If one to maximize thecriteria, say for example > 0.95 then it is very difficult to obtain the good coverage in theparameter range under the corresponding scan. Another reason is that for the Goldstoneboson-like DM, it is not possible to achieve the composition greater than 70%. This is due tothe fact that such composition is disfavored by electroweak precision test (EWPT) and canfurther lead to the tachyonic solution of the mass spectrum in dark scalar mixing matrix inEq. 3.19.

5.1 Dark Matter Properties in G2HDM and ExperimentalConstraints

In order to study the complex scalar dark matter in G2HDM, one needs to face the existingconstraint coming from the observed relic abundance of dark matter in the universe. Further-more, as already explained in chapter 4, the direct search of dark matter via its interactionwith the nucleon, the dark matter annihilation in the recent time or indirect search as wellas collider search need to be taken into account. In this section we will describe thoseconstraints used in analyzing dark matter phenomenology within G2HDM model.

5.1.1 Relic Density

The relic abundance of a typical weakly interacting massive particle (WIMP) is determinedby its interaction with other particles in early universe. When WIMP interacts effectivelywith the cosmic plasma such that it experiences thermal equilibrium with the plasma, itsnumber density is simply determined by its mass through Boltzmann factor e−

mT . However,

as the interaction rate become less effective than the Hubble expansion rate, the WIMP entersthe freeze-out regime. In order to study the WIMP abundance during the freeze-out, oneneeds to solve Boltzmann equation as explained in the previous chapter. The relevant quantityof interest is the thermally averaged cross section ⟨σv⟩ which encodes the WIMP interactionthat controls the WIMP number density.

The thermally averaged cross section ⟨σv⟩ contains all possible processes that affect theexistence of dark matter species. These include dark matter self annihilation as well as dark

5.1 Dark Matter Properties in G2HDM and Experimental Constraints 45

Fig. 5.1 The Feynman diagrams of dark matter annihilation into W+W− pair. The hi and Zidenote the three Higgs boson and three neutral gauge boson mediators. The t on the bottomleft panel stands for t-channel diagram.

matter interaction with heavier Z2 odd particles or coannihilation. In case of complex scalardark matter within G2HDM model, the relevant dark matter self annihilations includes allpossible annihilations into SM and non-SM final states. However, in this study, there areonly four final states that dominates the dark matter self annihilation: W+W−, h1h1, ZZ,and f f . The h1 is the usual SM Higgs while f f refers to SM fermion pair (cc, ττ , bb, tt).These four final states control the relic abundance of the complex scalar dark matter D. Thisapplies to all possible complex scalar dark matter candidate in this study, starting from inertdoublet-like, triplet-like, or Goldstone boson-like dark matter.

The first dominant annihilation cross section is given by W+W− final state. The contri-bution coming from this final state is more than 50%. It occurs when the dark matter massis above 130 GeV as will be explained more detail below. The relevant Feynman diagramsare listed in Fig. 5.1. The second important final state is given by SM Higgs pair h1h1. Thiscontribution is typically of order 25% or more and it is relevant for heavy dark matter mass.The next important final state for heavy dark matter regime is ZZ pair. This will take oforder 20% portion of annihilation cross section. For fermion pair, cc, ττ , and bb contributesignificantly for dark matter mass below 70 GeV. As for tt pair, the contribution is importantin the mass range between 170 GeV to 1 TeV. The percentage of this final states vary from


10% to 20% in this mass range. The relevant diagrams for these final states are shown inFigs. 5.2, 5.3 and 5.4.

Fig. 5.2 The Feynman diagrams of dark matter annihilation into h1h1 pair. The hi denote thethree Higgs boson mediators. The t (u) represents t (u)-channel diagram respectively.

Fig. 5.3 The Feynman diagrams of dark matter annihilation into ZZ pair. The hi denote thethree Higgs boson mediators. The t (u) represents t (u)-channel diagram.

The coannihilation of dark matter with other Z2 odd particles typically occurs when themass difference between them is less than 10%. In this case, the number density of the darkmatter is comparable with those of coannihilating partners. Therefore, one needs to includecoannihilation in determining the abundance of dark matter. In this study, the dark mattercandidate D can coannihilate with heavier complex scalar ∆, charged Higgs H±, new W ′

gauge boson, and heavy fermions f H . However, in the case of heavy fermion coannihilation,we restrict ourselves to exclude this possibility. This can be realized via the following set up


Fig. 5.4 The diagrams relevant for f f final states. The hi, Zi, and f Hi denote the three Higgs

boson, three neutral gauge boson as well as three heavy fermion mediators.

for the heavy fermion mass

m f H = max[1.5TeV,1.2mD] (5.2)

where m f H denotes heavy fermion mass. This means that the heavy fermion mass is alwaysbigger than 1.5 TeV and when it is bigger than this value, its mass is set to be 1.2 darkmatter mass in order to avoid the coannihilation. The 1.5 TeV coming from recent searchof new fermion at the LHC [44]. Note that even though the mass difference determinethe abundance of coannihilating particles, the coannihilation contribution to dark matterabundance is controlled by its cross section. Therefore, if the coannihilating particle has verysmall interaction with the dark matter, it will not affect the dark matter abundance.

As we use the micrOMEGAs code [45] in our relic density calculation, the criteria we usein determining the initial abundance of the coannihilating particles is given by the Boltzmannsuppression factor B as

B =K1((mi +m j)/T

)K1 (2mD/T )

≈ e−X f(mi+m j−2mD)

mD > Bε , (5.3)

where K1 is the modified Bessel function of the first kind. It is a function of the coannihilatingparticle masses mi and m j or the dark matter mass mD. The X f is the usual freeze-out param-eter which is defined as mD

T . The Bε is the minimum threshold to account for coannihilation.Its default value is 10−6. However in our calculation, we set Bε to be greater than 10−4 inorder to optimize the result.

Finally, to compare the relic density calculation against the experimental value, wewill consider the latest result from the PLANCK collaboration [46] for the relic density,Ωχh2 = 0.120±0.001. In particular, we will require that the model predicts this measuredvalue within 2σ accuracy.


5.1.2 Direct Detection

For generic WIMP dark matter, the relevant interaction in account of direct search is givenby DM-nucleon scattering. This is due to the fact that the mass of the DM lies between10 GeV to TeV regime. This will make the interaction between DM and nucleon becomesmore relevant than those with the electrons. Furthermore, the typical energy transfer in thisinteraction is of order a few keV. The most recent constraint for dark matter direct search isgiven by XENON1T experiment [47]. The null result from this search put the most stringentlimit on the dark matter nucleon spin independent cross section so far, especially for the darkmatter mass that lies between 10 GeV to 100 GeV. The current XENON1T excluded theDM nucleon elastic cross section above 10−46cm2 at the DM mass around 25 GeV. Thisresult was drawn by assuming that the interaction between dark matter-proton is identicalwith that of neutron. This framework is known as isospin conserved (ISC) interaction.

In general, the DM interaction between proton and neutron can be different so that theratio between the DM-neutron effective coupling fn and the DM-proton effective couplingfp can have any value dependent on the model parameter space. This fn/ fp = 1 is calledisospin violation (ISV). For instance, if DM interacts with the nucleon mediated by Z boson,the strength is characterized by the hypercharge quantum number of U(1)Y and the thirdgenerator T3 of SU(2)L group. Hence, the Z boson interacts with proton and neutron will notbe the same and the fn/ fp can be as larger as one order of magnitude.

In the case of inert doublet-like DM, we found the DM couple to proton or neutronstrongly via Zi bosons exchange which leads a strong ISV. On the other hand, for triplet-likeDM and Goldstone boson-like DM the ISV is still observed even though it is not very strongas in inert doublet-like DM case. In the limit of fn/ fp ∼−0.7 [48], one can find a maximumcancellation between proton and neutron if a XENON target is used in the detector. Generallyspeaking, this exact cancellation is located at a tiny region but the G2HDM complex scalarDM can have a much wider distribution of fn/ fp. In account of the ISV, we compute thedark matter direct search in nucleus level instead of nucleon level N as

σDN =4µ2

A

π[ fpZ + fn(A −Z )]2 , (5.4)

where we ignore all the isotopes of Xenon and simply fix the mass number A and protonnumber Z to 131 and 54, respectively. We obtain the effective couplings fp and fn by usingmicrOMEGAs code [45]. The DM-nucleon reduced mass is denoted as µA = mDmA /(mD +

mA ). One the other hand, the limit published by XENON1T is assumed to be isospinconserved fn = fp. Hence, our next step is to release this assumption and reconstruct the


limit in nucleus level as

σX1TDN = σ

SIp (X1T)×A 2 × µ2

A

µ2p, (5.5)

where µ2p is the DM-proton reduced mass. In this study, we use Eq. (5.5) to examine our

direct detection calculation and compare it with the reconstructed experimental limit inEq. (5.5).

D D

q q

hi

qq

D D

Zi

Fig. 5.5 The dominant Feynman diagrams for Higgs bosons (left) and Z bosons (right)exchange.

In addition, as we deal with the complex scalar dark matter, we need to consider theanti-dark matter interaction with the nucleon. The dark matter-nucleon interaction andanti-dark matter-nucleon interaction in general can be quite different. The spin independentinteraction for complex scalar dark matter can be written in terms of effective operator as[49]

LS = 2λN,eMDDD∗ψNψN + iλN,o((∂µD)D∗−D(∂µD∗))ψNγµψN (5.6)

where the ψN , λN,e, and λN,o denote the nucleon field operator, the coupling of even operator,and the coupling of odd operator respectively. The effective coupling of dark matter (anti-darkmatter) with the nucleon is given by

λN =λN,e ±λN,o

2(5.7)

where the plus (minus) sign stands for dark matter-nucleon (anti-dark matter-nucleon) interac-tion. The first term in the right hand side of Eq. (5.6) represents the even operator interactionbetween dark matter and the nucleon. It is called even operator because when one exchangesthe dark matter D with its anti-dark matter D∗, the interaction stays the same. On the other


hand, under the similar exchange between D and D∗ the second term flips sign. Thus, it iscalled the odd operator. As a result, the interaction strength between dark matter-nucleonand anti-dark matter-nucleon will not be the same and it is given by the Eq. (5.7).

In the study of complex scalar dark matter within G2HDM model, the dominant con-tribution to dark matter-nucleon interaction is given in Fig. 5.5. The left panel describesthe Higgses exchange interaction that leads to isospin conserved interaction. Moreover,this kind of diagram also describe the even operator in terms of dark matter and anti-darkmatter exchange. The right panel represents the neutral gauge bosons exchange which isisospin violating interaction. This type of interaction belongs to the odd operator which flipssign if we switch dark matter with anti-dark matter. There is another diagram that affectdark matter-nucleon interaction. They are given by the exchange of the heavy fermion f H

mediators. This kind of operator leads to both isospin violating interaction as well as mixedeven-odd operator. However, since the contribution of this heavy fermion is sub-dominant,we do not discuss this contribution in detail even though we include it in our analysis. As anadditional remark, the two dominant contribution coming from Higgses and neutral gaugebosons exchange can be mixed via interference terms. This will make the dark matter directsearch study in G2HDM model becomes quite interesting and contains very rich physicalphenomenology as we will discuss below in three different dark matter composition cases.

Finally, in doing the analysis of dark matter direct search study, we assume that theabundance of dark matter is equal to that of anti-dark matter. We sum up these two contribu-tions and then take the average to determine the cross sections between the dark matter andanti-dark matter with the nucleon before going further into nucleus level calculation.

5.1.3 Indirect Detection: Gamma-ray from dSphs

Excluding the early universe, DM at the present may also annihilate into the SM particlessignificantly at the halo center where DM density is dense enough to produce cosmic rays orphotons distinguishably from those standard astrophysical background. Such a measurementis known as DM indirect detection. As long as indirect detection constraints are concerned,the continuum gamma ray observations from dwarf spheroids (dSphs) can usually put arobust and severe limit on the DM annihilation cross section at the DM mass range greaterthan 10 GeV [50]. This is owing to two advantages of searching DM at the dSphs. First,the dSphs provide almost background-free system because it is faint but widely believe thatdSphs are DM dominant systems. Second, their kinematics can be precisely measured, hencethe systematical uncertainties from DM halo can be controlled. Therefore, in this work wewill only use the dSphs constraints implemented in LikeDM [51] to evaluate the statistical χ2

of our model based on Fermi Pass 8 data, recorded from 4 August 2008 to 4 August 2015.


The standard gamma-ray fluxes produced from DM annihilation at the dSphs halo

dΦγ

dEγ

=⟨σv⟩

8πm2D× J×∑

chBR(ch)×

dNchγ

dEγ

, (5.8)

where J =∫

dldΩρ(l)2 is the so-called J-factor which counts the amount of annihilationfrom a direction (angle Ω) given the DM density distribution ρ . Here, we take 15 dSphs andtheir J-factors as the default implementation in the LikeDM. The index ch presents the DM

annihilation channels. The annihilation branching ratio BR(ch) and energy spectradNch

γ

dEγare

computed by using micrOMEGAs code and PPPC4 [52] code, respectively.

5.1.4 Collider Search

Concerning the interaction between DM and SM, one might expect that DM could beproduced inside the collider. Unfortunately, DM is totally the missing energy and not able tobe seen directly inside the collider. Traditionally, one might look for an extra SM radiation Xplus missing energy, called mono-X searches where X can be lepton, jet, photon, W , Z, andh. Among these signatures, mono-jet search is the most statistically significant at the LHC.Still, null-signal had been reported by both ATLAS [53] and CMS [54]. Hence, a stringentlimit based on the monojet search shall be considered in our analysis.

Fig. 5.6 The Feynman diagrams of leading contributions for monojet plus missing energy.


As shown in Fig. 5.6, the G2HDM DM can be produced via Higgs and Z boson exchangein the leading contribution. As in the direct detection case, the contribution of heavy fermionsis sub-dominant. The couplings in Fig. 5.6 is also presented in the DM direct detection search(see as well as Fig. 5.5) but the DM DD constraints are usually stronger. Hence, we tookten benchmark points from mD ≈ 80 GeV to mD ≈ 2.3 TeV and their DM-neutron elasticscattering cross sections are just below the current XENON1T limit.

In Table B.1, we present the inputs of these ten benchmark points and their mass spectrumand cross sections. The tree-level and loop-level contributions refer to the two upper and thetwo bottom diagrams of Fig. 5.6, respectively. We computed their cross sections by usingMadGraph5 [55] in the parton level with default basic kinematic cuts. The latest CMS 13 TeVwith luminosity of 36.1fb−1 result [53] reported that the observed events are 255486 and theSM predicted the background are 245900±5800. Apparently, the most optimistic benchmark(A) predicts only ≈ 0.5 events before applying any sophisticate cut, much less than the SMuncertainties. Therefore, we do not consider the monojet limit as a global constraint inthis work. However, the monojet search might be a very important approach for the future27 TeV or 100 TeV hadron collider since the future DM direct detection experiments wouldbe contaminated from the atmospheric and solar neutrinos known as the neutrino floor.

Once one considers the DM mass is lighter than half of SM Higgs mass, the Higgs bosondecays into DM pair shall be constrained by current LHC data. At tree level in this model,the invisible Higgs decay width is given by

Γ(h → DD∗) =(OD

32)4

16πmh1

λ2hDD∗

√1− 4m2

D

m2h1

, (5.9)

where the λhDD∗ coupling is

λhDD∗ = O11λH∆v+O21λΦ∆vΦ −2O31λ∆v∆. (5.10)

The current LHC 95% upper limit of the branching ratio of the Higgs invisible decay isBR(h1 → inv)< 24% [11]. If mD ≪ mh1 together with SM Higgs decay width 13 MeV [11],the LHC limit implies that an upper bound,

(OD32)

2λhDD∗ < 5.099 GeV. (5.11)

However, we found this limit is not as stringent as DM direct detection. Unless mD ≲ 10 GeVwhere DM recoil energy is below the XENON1T threshold, the Higgs invisible decay limitcan play more important role.

5.2 Numerical Methodology 53

5.2 Numerical Methodology

In this section, we will explain our method to collect the data in studying the dark matterphenomenology of G2HDM model. We perform random scan by using Fortran code to getthe sample points consistent with the constraints and conditions from various theoreticaland phenomenological arguments. We first do the scan on the scalar sector by inputing λH ,λΦ, λ∆, λHΦ, λH∆, λΦ∆, λ ′

HΦ, λ ′

H , MH∆, MΦ∆, and v∆ as our input parameter. The value ofstandard model VEV v is fixed to 246 GeV. Furthermore, unlike in [26] where the value ofvΦ was fixed, we will also scan vΦ in the range between 20 TeV to 100 TeV. As vΦ providesthe highest mass scale in the scan, this parameter sets the typical energy scale of G2HDMmodel.

In order to get the vacuum expectation value (VEV), we use these scalar parametersto minimize the scalar potential according to equations (3.14), (3.15) and (3.16). Theseconditions will make the parameters µH , µΦ, and µ∆ that appear in the scalar potentialexpressible in terms of other scalar parameters. Next, we enter these 12 parameters inorder to satisfy unitarity condition. This is done by considering the 2 → 2 scalar-scalarscattering amplitude in the high energy limit such that we only need to consider four pointcontact interaction which is free from center of mass energy suppression. For instance, thecorresponding scalar particles in the initial and final states of these scatterings in the diagonalchannels can be ( hh√

2, G0G0√

2, H0∗

2 H02 , H+H−, φ2φ2√

2, G0

HG0H√

2, Gp

HGmH , δ3δ3√

2, ∆p∆m). The resulting

amplitudes of these diagonal channels as well as other off-diagonal channels are all directlyrelated to the scalar quartic couplings and they can be found in [26].

Next, we numerically diagonalize the scalar mixing mass matrices given in Eqs. (3.17)and (3.19). The output of this diagonalization procedure are the masses of the physical scalarHiggs bosons mh1 , mh2 , mh3 and the corresponding mixing angles components OH

i j as wellas the mass of the physical dark scalars mD and m

∆with their corresponding mixing angles

components ODi j. The resulting mass is ordered in ascending manner mh1 < mh2 < mh3 and

mD < m∆

. We also calculate the charged Higgs mass mH± and heavy fermion mass m f H . Asfor the observed standard model higgs, we identify this by mh1 and constraint it to have thevalue 125.09±0.24 GeV. Based on this, we further calculate the branching ratio of the Higgsdecay into γγ and ττ .

The points that satisfy all the theoretical and phenomenological constraints will be passedto calculate the mass spectrum in the gauge boson sector. In this sector we take additionalscan parameter as our input namely gH and gX . However, we fix the Stueckelberg mass MX

to 2 TeV. We then do the numerical diagonalization on the gauge boson mixing matrix inEq. (3.36). The outcome of this diagonalization are three neutral gauge boson masses inascending order similar to those of scalar Higgs mass mZ < mZ′ < mZ′′ and the gauge boson


Fig. 5.7 The typical diagram used to study perturbative unitarity in scalar-scalar scattering.The Si, S j (Sk, Sl) denote the scalar particles in the initial (final) states respectively. These

scalars can be one of the following pairs ( hh√2, G0G0√

2, H0∗

2 H02 , H+H−, φ2φ2√

2, G0

HG0H√

2, Gp

HGmH ,

δ3δ3√2

, ∆p∆m) as described in the text.

mixing angles components OGi j. The lightest gauge boson is identified as the SM Z boson

which has the value within 3σ accuracy as 91.1876±0.0021 GeV. The W ′ gauge boson is amember of the Z2 odd family and has the potential to be a viable dark matter candidate. Inorder to avoid this, we demand that W ′ is always heavier than D. This is realized by imposingthe minimum value of gHmin as

gHmin =2mD√

v2 + v2Φ+4v2

∆

. (5.12)

We also do a special scan to get the inert doublet-like dark matter candidate fH0∗2> 0.67.

This is due to the fact that the doublet dark matter solutions are highly suppressed comparedto those of triplet- and Goldstone-like. In order to get doublet-like dark matter, we have toset MH∆ ≪ v∆ and make the (2,2) entry in Eq. (3.19) less massive than the (3,3) one with thefollowing condition

λ′HΦ <

MΦ∆

2v∆

. (5.13)

This condition will make the scan range of λ ′HΦ

became very limited.We collect about 5 million points that satisfy all the conditions in scalar and gauge

sector constraints (SGSC) and further pass these points to MicrOMEGAS [45] to calculaterelic density, DM-nucleon cross section and annihilation cross section at present time.

5.2 Numerical Methodology 55

Parameter Doublet-like Triplet-like Goldstone-likeλH [0.12, 2.75] [0.12, 2.75] [0.12, 2.75]λΦ [10−4, 4.25] [10−4, 4.25] [10−4, 4.25]λ∆ [10−4, 5.2] [10−4, 5.2] [10−4, 5.2]

λHΦ [−6.2, 4.3] [−6.2, 4.3] [−6.2, 4.3]λH∆ [−4.0, 10.5] [−4.0, 10.5] [−4.0, 10.5]λΦ∆ [−5.5, 15.0] [−5.5, 15.0] [−5.5, 15.0]λ ′

HΦ[−1.0, 18.0] [−1.0, 18.0] [−1.0, 18.0]

λ ′H [−8

√2π , 8

√2π] [−8

√2π , 8

√2π] [−8

√2π , 8

√2π]

MH∆/GeV [0.0, 15000] [0.0, 5000.0] [0.0, 5000.0]MΦ∆/GeV [0.0, 5.0] [−50.0, 50.0] [0.0, 700]

v∆/TeV [0.5, 2.0] [0.5, 20.0] [14.0, 20.0]vΦ/TeV [20, 100] [20, 100] [20, 28.0]

gH [see text, 0.1] [see text, 0.1] [see text, 0.1]gX [10−8, 1.0] [10−8, 1.0] [10−8, 1.0]

Table 5.1 Parameter ranges used in the scans mentioned in the text.

Finally, the annihilation cross section for each DM composition is passed to LikeDM [51]for the calculation of indirect detection likelihood. Due to the different underlying physicalconsideration, each dark matter case will have different phenomenology. Therefore, wefurther analyze the resulting points calculated by MicrOMEGAS. From here, one can determinethe relevant couplings and parameters which are sensitive to the dark matter composition indoublet-, triplet- or Goldstone-like case. The complete parameter set that we scanned in thisstudy for each case is summarized in Table 5.1.

Chapter 6

Dark Matter in G2HDM: NumericalResults

Having explained our methodology and strategy in the previous chapter, we further discuss indetail the numerical results of complex scalar dark matter study in this chapter. We start ourdiscussion by imposing the relic density requirement and collect the surviving parametersto face the direct search constraint. Next, the viable parameters space are fed against theindirect search experimental data. We will discuss the phenomenology of each dominantcomponent of the dark matter candidate starting from the inert doublet-like DM, SU(2)H

triplet-like DM, and Goldstone boson-like DM in different sections.

6.1 Inert Doublet-like DM

The doublet-like dark matter is characterized by the dominant H02 component with the

composition (OD22)

2 > 2/3. In G2HDM model, the doublet-like dark matter is equal to theinert dark matter candidate in IHDM model in the limit of the mass degeneracy between thescalar (S) and pseudo-scalar (A) part of the inert scalar. This doublet-like dark matter is themain motivation behind the construction of G2HDM as already mentioned before. Therefore,in this part of result session, we will give a detail discussion in this doublet dark mattercandidate. We show the relic density as a function of the DM mass in the left panel of Fig. 6.1.Similar to the IHDM model, the cosmological interesting relic abundance ΩDh2 ∼ 0.1 occursin three different mass regions [56, 57]: (i) the low DM mass between 10 GeV to 40 GeV,(ii) the intermediate DM mass between 150 GeV to 500 GeV, and (iii) the heavy DM massgreater than 500 GeV. The underlying physics behind these three regimes are different.

58 Dark Matter in G2HDM: Numerical Results

100 101 102 103

mD (GeV)

10−7

10−5

10−3

10−1

101

103ΩDh

2

fH2> 2/3, all points SGSC

Ωh2 ≈ 0.12

h1 resonance

Z resonance

H± coannihil.

100 101 102 103

mD (GeV)

10−45

10−44

10−43

10−42

10−41

10−40

10−39

10−38

10−37

σSI

n(c

m2 )

fH2> 2/3, all points SGSC

XENON1

T×102

CRESST-III

SGSC+RD

Fig. 6.1 Left: the doublet-like DM relic density as function of the DM mass. Right: the DMmass versus the DM-neutron elastic scattering cross section. The gray scatter points agreewith the SGSC constraints. The blue scatter points agree with PLANCK data within 2σ

region.

For dark matter mass that lies between 1 GeV to 10 GeV, the dominant contribution ofdark matter annihilation cross section is given by DD∗ → cc and τ+τ−. These two processesproceed via three Higgses exchange, three neutral gauge bosons exchange as well as threeheavy fermions exchange as shown in Fig. 6.2. The most dominant contribution is given

Fig. 6.2 Feynman diagrams of dark matter annihilation into ττ , cc and bb final states.

by the SM Higgs exchange h1. Therefore, the corresponding amplitude is proportional tothe product of λDD∗h1 and the Yukawa couplings of τ and c which are proportional to thecorresponding fermion masses. Due to the smallness of these Yukawa couplings, the resultingannihilation cross sections will be very small. This will cause the relic density in this regimebecomes very large as was shown clearly in the left panel of Fig. 6.1. At about 10 GeV, theannihilation into bb channel opens. When this happens, the bb final state starts to dominatethe annihilation cross section due to its larger Yukawa coupling. As a result, the relic densityis reduced further. The observed relic density by PLANCK data within 2σ region is achievedwhen the dark matter mass is located between 11 to 40 GeV. Going further, as the dark matter

6.1 Inert Doublet-like DM 59

mass reaches 1/2 of the Z boson mass at 45 GeV, the SM Z mediated cross section becomesimportant. The relic abundance is suppressed up to 10−4 which is far from the observedexperimental data. This effect is depicted by the orange dip in the same figure. The relicdensity suppression is also seen at 1/2 of the Higgs mass around 63 GeV. At this point, theSM Higgs exchange dominates the annihilation cross section. As a consequence, the relicdensity become vanishingly small as described by the blue points in the same figure.

In the intermediate mass range between 150 GeV to 500 GeV, the dominant annihilationcross sections are given by the W+W−, h1h1, ZZ and tt final states. The W+W− finalstate occurs via four point contact interaction (p-channel), s-channel of three Higgses andthree neutral gauge bosons exchange, and t-channel of charged Higgs exchange. If onetakes the non-relativistic approximation, the scalars exchange diagrams lead to the s-waveannihilation cross section while the gauge bosons exchange diagrams will contribute to thep-wave part. One expects that due to their heavy masses, the h3 and Z′′ exchange will besub-dominant compared to the other mediators. Next, the h1h1 final state is mediated viap-channel contact interaction, three Higgses exchange in the s-channel, D, ∆,W ′ exchangein the t and u-channel. In this case, one expects that the most dominating diagram is givenby the p-channel interaction as there is no propagator and hence no mass suppression. Thenext dominant amplitudes are given by h1 and h2 exchange, as well as D and ∆ exchange.The contributions from the W ′ and h3 exchange are small due to their large mass suppression.It is interesting that the heavier dark scalar ∆ gives comparable contribution with that of Din the t/u-channel. This happens because of the mass splitting between them is small. Wewill discuss this further in the coannihilation case below. The same argument also holds inthe case of ZZ final state which has exactly similar diagrams as the h1h1 final state. Finally,for the tt final state, the diagram is similar to those of low mass dark matter case. Theheavy fermion contribution is suppressed due to its typical TeV mass scale. All of thesecontributions will make the relic density to have small value in this intermediate mass range.Unfortunately, one can not obtain the points that satisfy the PLANCK data as can be seenfrom the left panel of Fig. 6.1. This happens because the doublet-like dark matter is themember of the usual SU(2)L doublet. Its interactions with the SM particles such as h1 and Zare controlled the SM couplings. Therefore, one would expect that these kind of interactionsare unsuppressed by any mixing angles and hence make the corresponding annihilation crosssections larger.

Before moving further to higher dark matter mass > 500 GeV, let us discuss about thecoannihilation in doublet-like dark matter. As one can see from the left panel of Fig. 6.1.The relevant contribution of coannihilation is given by the charged Higgs. The mass splitting


between charged Higgs and doublet dark matter is approximately given by

m2H± −m2

D ≈−12

λ′Hv2 . (6.1)

This tells us that the initial abundance of charged Higgs H± compared to the D abundance isdetermined by the λ ′

H parameter. This coannihilation contribution starts to become importantwhen the dark matter mass is larger than 100 GeV. The possible coannihilation cross sectionin this case is the annihilation between D and H± as well as H± with themselves. In thisstudy, we learn that it is the second interaction which gives us the dominant coannihilationcontribution. This is due to the fact that the interaction of H± between themselves also occurvia the electromagnetic interaction. As we already know that the electromagnetic interactionis stronger than the typical WIMP dark matter interaction. As an additional remark, wemention that the doublet dark matter solution which is characterized by the (2,2) componentof mixing matrix in Eq. (3.19) is very hard to find and one needs to do a dedicated scanin order to get the desired points. This lies on the fact that the (3,3) element of the massmatrix in Eq. (3.19) is smaller than the (2,2) element due to the presence of the large v∆ inthe denominator of the (3,3) element. This will make the (3,3) element always smaller thanthe (2,2) element and hence the natural DM candidate in this model behaves like the triplet∆p component. If one insists to have doublet-like DM, one needs to go to particular directionin the parameter space. However, by doing this, the mass splitting between D and ∆ willalways be less than 10 % as the argument inside the square root of Eq.( 3.21) is very small.With this setup in mind, one expects that the coannihilation between D and ∆ will appeareverywhere in the dark matter mass. However this is not the case. One needs to rememberthat the coannihilation cross section is controlled by the interaction between the dark matterand coannihilating particles. We found that the coannihilation with ∆ is very small comparedto the charged Higgs case. This is the reason we do not plot the ∆ coannihilation in the leftpanel of Fig. 6.1.

Finally, in the heavy mass region mD > 500 GeV, the annihilation cross sections aredominated by the SM Higgs h1h1 (Fig. 6.3) as well as the longitudinal components of thegauge bosons ZLZL (Fig. 6.4) and W+

L W−L (Fig. 6.5). For the SM Higgs final state, the

important contribution is given by four point like interaction, three Higgses exchange ins-channel, t and u channel exchange of D and ∆ as shown in Fig. 6.3. The W ′ exchangeis suppressed by its heavy mass and hence its contribution is sub-dominant. The typicalamplitudes for the longitudinal gauge bosons scattering scale with the energy. Thus, in orderto preserve the unitarity, one needs to cancel this energy dependence. For the ZLZL finalstate the relevant diagram is described in Fig. 6.4. There is an exact cancellation in the


Fig. 6.3 Relevant diagrams for h1h1 final state. For the doublet case, both D and ∆ exchangein t- and u-channels are important. For the triplet-like and Goldstone-like dark matter, onlyD exchange is relevant.

Fig. 6.4 Relevant diagrams for ZLZL final state. The first three diagrams scale with thecenter-of-mass energy. The sum of these three diagrams cancel the energy dependence.


energy dependence between four points contact interaction with the t and u-channels. Thecontribution of W ′ exchange is suppressed and therefore not relevant for the annihilationcross section. On the other hand, the diagram with ∆ mediator is suppressed by the mixingangle (OD

32)2. Thus, the only relevant diagrams for ZLZL final state are given by the ones

with Higgses exchange.The amplitudes that scale with the center-of-mass energy also observed in the W+

L W−L

final state. The corresponding diagrams for this process are given in Fig. 6.5. The cancellation

Fig. 6.5 Relevant diagrams for W+L W−

L final state.

occurs between the p-channel contact interaction and t-channel charged Higgs exchange.However, unlike the ZLZL case, the cancelation in this case is not exact. The sum of thesetwo amplitudes is given by

i(Mp +MH−)L = ie2(OD

22)2

2m2W s2

W×[(s−2m2

W )

2+

(t −m2D)

2

(t −m2H−)

], (6.2)

where t = m2D +m2

W − s2 . If mD = mH− , the second term on the right hand side is equal to

t −m2D and the exact cancelation can be realized. Thus, the cancellation depends on the mass

splitting between D and charged Higgs H− which is controlled by λ ′H as in Eq. (6.1). As one

goes to higher dark matter mass, this mass splitting become smaller and the cancelation canbe realized. The next relevant diagram is given by the Zi mediator. In the non-relativisticlimit, these amplitudes lead to the p-wave annihilation cross section which proportionalsto the square of the dark matter velocity v2. In addition, in the case of Z′ (Z′′) exchange,


these two diagrams are further suppressed by the off-diagonal elements of gauge bosonmixing matrix OG

12 (OG13) leaving only the SM Z boson exchange as the leading order diagram.

Moreover, due to its SM nature, the amplitude of SM Z boson exchange is fixed by the SMcoupling. It does not change as one varies the dark matter mass. The Higgses exchange onthe other hand, is proportional to the λDD∗hi that changes its value as the dark matter massvaries. Therefore, the relevant diagram in determining the relic density at higher dark mattermass is governed by the Higgs exchange diagrams. The expression of λDD∗hi coupling fordoublet-like DM is given by

λDD∗hi = i[−2λHvOH

1i −λHΦvΦOH2i +λH∆v∆OH

3i](OD

22)2 , (6.3)

where the index i runs from 1 to 3. The coupling λDD∗hi is a function of the scalar parametersλ s which are bounded and can not be arbitrarily large. As a result, when dark matter massgetting bigger the annihilation cross section becomes smaller due to the dark matter masssuppression in the denominator. Therefore, the resulting relic density will be bigger in thehigh mass scale as one can see from the left side of Fig. 6.1. This is consistent with theunitarity constraint.

The next experimental constraint to be considered is the direct detection experimentgiven by the XENON1T 2018 [47] and CRESST-III 2017 [58] data. In the right panelof Fig. 6.1, the grey points represent all the points we scan that satisfy SGSC constraints.One sees that for the dark matter mass larger than 10 GeV, the doublet-like dark matter isexcluded by the XENON1T and CRESST-III experiments. There are points that escapethese two experimental constraints which are situated below 10 GeV, however in this regime,the observed relic abundance will be too large compared to the PLANCK observation andtherefore can not be a suitable dark matter candidate.

The interaction between DM and the nucleon is mediated by three Higgses exchange inthe t-channel, three neutral gauge bosons exchange in the t-channel, as well as sub-dominantheavy fermion exchange. For the doublet-like dark matter, the dominant contribution of thedark matter-nucleon interaction is given by the SM Z boson exchange. The coupling betweendoublet-like dark matter and the Z boson gµ

ZDD is given by

gµ

ZDD = i[

gcW

2+

g′sW

2

](OD

22)2OG

11(pD∗ − pD)µ (6.4)

where pµ

D is the dark matter four momentum. We follow the convention that all momentaare pointed into the interaction vertex. The Z boson-nucleon coupling which can be ex-tracted from its quark coupling, is dominated by the typical SM Z-quark coupling and it is


proportional to OG11 ∼ 1. This can be seen from

gVqqZ =

i2

[g

cW

(T3 −2Qs2

W)OG

11 +gHT ′3O

G21 +gX XOG

31

], (6.5)

where gVqqZ is the vector coupling of Z boson and quark field. The T3, T ′

3, and X denote thethird component of SU(2)L, SU(2)H , and generator of U(1)X group respectively. The firstterm on the right hand side of this equation is the SM coupling while the second and thirdterms are suppressed by off diagonal component of the gauge boson mixing matrix OG. Inthe limit of SM case (gH = gX = 0), the ratio between neutron and proton coupling to Z isgiven by

fn

fp=

−12

12 −2s2

W≈−12.5 , (6.6)

where we have used the fact that for proton T3 =12 and Q = 1 while the neutron has T3 =−1

2and Q = 0. In this case, dark matter interacts differently with proton and neutron and hencethe resulting cross section will violate isospin conservation. Since the inert doublet-like darkmatter is a member of SU(2)L doublet, the dominant contribution comes from the SM Zexchange.

One notes that there is a difference between the dark matter-neutron cross section anddark matter-proton cross section. This is the effect of the ISV mentioned before. Thiscan be easily seen from the Eq. (6.5). At quark level, the SM Z boson couples differentlywith up-type and down-type quark and the difference follows from the different values ofT3 and Y assignments for each type of quark. Furthermore, the proton and neutron havedifferent assignments under SU(2)H and U(1)X as can be seen from Eq. (6.5). Since theproton contains two u-quarks and one d-quark as opposed to the neutron with one u-quarkand two d-quarks, the ratio of the dark matter-neutron coupling with respect to the protoncoupling fn/ fp is different. In doublet-like dark matter, the typical value of fn/ fp is oforder ∼10. This is very far from the maximum cancellation condition fn/ fp ≈−0.7. As aconsequence, the resulting DM-nucleon cross section is quite large with the typical valueof the order 10−38cm2 (10−41cm2) for the neutron (proton). Therefore, one expects that thedark matter-neutron cross section is much larger than the proton cross section. Furthermore,to confront with the current XENON1T data, it is adequate to use the spin independentneutron cross section. Unfortunately, there is no surviving parameter space that agrees bothconstraints from PLANCK and XENON1T experiment. This will make the doublet-like DMin G2HDM completely ruled out by those two experiments.

6.2 SU(2)H Triplet-like DM 65

6.2 SU(2)H Triplet-like DM

The triplet-like dark matter is mainly composed of ∆p component with the composition(OD

32)2 > 2/3. With such large triplet fraction, one expects that the corresponding coupling

is different from those of doublet-like dark matter. The relic density of triplet-like darkmatter is shown in the left panel of Fig. 6.6. Unlike the doublet case, the suitable relicdensity consistent with the PLANCK data occurs almost everywhere in the dark matter massespecially for the mass above 10 GeV.

For dark matter mass between 1 to 10 GeV, the dominant annihilation cross section isgiven by the cc and ττ final states. The full diagrams of this process are given by Fig. 6.2.However, only the s-channel Higgs exchange is relevant for this mass range especially theone mediated by h1 and h2. This comes from the dark matter couplings with those twoHiggses which are given by

λDD∗hi = i[−λH∆vOH

1i −λΦ∆vΦOH2i +2λ∆v∆OH

3i](OD

32)2 . (6.7)

One expects that for i equals to 1, the first term on the right hand side would be dominant andunsuppressed by the Higgs mixing matrix. Thanks to the large value of vΦ and v∆ comparedto v, the second and third term could have comparable value with the first term. The sameargument also hold for i equals to 2 which describes the h2 exchange. Thus, as long as themh2 is not too far from mh1 , these two contribution need to be taken into account. The h3

mediator on the other hand, is sub-dominant due to its heavy mass. As before, the smallvalue of c quark and τ lepton Yukawa couplings will make the corresponding annihilationcross section becomes small such that they enhance the relic density in this low mass regime.As a side note, the right hand side of Eq. (6.7) consists of three different terms that can becancelled or enhanced each other depending on the relative signs of these terms. Furthermore,the interference between h1, h2 and another possible mediator in general, could make theresulting cross section to have spreading values as one can see in Fig. 6.6.

Similar to the inert doublet-like case, the bb channel dominates the annihilation crosssection when the dark matter mass is larger than 10 GeV. This will make the cross sectionbecome larger and reduce the relic density. As a result, the observed relic abundance fromPLANCK data can be satisfied in this range. The next final state to be considered is theW+W− which opens at the dark matter mass around 40 GeV. This channel occurs via p-channel contact interaction, s-channel Higgses and gauge bosons exchange, and the t-channelcharged Higgs exchange as depicted in Fig. 6.5 with the relevant replacement W+

L →W+.The p-channel contact interaction is suppressed by (OD

22)2. The charged Higgs exchange is

sub-dominant due to its heavy mass as will be explained later when we discuss coannihilation.


100 101 102 103

mD (GeV)

10−7

10−5

10−3

10−1

101

103

105

107ΩDh

2f∆p

> 2/3, all points SGSC

Ωh2 ≈ 0.12

h1 resonance

h2 resonance

W ′ coannihil.

HF coannihil.

100 101 102 103

mD (GeV)

10−48

10−47

10−46

10−45

10−44

10−43

10−42

10−41

10−40

σSI

n(c

m2 )

f∆p> 2/3, all points SGSC

XENON1

T(fn/fp=

-0.5)

XENON1

T(fn/fp=

1)

SGSC+RD

SGSC+RD+DD

Fig. 6.6 Triplet-like SGSC allowed regions projected on the (mD, ΩDh2) (left) and (mD, σSIn )

(right) planes. The gray area on the left has no coannihilation or resonance. The gray area onthe right is excluded by PLANCK data at 2σ . Some orange squares are above the XENON1Tlimit due to ISV cancellation at the nucleus level.

For the neutral gauge boson mediator, the dominant amplitude is given by the Z′ exchangeand its proportional to dark matter coupling with the Z′ which is given by

gµ

DD∗Z′ = i[

gH(OD32)

2OG22 +

12

gH(OD12)

2OG22 +gX(OD

12)2OG

32

](pD∗ − pD)

µ , (6.8)

which is proportional to OGi j gauge boson mixing matrix. Note that the second and third

terms on the right hand side come from Goldstone-like contribution which can give non-negligible role in Z′ exchange. In our scan, the dominant OG

2i component is given by OG23

which corresponds to Z′′ exchange. However, due to its larger mass, this channel wouldbe sub-dominant. The SM Z exchange on the other hand, is suppressed by OG

21. In non-relativistic approximation, this will contribute to the p-wave annihilation cross section.Another contribution comes from the three Higgses exchange with the first two Higgsesh1 and h2 dominating the cross section. Thanks to both cancellation (enhancement) in thecoupling in Eq. (6.7) and also the destructive (constructive) interference between h1 andh2 diagrams, the corresponding cross section will be small (large). In the non-relativisticapproximation, these Higgses exchange will lead to the velocity independent or s-waveannihilation cross section. Taking into account the similar enhancement (cancellation) in Z′

mediator coupling of Eq. (6.8), these three amplitudes control the dark matter relic abundance.


The next final state is ZZ pair that occurs via p-channel contact interaction, t and u-channel of D, ∆,W ′ exchange, and the s-channel of three Higgses interaction. This channelis also controlled by the h1 and h2 exchange while the other contributions from the t- andu-channels are suppressed not only by the masses of D, ∆,W ′ but also by the mixing matrixelements. For the D mediator, the amplitude is proportional to the (OG

21)2 and hence reduces

the associated amplitude. The suppression of ∆ is originated from the vanishing small valuesof OD

22 and also of OG21. On the other hand, the W ′ exchange contributes sub-dominantly due

to its dependence on (OD22)

2. The p-channel suffers from large (OG21)

2 suppression. Movingfurther, the h1h1 final state is dominated by four point contact interaction, s-channel h1 andh2 exchange, t- and u-channel D exchange as given in Fig (6.3). As expected the p-channelgives the most important contribution while the other three contributions are comparable toeach other. For the p-channel interaction, the relevant coupling is

λ3DD∗h1h1 = i[λH∆(OD

32)2(OH

11)2] , (6.9)

while another relevant coupling for h1h1h1 which accounts for h1 exchange is given by

λh1h1h1 = i[6λHv(OH

11)3 −3λH∆v∆(OH

11)2OH

31 +3λHΦvΦ(OH11)

2OH21], (6.10)

and the corresponding h1h1h2 coupling is written as

λh1h1h2 = i[λHΦvΦ(OH

11)2OH

22 −λH∆v∆(OH11)

2OH32]. (6.11)

From the left panel of Fig. 6.6, one can see a very clear h1 resonance near mD ≈ 63 GeV.As in typical resonance effect, the corresponding relic density will be too small. However,the observed relic density from the PLANCK data can still be accommodated in tripletdark matter case thanks to the suitable adjustment via cancellation among different terms inEq. (6.7) for i equals to 1.

When the dark matter mass is above 100 GeV, the longitudinal components of the W andZ boson start to dominate the annihilation cross section. The SM Higgs boson h1 final statealso gives important contribution. On average, in this mass regime, the most dominating finalstates are the W+

L W−L (≥ 50%), h1h1 (∼ 25%) and ZLZL (∼ 20%). As before, the diagrams

relevant to these three final states are described in Figs. 6.5, 6.3, 6.4 for W+L W−

L , h1h1, andZLZL respectively. There is an exact cancellation between the p-channel contact interaction,t-channel as well as u-channel D exchange in ZLZL final state. This will leave the s-channelHiggses exchange to control the corresponding annihilation cross section. In the case ofW+

L W−L the situation is quite different as there is a big mass difference between D and


charged Higgs. However, the energy dependence of the amplitude is still vanishing small viasimilar cancellation between p-channel contact interaction and the t-channel charged Higgsexchange. This is realized by taking the limit of s >> mD,mH− . In this limit, Eq. (6.2) ismodified as

i(Mp +MH−)L = ie2(OD

22)2

2m2W s2

W×[

s2+

(− s2)

2

(− s2)

], (6.12)

where we have taken t = − s2 in this high energy limit. Since we only scan for the dark

matter mass below 10 TeV, we can not see this cancellation as in doublet-like dark matter.As a result, the points in high energy regime are spreading everywhere and do not have thetendency of having larger relic abundance. This can be seen from the left panel of Fig. 6.6.

Another mechanism that comes into play in high energy dark matter mass is given by theappearance of new resonances and coannihilations. At mD > 100 GeV, there are a lot pointsthat satisfy 2mD ≈ mh2 . This will generate the h2 resonance as depicted by the red points inthe left panel of Fig. 6.6. This resonance contribution occurs via the s-channel h2 exchangethat appears in any dominant final states mentioned before. As opposed to the SM Higgsresonance which located around 63 GeV, the h2 resonance spreads in a wide range of the darkmatter mass. This is due to the fact that mh2 can take any value greater than mh1 and less thanmh3 . In the case of coannihilation, one needs to remember that the triplet-like dark matter ischaracterized by the ∆p component. This fraction is dominated by the (3,3) component ofthe mixing matrix in Eq. (3.19). Therefore, the mass splitting between D and charged HiggsH± is no longer described by Eq. (6.1). In fact, from our scan result, the charged Higgsmass is always more than twice heavier than the D mass. In addition, the mass splittingbetween D and ∆ is also large. This is due to the choice of large v∆ that makes the (3,3)component always smaller than (2,2) component of dark scalar mixing matrix in Eq. (3.19).Thus, the triplet-like dark matter provides the natural dark matter candidate in G2HDMmodel. Based on these two mass splittings, there is no coannihilation between triplet -likedark matter and either H± or ∆. Interestingly, at above 400 GeV, there is coannihilation effectcoming from W ′ and heavy fermion f H as depicted by the gold square and green points inthe left panel of Fig. 6.6. These two coannihilations are important when the annihilationcross section becomes ineffective. This conclusion can be drawn based on the location ofthese coannihilation points which are situated above the 2σ red line.

The important channel for triplet-like dark matter-nucleon interaction is given by thet-channel hi and Zi exchange. The heavy fermion exchange is not relevant due to its heavymass suppression in the propagator. In particular, the dominant contribution to the spinindependent cross section is given by the h1, Z and Z′ mediator. Out of these three dominant


contributions, the h1 gives the most significant impact on the cross section. The upper boundof this Higgs exchange can even reach 10−41cm2 which is 5 order of magnitude abovethe current XENON1T limit for the dark matter mass less than 500 GeV. This is due tothe big range of the λDD∗h1 coupling given in Eq. (6.7). Above 500 GeV, the h1, Z and Z′

contributions become comparable to each other. The typical cross section in this mass rangeis below 10−45cm2 which can escape the current XENON1T data. Unlike the doublet-likecase, the Z boson exchange is no longer characterized by the SM coupling as one can seefrom Eq. (6.8). From this equation, we see that the Zi exchange is controlled by the SU(2)H

gauge coupling as well as OG2i components. This will make the SM Z boson exchange become

much smaller compared to the similar exchange in the doublet-like case. Further commentneeds to be made regarding the comparable contribution between Z and Z′. As one naivelyexpects, the Z exchange is more important than the Z′ one due to the heavier mass of the latercontribution. This comparable contribution can be understood from the mixing matrix givenin Eq. (3.36) especially the lower 3x3 mixing part. Thanks to the large value of vΦ comparedto the SM VEV, the (2,2) component will strongly mix with the (3,3) component while themixing between the (1,1) components are quite small. Thus, one expects that OG

21 will bemuch smaller than OG

22. In addition, as long as the Z′ mass is not much heavier than the Zmass, the resulting cross sections between these two contributions would be comparable asobserved in our case.

The ISV effect is still observed even though it becomes milder than the doublet-likecase thanks to the h1 contribution. The dark matter-neutron cross section is slightly largerthan the proton one. This can be understood from the value of | fn/ fp| ∼ O(1). Moreover,there is a variation of the sign between fp and fn for each points in our scan. In somecases, fp will have the same sign with fn while in other case they have opposite signs. Thus,the maximal cancellation fp/ fn ≈−0.7 can not be realized in this case. Due to the Z andZ′ exchange, the dark matter-nucleon and the anti-dark matter-nucleon cross sections aredifferent. As mentioned before, we take the average value between these two contributions.The result of our calculation for direct detection is shown in the right panel of Fig. 6.6.After calculating the averaged cross section in the nucleus level, we project the result on thedark matter-neutron cross section. The blue points denote the allowed region that satisfythe SGSC constraints as well as 2σ relic density. The orange region describes the pointsthat survive the SGSC, relic density, and XENON1T constraints. Note that there are feworange points located above the XENON1T exclusion line. This is due to the ISV effectthat weakens the XENON1T exclusion limit. From this figure, the allowed dark matter masswhich satisfies all the constraints is located above 400 GeV.


101 102 103

mD (GeV)

10−29

10−28

10−27

10−26

10−25

10−24〈σv〉(

cm3 s−

1 )f∆p

> 2/3, All points SGSC+RD

CTA W+W−

CTA bb

W+W− (ID exc.)

bb (ID exc.)

W+W−

bb

101 102 103

mD (GeV)

10−47

10−46

10−45

10−44

10−43

10−42

10−41

10−40

σSI

n(c

m2 )

f∆p> 2/3

XENON1

T

SGSC+RD (ID exc.)

SGSC+RD+ID

SGSC+RD+ID+DD

Fig. 6.7 The annihilation cross section at the present universe (left) and DM-neutron elasticscattering cross section (right) for f∆ > 2/3 case. In the left panel, the annihilation final stateis classified to be three main types: W+W− (blue), bb (green), and h1h1 (orange). However,the exclusion by ID is marked by unfilled and light colors. In the right panel, the regionallowed by SGSC+RD+ID+DD constraints is marked by filled dark blue squares. However,the region excluded by SGSC+RD+ID and SGSC+RD+DD is marked in orange crossesand light blue squares. Projected sensitivities from the CTA experiment for the W+W− andbb final states are also shown.

The indirect dark matter search constrains the allowed thermally averaged cross sectionin the recent universe. Since the temperature of the current universe is very low ∼ 2.7 K,the p-wave cross section is extremely suppressed compared to the s-wave part. In addition,there is no coannihilation is present time as the heavier Z2 particles are expected to havedecayed into the lightest one. Thus, it is sufficient to consider the dark matter annihilationcross section alone. The corresponding Feynman diagrams of indirect detection are similarto the relic density diagrams. The indirect detection (ID) constraints for triplet-like DM isgiven in Fig. 6.7. All the points in this figure satisfy both the SGSC and 2σ relic densityconstraints. The left panel shows the present time of dark matter annihilation as a functionof the dark matter mass. We only present the most dominant final states which are givenby bb and W+W− pairs. As the dark matter mass get less than ∼ 70 GeV, the dark matterpair dominantly annihilates into bb. The allowed region coming from this annihilationis depicted by the green triangle points while the excluded ones are represented by theorange triangle points. At the region near SM Higgs resonance, it is shown that the shapeof the thermally averaged cross section forms a dip. This seems to contradict the typicalcross section which has the spiky peak in the appearance of the resonance. This happens

6.3 SU(2)H Goldstone Boson-like DM 71

because of the 2σ relic density constraints imposed on this region. Near the resonance, thecorresponding annihilation cross section would have a very large value and hence suppressedthe relic density. In order to satisfy the observed relic density, the λDD∗h1 coupling needs tobe extremely fine tuned to the small value such that the associated cross section becomessmaller. Thus, this fine tuned coupling will make the recent dark matter annihilation crosssection to have smaller value. Above 100 GeV, the most dominant contribution is given byW+W− final state. The ID allowed region in this final states are shown by the blue pointswhile the ID exclusion region is represented by the red square area. Note that the current DMID sensitivity can only apply strongly for the DM mass located between 10 GeV and fewhundred GeV. Furthermore, the future CTA sensitivity [59] described by the red (purple)line for the W+W− (bb) final states might be able to put the limit in the TeV region. In theright panel of Fig. 6.7 we compare the exclusion limit given by recent XENON1T data (bluesquares) and Fermi gamma-ray constraints (orange crosses). One can see the XENON1Texclusion (unfilled squares) power is much stronger than Fermi gamma-ray exclusion. Theblue filled region describes the area that passes all the constraints discussed so far: SGSC,2σ PLANCK relic density, XENON1T data and Fermi gamma-ray exclusion.

6.3 SU(2)H Goldstone Boson-like DM

It is easy to note that the (1,1) and (3,3) elements of Eq. (3.19) have a see-saw behaviourcontrolled by the value of v∆. The (2,2) element remains almost unaffected thanks to theterm proportional to v2

Φ. Goldstone boson-like DM is characterized by a large value in the

element (1,1) of Eq. (3.19) when compared to the (1,3) element, −MΦ∆vΦ/2. The size ofthe (1,2) element is not relevant since the (2,2) element remains much larger. The differencein size between the (1,1) and (1,3) elements is best measured by taking the ratio betweenthem which is roughly approximated by 2v∆/vΦ. In other words, the v∆/vΦ ratio controls theGoldstone boson composition of the DM mass eigenstate. This is illustrated in Fig. 6.8. Notethat when the ratio v∆/vΦ grows close to 1, which means vΦ ∼ 20 TeV in our scan range,EWPT disfavour the presence of light Z′ states and larger mixing with the SM Z. Values offGp larger than ∼ 0.8 are accessible only through negative MΦ∆ resulting in tachyonic DMmasses.

Thus, in the case of Goldstone boson-like DM, there will be a mixture coming fromthe pure triplet component, Goldstone-doublet and triplet-doublet components. There is achange in the dominant DM annihilation cross section in Goldstone boson-like DM. Themost dominant channel is still given by the W+

L W−L final state which contributes around 50%.

The next important process is given by Z′Z′ final state. In both two cases, the s-channel h1


0.0 0.2 0.4 0.6 0.8 1.0v∆/vΦ

0.0

0.2

0.4

0.6

0.8

1.0

f GP

After SGSC

Tachyonic DM

EW

PT

dis

favou

red

Fig. 6.8 Correlation between the ratio v∆/vΦ and the mixing parameter fGP after applyingthe constraints from the scalar and gauge sectors.

and h2 exchange affect the whole amplitude. However, unlike in previous two scenarioswhere doublet or triplet component fully controls the corresponding dominant couplings(λDD∗h1 and λDD∗h2), there is negative effect coming from the "impurity" contributions in theλGDD∗h1 and λGDD∗h2 . The expression of λGDD∗h1 is given as

λGDD∗h1 = i[(λHΦ +λ

′HΦ)vOH

11 −λΦ∆v∆OH31 +2λΦvΦOH

21](OD

12)2 (6.13)

+ i[MH∆OD

12 +λ′HΦvΦOD

32]

OD22OH

11

+ i[λH∆vOH

11 −2λ∆v∆OH31 +λΦ∆vΦOH

21](OD

32)2 ,

where the first term inside the bracket on the right hand side denotes the pure Goldstonecontribution, the second term describes the mixing between Goldstone-doublet and doublet-triplet, and the last term represents the pure triplet contribution. The last two lines denote the"impurity" terms. In the case of λGDD∗h2 , the coupling is given by

λGDD∗h2 = i[−λΦ∆v∆OH

32 +2λΦvΦOH22](OD

12)2 (6.14)

+ i[−2λ∆v∆OH

32 +λΦ∆vΦOH22](OD

32)2 ,

where the first (second) line denotes Goldstone (triplet) contribution.As a result, these dominant couplings will have smaller value and make the corresponding

cross section decreased. This can be easily seen from the left panel of Fig. 6.9 where thereare a lot of points reside above the observed relic density band. In addition, the presence of

6.3 SU(2)H Goldstone Boson-like DM 73

100 101 102 103

mD (GeV)

10−5

10−3

10−1

101

103

105ΩDh

2

fGP > 2/3, all points SGSC

Ωh2 ≈ 0.12

h1 resonance

W ′ coannihil.

HF coannihil.

100 101 102 103

mD (GeV)

10−47

10−46

10−45

10−44

10−43

10−42

σSI

n(c

m2 )

fGP > 2/3, all points SGSC

XENON1T(fn/fp=-0.5)

XENON1T(fn/

fp=1)

SGSC+RD

SGSC+RD+DD

Fig. 6.9 Goldstone boson-like SGSC allowed regions projected on the (mD, ΩDh2) (left) and(mD, σSI

n ) (right) planes. The gray area on the left has no coannihilation or resonance. Thegray area on the right is excluded by PLANCK data at 2σ . The orange squares above theXENON1T limit present the ISV cancellation at the nucleus level.

new dominant Z′Z′ final state stems from the fact that the coupling λGDD∗Z is suppressedby the off-diagonal rotation matrix elements OG

21 while λGDD∗Z′ contains the somewhatlarger OG

22. One expects small OG21 due to the SM-likeness of Z in this model. Let’s write

λGDD∗Z(′) = gGDD∗Z(′)(pD∗ − pD)µ . The expression of gGDD∗Z is given by

gGDD∗Z =− i2

gH[(OD

12)2 +2(OD

32)2]OG

21 − icW e2sW

(OD22)

2OG11 , (6.15)

where the first (second) term on the right hand side coming from Goldstone (triplet) com-ponent which is suppressed by OG

21 while the last term suppressed by (OD22)

2 denotes thedoublet contribution. Furthermore, the coupling of gGDD∗Z′ is written as

gGDD∗Z′ =− i2

gH[(OD

12)2 +2(OD

32)2]OG

22 , (6.16)

where only Goldstone and triplet component are present in this case. In contrast to theλGDD∗h1 and λGDD∗h2 , the existence of non-Goldstone contribution gives an enhancementto both of the λGDD∗Z and λGDD∗Z′ couplings. In addition, the p-channel contact interactionbetween D and Z′ also contributes significantly to the Z′Z′ final state. This contact coupling


is expressed as

gGDD∗Z′Z′ =

i2

[g2

H(OG22)

2 +4gHgXOG22O

G32 +4g2

X(OG32)

2](OD

12)2 (6.17)

+ 2ig2H(O

D32)

2(OG22)

2 ,

where one can also see the enhancement effect coming from the triplet contribution. Thusin the case of Z′Z′ final state, not only h1 and h2 s-channel exchange is important, but alsothe t-channel of D exchange which is controlled by λGDD∗Z′ as well as p-channel contactinteraction even though the h1 and h2 exchange control the range of the cross section thanksto a big range in the corresponding couplings in Eq. (6.13) and Eq. (6.14).

Coannihilation in this case is very similar to the triplet-like DM case. The most relevantcoannihilations happen with W ′ and heavy fermions for large masses and large relic density.Coannihilation with W ′ starts close to DM mass of 300 GeV and mostly above relic density of10−1. As before, this is where the usual DD∗ annihilation channels become smaller leavingmore room for coannihilations that, otherwise, would be negligible. For the case of heavyfermions, coannihilation happens for DM masses above 1 TeV and mostly for the upperbound of relic density, where DD∗ coannihilation is even more suppressed than for the W ′

case.As one can see in Fig. 6.9, the DM-neutron cross section is between the orders of

magnitude 10−46 and 10−42 cm2. The most dominant contribution comes from h1 exchangepeaking for DM masses of order 30 GeV. The next dominant channel is given by Z and Z′

bosons exchange. The contributions from these two gauge bosons result in the base DM-neutron cross section that sits just below 10−45 cm2. The interference between h1, Z, and Z′

exchange makes the spin independent cross section varies in a wide range. Furthermore, theGoldstone-like dark matter provides us some interesting result regarding the ISV. All the DMcompositions discussed have some amount of ISV. But Goldstone boson-like DM is the onewhere fn/ fp is closer to the maximally cancelling value of ∼−0.7. In our analysis, most ofthe times the D∗-nucleus cross section is larger than that of D-nucleus, resulting in an averageDM-nucleus cross section dominated by D∗-nucleus. For the cases where fn/ fp ≈−0.7 forD∗-nucleus this would result in noticeable ISV cancellation. In the right panel of Fig. 6.9,the orange points pass the RD constraint from PLANCK and the DD limit set by XENON1Tat nucleus level thanks to ISV cancellation.

In the ID side, there is no relevant constraints for this Goldstone boson-like case. Most ofthe points with relic density in agreement with PLANCK have a very low annihilation crosssection in the present and are far beyond the reach of current observations. For DM massesbelow 100 GeV, the annihilation is dominated by bb final state with 90% of the total cross

6.4 Constraining Parameter Space in G2HDM 75

102

mD (GeV)

10−34

10−32

10−30

10−28

10−26〈σv〉(

cm3 s−

1 )

fGP > 2/3, All points SGSC+RD

CTA W+W−CTA bb

W+W−

bb

102

mD (GeV)

10−46

10−45

10−44

σSI

n(c

m2 )

fGP > 2/3

XENON1

T(fn/f

p=-0.

5)

XENON1

T(fn/f

p=1)

SGSC+RD+ID

SGSC+RD+ID+DD

Fig. 6.10 The present time total annihilation cross section by dominant annihilation channel(left) and the DM-neutron elastic scattering cross section (right) for fGP > 2/3. Some bluefilled squares above the XENON1T limit are due to the ISV cancellation at the nucleus level.Projected sensitivities from the CTA experiment for the W+W− and bb final states are alsoshown.

section in average. For DM mass above the mass of the W±, the W+W− final state dominatescompletely with an average of 50% of the total cross section, However, W+W− final statemay compose the total cross section almost completely for some points while it may goes aslow as 17% for others. Unlike triplet-like DM, ID alone does not further constrain the pointsallowed by PLANCK. Fig. 6.10 shows the final result for this section including ID. The rightpanel shows the zoomed in region of points allowed by the RD constraint. As mentionedbefore, ISV cancellation ( fn/ fp ≈−0.7) reduces the sensitivity of the XENON1T result andsome points pass all the constraints (SGSC+RD+ID+DD) even though they are above thedirect detection limit at nucleon level.

6.4 Constraining Parameter Space in G2HDM

From previous sections, we have learned that the doublet-like DM scenario cannot fulfillthe DM constraints and that Goldstone boson-like DM requires ISV cancellations close tomaximal to pass the XENON1T limit. Therefore, in this section we are going to discuss theallowed G2HDM parameter space based on the triplet-like DM.

In Fig. 6.11, we present the λi allowed region of SGSC constraints (green region) andSGSC+RD+DD constraints (red scatter points). Comparing the green region with the redscatter points, one can easily see that λΦ, λH∆, and λΦ∆ are mostly constrained by RD+DD


10−3

1

λΦ

10−3

1

λ∆

−3

0

3

λH

Φ

0

5

λH

∆

0

5

10

λΦ

∆

0

5

10

λ′ H

Φ

10−1 1λH

−20

0

20

λ′ H

10−3 1λΦ

10−3 1λ∆

−3 0 3λHΦ

0 5λH∆

0 5 10λΦ∆

0 5 10λ′HΦ

10−1 1

λH1.29e-01,2.80e+00

10−3 1

10−3

1

λΦ

6.35e-03,4.09e+00

10−3 1

10−3

1λ∆

1.01e-04,4.99e+00

−3 0 3

−3

0

3

λHΦ

-5.67,3.41

0 5

0

5 λH∆

-3.39,4.07

0 5 100

5

10λΦ∆

-0.07,6.62

0 5 100

5

10 λ′HΦ

-0.01,15.90

−20

0

20λ′H

-22.74,9.57

Fig. 6.11 A summary plot for the scalar parameter space allowed by the SGSC constraints(green region) and SGSC+RD+DD constraints (red scatter points). The numbers writtenin the first block of each column are the 1D allowed range of the parameter denoted in thehorizontal axis after SGSC+RD+DD cut.

constraints. As discussed in previous section, the dark matter relic abundance is mainlycontrolled by three dominant contributions coming from W+W−, h1h1, and ZZ final states.Furthermore, we already see that only dark matter mass larger than 400 GeV can satisfy allthe DM constraints. The dominant contribution of the gauge boson final state in this case isgiven by its longitudinal component. In this range, those three dominant annihilation crosssections are determined by λDD∗h1 and λDD∗h2 originated from s-channel h1 and h2 exchange.Thus, to understand which parameters are sensitive to the dark matter phenomenological

6.4 Constraining Parameter Space in G2HDM 77

constraints in general, it is sufficient to look at these two couplings. One can see fromEq. (6.7) that there are three dominant terms that contribute to the DM-DM-Higgs couplings,λH∆vOH

11, λΦ∆vΦOH22, and λ∆v∆OH

33. Clearly, λH∆, and λΦ∆ are restricted by the allowedHiggs coupling sizes. In addition, the p-channel in the h1h1 final state also depends stronglyon λH∆. However, λ∆ and v∆ are not constrained because the cross section is suppressed bythe heavy mediator mh3 and the condition v∆ < vΦ. Next, due to a rather loose requirementon the triplet-like dark matter OD

32 > 2/3 one may expect the contribution from anotherdark matter component. For triplet case, another dominating contribution comes fromGoldstone-like part and can be as large as (OD

12)2 ≈ 1/3 while the doublet component is

strongly suppressed. Therefore, one needs also to consider the first lines of both Eq. (6.13)and Eq. (6.14) to account this impurity effect. Due to the large value of vΦ and v∆, the firstline of Eq. (6.14) will put constraint on the λΦ∆ and λΦ respectively. Moreover, from thefirst line of Eq. (6.13), one can see the additional constraint on λΦ∆ and λΦ as this couplingdirectly depend on these two parameters. In addition, there is also two additional termsλHΦ and λ ′

HΦthat only appear in the first two terms of the first line in Eq. (6.13). These

two terms are not suppressed by any mixing angle and only account for the impurity effect.As a consequence, one expects that the λHΦ and λ ′

HΦwill be mildly constrained. As an

additional remark, there is also a dependence on the λ s in the components of the orthogonalmixing matrices OD

i j and OHi j . This will induce indirect constraints on the λ s inside these two

matrices even though it is very difficult to see this dependence unlike the λ s appear explicitlyin the relevant couplings.

Next, we project the allowed G2HDM parameter space to the two VEVs vΦ and v∆, twomass scales MH∆ and MΦ∆, and two new gauge couplings gH and gX . Importantly, onlygauge coupling gH and the VEV vΦ can be further constrained. Interestingly, we found sucha exclusion comes from the lower allowed DM mass. The allowed DM mass values rangefrom hundreds of GeV to TeV. This range is reflected in gH since the minimal value wechoose for gH is given by Eq. (5.12) and depends directly on the DM mass. In sum, a goodscalar DM candidate in G2HDM requires gH > 7.09×10−3 and vΦ > 22.7 TeV.


0

25

50

MΦ

∆/G

eV

103

104

v ∆/G

eV

50

100

v Φ/G

eV

10−2

10−1

g H

1000 3000

MH∆/GeV

10−7

10−5

10−3

g X

0 25 50

MΦ∆/GeV103 104

v∆/GeV50 100

vΦ/GeV10−2 10−1

gH

1000 3000

MH∆/GeV2.75,

4999.70

0 25 500

25

50

MΦ∆/GeV0.01,49.95

103 104

103

104v∆/GeV

5.00e+02,2.00e+04

50 100

50

100

vΦ/GeV22685.96,99999.40

10−2 10−1

10−2

10−1 gH7.09e-03,1.00e-01

10−7

10−5

10−3 gX1.01e-08,5.50e-02

Fig. 6.12 A summary plot table of the parameter space of the two VEVs vΦ and v∆, two massscales MH∆ and MΦ∆, and two new gauge couplings gH and gX . The color scheme is thesame as Fig. 6.11.

Chapter 7

Summary

The G2HDM is a novel two Higgs doublet model with a DM candidate arises naturallywithout imposing any ad hoc discrete symmetry. After SU(2)H symmetry breaking, one canfind three potential DM candidates: the lightest new dark scalar, heavy neutrino, and theSU(2)H gauge boson W ′(p,m). Though these three candidates are all interesting, we focusedthis paper on the most popular one, the new scalar DM, which is also well discussed inthe inert doublet Higgs DM model. Different to the inert doublet Higgs DM model, themixing between Z -odd scalars adds a touch of complexity since DM in G2HDM not onlycomes from the inert doublet but may also be Goldstone boson-like and SU(2)H triplet-like.We took the dominant composition ( f j > 2/3) as a criteria to classify them but the mixturebetween them can be simply inferred. In this paper, we have discussed these three typesindividually with two assumptions: that all the new non-SM heavy fermion are heavy enoughto have mostly negligible contributions and that DM shall be thermally produced before thefreeze-out temperature. We have comprehensively shown their detectability and exclusionsby the current SGSC and DM constraints (mainly RD+DD).

For the inert doublet-like DM, we found some interesting features. First, the maindifference between the inert doublet DM in IHDM and G2HDM is that in IHDM there isa mass splitting between scalar and pseudoscalar while they are completely degenerated inG2HDM. As long as the mass splitting in IHDM remains larger than the exchanged energybetween DM and nucleon in the direct detection experiment, the interaction mediated by theZ gauge boson remains suppressed. Since this mass splitting does not exist in G2HDM, suchinteractions are unsuppressed and bring the spin independent cross section up to ∼ 10−38 cm2,which is significantly above the XENON1T 95% C.L. limit for masses above ∼10 GeV andabove the CRESST-III result down to 2 GeV.

80 Summary

On the other hand, at the mass region mD ≲ 10 GeV, DM is over abundant because ofoff-shell annihilation channels. Hence, we confirmed that the inert doublet-like DM can becompletely excluded by RD+DD constraints.

Next, a SU(2)H triplet-like scalar DM was discussed. Since the composition fH2 hasto be tiny in order to avoid the tension with DM DD, the triplet-like DM can mostly mixwith the Goldstone boson Gp,m

H . There is no Z-resonance region in the triplet-like DM forDM annihilation and the parameter space is more or less consistent with Higgs portal DM.However, DD is still the most stringent constraint comparing with ID and collider constraints.The allowed DM mass mD by SGSC+RD+DD is required to be heavier than 300 GeV.Despite weaker constraints from ID and collider constraints, it might be possible to detectthe heavy DM mass region by the future CTA and 100 TeV colliders even if a DM signal isnot found at direct detection experiments before hitting the neutrino floor.

We explored the Goldstone boson-like DM but we found it is not possible to find a pureGoldstone boson-like DM. The non-tachyonic DM condition and EWPT constraints prohibitthe composition fGP > 0.75 unless one would like to move to a more fine-tuned region wherev∆/vΦ ≫ 1 and v∆ ≫ 20 TeV. This will cause the Goldstone-like dark matter to receive asignificant component coming from the triplet. Furthermore, this triplet component wouldtypically reduce the coupling strength λGDD∗h1 and λGDD∗h2 relevant to determine the relicabundance. As a result, the annihilation cross section is smaller than the triplet-like case,resulting in larger relic density and less points within the PLANCK relic density measurement.Furthermore, XENON1T measurement excludes almost all the points with appropriate relicdensity, except for those close to maximal cancellation due to isospin violation ( fn/ fp ≈ 0.7)where direct detection sensitivity is notably reduced. Therefore, a small region with massabove 100 GeV can pass all the DM constraints applied in this work.

Finally, we presented the impact of DM constraints on the G2HDM parameter space.Based on the triplet-like DM case, we found λΦ, λH∆, λΦ∆, gH , and vΦ are significantlyconstrained by DM constraints, mainly RD+DD. Interestingly, the lower limit gH > 7.09×10−3 for vΦ < 100 TeV is within reach for the future linear or circular lepton-antileptonmachines and 100 TeV hadron colliders.

Appendix A

Relevant Couplings

In the following, we list the relevant couplings contribute to the dark matter analysis invarious processes discussed in the text. We use conventional notation g and g′ to denotethe Standard Model SU(2)L and U(1)Y coupling. The cW and sW denote the usual cosineand sine of the Weinberg angle. In addition, for the scalar-scalar-gauge vertex, we adopt theconvention that all momentum are incoming as shown in the vertex below.

Fig. A.1 The DD∗W+W− coupling for p-channel interaction.

A.1 Dominant Couplings for Dark Matter

82 Relevant Couplings

Fig. A.2 The dominant DD∗Zi and DD∗hi couplings for the inert doublet-like dark matter.

Fig. A.3 The dominant DD∗Zi and DD∗hi couplings for the SU(2)H triplet-like dark matter.

A.1 Dominant Couplings for Dark Matter 83

Fig. A.4 The dominant DD∗Zi and DD∗hi couplings for the SU(2)H Goldstone boson-likedark matter.

Appendix B

Benchmark Points for Monojet

Although in our study we do not impose collider search as our global dark matter constraint,we take 10 benchmark points to study dark matter production at the collider. These 10 pointsare located just below the XENON1T exclusion line.

86 Benchmark Points for Monojet

TableB

.110

benchmark

pointsforthe

mono-jetofthe

SU(2)H

triplet-likeD

M.

Benchm

arkpoint

AB

CD

EF

GH

IJ

λH

0.2480.311

0.7101.621

0.2120.535

0.5750.182

0.3640.141

λΦ

3.5150.620

2.4432.638

0.9241.450

1.6551.931

0.4420.930

λ∆

0.7353.129

4.0070.279

1.2740.772

0.4721.821

4.0820.045

λH

Φ−

0.964−

0.303−

1.975−

0.652−

0.297−

0.8201.594

−0.177

−0.414

0.127λ

H∆

0.340−

0.0431.464

−0.939

−0.313

−0.913

0.5370.169

−1.358

1.069λ

Φ∆

0.4292.525

0.7061.298

2.0780.936

1.0563.110

1.2381.067

λ′H

Φ1.213

9.7895.181

5.0696.040

2.9723.886

3.8221.909

0.910λ′H

−21.179

−1.930

−1.205

1.454−

3.071−

1.627−

3.097−

12.631−

2.870−

5.374M

H∆(G

eV)

178.7742.82

760.38684.84

1.83×10

31.60×

103

2.42×10

33.26×

103

3.81×

103

2.52×10

3

MΦ

∆(G

eV)

9.78×

10 −2

0.8155.445

8.81813.391

16.02523.543

6.01719.202

5.523v

∆(100G

eV)

125.4753.94

69.0748.26

47.3336.21

37.3922.95

32.389.99

vΦ(1TeV

)62.10

29.4030.25

24.5032.00

26.6127.19

63.3846.58

59.93g

H ×10

26.876

2.7516.498

5.3786.066

8.1398.420

8.4518.747

9.376g

X6.26×

10 −7

5.20×10 −

51.66×

10 −8

4.70×10 −

72.90×

10 −2

3.60×

10 −7

7.24×

10 −7

3.58×10 −

23.31×

10 −2

3.62×

10 −6

MX(G

eV)

2000m

D(G

eV)

94.83192.82

468.96565.00

890.98921.35

1123.491633.27

1815.832238.29

mh

2 (TeV)

150.7853.48

194.2224.45

23.2041.35

31.1129.43

83.7221.18

mh

3 (10TeV)

164.6835.016

66.9056.34

44.0945.36

49.52124.62

44.0281.72

mZ′(G

eV)

2000.00404.36

982.74658.80

863.191083.11

1144.831415.40

1389.162000

mZ′′(G

eV)

2135.152000.00

2000.002000.00

2248.862000.00

2000.003784.11

2933.292809.32

Ωh

2×10

1.1931.180

1.1861.188

1.2031.203

1.1921.207

1.2041.195

σSI×

10 −46(cm

2)1.058

4.0656.622

9.64712.275

12.70615.688

16.81313.455

17.866Tree

lvl.(pb)1.55×

10 −5

2.38×10 −

48.02×

10 −9

6.08×10 −

51.49×

10 −5

7.80×

10 −6

9.51×10 −

118.07×

10 −11

2.46×10 −

76.47×

10 −11

Loop

lvl.(pb)1.77×

10 −6

2.73×10 −

84.36×

10 −6

3.69×10 −

91.49×

10 −8

4.44×10 −

107.11×

10 −11

6.49×

10 −12

9.62×

10 −13

1.32×10 −

13

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