Complex Scalar Dark Matter in Gauged Two Higgs Doublet Model

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(1)Complex Scalar Dark Matter in Gauged Two Higgs Doublet Model (G2HDM). Chrisna Setyo Nugroho Supervisor: Professor Chuan-Ren Chen Professor Tzu-Chiang Yuan Department of Physics National Taiwan Normal University. This dissertation is submitted for the degree of Doctor of Philosophy. July 2019.

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(3) This work is presented to my parents!.

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(5) Declaration. I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other University. This work was done wholly or mainly while in candidature for a Ph.D degree at this University. This dissertation is my own work and jointly with others. Chrisna Setyo Nugroho July 2019.

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(7) Acknowledgements. Alhamdulillah, thanks to the most Merciful and Benelovent i have been able to finish this work. I offer my deepest gratitude to Prof. Tzu-Chiang Yuan for his patience and support during 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 as guided me to become a professional physicist. My sincere thank is addressed to my co-supervisor, Prof. Chuan-Ren Chen for his help and advices during my stay at NTNU. I also deeply thank Prof. Chiang-Hung Vincent Chang who has provided me a lot of support and help, especially during the early years of my graduate school life. I would like to thank Prof. Pham Quang Hung for his help and collaborations. Many thanks to my friends Dr. V. Q. Tran, Mr. Yu-Xiang Lin and Mr. Chia-Feng Chang for providing me a lot of ideas and help in research and daily life in Taiwan. I also offer my special thanks to Dr. Raymundo Ramos and Dr. Yue-Lin Sming Tsai for their important advices and help. Finally, i thank my parents in Indonesia and my family in Taipei for their endless support to me..

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

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(11) Table of contents List of figures. xiii. List of tables. xvii. 1. Introduction. 2. Short Review of the Standard Model 2.1 The Weak Interaction and the SM Particle Content . 2.2 The SM Gauge Interactions and Symmetry Breaking 2.2.1 The SM Field Contents . . . . . . . . . . . . 2.2.2 The SM Lagrangian . . . . . . . . . . . . . 2.2.3 The Higgs Mechanism . . . . . . . . . . . .. 3. 4. 1. . . . . .. 5 5 8 8 10 13. . . . . . . . . . .. 17 17 19 20 22 22 23 25 27 28 29. WIMP Dark Matter and Its Constraints 4.1 Astrophysical Evidence of the Dark Matter . . . . . . . . . . . . . . . . . 4.1.1 Galactic Rotation Curves . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . .. 33 33 33 34. The G2HDM Model 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . 3.2 The Particle Contents . . . . . . . . . . . . . . . 3.3 Higgs Potential . . . . . . . . . . . . . . . . . . 3.4 The SSB of the Potential and the Mass Spectrum 3.4.1 Spontaneous Symmetry Breaking . . . . 3.4.2 Scalar Mass Spectrum . . . . . . . . . . 3.4.3 Gauge Boson Mass Spectrum . . . . . . 3.4.4 Fermionic Mass Spectrum . . . . . . . . 3.5 The Accidental Z2 Symmetry . . . . . . . . . . . 3.6 Theoretical Constraints on the Scalar Sector . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . ..

(12) xii. Table of contents 4.2 4.3. 5. 6. 7. WIMP as Thermally Produced Dark Matter Dark Matter Searches . . . . . . . . . . . . 4.3.1 Dark Matter Direct Search . . . . . 4.3.2 Dark Matter Indirect Search . . . . 4.3.3 Dark Matter Collider Search . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. Dark Matter in G2HDM: Constraints and Methodology 5.1 Dark Matter Properties in G2HDM and Experimental Constraints 5.1.1 Relic Density . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Direct Detection . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Indirect Detection: Gamma-ray from dSphs . . . . . . . 5.1.4 Collider Search . . . . . . . . . . . . . . . . . . . . . . . 5.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . Dark Matter in G2HDM: Numerical Results 6.1 Inert Doublet-like DM . . . . . . . . . . 6.2 SU(2)H Triplet-like DM . . . . . . . . . 6.3 SU(2)H Goldstone Boson-like DM . . . . 6.4 Constraining Parameter Space in G2HDM. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . .. . . . . . .. . . . .. . . . . .. . . . . . .. . . . .. . . . . .. . . . . . .. . . . .. . . . . .. . . . . . .. . . . .. . . . . .. 35 37 38 39 40. . . . . . .. 43 44 44 48 50 51 53. . . . .. 57 57 65 71 75. Summary. Appendix A Relevant Couplings A.1 Dominant Couplings for Dark Matter. 79. . . . . . . . . . . . . . . . . . . . .. 81 81. Appendix B Benchmark Points for Monojet. 85. References. 87.

(13) List of figures 2.1. The fundamental particles in the Standard Model. (Source: wikipedia.org) . . . .. 3.1. A summary of the parameter space allowed by the theoretical and phenomenological constraints. 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]. 30. 4.1. Rotation curves of spiral galaxies as observed by Rubin et al. [31]. At large radial distance from the center, most of galaxies exhibit constant circular velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Dark matter number density per comoving volume as function of its mass over the temperature of the universe. Figure is taken from [33]. . . . . . . 37 The current (solid) and projected (dotted/dashed) bounds on the spin-independent WIMP DM-nucleon cross section. The orange band indicated the neutrino floor which is a typical extraterrestrial background. This plot is adopted from [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Indirect DM constraints for few final states: from IceCube [37], AMS02 [38], H.E.S.S. [39], PLANCK [40], CTA projected sensitivity [41], and FermiMAGIC collaborations [42]. The black dotted-dashed line denotes the typical annihilation cross section ⟨σ v⟩ = 3 × 10−26 cm3 s−1 . Figure adopted from [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40. 4.2 4.3. 4.4. 5.1. 5.2. 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. . . . . . . . . The Feynman diagrams of dark matter annihilation into h1 h1 pair. The hi denote the three Higgs boson mediators. The t (u) represents t (u)-channel diagram respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 45. 46.

(14) xiv. List of figures 5.3. 5.4. 5.5 5.6 5.7. The Feynman diagrams of dark matter annihilation into ZZ pair. The hi denote the three Higgs boson mediators. The t (u) represents t (u)-channel diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The diagrams relevant for f¯ f final states. The hi , Zi , and fiH denote the three Higgs boson, three neutral gauge boson as well as three heavy fermion mediators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dominant Feynman diagrams for Higgs bosons (left) and Z bosons (right) exchange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Feynman diagrams of leading contributions for monojet plus missing energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) 0 0 hh G√ states respectively. These scalars can be one of the following pairs ( √ , G2 , 2 H20∗ H20 , H + H − ,. 6.1. 6.2 6.3. 6.4. 6.5 6.6. 0 G0 φ√ 2 φ2 GH √ H, , 2 2. GHp Gm H,. δ√ 3 δ3 , 2. ∆ p ∆m ) as described in the text. .. 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. The gray scatter points agree with the SGSC constraints. The blue scatter points agree with PLANCK data within 2σ region. . . . . . . . . . . . . . . . . ¯ final states. ¯ cc Feynman diagrams of dark matter annihilation into ττ, ¯ and bb Relevant diagrams for h1 h1 final state. For the doublet case, both D and e exchange in t- and u-channels are important. For the triplet-like and ∆ Goldstone-like dark matter, only D exchange is relevant. . . . . . . . . . . Relevant diagrams for ZL ZL final state. The first three diagrams scale with the center-of-mass energy. The sum of these three diagrams cancel the energy dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relevant diagrams for WL+WL− final state. . . . . . . . . . . . . . . . . . . Triplet-like SGSC allowed regions projected on the (mD , ΩD h2 ) (left) and (mD , σnSI ) (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 cancellation at the nucleus level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 47 49 51. 54. 58 58. 61. 61 62. 66.

(15) List of figures The annihilation cross section at the present universe (left) and DM-neutron elastic 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 h1 h1 (orange). However, the exclusion by ID is marked by unfilled and light colors. In the right panel, the region allowed 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 crosses and light blue squares. Projected sensitivities from the CTA experiment for the W +W − and bb¯ final states are also shown. . . 6.8 Correlation between the ratio v∆ /vΦ and the mixing parameter fGP after applying the constraints from the scalar and gauge sectors. . . . . . . . . . 6.9 Goldstone boson-like SGSC allowed regions projected on the (mD , ΩD h2 ) (left) and (mD , σnSI ) (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 . The orange squares above the XENON1T limit present the ISV cancellation at the nucleus level. . . . . . . . . . . . . . . . . . . . . . . . 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 blue filled 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 also shown. . . . . 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 written in the first block of each column are the 1D allowed range of the parameter denoted in the horizontal axis after SGSC+RD+DD cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . .. xv. 6.7. A.1 The DD∗W +W − coupling for p-channel interaction. . . . . . . . . . . . . A.2 The dominant DD∗ Zi and DD∗ hi couplings for the inert doublet-like dark matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 The dominant DD∗ Zi and DD∗ hi couplings for the SU(2)H triplet-like dark matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 The dominant DD∗ Zi and DD∗ hi couplings for the SU(2)H Goldstone bosonlike dark matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70 72. 73. 75. 76. 78 81 82 82 83.

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(17) List of tables 1.1. Table of acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.1. Fermion and Higgs fields and the corresponding SM quantum numbers. The index i denotes the fermion generation. The numbers in the last column are the hypercharge Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 3.1 3.2. Matter contents and their quantum number assignments in G2HDM. . . . . The Z2 assignments in G2HDM model. . . . . . . . . . . . . . . . . . . .. 19 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.

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(19) Chapter 1 Introduction The Standard Model (SM) [1–3] provides a very good explanation about the fundamental interaction between particles observed in nature. Except for gravitational interaction, the SM has been able so far to explain the strong force which is responsible to put the neutron and proton together inside the nucleus of an atom, the electromagnetic force which is involved in the interaction between charged particles, and the weak force which explains the radiative beta decay. In addition, it also predicts the existence of the new massive particles W and Z 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, were believed to be originated from the Higgs mechanism [4–6]. The Higgs mechanism predicts the 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 fundamental particle interactions in nature, the SM can not explain the following puzzles: • The observed neutrino masses which can be inferred by the neutrino oscillation experiments. In SM, neutrinos are assumed to be massless. • The Baryon Asymmetry of the Universe (BAU) that accounts for the dominance of the matter over anti-matter in the universe. • The existence of the dark matter which accounts for about 80% of the total matter and 30% of the total energy density of the universe. • The observed dark energy which is the dominant form of the energy density budget in the universe..

(20) 2. Introduction • etc.. To explain these issues one needs to go Beyond the Standard Model (BSM). There are a lot of BSM models in the literature constructed to attack different open problems in physics. For instance, there is the Minimal Superymmetric Standard Model (MSSM) based on supersymmetry (SUSY) which predicts that every bosonic degree in the SM has its fermionic partner and vice versa. In this model, there exists the lightest supersymmetric particle (LSP) that provides a viable dark matter candidate whose stability protected by the R symmetry. Two Higgs Doublet model (2HDM) is the next non-minimal version of the SM. Instead of having only one Higgs doublet, it has two Higgs doublets which renders the scalar potential more complicated than that in the SM. In the case that the additional Higgs doublet does not acquire the vacuum expectation value (VEV), this becomes the Inert Higgs Doublet Model (IHDM) which provides viable dark matter candidate called the inert dark matter. However, one needs to impose by hand the additional ad hoc Z2 symmetry in the scalar 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) which embeds the two Higgs doublets in the popular 2HDM into a doublet of a non-abelian gauge group SU(2)H . The neutral component of the second doublet can be a dark matter candidate which 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 phenomenology in 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, scalar potential, 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 dark matter and methodology used in our analysis. • Chapter 6 summarizes the numerical results of our dark matter study and the allowed parameter space of G2HDM. • Finally, we conclude this thesis in Chapter 7. Some relevant Feynman rules involving the dark matter and 10 monojet benchmark results are listed in two appendices..

(21) 3 The metric and units used throughout in this thesis are gµν = (+1, −1, −1, −1) and h¯ = 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..

(22) 4. Introduction Acronym 2HDM ATLAS BAU BR BSM CKM CMB CMS CP CTA DD DM dSphs EM EW EWPT FCNC Fermi-LAT G2HDM HP ID IHDM ISC ISV LHC MSSM PandaX PLANCK PU QCD QED RD SGSC SM SSB SUSY VEV VS WIMP XENON1T. Description Two Higgs Doublet Model A Toroidal LHC ApparatuS Baryon Asymmetry of the Universe Branching Ratio Beyond the Standard Model Cabibbo-Kobayashi-Maskawa Cosmic Microwave Background Compact Muon Solenoid Charge Conjugation and Parity Cherenkov Telescope Array Direct Detection Dark Matter dwarf Spheroids Electromagnetism Electroweak Electroweak Precision Test Flavor Changing Neutral Current Fermi Large Area Telescope Gauged Two Higgs Doublet Model Higgs Physics Indirect Detection Inert Higgs Doublet Model Isospin Conserved Isospin Violation Large Hadron Collider Minimal Supersymmetric Standard Model Particle and Astrophysical Xenon Detector https://www.cosmos.esa.int/web/planck/home Perturbative Unitarity Quantum Chromodynamics Quantum Electrodynamics Relic Density Scalar and Gauge Sector Constraints Standard Model Spontaneous Symmetry Breaking Supersymmetry Vacuum Expectation Value Vacuum Stability Weakly Interacting Massive Particle http://www.xenon1t.org Table 1.1 Table of acronyms..

(23) Chapter 2 Short Review of the Standard Model The gauge theory of the electroweak interaction or the Standard Model (SM) will be summarized in this chapter. We will list some of the important properties in SM and their roles in particle physics.. 2.1. The Weak Interaction and the SM Particle Content. The Standard Model (SM) was motivated by the weak interaction observed in many decay processes starting from β decay, inverse β decay, muon decay, and many other fundamental processes in nuclear physics. The first attempt to understand the nature of weak interaction in neutron β decay was done by Fermi by taking analogous current-current interaction in Quantum Electrodynamics (QED). The neutron β decay can be explained quite well at energy scale around few GeV by using a particular form of the Fermi Theory in the picture of current-current interaction Hamiltonian density as GF HI = √ Jµ† J µ , 2. (2.1). where the current is written in terms of hadronic and leptonic currents of the form Jµ = Jµhad + Jµlept ¯ µ (1 − γ5 )e , = pγ ¯ µ (1 − γ5 )n + νγ. (2.2) (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.

(24) 6. Short Review of the Standard Model. its value has been determined from low energy experiment as [11] GF = 1.1663787 × 10−5 GeV−2 .. (2.4). From the magnitude of Fermi constant above, the weak interaction indeed quite weak, and extremely short ranged. All particles that exist in nature (hadrons and leptons) were known to interact weakly. Furthermore, their interactions are represented by SU(2)L multiplets of weak isospin group. As a consequence, the currents in Eq. (2.3) encodes the weak isospin structure within it, and the corresponding currents are called weak isospin currents. The V − A (vector minus axial vector) structure of the weak currents stems from the fact that the parity is violated in weak processes. Moreover, this also implies that only left-handed component of the fermion fields contribute in the weak interaction. The electric charge of any particles involved in weak interaction can be realized by assigning additive quantum number 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 of weak isospin generator, and the weak hypercharge of the particle. This assignment was result from the experimental fact that both weak isospin and weak hypercharge quantum number are violated in weak interaction but leave the electric charge intact. The shortcoming of the four fermion interaction described above is that they are nonrenormalizable. This can easily be seen from the existence of inverse square dimension of the Fermi coupling constant GF . At the same era, it was shown that the gauge theory can be 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 it takes the form of V − A structure in the low energy limit. The gauge theory dictates the need of the mediators in every interaction. This so called gauge boson acts as a messenger that brings the information being exchanged during the interaction. Due to the short ranged nature of the weak interaction, one expects that the gauge boson to be massive. This is where the Higgs mechanism enters the game. Utilizing the Higgs mechanism will cause the gauge symmetry to be spontaneously broken, but as an advantage, one obtains massive gauge bosons. This is exactly what we want, since the weak interaction violates both of weak isospin and weak hypercharge quantum number, one can employ the Higgs mechanism to spontaneously break the SU(2)L and U(1)Y symmetry to get the corresponding massive gauge bosons while preserving the electromagnetic gauge symmetry. Diagrammatically, this.

(25) 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 the gauge theory of the electroweak interaction or known as the Standard Model (SM). The Standard Model (SM) was developed to explain the weak and electromagnetic interaction of all particles observed in nature. All fundamental particles known so far are listed in Fig. 2.1. One notices that instead of lepton and hadron that appear in the V − A current, 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 1/2 elementary particles that respect Fermi-Dirac statistics [12] widely known as fermions. The main difference between those two species is that quarks are involved in strong interaction which is responsible to hold protons and neutrons inside the nucleus, while leptons are inert to it. According to the weak isospin quantum number (flavor), there are 6 types of quarks and leptons, 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), electron neutrino (νe ), muon (µ), muon neutrino (νµ ), tau (τ) and tau neutrino (ντ ). They are arranged into three groups with similar quantum number assignments even though their masses are not 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 + 23 , and down-type quarks (d, s, b) have electric charges − 13 . For leptonic groups, e, µ, τ have identical electric charges of −1 and the neutrinos are electrically neutral. Moreover, for each fermion listed on the Fig. 2.1, there is a corresponding antiparticle which has the same properties as the original fermion except its opposite electric.

(26) 8. Short Review of the Standard Model. charge. The antiparticle of u quark is anti u quark (u), ¯ the antiparticle of electron (e− ) is the positron (e+ ) etc. It is known that leptons exist as free particles in nature while quarks always appear 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 consists of quark and antiquark pair. Moving forward to the messenger of the SM interaction, there are gauge bosons that follow the Bose-Einstein statistics [13]. The bosons are characterized by their integral spin quantum number. The gauge bosons are spin 1 particles that mediate every interaction described by the gauge interaction. The weak interaction is mediated by three massive gauge bosons W + , W − and Z. The electromagnetic interaction occurs by exchanging massless spin one photon (γ). Furthermore, these two interactions belong to the unified framework of the electroweak interaction. Finally, the strong interaction which is responsible for the nuclear forces requires massless colored gluons as their messenger. Lastly, the SM requires a special boson to generate masses for all the fundamental particles listed above. This boson is called the Higgs boson which is a spin zero scalar particle responsible for the spontaneously symmetry breaking in SM. This boson was discovered in 2012 [7, 8] at the large hadron collider (LHC). It was shown that this particle has a mass around 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 number assignments under the corresponding gauge group, SM Lagrangian density which describes how particles interact to each other, and finally explain the mass generation via spontaneously symmetry breaking (SSB) or Higgs mechanism.. 2.2.1. The SM Field Contents. Gauge theory describes the interaction of all fundamental particles in nature. This theory controls the behaviour of each particle under the corresponding gauge transformation which is internal transformation on the field. This transformation is also known as local phase rotation and it depends on some parameters similar to the Euler angles that parameterized three dimensional rotation in Euclidean space. The particle interaction can be written in terms of the Lagrangian density analogous to those of classical mechanics formalism. The main difference is that the Lagrangian density is the function of the fields that represent the elementary particles involving in the interaction. Gauge symmetry is reached when the.

(27) 2.2 The SM Gauge Interactions and Symmetry Breaking. 9. Lagrangian density is invariant under the gauge transformation. Taking into account the strong 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 the matter 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. The electron 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 =. 1 (1 ∓ γ5 ) ψ. 2. (2.9). The left-handed multiplets of the Standard Model correspond to the SU(2)L doublet of weak interaction, while the right-handed particles belongs to singlets. Each multiplet has the unique quantum number assignment corresponds to its respective representation of the group. In case of SU(2)L group, the left-handed quantum numbers are +1/2 for the electron neutrino νe 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 leptonic family is −1/2 while the right-handed singlet carries a hypercharge −1. These assignments follow from the condition that the electric charge of each particle satisfies the relation in Eq. (2.5). The quark sector can also be written in a similar way. The difference between leptonic families and quark families is that the latter one always involve massive particles while the former one is not. This is due to the fact that the neutrinos are massless in the Standard Model. Another feature that does not possessed by the leptonic families is that the quark families transform non-trivially under SU(3)C gauge group. As a consequence, the quark particles also carry SU(3)C quantum number. Finally, the Higgs boson is introduced as a doublet with respect to SU(2)L and has the hypercharge +1/2. In Table 2.1, we collect the fermionic (quarks and leptons) and Higgs fields as well as their quantum number assignments under the full Standard Model gauge group..

(28) 10. Short Review of the Standard Model Fields νL LLi = eL i eRi uL QLi = dL i uRi dRi φ+ H= φ0. SU(3)C. SU(2)L. U(1)Y. 1. 2. − 12. 1. 1. −1. 3. 2. + 16. 3 3. 1 1. + 23 − 13. 1. 2. + 12. Table 2.1 Fermion and Higgs fields and the corresponding SM quantum numbers. The index i 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 Lagrangian density of the SM. First, let us write down the kinetic terms for the gauge fields. a where Gaµν , Fµν. 1 a µν 1 1 Fa − Bµν Bµν LGauge = − Gaµν Gaµν − Fµν 4 4 4 and Bµν are the gauge field strengths defined as Gaµν ≡ ∂µ Gaν − ∂ν Gaµ + gs f abc Gbµ Gcν , a Fµν ≡ ∂µ Wνa − ∂ν Wµa + gε abcWµbWνc ,. Bµν ≡ ∂µ Bν − ∂ν Bµ ,. (2.10). (2.11) (2.12) (2.13). where f abc and ε abc denote the structure constants of SU(3)C and SU(2)L gauge groups. The constants gs and g stand for the coupling constants of gauge interaction associated with the groups SU(3)C and SU(2)L respectively. The coupling constant of U(1)Y g′ does not appear yet as there is no self interacting gauge fields in Abelian group. However, as we will see later on, it will appear in the interaction between the gauge field and the fermions. The Lagrangian density in Eq. (2.10) is invariant under infinitesimal gauge transformations of SU(3)C , SU(2)L as well as U(1)Y given by.

(29) 2.2 The SM Gauge Interactions and Symmetry Breaking. a 1 1 Dµ εs (x) = ∂µ εsa (x) − gs f abc Gbµ (x)εsc (x) , gs gs 1 1 a ∂µ ε a (x) − gε abcWµb (x)ε c (x) , δWµa (x) = Dµ ε(x) = g g 1 δ Bµ (x) = ′ ∂µ ε(x) , g δ Gaµ (x) =. 11. (2.14) (2.15) (2.16). where εsa (x), ε a (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 of the 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 = iL¯ Li D / LL LLi + il¯Ri D / lR lRi + iQ¯ Li D / QL QLi + iq¯Ri D/ qR qRi ,. (2.17). where D / ≡ Dµ γµ and Dµ is the covariant derivative. The index i runs over the family of the quarks and leptons. The four covariant derivatives arise due to the fact that the left-handed and right handed fermions belong to different representations of the gauge groups as can bee seen from Table 2.1. Their expressions are given by. µ. σi YLL ∂µ + ig Wµi + ig′ Bµ LL , 2 2 YlR = ∂µ + ig′ Bµ lR , 2 λa a σi i ′ YQL = ∂µ + igs Gµ + ig Wµ + ig B µ QL , 2 2 2 YqR λa = ∂µ + igs Gaµ + ig′ Bµ qR , 2 2. DLL LL = µ. DlR lR µ. DQL QL µ. DqR qR. (2.18) (2.19) (2.20) (2.21). with λ a (a = 1, · · · , 8) are the eight Gell-Mann matrices correspond to SU(3)C generators and σ i are the Pauli matrices of SU(2)L generators. The LL , lR , QL , and qR denote the left-handed leptonic doublets, right-handed leptonic singlets, left-handed quark doublets and right-handed quark singlets respectively. This minimally coupled Lagrangian is invariant under the infinitesimal local transformation given by.

(30) 12. Short Review of the Standard Model. σi i i ε (x)LL (x) − ε(x)LL (x) , 2 2 i δ lR (x) = − ε(x)lR (x) , 2 λa a σi i i δ QL (x) = −i εs (x)QL (x) − i ε (x)QL (x) − ε(x)QL (x) , 2 2 2 λa a i δ qR (x) = −i εs (x)qR (x) − ε(x)qR (x) , 2 2 δ LL (x) = −i. (2.22) (2.23) (2.24) (2.25). The scalar Higgs Lagrangian density which gives the spontaneous symmetry breaking LH is given by † µ LHiggs = DHµ H DH H −V (H, H † ) (2.26) with V (H, H † ) denotes the Higgs potential for the Higgs doublet H. We will write down the explicit form of the minimal Higgs potential V (H, H † ) in the next subsection. The covariant derivative of the Higgs with minimal coupling is given by µ DH H. σi i ′ YH = ∂µ + ig Wµ − ig Bµ H , 2 2. (2.27). which is invariant under the infinitesimal transformation of the form δ H(x) = −i. σi i i ε (x)H(x) + ε(x)H(x) . 2 2. (2.28). Finally, we can write down the Yukawa Lagrangian LYukawa density which describes the interaction between matter fields and the Higgs field as ij LYukawa = −Q¯ LiYd dR j H − Q¯ LiYui j uR j H˜ − L¯ LiYei j eRi H + h.c.. (2.29). where H˜ = iσ2 H ∗ . The i, j denote the family indices. Here we expand the qR explicitly since the Yukawa couplings between uR and dR are different. The full SM Lagrangian density therefore can be expressed as LSM = LGauge + LMatter + LHiggs + LYukawa .. (2.30). This Lagrangian density describes all the interaction known in nature except the gravitational one. Notice that all the particles that we have now are still massless..

(31) 13. 2.2 The SM Gauge Interactions and Symmetry Breaking. 2.2.3. The Higgs Mechanism. The minimal Higgs potential that (minimal in the sense that only one Higgs doublet is involved) invariant under the SM gauge transformation is given by V (H, H † ) = −m2 H † H +. λSM † 2 (H H) , 4. (2.31). with λSM > 0 to ensure vacuum stability and such that the extremum of the potential located at ∂V λSM † † 2 = ⟨H ⟩ −m + (H H) = 0 , (2.32) ∂ ⟨H⟩ 2 λSM † ∂V 2 −m + (2.33) = (H H) ⟨H⟩ = 0 . ∂ ⟨H † ⟩ 2 This leads to two solutions of the extremum point as ⟨H⟩ = ⟨H † ⟩ = 0 , 2m2 † ⟨H H⟩ = . λSM. (2.34) (2.35). The first solution corresponds to the local maximum while the second one is the true minimum 2 1/2 of the potential. The vacua are degenerate and form a circle of radius λ2m . The stability SM of the theory requires us to take the second solution. We choose the minima of the form ⟨H⟩ =. 0. !. √v 2. 2m , v= p λSM. (2.36). and furthermore expand the theory around the ground state (minima of the potential) by shifting the Higgs field as H → H + ⟨H⟩ = H +. 0 √v 2. ! .. (2.37). By doing this we fix the vacuum and therefore single out a particular direction in the ground state. Thus, there is a special direction in the vacuum of our theory and this particular point clearly breaks the corresponding symmetry that we have in the beginning. This mechanism refers to the spontaneously breakdown of the symmetry (SSB) or the Higgs mechanism. This occurs when the symmetry of the theory is broken by the ground state..

(32) 14. Short Review of the Standard Model. The mass spectrum of the Standard Model can be obtained by substituting the Higgs field as in Eq. (2.37) and retain only the quadratic terms of the fields (taking into account the mixing terms as well). This will give three massive gauge bosons as Wµ±. (Wµ1 ∓ iWµ2 ) √ = 2. with mass. Zµ = − sin θW Bµ + cos θW Wµ3. v mW ± = g , 2 with mass. mZ =. (2.38) p v g2 + g′2 , 2. (2.39). where the weak mixing angle θW is related to the weak couplings as sin θW = p One can immediately see that cos θW = been redefined as. mW ± mZ .. g′ g2 + g′2. .. (2.40). Note that after the SSB, our gauge fields have. i ∂µ φ ± → Wµ± , (2.41) mW 1 Zµ − ∂µ χ → Zµ , (2.42) mZ where χ is defined as χ ≡ √i2 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 said to have eaten the Goldstone bosons during the SSB and hence become massive. In addition to the massive gauge fields, there is one massless degree of freedom written as Wµ± ∓. Aµ = cos θW Bµ + sin θW Wµ3 ,. (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 the √ generic 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 neutrinos are massless in this SM framework. As an additional remark, the general form of the charged fermion mass matrices M f are given by 3 × 3 complex matrices which can be brought into.

(33) 2.2 The SM Gauge Interactions and Symmetry Breaking. 15. diagonal forms by using two unitary matrices via the following transformations VLu Mu (VRu )† = diag(mu , mc , mt ) ,. (2.44). VLd Md (VRd )† = diag(md , ms , mb ) ,. (2.45). VLe Me (VRe )†. (2.46). = diag(me , mµ , mτ ) ,. where VLu , VRu , VLd , VRd , VLe and VRe are different unitary matrices operate in the associated flavor vector space, i.e. uL = VLu u′L , uR = VRu u′R , dL = VLd dL′ , dR = VRd dR′ , eL = VLe e′L , eR = VRe e′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. The expression of the first two currents are described by Lem = − e Qu (u¯L γ µ uL + u¯R γ µ uR ) + Qd d¯L γ µ dL + d¯R γ µ dR µ µ + Qe (e¯L γ eL + e¯R γ eR ) Aµ ,. (2.48). and h1 1 1 e 1 LNC = − u¯L γ µ uL − d¯L γ µ dL + ν¯ L γ µ νL − e¯L γ µ eL sin θW cos θW 2 2 2 2 2 µ µ µ − sin θW Qu (u¯L γ uL + u¯R γ uR ) + Qd d¯L γ dL + d¯R γ µ dR i + Qe (e¯L γ µ eL + e¯R γ µ eR ) Zµ .. (2.49). The electric charges of up-type quarks, down-type quarks and charged leptons are denoted as Qu , Qd and Qe respectively. These interaction currents are unchanged under the transformations given in Eq. (2.47). Finally, under the transformation in Eq. (2.47), the charged current is modified into the following form e LCC = − √ u¯′ L γ µ Wµ+ VCKM dL′ + ν¯ L γ µ Wµ+ (VLe )† e′L + h.c. 2 sin θW. (2.50). where VCKM = VLu (VLd )† is the usual Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix in the quark sector [15, 16]. Due to the massless neutrinos, there is no corresponding mixing matrix in the lepton sector. We note that the neutral current interaction of the neutrinos with.

(34) 16. Short Review of the Standard Model. the Z boson is invariance under this field redefinition. Thus we can drop all the primes in the physical fermion fields in the SM Lagrangian..

(35) Chapter 3 The G2HDM Model The Gauged Two Higgs Doublet Model (G2HDM) proposed by Huang et al. [9] will be discussed in this chapter. The model was constructed to explain the observed dark matter relic abundance 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 the usual ad hoc Z2 discrete group. First, we will discuss the important motivation behind the creation of the G2HDM model. We further discuss the particle contents in this model and their role. The potential consistent with the underlying symmetry of the gauge group as well as its SSB breaking pattern will be briefly reviewed. As a bonus, the existence of the accidental Z2 in the model will be elaborated. Lastly, we will summarize the corresponding theoretical constraints coming from the vacuum stability, perturbative unitarity as well as Higgs phenomenology on the scalar sector.. 3.1. Motivation. The discovery of the 125 GeV Higgs boson at the LHC was certainly a milestone in the history of particle physics. It completes, at least, the Standard Model particle contents discussed in the chapter 2. However, the SM alone can not answer the remaining real open problems in physics such as the origin of the neutrino mass, the observed dark energy that dominates our universe, and the existence of the unknown matter, which constitutes almost one 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 models by 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, the general two Higgs doublet model (2HDM) is a simple extension [17] by just adding one.

(36) 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 doublet to be the DM candidate. The stability of the DM candidate in this model is achieved by assigning a discrete Z2 symmetry on the scalar potential of the model. However, according to the study of reference [22, 23], the discrete and continuous global symmetry is strongly disfavored by the gravitational effects. In order to avoid these unpleasant features as well as providing the DM candidate, a recent study in [9] has constructed a model called Gauged Two Higgs Doublet model (G2HDM) in which the two Higgs doublets H1 and H2 are put together into a doublet H = (H1 , H2 )T of a new non-abelian SU(2)H gauge group. The neutral component of H2 is stable thanks to the SU(2)H gauge symmetry and hence it is a viable DM candidate. Other particles are incorporated in the G2HDM model including a SU(2)H doublet, a SU(2)H triplet, and heavy SU(2)L singlet Dirac fermions. Their roles will be explained in the following session. In addition, The SM right-handed fermions are paired with new heavy right-handed fermions to form SU(2)H doublets. After the SSB, the VEV of SU(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 belong to the Z2 even particles while some of the new scalars, W ′ and new heavy fermions are odd. The lightest odd particle if neutral can be a DM candidate whose stability is protected 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 generate masses 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, systematic studies on theoretical and phenomenological constraints for both the Higgs and gauge sectors in G2HDM have been presented in [26] and [27] respectively..

(37) 19. 3.2 The Particle Contents. 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 also another 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 SM gauge group. Furthermore, all the scalars in this model carry a particular U(1)X quantum number. Moving to the fermion sector, the SM left-handed SU(2)L doublets transform trivially under SU(2)H , while the SM right-handed SU(2)L singlets along with the new right-handed singlets are paired together to form SU(2)H doublets. Due to the anomaly cancellation, the existence of new heavy left-handed fermions is a must. These new heavy fermions belong to the singlet representation of both SU(2)L and SU(2)H gauge groups. The full matter contents of the G2HDM model together with their respective quantum number assignments are collected in Table 3.1. Matter Fields T. QL = (uL dL ) T UR = uR uH R T DR = dRH dR uH L dLH LL = (νL eL )T T NR = νR νRH T ER = eH e R R νLH eH L. SU(3)C. SU(2)L. SU(2)H. U(1)Y. U(1)X. 3 3 3 3 3 1 1 1 1 1. 2 1 1 1 1 2 1 1 1 1. 1 2 2 1 1 1 2 2 1 1. 1/6 2/3 −1/3 2/3 −1/3 −1/2 0 −1 0 −1. 0 1 −1 0 0 0 1 −1 0 0. T 1 2 2 1/2 1 H = (H1 H2 ) √ ∆3 /2 √ ∆ p/ 2 ∆H = 1 1 3 0 0 ∆m / 2 −∆3 /2 ΦH = (Φ1 Φ2 )T 1 1 2 0 1 Table 3.1 Matter contents and their quantum number assignments in G2HDM..

(38) 20. 3.3. The G2HDM Model. Higgs Potential. The most general Higgs potential which is invariant under the full G2HDM gauge group can be 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 corresponding scalar fields while the last term denotes the mixing interaction among them. The self interaction of the SU(2)L and SU(2)H scalar doublet H is given by 2 1 V (H) = µH2 H αi Hαi + λH H αi Hαi + λH′ εαβ ε γδ H αi Hγi H β j Hδ j , 2 2 † † † † † † † † 2 ′ = µH H1 H1 + H2 H2 + λH H1 H1 + H2 H2 + λH −H1 H1 H2 H2 + H1 H2 H2 H1 , (3.2) where (α, β , γ, δ ) and (i, j) stands for the SU(2)H and SU(2)L indices respectively. These ∗ . Note that indices run from 1 to 2. The upper and lower indices are related as H αi = Hαi V (H) in Eq. (3.2) consists of all the possible renormalizable terms involving H1 and H2 and it is automatically invariant under H1 → H1 and H2 → −H2 . Unlike IHDM, this Z2 symmetry is not imposed by hand. The gauge invariance of SU(2)L × SU(2)H implies this symmetry automatically! The self interaction of ΦH can be written as 2 V (ΦH ) = µΦ2 Φ†H ΦH + λΦ Φ†H ΦH , = µΦ2 (Φ∗1 Φ1 + Φ∗2 Φ2 ) + λΦ (Φ∗1 Φ1 + Φ∗2 Φ2 )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 2 V (∆H ) = − µ∆2 Tr ∆2H + λ∆ Tr ∆2H , 2 1 2 2 1 2 = − µ∆ ∆ + ∆ p ∆m + λ ∆ ∆ + ∆ p ∆m , 2 3 2 3. (3.4). where the triplet fields ∆H is written in the following matrix form ∆H =. √ ! ∆3 /2 ∆ p / 2 √ = ∆†H with ∆m = (∆ p )∗ and (∆3 )∗ = ∆3 . ∆m / 2 −∆3 /2. (3.5).

(39) 21. 3.3 Higgs Potential. Note that V (∆H ) in Eq. (3.4) is invariant under ∆3 → ∆3 and ∆ p,m → −∆ p,m . Finally, the mixing terms describing all possible interaction between the scalar fields can be expressed as † † Vmix (H, ∆H , ΦH ) = + MH∆ H ∆H H − MΦ∆ ΦH ∆H ΦH ′ + λHΦ H † H Φ†H ΦH + λHΦ H † ΦH Φ†H H † † 2 + λH∆ H H Tr ∆H + λΦ∆ ΦH ΦH Tr ∆2H .. (3.6). Furthermore, Eq. (3.6) can also be expressed in terms of their fundamental fields H, ∆H and ΦH as the following 1 † 1 † 1 † 1 † Vmix (H, ∆H , ΦH ) = + MH∆ √ H1 H2 ∆ p + H1 H1 ∆3 + √ H2 H1 ∆m − H2 H2 ∆3 2 2 2 2 1 ∗ 1 ∗ 1 ∗ 1 ∗ − MΦ∆ √ Φ1 Φ2 ∆ p + Φ1 Φ1 ∆3 + √ Φ2 Φ1 ∆m − Φ2 Φ2 ∆3 2 2 2 2 + λHΦ H1† H1 + H2† H2 (Φ∗1 Φ1 + Φ∗2 Φ2 ) ′ + λHΦ H1† H1 Φ∗1 Φ1 + H2† H2 Φ∗2 Φ2 + H1† H2 Φ∗2 Φ1 + H2† H1 Φ∗1 Φ2 1 † † 2 ∆ + ∆ p ∆m + λH∆ H1 H1 + H2 H2 2 3 1 2 ∗ ∗ + λΦ∆ (Φ1 Φ1 + Φ2 Φ2 ) ∆ + ∆ p ∆m . (3.7) 2 3 . 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 is invariant under this discrete Z2 symmetry automatically. As an additional remark, one notes that the coefficient of µ∆2 in V (∆H ) has a negative sign as opposed to that of µH2 and µΦ2 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 1 1 1 µH2 − MH∆ · v∆ + λH∆ · v2∆ + λHΦ · v2Φ , 2 2 2 1 1 1 ′ µH2 + MH∆ · v∆ + λH∆ · v2∆ + (λHΦ + λHΦ ) · v2Φ , 2 2 2. (3.8) (3.9). respectively. Since these coefficients consist of several terms, the total value of these coefficients can have either positive or negative values, even if one choose the positive value of µH2 . Thus, if one sets Eqs. (3.8) and (3.9) to be negative and positive respectively, one can then obtain ⟨H1 ⟩ ̸= 0 and ⟨H2 ⟩ = 0 which will trigger the spontaneous breakdown of SU(2)L symmetry..

(40) 22. The G2HDM Model. By expanding the potential, one can also obtain the coefficients of the quadratic terms for Φ1 and Φ2 which are given by 1 1 1 ′ ) · v2 , µΦ2 + MΦ∆ · v∆ + λΦ∆ · v2∆ + (λHΦ + λHΦ 2 2 2 1 1 1 µΦ2 − MΦ∆ · v∆ + λΦ∆ · v2∆ + λHΦ · v2 , 2 2 2. (3.10) (3.11). respectively. Analogous to the previous case of H1 and H2 , with the positive value of µΦ2 , we can obtain ⟨Φ1 ⟩ = 0 and ⟨Φ2 ⟩ = ̸ 0 by choosing some particular combinations on the parameter space. In (3.4), if µ∆2 > 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 the other gauge symmetries in this model. Note that the scalar potential in G2HDM is CP-conserving due to the fact that all terms in V (H), V (ΦH ), V (∆H ) and Vmix (H, ∆H , ΦH ) are Hermitian, implying all the coefficients are necessarily real. Therefore, there is no CP violation in the scalar sector of this model.. 3.4 3.4.1. The SSB of the Potential and the Mass Spectrum Spontaneous Symmetry Breaking. Similar to the case of the SM in chapter 2, the spontaneous symmetry breaking can be realized by expressing the scalar fields in terms of their fluctuation around the VEVs as H1 =. G+ v+h G0 √ + i√ 2 2. ! , H2 =. ! H+ , ΦH = H20. GHp vΦ√+φ2 2. ! G0. + i √H2. , ∆H =. −v∆ +δ3 2 √1 ∆m 2. √1 ∆ p 2 v∆ −δ3 2. ! . (3.12). where v, vΦ and v∆ are VEVs which can be expressed as a function of the scalar potential parameters after minimizing the scalar potential. The set ΨG ≡ {G0 , G+ , G0H , GHp } are Goldstone bosons to be absorbed later by the gauge fields. We note that only the Z2 even fields are getting VEVs. Thus the accidental Z2 symmetry is not spontaneously broken which may lead to cosmological domain wall problem..

(41) 23. 3.4 The SSB of the Potential and the Mass Spectrum. By putting the value of the VEVs, v, vΦ , v∆ into the potential VT in Eq. (3.1), the total potential becomes VT (v, v∆ , vΦ ) =. 1 λH v4 + λΦ v4Φ + λ∆ v4∆ + 2 µH2 v2 + µΦ2 v2Φ − µ∆2 v2∆ 4 − MH∆ v2 + MΦ∆ v2Φ v∆ + λHΦ v2 v2Φ + λH∆ v2 v2∆ + λΦ∆ v2Φ v2∆ . (3.13). Next, after minimizing the potential in Eq.(3.13), we will have the following equations: 2λH v2 + 2µH2 − MH∆ v∆ + λHΦ v2Φ + λH∆ v2∆. . = 0, 2λΦ v2Φ + 2µΦ2 − MΦ∆ v∆ + λHΦ v2 + λΦ∆ v2∆ = 0 , 4λ∆ v3∆ − 4µ∆2 v∆ − MH∆ v2 − MΦ∆ v2Φ + 2v∆ λH∆ v2 + λΦ∆ v2Φ = 0 .. (3.14) (3.15) (3.16). By solving this set of coupled equations, one can get solutions for v, vΦ and v∆ in terms of other parameters in the potential. In addition, one can see the effects of triplet’s VEV v∆ 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, we obtain the following three diagonal blocks in the mass matrix. The first 3 × 3 block in the basis of S = {h, φ2 , δ3 } is given by . 2λH v2  MH2 =  λHΦ vvΦ v 2 (MH∆ − 2λH∆ v∆ ). λHΦ vvΦ 2λΦ v2Φ vΦ 2 (MΦ∆ − 2λΦ∆ v∆ ). . 1 4v∆. v 2 (MH∆ − 2λH∆ v∆ )  vΦ  2 (MΦ∆ − 2λΦ∆ v∆ ) 3 2 2 8λ∆ v∆ + MH∆ v + MΦ∆ vΦ. .. (3.17). This symmetric matrix can be diagonalized by an orthogonal matrix OH , (OH )T · MH2 · OH = Diag(m2h1 , m2h2 , m2h3 ) .. (3.18). The lightest eigenvalue mh1 is the mass of h1 which is identified as the 125 GeV Higgs boson observed at the LHC, while mh2 and mh3 are the masses of heavier Higgses h2 and h3 respectively. The physical Higgs hi (i = 1, 2, 3) is a mixture of the three components of S: hi = OHji S j . Thus the SM-like Higgs boson in this model is a linear combination of the.

(42) 24. The G2HDM Model. neutral components of the two SU(2) doublets H1 and ΦH and the real component of the SU(2)H triplet ∆H . The second block of 3 × 3 matrix in the basis of D = {GHp , H20∗ , ∆ p } is given by 1 ′ ′ v2 MΦ∆ v∆ + 12 λHΦ 2 λHΦ vvΦ  2 1 ′ ′ v2 MD =  MH∆ v∆ + 12 λHΦ Φ 2 λHΦ vvΦ 1 1 − 2 MΦ∆ vΦ 2 MH∆ v. . 1 4v∆.  − 12 MΦ∆ vΦ  1 . 2 MH∆ v MH∆ v2 + MΦ∆ v2Φ. (3.19). This matrix can also be diagonalized by an orthogonal matrix OD (OD )T · MD2 · OD = Diag(m2G˜p , m2D , m2∆˜ ) .. (3.20). One eigenvalue of Eq. (3.19) is zero (i.e. mG˜p = 0) and identified as the unphysical Goldstone boson G˜ p . The mD and m∆˜ (mD < m∆˜ ) are masses of two physical fields D and ∆˜ respectively. Their analytical expression is given by the following 2 MD, e ∆. √ −B ∓ B2 − 4AC = , 2A. (3.21). where the A, B,C parameters read A = 8v∆ , ′ B = −2 MH∆ v2 + 4v2∆ + MΦ∆ 4v2∆ + v2Φ + 2λHΦ v∆ v2 + v2Φ , ′ ′ C = v2 + v2Φ + 4v2∆ MH∆ λHΦ v2 + 2MΦ∆ v∆ + λHΦ MΦ∆ v2Φ .. (3.22). The physical field D can be a DM candidate in G2HDM. Furthermore, in a particular parameter 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 in calculating the thermal relic of the D. There are other neutral fields νLH , νRH 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 1 1 ′ 2 m2H ± = MH∆ v∆ − λH′ v2 + λHΦ vΦ , 2 2 m2G± = m2G0 = m2G0 = 0 ,. (3.23) (3.24). H. where mH ± is mass of the physical charged Higgs H ± , and mG± , mG0 , mG0 are masses of H the four Goldstone boson fields G± , G0 and G0H , respectively. Note that we have used the.

(43) 25. 3.4 The SSB of the Potential and the Mass Spectrum. minimization conditions Eqs. (3.14), (3.15) and (3.16) to simplify various matrix elements of e p,m will be absorbed the above mass matrices. The six Goldstone particles G± , G0 , G0H and G by the longitudinal components of the massive gauge bosons W ± , Z, Z ′ and W ′(p,m) after SSB.. 3.4.3. Gauge Boson Mass Spectrum. The gauge bosons mass can be extracted after the spontaneously symmetry breaking of the Higgs fields as discussed in the previous session. This can be read directly from the kinetic terms of the scalars ∆H , Φ and H as † † † ′ ′µ L ⊃ Tr Dµ ∆H D ∆H + D′µ Φ D′µ Φ + D′µ H D′µ H , (3.25) where the covariant derivatives of each fields are given by D′µ ∆H. h i ′ = ∂µ ∆H − igH Wµ , ∆H ,. (3.26). gH ′p p ′m m ′3 3 ∂µ − i √ Wµ T +Wµ T − igH Wµ T − igX Xµ · Φ , 2. (3.27). gH D′µ H = Dµ · 1 − i √ Wµ′p T p +Wµ′m T m − igH Wµ′3 T 3 − igX Xµ ·H , 2. (3.28). D′µ Φ = and. 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 Wµ′. 3. 1 ′p p ′m m = ∑ W T = √ Wµ T +Wµ T +Wµ′3 T 3 , 2 a=1 ′a a. (3.29). in which the SU(2)H generators is equal to the half of Pauli matrices T a = τ a /2 acting on √ ′ (p,m) the SU(2)H space, Wµ = (Wµ′1 ∓ iWµ′ 2 )/ 2, and 1 T p = τ 1 + iτ 2 = 2. ! 0 1 1 , T m = τ 1 − iτ 2 = 2 0 0. are the usual SU(2) fundamental representation ladder operators.. 0 0 1 0. ! ,. (3.30).

(44) 26. The G2HDM Model The SM charged gauge boson W ± obtained its mass entirely from v, so it is given by 1 MW ± = gv , 2. (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 from the doublets are similar with that of the standard model. Since ∆H transforms as a triplet under SU(2)H , i.e., in the adjoint representation, the contribution to the W ′a masses arise from the term L. ⊃ g2H Tr. i † h ′ ′µ W , ∆H Wµ , ∆H .. (3.32). Therefore, the W ′(p,m) receives a mass from ⟨∆3 ⟩, ⟨Φ2 ⟩ and ⟨H1 ⟩ 1 2 2 2 gH v + v2Φ + 4v2∆ , mW ′(p,m) = 4. (3.33). while gauge bosons X and W ′3 , together with the SM W 3 and U(1)Y gauge boson B, acquire their masses from ⟨Φ2 ⟩ and ⟨H1 ⟩ only but not from ⟨∆H ⟩: 2 2 1 2 ′3 3 ′ 2 ′3 v 2gX Xµ + gH Wµ − gWµ + g Bµ + vΦ −2gX Xµ + gH Wµ , 8. (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 = +. 2 1 ∂µ a + 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 and U(1)Y respectively, and a is the axion field, we have the following matrix in the basis of . V ′ = B,W 3 ,W ′3 , X :     2 MG =   . g′2 v2 4. + MY2 ′. − g g4v. 2. g′ gH v2 4. g′ gX v2 2. + MX MY. ′. − g g4v. 2. g2 v2 4 ggH v2 − 4 2 − ggX2 v. g′ gH v2 4 ggH v2 − 4 g2H (v2 +v2Φ ) 4 gH gX (v2 −v2Φ ) 2. g′ gX v2 2. + MX MY. .  2  − ggX2 v  . gH gX (v2 −v2Φ )  2  2 2 2 2 gX v + vΦ + MX. (3.36).

(45) 27. 3.4 The SSB of the Potential and the Mass Spectrum. As a result, this mass matrix has only one zero mode corresponding to the photon, and three massive modes Z, Z ′ , Z ′′ . This mass matrix can be diagonalized by general 4 × 4 orthogonal matrix. After making a rotation in the 1 − 2 plane by the usual Weinberg angle θw , the mass matrix MG2 will further transform into a block diagonal matrix with the vanishing first column and first row. The nonzero 3-by-3 block matrix can be further diagonalized by an orthogonal matrix O G , .    Z ZSM  ′ G  ′  W3  = O ·  Z  , X Z ′′. (3.37). where ZSM is the SM Z boson without the presence of the W3′ 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 the most 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 singlet H and including additional SU(2)L singlets uH R and dR 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., T H URT = (uR uH R )2/3 and DR = (dR dR )−1/3 , where the subscript represents hypercharge, we have 1 ≈ LYuk ⊃ yd Q¯ L (DR · H) + yu Q¯ L UR · H + H.c., ˜ = yd Q¯ L dRH H2 − dR H1 − yu Q¯ L uR H˜ 1 + uH R H2 + H.c.,. (3.38). ≈. ∗ . After the EW symmetry breaking ⟨H ⟩ ̸= 0, u where H ≡ (H˜ 2 − H˜ 1 )T with H˜ 1,2 = iτ2 H1,2 1 H H and d obtain their masses but uR and dR 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 H the left-handed SU(2)L,H singlets uH L and dL and the SU(2)H doublets UR and DR ,. H ˜ LYuk ⊃ − y′d d L (DR · ΦH ) + y′u uH L UR · ΦH + H.c., H ∗ H ∗ = − y′d d L dRH Φ2 − dR Φ1 − y′u uH u Φ + u Φ R L R 2 + H.c., 1 1A · B. (3.39). is defined as εi j Ai B j where A and B are two 2-dimensional spinor representations of SU(2)H ..

(46) 28. The G2HDM Model. ˜ H = (Φ∗ − where Φ has Y = 0, Y (uH ) = Y (UR ) = 2/3 and Y (dLH ) = Y (DR ) = −1/3 with Φ L√ 2√ √ H H H H ′ ′ ∗ T Φ1 ) . With ⟨Φ2 ⟩ = vΦ / 2, uR (uL ) and dR (dL ) obtain masses yu vΦ / 2 and yd vΦ / 2, respectively. Note that there is no contribution from v∆ for the masses for both the SM and new fermions. The lepton sector is similar to the quark sector as ≈ ′ H ˜ LYuk ⊃ ye L¯ L (ER · H) + yν L¯ L NR · H − y′e eH L (ER · ΦH ) + yν ν L NR · ΦH + H.c., H ˜ ¯ ˜ = ye L¯ L eH R H2 − eR H1 − yν LL νR H1 + νR H2 ′ H H ∗ H ∗ (3.40) − y′e eH L eR Φ2 − eR Φ1 − yν ν L νR Φ1 + νR Φ2 + H.c., T H H where ERT = (eH R eR )−1 , NR = (νR νR )0 in which νR and νR are the right-handed neutrinos H H and their SU(2)H partner respectively, while eH L and νL are SU(2)L,H singlets with Y (eL ) = −1 and Y (νLH ) = 0 respectively. Notice that neutrinos are purely Dirac in this setup, i.e., νR √ paired up with νL having Dirac mass MDν = yν v/ 2, while νRH paired up with νLH having √ H Dirac mass MDν = y′ν vΦ / 2. As a result, the lepton number is conserved, implying vanishing neutrino-less double beta decay. We note that the accidental Z2 symmetry in the scalar sector can be extended to the fermion 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 model is protected by the accidental discrete Z2 symmetry which is automatically implied by the SU(2)L × U(1)Y × SU(2)H × U(1)X gauge symmetry. However, as the gauge symmetry experiences 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, this is not the case. Thanks to its special vacuum alignment where H2 fields do not acquire a VEV, the accidental Z2 symmetry remains intact after SSB. Thus besides the two well-known accidental global symmetries of baryon number and lepton number in the SM, there is also a accidental discrete Z2 symmetry in G2HDM. All the field content arrange themselves into SM ) belong to Z even, while a particular element of this Z2 group. All the SM fermions ( fL,R 2 H the new heavy fermions ( fL,R ) classified as Z2 odd. The SM Higgs boson (h1 ) as well as its heavy partners (h2 , h3 ) are members of Z2 even due to their mixings. On the other hand, the ˜ is a part of Z2 odd. The SM scalars that mix together in the second block of Eq.(3.19) (D, ∆) gauge boson (Z) along with its heavy partners (Z ′ , Z ′′ ) align themselves into Z2 even. Lastly,.

(47) 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, they can be separated into opposite sides of Z2 members. The summary of the Z2 assignments for all the particles in G2HDM is collected in the table 3.2 below. SM Z2 Even h1 , h2 , h3 W ± , Z, Z ′ , Z ′′ fL,R H ˜ H± Z2 Odd D, ∆, W ′(p,m) fL,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 the scalar sector parameters space of G2HDM model discussed in [26]. This study focused on constraining the parameter space of the potential under several physical arguments and phenomenological results. The first physical consideration is the boundedness of the scalar potential. This relies on the fact that the potential must be bounded from below to ensure the vacuum stability (VS) of the theory. Second, one needs to examine whether the theory satisfies the perturbative unitarity (PU) or not. This kind of check can be realized by calculating the scattering amplitudes at very high energy. If the amplitudes are well behaved and do not go to infinity as the energy gets higher, then the unitarity of the theory is maintained. In the scalar sector of G2HDM model, this can be achieved by considering all possible 2 → 2 scattering amplitudes in the scalar sector and evaluate them in the high energy limit. Thus, all scalars appear in this model need to be taken into account for these scattering amplitudes. Finally, the experimental constrains coming from the LHC Higgs 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 the theoretical constraints from vacuum stability (VS), perturbative unitarity (PU) and the 125 GeV Higgs physics (HP) data including the Higgs boson mass and signal strengths of Higgs boson decays into diphoton and τ + τ − from the LHC. The study showed that out of the eight λ −parameters, only two of them λH and λHΦ are essentially constrained by (VS+PU+HP). ′ Other couplings like λH′ , λHΦ and λΦ∆ are less constrained. This study also concluded that some of the parameters such as MH∆ , MΦ∆ and the VEVs are constrained only by HP but not by (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 246 GeV and 10 TeV respectively..

(48) 30. The G2HDM Model. λH λΦ λ∆ 10−4 10−2 1 10−4 10−2 1 10−4 10−2 1 λH. 1 10−2. −20. λHΦ 0. 20 −20. λH∆ 0. 20 −20. λΦ∆ 0. 20 −20. λ0HΦ 0. 20 −20. λ0H 0. 1. λH 0.13,2.70. 10−2. λΦ. 10−4 1 10−2. 10−4 1. λΦ < 4.19. 10−2. λ∆. 10−4 1. 10−4 1. λ∆ < 5.03. 10−2. 10−2. λHΦ. 10−4 20. 10−4 20. λHΦ -5.69,3.52. 0. 0. λH∆. −20 20. −20 20. λH∆ -3.90,9.25. 0. 0. λΦ∆. −20 20. −20 20. λΦ∆ -5.41,13.25. 0. 0. λ0HΦ. −20 20. −20 20. λ0HΦ -0.38,17.11. 0. 0. λ0H. −20 20. −20 20. λ0H -23.82,2.54. 0 −20 10−4 10−2. 20. 1 10−4 10−2. 1 10−4 10−2. 1. −20. 0. 20 −20. 0. 20 −20. 0. 20 −20. 0. 20 −20. 0. 0 −20 20. Fig. 3.1 A summary of the parameter space allowed by the theoretical and phenomenological constraints. 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]..

(49) 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 red triangular block corresponds to (VS+PU) constraints, while the lower magenta triangular block corresponds to the (VS+PU+HP) constraints. The diagonal panels indicate the allowed ′ under the combined constraints ranges of the eight couplings λH,Φ,∆ , λH′ , and λHΦ,H∆,Φ∆ , λHΦ of (VS+PU+HP)..

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