Thesis: Darvishi, SM with a Doublet and a Complex Singlet

Extension of the Standard Model with a Doublet and a Complex Singlet

Neda Darvishi

Doctoral dissertation prepared under the supervision of prof. dr hab. Maria Krawczyk at the Institute of Theoretical Physics, Faculty of Physics, University of Warsaw

April 2017

This thesis is dedicated to my father, for his endless love, support and encouragement.

ii Acknowledgments

I would like to express my deepest gratitude to my supervisor Prof. Maria Krawczyk for her endless support, sharing her experience, teaching a lot of useful computational technique and guiding me at all the stages. It would never have been possible for me to take this work to completion without her incredible support and encouragement.

I am also thankful to Dr Saereh Najjari, Dr Aqeel Ahmed, Dr Dorota Sokolowska and Dr Bogumila Swiezewska, who offered me their time for scientific discussions, and of course for non-scientific conversations making everyday work more cheerful. I especially appreciate the company of Dr M.R. Masouminia who were helping with computer codes and other technical issues and also collaborating to obtain some of the results presented in this thesis.

I am also grateful to administrative staﬀ, especially to Mrs. Anna Kaczor, who made my stay easier and helped with numerous procedures that arise during my stay in Warsaw.

Finally, I express a deep sense of gratitude to my parents, who have always stood by me like a pillar in times of need.

iii

Abstract

The Standard Model (SM) of particle physics is very successful, yet it leaves many basic questions unanswered. This thesis is focused on some of these open problems. The aim of this study is to explore and exploit the origins of the matter-antimatter asymmetry in the Universe and the origins of dark matter (DM). Baryogenesis, the creation of the baryon asymmetry in the Universe, is a long-standing problem in cosmology. Sakharov formulated his well-known conditions for baryogenesis: baryon number violation, C and CP violation, and a departure from thermal equilibrium. Among the many particle physics scenarios that have been proposed in the past decades, electroweak baryogenesis is interesting. It has become apparent that the SM of electroweak interactions is unable to account for the observed magnitude of the BAU for at least two reasons. Firstly, the electroweak phase transition is not strongly ﬁrst-order and therefore, any baryon asymmetry created during the transition would subsequently be washed out by unsuppressed baryon violating processes in the broken phase. Secondly, there is not enough CP violation from the CKM matrix to generate the baryon asymmetry.

With the motivation of investigating these problems, we extend the SM by a neutral complex scalar singlet, and a pair of heavy iso-doublet vector quarks (VQ). We consider the potential with a softly broken global U(1) symmetry, which we call the Constrained SM+CS model (cSMCS). Assuming nonzero vacuum expectation value for the complex singlet, we analyze the physical conditions for spontaneous CP violation. The mixing of SM quarks with heavy VQ pairs result in the appearance of additional CP violation. This model provides the strong enough ﬁrst order EW phase transition and leads to a proper description of baryogenesis. The scalar spectrum of the cSMCS includes three neutral Higgs particles with the lightest one considered to be the 125 GeV SM-like Higgs boson found in 2012 at the LHC. In the considered model, the SM-like Higgs boson comes mostly from the SM-like SU(2) doublet, with a small correction from the singlet.

We present a prediction of the production rates of the cSMCS model Higgs bosons at the LHC, using a conventional effective LO QCD framework and the unintegrated parton distribution functions (UPDF) of Kimber-Martin-Ryskin (KMR). We first compute the SM Higgs production cross-section and compare the results to the existing calculations from different frameworks as well as to the experimental data from the CMS and ATLAS collaborations. It is shown that our framework is capable of producing sound predictions in this case. There- fore we use it for calculation of the cSMCS predictions for the Higgs boson production at the LHC. These predictions for yet undetected Higgs bosons of the cSMCS model may provide some clues for the future discovery.

v On the other hand, the gravitational eﬀects of DM have been observed in galaxies, clusters of galaxies, the large-scale structure of the Universe and the Cosmic Microwave Background Radiation. Since none of the particles in the cSMCS model satisﬁes these conditions, we introduce, in addition a SU(2) doublet with zero vacuum expectation value (The Inert Dou-

blet). This model, that we call cIDMS has an exact Z2 symmetry and provides correct relic density of DM while fulﬁlling direct and indirect DM detection limits and simultaneously agree with the LHC results.

The last part of this thesis is devoted to the applying Veltman condition in the Two Higgs Double Model (2HDM) in order to ﬁnd the masses of the heavy scalars. The 2HDM is one of the simplest extensions of the SM, providing rich phenomenology. It contains an extended scalar sector which instead of one complex SU(2) doublet (present in the SM), contains two of the doublets, with weak hypercharges equal to 1. We analyze the soft Z2 symmetry breaking version of the 2HDM with non-zero vacuum expectation values for both Higgs doublets (Mixed vacuum). We assume that CP is conserved in the scalar sector. In the particle spectrum of this model, there are two neutral CP-even scalars h and H, h being lighter than H. These scalars are two possible candidates for the SM-like Higgs boson, forming two possible scenarios. In the model, there are also a CP-even scalar (pseudoscalar) A and the

charged Higgs bosons H±. The results are constrained by comparing the properties of the light Higgs particle with the corresponding LHC data. We have found that the consistent solution exists only for the SM-like h scenario, with properties in agreement with the recent experimental data.

vi Contents

1 Introduction 3

1.1 The Standard Model ...... 5

1.1.1 The SM Lagrangian ...... 5

1.1.2 The Cabibbo-Kobayashi-Maskawa Matrix ...... 10

1.2 Baryon Asymmetry in the Universe and Sakharov Condition ...... 11

1.3 Electroweak Baryogenesis ...... 14

1.3.1 Sakharov Conditions in the EW Theory ...... 14

1.4 Dark matter and its constraints ...... 21

1.5 Content of the thesis ...... 25

2 The Extended Standard Model with a Complex Singlet 27

2.1 Explicit and spontaneous CP violation ...... 28

2.2 The cSMCS: The SM plus a complex singlet ...... 29

2.2.1 Potential ...... 30

2.2.2 Positivity conditions ...... 31

2.2.3 Extremum conditions ...... 31

2.2.4 The CP violating vacuum ...... 32

2.3 Physical states in the Higgs sector ...... 33

2.4 Allowed regions of parameters for CP violating vacuum ...... 36

2.5 J type-invariants ...... 40

vii 2.6 Comparison with data ...... 41

2.6.1 Properties of h1 Higgs boson in the light of LHC data ...... 41 2.6.2 Oblique parameters S,T,U ...... 43

2.6.3 Benchmarks ...... 44

2.7 The electroweak phase transition ...... 45

2.8 Baryogenesis ...... 48

2.9 Conclusion and outlook ...... 51

3 A phenomenological study on the production of Higgs bosons in the cSMCS model at the LHC 53

3.1 Calculation of the Higgs production cross-section ...... 55

3.2 Numerical analysis ...... 58

3.3 Results and discussions ...... 59

3.4 Conclusions ...... 64

4 IDMS: Inert Doublet Model with a complex singlet 67

4.1 The IDMS: The IDM plus a complex singlet ...... 68

4.2 The constrained IDMS: cIDMS ...... 70

4.2.1 Parameter choice ...... 71

4.2.2 Positivity conditions ...... 72

4.2.3 Extremum conditions ...... 72

4.2.4 Vacuum stability ...... 73

4.2.5 Mass eigenstates ...... 74

4.3 Scanning and experimental constraints ...... 75

4.3.1 LEP bounds ...... 76

4.3.2 Oblique parameters S,T,U ...... 77

4.3.3 LHC constraints on Higgs parameters in the cIDMS ...... 78

viii 4.3.4 DM constraints ...... 83

4.4 Dark Matter in the cIDMS ...... 84

4.4.1 Benchmarks ...... 85

4.4.2 Light DM ...... 87

4.4.3 Medium DM ...... 88

4.4.4 Heavy DM ...... 91

4.5 Conclusions and Outlook ...... 92

5 Implication of Quadratic Divergences Cancellation in the Two Higgs Dou- blet Model 95

5.1 Mixed Model with a soft Z2 symmetry breaking ...... 96 5.2 Cancellation of the quadratic divergences ...... 98

5.3 Approximate solution of the cancellation conditions ...... 100

5.4 Solving the cancellation conditions ...... 101

5.5 Experimental constraints ...... 103

5.6 Benchmarks ...... 103

5.7 Summary and Conclusion ...... 105

6 Summary and Conclusion 109

7 Appendix 113

7.1 Appendix A: Experimental constraints on the Higgs sector ...... 113

7.2 Appendix B ...... 115

7.2.1 Coupling of scalars with gauge bosons and fermions ...... 115

7.2.2 Possible minima ...... 116

7.2.3 Oblique parameters for cSMCS ...... 117

7.2.4 Decays h γγ in the cSMCS ...... 119 → 7.2.5 Higgs trilinear couplings ...... 120

ix 7.2.6 J-invariant for κ =0...... 120 4 6 7.3 Appendix C ...... 120

7.3.1 Evaluation of the integral ...... 121

7.3.2 The ﬁeld-dependent mass mi ...... 121 7.4 Appendix D: LO Higgs Production Matrix Element ...... 122

7.5 Appendix E ...... 123

7.5.1 Decays h γγ and h Zγ in cIDMS ...... 123 → → 7.5.2 Oblique parameters for cIDMS ...... 125

7.5.3 Benchmarks ...... 126

7.5.4 S,T and , for the benchmarks of the cIDMS ...... 127 Rγγ RZγ 7.6 Appendix F: Oblique parameters for 2HDM ...... 131

x Preface

This dissertation is based on the following publications, references [1, 2, 3, 4, 5, 6, 7],

1. M. Krawczyk, N. Darvishi and D. Sokolowska, “The Inert Doublet Model and its extensions,” Acta Phys. Polon. B 47 (2016) 183 doi:10.5506/APhysPolB.47.183, arXiv:1512.06437[hep-ph].

2. N. Darvishi and M. Krawczyk, “CP violation in the Standard Model with a complex Singlet,” arXiv:1603.00598v2[hep-ph] [submitted in Journal of Physics G].

3. N. Darvishi, “Baryogenesis of the Universe in cSMCS Model plus Iso-Doublet Vector Quark,” arXiv:1608.02820, JHEP 1611 (2016) 065, DOI: 10.1007/JHEP11 (2016)065.

4. N. Darvishi, M.R. Masouminia, “A phenomenological study on the production of Higgs bosons in the cSMCS model at the LHC,” arXiv:1611.03312v1.

5. C. Bonilla, D. Sokolowska, N. Darvishi, J.L. Diaz-Cruz, M. Krawczyk, “IDMS: Inert Dark Matter Model with a complex singlet,” 2016, Journal of Physics G, Volume 43 Number 065001.

6. N. Darvishi, “Extension of Standard Model with a Complex Singlet and Iso-Doublet Vector Quarks,” presented at the DISCRETE 2016, to appear in Journal of Physics: Conference Series (IOP).

7. N. Darvishi and M. Krawczyk, “Implication of cancellation quadratic divergences in the Two Higgs Doublet Model,” [to be submitted for publication April 2017].

xi Acronyms

2HDM : Two Higgs Doublet Model

ABJ : Adler-Bell-Jackiw

BAU : Baryon Asymmetry in the Universe

BBN : Big Bang Nucleosynthesis

CKM : Cabibbo-Kobayashi-Maskawa

CMB : Cosmic Microwave Background

COBE : Cosmic Background Explorer

DGLAP : Dokshitzer-Gribov-Lipatov-Altarelli-Parisi

DM : Dark Matter

EDM : Electric Dipole Moment

EWPT : Electrowaek Precision Test

EW : Electrowaek

IDM : Inert Doublet Model

KMR : Kimber-Martin-Ryskin

LEP : Large ElctronPositron collider

LHC : Large Hadron Collider

MACHO : Massive Astrophysical Compact Halo Objects

PDF : Parton Distribution Functions

QED : Quantum Field Theory

QFT : Quantum electrodynamics

SFOPT : Strong enough First Order Phase Transition

SM : Standard Model

1 SSB : Spontaneously Symmetry Breaking

UPDF : Unintegrated Parton Distribution Functions

VEV : Vacuum Expectation Value

VQ : Vector Quarks

WIMP : Weakly Interacting Massive Particles

2 Chapter 1

Introduction

The Standard Model (SM) of particle physics is very successful, yet it leaves many basic questions unanswered. In my thesis, I focus on some of these open problems. The main aim of my study is to explore and exploit the origins of the matter-antimatter asymmetry in the Universe and the origins of dark matter (DM). Among the various scenarios that try to explain the baryon asymmetry of the Universe, the mechanism of electroweak (EW) baryogenesis is one of the most attractive [8]. One of the motivations to consider the EW scenario lies in the fact that with the advent of new high-energy colliders especially the Large Hadron Collider (LHC), physics at the EW scale becomes more accessible and testable. Furthermore, all the necessary ingredients for a successful baryogenesis [9] (i.e. baryon number violation, C and CP violation and departure from thermal equilibrium) can be found in quantum theories of particle interactions. It has become apparent that the SM of EW interactions is unable to account for the observed magnitude of the BAU for at least two reasons. Firstly, the EW phase transition is not strongly ﬁrst-order and therefore, any baryon asymmetry created during the transition would subsequently be washed out by unsuppressed B-violating processes in the broken phase [10, 11, 12, 13, 14, 15, 16, 17, 18]. Secondly, there is not enough CP violation from the CKM matrix to generate the B asymmetry [19]. Various possible solutions of these problems, usually involving extensions of the SM, have been suggested. In order to have an extra source of CP violation, that could allow to address this important issue, various extensions of the SM are considered [20, 21, 22, 23, 24, 25].

In this thesis, we consider the possibility of the additional source of CP violation that could provide in the model with a neutral complex scalar singlet χ, which accompanies the

SM-like Higgs doublet Φ, and a pair of iso-doublet vector quarks (VQ) VL + VR. This kind of extension of the SM was discussed in the literature with various motivations, see e.g.

3 [26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. We consider the potential with a softly broken global U(1) symmetry, in the Constrained SM+CS model (cSMCS). Assuming nonzero vacuum expectation value for the complex singlet, we analyze the physical conditions for spontaneous CP violation. We found that this model provides the strong enough ﬁrst-order EW phase transition, that is needed to suppress B-violating processes in the broken phase and leading to a proper description of baryogenesis.

On the other hand, the gravitational effects of DM have been observed in galaxies, clusters of galaxies, the large-scale structure of the Universe and the Cosmic Microwave Background Radiation. DM constitutes 85% of the matter density in the Universe and for 23% of its total energy density [36] and therefore, it is one of the essential ingredients of our Universe. The DM particle should be neutral, stable, weakly interacting and leading to the observed large-scale structure formation of the Universe. Since none of the particles in the cSMCS model satisfies these conditions, we introduce an additional SU(2) doublet with zero vacuum expectation value (the Inert Doublet). This model has an exact Z2 symmetry, that we call cIDMS, provides correct relic density of DM while fulfilling direct and indirect DM detection limits and simultaneously agree with the LHC results.

The other issue that is included in this thesis is related to the basic problems of the SM, the naturalness problem of the Higgs mass. From experimental data, the Higgs mass (125 GeV) is in the order of the EW scale, yet from the naturalness perspective, this mass must be much larger than the EW scale. This is because of the large radiative corrections to the Higgs mass, that implies an unnatural ﬁne-tuning between the tree-level Higgs mass term and the radiative corrections. These radiative corrections diverge, showing a quadratic sensitivity to the largest scale in the theory. Solutions to this hierarchy problem imply new physics beyond the SM. This new physics must be able to compensate the large corrections to the Higgs mass, which can be obtained due to of new symmetries and particles. The quadratic divergences were studied within the SM by Veltman [37]. He suggested that the radiative

4 corrections to the scalar mass vanish (or are kept at a manageable level). This is popularly known as the Veltman condition. In this thesis, we apply Veltman condition in Two Higgs Double Model (2HDM) in order to ﬁnd the mass of the heavy scalars. 2HDM is one of the simplest extensions of the SM, providing interesting phenomenology, with an extended scalar sector which instead of one complex SU(2) doublet present in the SM contains two doublets with weak hypercharges equal to 1. We consider the soft Z2 symmetry breaking version of the 2HDM with non-zero vacuum expectation values for both Higgs doublet (Mixed vacuum). We assume that CP is conserved in the scalar sector. In the particle spectrum of this model, there are two neutral CP-even scalars h and H, h being lighter than H. These scalars are two possible candidates for the SM-like Higgs boson, forming two possible scenarios. In the

model, there are also a CP-even scalar (pseudoscalar) A and the charged Higgs bosons H±. Solutions of two relevant Veltman conditions were found in the considered SM-like scenarios, and the results are constrained by comparing the properties of the light Higgs particle with the LHC data. We have found that the solution exists only for the SM-like h scenario.

In this chapter, we review the current notion of the SM, the baryon asymmetry of the Universe and EW baryogenesis in the chapters 1.1, 1.2 and 1.3, respectively. In section 1.4, we review the issue of DM and giving evidence for the existence of DM, describing DM detection experiments and the constraints from the measurements of the DM relic density.

1.1 The Standard Model

There are four known fundamental interactions (i.e. strong, weak, electromagnetic and gravitational) that govern our Universe. The strong, the weak and the electromagnetic forces are described by the SM of Particle Physics. On the other hand, the gravity is described by the SM of Cosmology based on the General Relativity. Since in the present thesis, we intend to work with the extensions to the SM, here we brieﬂy review the properties of the known and currently accepted SM of Particle Physics. The content of this section is based on the following references [49, 50, 51, 52, 53].

1.1.1 The SM Lagrangian

The SM of Particle Physics is the theory of fundamental interactions (i.e. strong, weak, electromagnetic) between fundamental particles. These interactions are viewed as exchanges of particles in a relativistic quantum field [54]. Quantum field theory (QFT) was first applied to the U(1) gauge invariant theory of electrodynamics (QED) in the 1940s by Feynman,

5 Schwinger, Dyson, and Tomonaga [55]. By the late 1970s, QFT was used to describe both the strong and the unified weak and electromagnetic interactions. The SM is based on gauge structure SU(3) SU(2) U(1) , where c stands for color, the subscript L indicates that c × L × Y they only act on left fermions, and Y stands for weak hypercharge. The strong sector SU(3)c describes the interaction between quarks and gluons, while the EW sector SU(2) U(1) L × Y is spontaneously broken to the electromagnetic field that acts on particles with a non-zero charge Q = I3L + Y (I3L is the third isospin component of SU(2)L). The SM is defined by the Lagrangian that contains all renormalizable terms that are gauge and Lorentz invariant.

Fermions νLi eLi eRi uLi dLi uRi dRi 1 1 1 1 2 1 Y - 2 - 2 -1 6 6 3 - 3 1 1 1 1 I3L 2 - 2 0 2 - 2 0 0

Table 1.1: The SM fermions with their hypercharge charges under the SM gauge group. 1 5 The fermions are separated into left-handed (ψL = 2 (1 γ )ψ)) and right-handed (ψR = 1 5 − 2 (1 + γ )ψ)) ﬁelds.

The SM fermions are listed in table 1.1 along with their hypercharge charges and their a corresponding iso-spins. There are also eight gluons (Gµ, corresponding to the eight gener- α ators of SU(3)c), three Wµ bosons (corresponding to the three generators of SU(2)L), one

Bµ boson (corresponding to the generator of U(1)Y ), and one SU(2) Higgs doublet. These particles and their notations are listed below:

Fermions ! νeL – Left-handed lepton doublets: lLi = eL i – Right-handed charged leptons: eRi ! uL – Left-handed quark doublets: QLi = dL i – Right-handed quarks: uRi, dRi

Gauge ﬁelds

a – SU(3)c bosons (gluons): Gµ, a = 1,..., 8 α – SU(2) bosons: Wµ , α = 1, 2, 3

– U(1) boson Bµ

6 + ! ϕ ? Higgs boson doublet: Φ = , (we use also Φe = iσ2Φ ) ϕ0

The SM Lagrangian, which describes the interactions between fundamental particles, is given by:

= + + + . (1.1) L Lgauge LScalar LYukawa LFermion Where L describes the pure gauge bosons interaction, describes the scalar sector gauge LScalar of the model with SU(2) doublet, represents the Yukawa interaction of Φ with the LYukawa fermions, and is the gauge boson-fermion interaction. LFermion The Lagrangian for the gauge interactions is given by:

1 1 1 = Ga Gaµν W α W αµν B Bµν, (1.2) Lgauge −4 µν − 4 µν − 4 µν with the ﬁeld strength tensors as

Ga = ∂ Ga ∂ Ga + g f Gb Gc , µν µ ν − ν µ s abc µ ν W α = ∂ W α ∂ W α + gε W βW γ, µν µ ν − ν µ αβγ µ ν B = ∂ B ∂ B . (1.3) µν µ µ − ν µ a Here fabc and εαβγ are structure constants for SU(3)c and SU(2)L groups, respectively, λ α and τ are their generators in triplet and doublet representations and gs, g, g0 are coupling

constants for SU(3)c, SU(2)L and U(1)Y respectively. The Lagrangian for scalar sector is given by

µ 2 2 = (D Φ)†(D Φ) µ Φ†Φ + λ Φ†Φ , = T V (Φ). (1.4) Lscalar µ − Lscalar − The neutral component of the scalar doublet Φ acquires a non-zero vacuum expectation value (VEV), v = pµ2/λ ( 246 GeV), which causes of spontaneous breaking of the SU(2) ∼ L × U(1)Y gauge symmetry (SSB). It leads to eﬀective mass terms for gauge bosons and fermions. Expanding around the vacuum state gives ! 1 φ+ Φ = , (1.5) √2 v + H + iχ

where H is Higgs particle. The Higgs boson mass mH obtain from the potential V (Φ) in the Lagrangian density (1.4), is equal to

m = √2λv 125 GeV. (1.6) H ≈ 7 The covariant derivatives are deﬁned as

1 α α D = ∂ ig τ W ig0YB . (1.7) µ µ − 2 µ − µ

The Higgs doublet does not couple to the SU(3)c gauge fields. The VEV in Eq. (1.5) leads µ to quadratic terms for the gauge fields from the kinetic term (DµΦ)†(D Φ) in Eq. (1.4). These terms mix the different gauge fields, but they can be diagonalized by transforming the gauge fields to their mass eigenstates:

3 Aµ = cos θW Bµ + sin θW Wµ , (1.8)

Z = sin θ B + cos θ W 3, (1.9) µ − W µ W µ

1 1 2 W ± = (W iW ), (1.10) µ √2 µ ∓ µ

g0 and their masses are (tan θW = g ):

1 mW mW± = gv, mZ = , mA = 0, (1.11) 2 cos θW the massless boson Aµ is identiﬁed with the photon. The interactions between the Higgs doublet and the fermions, the so-called Yukawa interaction, have to be added by hand:

= ye l¯ Φe yu Q¯ Φ˜u yd Q¯ Φd + h.c, (1.12) LY ukawa − ij Li Rj − ij Li Rj − ij Li Rj where ye, yd and yu are 3 3 arbitrary complex matrices that operate in the ﬂavour space. × Since the quark sector is the crucial part of ﬂavour physics, we spell out these terms explicitly. After the spontaneous symmetry breaking of the EW symmetry, the following mass terms for the fermions arise:

quark = v yu u¯ u v yd d¯ d + h.c + interaction terms. (1.13) LY ukawa − √2 ij Li Rj − √2 ij Li Rj = M u u¯ u M d d¯ d + h.c + interaction terms. − ij Li Rj − ij Li Rj The terms describing the interaction of the fermions to the Higgs h, dQh¯ , are omitted. To u d obtain proper mass terms for fermions, the matrices Mij and Mij should be diagonalized.

We do this with unitary matrices Vu and Vd for the up and the down fermions, respectively, as follows:

u u u u Mdiag = VL M VR † (1.14)

d d d d† Mdiag = VL M VL

8 u u Using the requirement that the matrices V are unitary (VL VL † = 1) the Lagrangian can now be expressed as follows:

quark u u = u¯ M u0 d¯ M d0 + . LY ukawa − 0Li ij Rj − 0Li ij Rj ··· The Lagrangian for fermion is given by: = i(Q¯ γµD Q +u ¯ γµD u + d¯ γµD d + ¯l γµD l +e ¯ γµD e ), (1.15) Lfermion L µ L R µ R R µ R L µ L R µ R where the EW covariant derivatives of the fermions are:

1 α α D (L ,Q ) = (∂ ig τ W ig0YB )(L ,Q ). (1.16) µ L L µ − 2 µ − µ L L

D (u , d , e ) = (∂ ig0YB )(u , d , e ). (1.17) µ R R R µ − µ R R R Expressing the Lagrangian in terms of the quark mass eigenstates Q0 instead of the weak interaction eigenstates Q, results in the quark mixing between families (i.e. the oﬀ-diagonal elements) which appears in the charged current interaction: g ¯ µ + ¯ µ cc = [u Liγ (VCKM )ijd0 W + d Liγ (VCKM )† u0 W − L −√2 0 Lj µ 0 ji Lj µ µ + µ ν¯ γ e W e¯ γ ν W −], (1.18) − Li Li µ − Li Li µ u d† The unitary matrix VCKM = VL VL is known as the CKM matrix and it represents the generalization of the Cabibbo-Kobayashi-Maskawa mechanism from two to three generations.

We can notice that in the SM there is no mixing in the neutral sector (FCNC) since the u u† matrix VL is unitary VL VL = 1. CP violation shows up in the complex Yukawa couplings, let us examine the Yukawa part of the Lagrangian:

= y ψ¯ Φψ + y∗ ψ¯ Φ†ψ . (1.19) −LY ukawa ij Li Rj ij Ri Lj The CP operation transforms the spinor ﬁelds as follows:

¯ 1 ¯ CP (ψLiΦψRj)CP − = ψRiΦ†ψLj (1.20)

Therefore, remains unchanged under the CP operation if y = y∗ , i.e. parameters LY ukawa ij ij are real. Similarly, if we look at the charged current coupling in the basis of quark mass eigenstates, g ¯ µ + ¯ µ cc = [u Liγ (VCKM )ijd0 W + d Liγ (VCKM )† u0 W −], (1.21) L −√2 0 Lj µ 0 ji Lj µ and the CP-transformed expression, g CP ¯ µ T ¯ µ + = [d Liγ (VCKM ) u0 W − + u Liγ (VCKM )∗ d0 W ]. (1.22) Lcc −√2 0 ji Lj µ 0 ij Lj µ

We see that, remains unchanged under the CP operation if (V ) = (V )∗ . Lcc CKM ij CKM ij 9 1.1.2 The Cabibbo-Kobayashi-Maskawa Matrix

The CKM matrix connects the quark weak interaction eigenstates to the mass eigenstates. A 3 3 matrix is parameterizable, in general, by 18 parameters. The unitarity condition × reduces the parameters to nine: three angles and six phases. Indeed, ﬁve of these phases can be absorbed in the wave function deﬁnitions of the quarks, therefore we are left with three angles and one phase. CP Violation in the SM is due to this irreducible complex phase in the Lagrangian, since each term of the Lagrangian is transformed into its hermitian conjugate under CP, and therefore, if the Lagrangian contains complex parameters, it will not be invariant under CP. The CKM matrix can be written as:   Vud Vus Vub V = V uV d† =   . (1.23) CKM L L Vcd Vcs Vcb  Vtd Vts Vtb

The CKM matrix element Vij describes the probability of a transition from one quark i to another quark j. Of the many possible characterizations, a standard choice of CKM matrix has become

  iδ13   1 0 0 c13 0 s13e− c12 s12 0

VCKM = 0 c s  0 1 0  s c 0  23 23 − 12 12  iδ13 0 s23 c23 s13e 0 c13 0 0 1 − − (1.24)  iδ13  c12c13 s12c13 s13e− =  s c c s s eiδ13 c c s s s eiδ13 s c . − 12 23 − 12 23 13 12 23 − 12 23 13 23 13  s s c c s eiδ13 c s s c s eiδ13 c c 12 23 − 12 23 13 − 12 23 − 12 23 13 23 13 where sij = sin θij, cij = cos θij and δ is the CKM phase [56]. The unitarity of the CKM P P matrix imposes i VijVik∗ = δjk and j VijVkj∗ = δik. The complex nature of the CKM matrix is only the origin of a CP violation in the SM.

The amount of CP violation in the SM can be represented in terms of the Jarlskog invariant [57]. This quantity denoted as J, can be derived in a simple way from the CKM matrix. Remove one column and one row from the CKM matrix and take the product of the diagonal elements with the complex conjugate of the non-diagonal elements. The imaginary part of the product is then equal to J. In total there will be nine possible expressions for J which all give the same result:

kl X Φij = J ikmjln = Im(VijVkj∗ VklVil∗). (1.25) m,n

10 kl It follows directly from Eq. (1.25), where Φij are antisymmetric with respect to the up-type quark labels i, k and similarly for the j and l labels from down-type quarks,

Φil = Φkl, Φkj = Φkl (1.26) kj − ij il − ij 2 J = c12c13c23s12s13s23 sin δ. (1.27) +0.20 5 The current evaluation gives J = 2.96 0.16 10− , that is too small to generate baryon − × asymmetry [42].

1.2 Baryon Asymmetry in the Universe and Sakharov Condition

The Universe is made up almost exclusively of baryons with no signiﬁcant amount of antibaryon [40]. The trace amounts of antimatter detected in satellite experiments are consistent with all the antimatter in our neighborhood being produced by collisions of cosmic rays or in astrophysical sources [41]. Cosmic rays are the most important way for the detection of antimatter, as the annihilation of a particle-antiparticle pair produces γ-rays. A small fraction of antiprotons is found in the cosmic rays coming from galaxy, but they could be produced by the process of proton-antiproton pair creation coming from the collision of high energy γ-rays or nuclei with interstellar matter, in processes like

γ + γ p +p. ¯ (1.28) ↔ We can characterize the asymmetry between matter and antimatter in terms of the baryon-to-photon ratio η, n n¯ n η b − b = B , (1.29) ≡ nγ nγ

where nb is the number density of baryons, n¯b the number density of anti-baryons, nB is

the resulting diﬀerence and nγ the photon density. The baryon-to-photon value η is not constant throughout the history of the Universe. Therefore it is more convenient to relate

the baryon-number density nB to the entropy density s, thus deﬁning the baryon-to-entropy ratio as nB/s. This is related to the value of the present baryon-to-photon ratio by [42] n η B = . (1.30) s 7.04

It is possible to determine the baryon relic density, ρ 8πG Ω = = ρ, (1.31) b ρ 3 2 crit H 11 where ρ is the density and ρcrit is the critical density corresponding to a ﬂat Universe. The r.h.s of the Eq. (1.31) arises from Einstein’s ﬁeld equations, where G is Newton’s gravitational constant. The density parameter is often quoted multiplied by the Hubble 2 1 1 parameter squared, Ω h (h = ( /100) km s− Mpc− , where is the present-day Hubble b H H expansion rate).

There are two ways in which the relic density can be determined. The ﬁrst one comes from the primordial abundances of light elements in the Big Bang Nucleosynthesis (BBN). The second one is based on the temperature anisotropies present in the Cosmic Microwave Background (CMB) radiation. BBN predicts the primordial abundances of the light cosmological elements: 4He, D,3 He and 7Li that are produced during the ﬁrst 20 minutes after the Big-Bang1 when the Universe was dense and hot enough for nuclear reactions to take place. Comparing the calculated and observed abundances, there is an overall good agreement except for the 7Li [44]. The essential cosmological parameter of the model is the baryon to photon ratio, η n /n , where the photon number density is determined from ≡ B γ the CMB temperature and nB is related the baryonic density. Ωb is now well measured from the angular power spectrum of the CMB temperature anisotropies. The observations from

WMAP experiment gives the following value for cosmological density parameter and nB/s ratio [45, 46] Ω h2 = 0.02226 0.00016, (1.32) b ±

translates to the following value for the nB/s ratio:

nB 11 = 8.7 0.3 10− . (1.33) s ± ×

The smallness of the nB/s ratio tells us that only a small fraction of the total matter density present in the early Universe has survived annihilation.

Once the existence and amount of baryon-antibaryon asymmetry is experimentally con- ﬁrmed, a natural question arises about the reasons for this. In 1967, Andrei Sakharov de- lineated three necessary conditions to dynamically generate the observed baryon asymmetry from an initially symmetric state, regardless of the exact mechanism. His three conditions for baryogenesis are now known as the Sakharovs conditions [9], and are:

Baryon-number violation process;

C and CP violation;

Departure from thermal equilibrium.

1Nucleosynthesis occurred between 3 minutes and 20 minutes after the Big Bang.

12 Demand on the baryon-number violation is the most obvious of Sakharovs conditions, as without it a baryon asymmetry cannot be generated from initially symmetric conditions. The baryon number B is deﬁned as

n n B = q − q¯, (1.34) 3 where nq and nq¯ are the number of quarks and antiquarks which form a particle, respectively. Baryons therefore have baryon number 1, mesons and leptons 0 and quarks 1/3, while the respective antiparticles have baryon numbers of opposite sign. Baryon-number violation has not been observed in our cool, present Universe and this is actually a conserved quantity, which is responsible for the stability of the proton.

The charge-conjugation (C) operator transforms particles into their antiparticles, while the parity (P) operator performs a spatial reﬂection with respect to the coordinate origin. Therefore C and CP symmetries mean, respectively, the invariance of physical laws under charge conjugation, and charge conjugation combined with parity. The presence of baryon- number violating processes alone would not generate a net baryon number, because the baryon excess would be produced in equal amount as the antibaryon excess, in absence of a C and CP violation processes.

Consider a B-number violating process X Y + B. If C is conserved, then every → B-number violating decay has the same decay width as the C-conjugate one:

Γ(X Y + B) = Γ(X¯ Y¯ + B¯) (1.35) → → Since both processes proceed at the same rate, B- number is conserved over long periods of time. So C violation is a Sakharov condition. However, this is not quite enough. Consider a hypothetical B-number violating process X q q which creates left-handed baryons. If → L L CP is a symmetry of nature, then this process proceeds at the same rate as the CP-conjugate process X q q , and thus → R R Γ(X q q ) + Γ(X q q ) = Γ(X¯ q¯ q¯ ) + Γ(X¯ q¯ q¯ ). (1.36) → L L → R R → L L → R R The C-conjugate reactions have a different rate, but the sum of the two will still preserve baryon number. Thus, CP needs to be violated as well, so that the rate of baryon generation exceeds that of anti-baryon generation. With C and CP violation, the rate of B production can exceed that of B¯-production. C violation in the weak interaction was first demonstrated in 1957 by physicists studying the decay of muons and anti-muons [47]. Indirect violation of CP symmetry in the SM was first observed in the decay of neutral K-mesons in 1964 [48].

13 The departure from equilibrium is the last of Sakharovs conditions. If there exists a B-violating decay X Y + B, (1.37) → where the particles X and Y have zero baryon number, while B represents the particle with B = 0, its inverse process 6 Y + B X (1.38) → must have the same reaction rate, in the thermal equilibrium. Therefore, if the first process produced a net baryon number, the inverse process would bring it back to zero. Thus, any baryogenesis must happen under conditions outside of the thermal equilibrium. After baryon asymmetry takes place and the Universe returns to thermal equilibrium, the conditions must have changed such that the generated asymmetry cannot be disappear. In the EW phase transition, a sufficient departure from equilibrium is achieved if the transition is strongly first order. This point will be studied in section 1.3.

1.3 Electroweak Baryogenesis

The goal of any baryogenesis mechanism is to explain the observed asymmetry between matter and anti-matter nb n¯b 10 η − 10− . (1.39) ≡ nγ ∼ The special appeal of EW baryogenesis is due to the fact that it only involves the physics of EW scales. This makes the scenario in principle testable. In order to produce EW baryogenesis, we need to satisfy the Sakharov conditions within the EW theory.

1.3.1 Sakharov Conditions in the EW Theory

Baryon number violation

There is baryon number violation in the SM, but this is a subtle, non-perturbative effect which is completely negligible for particle reactions in the laboratories at present-day collision energies, but very significant for the physics of the early Universe. Here is has been outlined that how this effect arises. It is known that baryon and lepton number, which are conventionally assigned to quarks and leptons as shown in the table,

14 Q Q¯ l ¯l B 1/3 -1/3 0 0 L 0 0 1 -1 are conserved to high accuracy quantum numbers in the particle physics. Baryon number B L and lepton number currents are deﬁned as Jµ and Jµ , which are conserved at the Born level. By applying the Noether’s theorem at the Born level we have the current conservation, 1 X ∂ J µ = ∂ (Q¯ γµQ +u ¯ γµu + d¯ γµd ) = 0, (1.40) µ B 3 µ L L R R R R and µ X ¯ µ µ ∂µJL = ∂µ (lLγ lL +e ¯Rγ eR) = 0. (1.41) Thus, the associated time-independent B and L charge operators are Z ˆ 3 0 B = d xJB(x), (1.42)

and Z ˆ 3 0 L = d xJL(x). (1.43) Beyond the Born approximation, these symmetries are explicitly broken and Eqs. (1.40) and (1.41) no longer hold. Decomposition of the vector current into its left- and right-handed parts (axial vector current) gives for fermions (f = Q, `):

¯ µ ¯ µ ¯ µ fγ f = fLγ fL + fRγ fR. (1.44)

However, in the presence of W boson ﬁeld, the axial vector current is not conserved and the current conservation laws are violated. This phenomenon is called ABJ anomaly [58].

In a gauge theory based on a gauge group G, which is a simple Lie group of dimension dG, ¯ ¯ the anomaly equations for the L- and R-chiral currents fLγµfL and fRγµfR read g2 ∂ f¯ γµf = ς F a Fãµν, (1.45) µ L L − L 32π2 µν g2 ∂ f¯ γµf = +ς F a Fãµν, (1.46) µ R R R 32π2 µν aµν ãµν µναβ a where F is the (non)abelian field strength tensor (a = 1, ..., dG) and F = Fαβ/2 is the dual tensor, g denotes the gauge coupling and the constants ςL and ςR depend on the representation which the fL and fR form. In total, the electroweak anomaly equation is [59]

µ µ nF 2 αβµν a aµν 1 αβµν 2 µν ∂ J = ∂ J = (g W W˜ g0 B B˜ ), (1.47) µ B µ L 32π2 αβ − 2 αβ 15 where nF is the number of generations. The lepton-number current anomaly is found to be the same as Eq. (1.47), so that ∂ (J µ J µ) = 0, (1.48) µ B − L this implies that the quantity B L is strictly conserved, while the non-conservation of − µ µ nF 2 αβµν a aµν 1 αβµν 2 µν ∂ (J + J ) = (g W W˜ g0 B B˜ ), (1.49) µ B L 16π2 αβ − 2 αβ means that B + L is violated by

∆(B + L) = 2nF NCS,

where NCS is topological winding number. In what follows, it will be shown that the topo-

logical winding number NCS is equal to the Chern-Simons number.

a As only the SU(2) gauge fields Wµ enter the Lagrangian density, while the U(1) coupling constant g0 = 0, the anomaly Eq. (1.47) transforms to n ∂ J µ = F g2αβµνW a W˜ aµν. (1.50) µ B 32π2 αβ Defining g2 ∂ J µ = αβµνW a W˜ aµν, (1.51) µ CS 32π2 αβ using 0123 = 1, the Chern-Simons current will be − g2 g J µ = ναβT r W W + i W W W . (1.52) CS −64π2 ν αβ 3 ν α β Therefore, the Chern-Simons number can be defined as [59]

Z g2 Z g N = d3xJ 0 = d3xναβT r W W + i W W W . (1.53) CS CS −64π2 ν αβ 3 ν α β So, the non-conservation of baryon number becomes manifest when considering the total derivatives of baryon current and Chern-Simons current. Hence, every change in the Chern-

Simons number is reﬂected into a change in the baryon number B by nF = 3 units. This translates into the creation or the destruction of nine quarks and three leptons, with each generation equally represented, as shown in ﬁgure 1.1.

The topological nature of the Chern-Simons number NCS becomes manifest when con-

sidering static gauge field configurations with varying NCS. In figure 1.2, minima represent the vacuum states of the SU(2) gauge theory, the n-vacua. As the field theories constructed about these vacua are completely equivalent and can be obtained from one another by gauge transformation, the vacuum structure of the EW theory is degenerate. The energy of barrier

16 푠퐿 푠퐿 푡퐿

푐퐿 푏퐿

푑퐿 푏퐿

푑퐿 휈퐿

푢퐿 휈휇 휈푒

Figure 1.1: An example of a (B+L)-violating SM amplitude. Each sphaleron violates baryon and lepton numbers each by nF units, where nF is the number of families, with subsequent production of 9 quarks and 3 leptons, equally distributed between generations.

E T=0/

Esphaleron

-1 0 1 a fields Wµ , Φ T=0

Figure 1.2: Vacuum structure of SU(2) gauge theory. The energy of gauge ﬁeld conﬁgurations is displayed as a function of Chern-Simons number.

between two minima (with non vanishing ﬁeld strength and energy), is called sphaleron [42],

4π λ E (T ) = v f( ), (1.54) sph g T g

where vT is the VEV of the SM Higgs doublet ﬁeld Φ(x) at temperature T. At T = 0 we

have vT =0 = 246 GeV. The parameter f varies between 1.6 < f < 2.7 depending on the value of the Higgs self-coupling λ, i.e., on the value of the SM Higgs mass. This yields E (T = 0) 8 13 TeV. sph ' − 17 The Sphaleron Rate

The rate of Baryon-number violation transitions will be proportional to the Boltzmann factor

exp(Esph(T )/T ) as long as the energy of the thermal excitations is smaller than that of the barrier, while unsuppressed transitions will occur above that barrier. At temperature below critical temperature T (i.e, T < T 100 GeV), where the EW gauge symmetry is broken, c c ∼ the sphaleron B + L transition rate per volume V is given by (see, e.g. [60, 61])

Γsph 3 B+L¡ mW 4 = µ mW exp( Esph(T )/T ), (1.55) V αωT − where µ 1 is a dimensionless constant and m (T ) = gv /2 is the temperature-dependent ∼ W Tc mass of the W boson. As the temperature rises, the sphaleron barrier becomes lower and lower until it disappears at extremely high temperatures.

The sphaleron rate per unit volume, at temperatures T > T 100 GeV is proportional c ∼ to T 4 and has the following form [62, 63]

Γsph B+L¡ = Kα5 T 4, (1.56) V ω where K is the numerical factor reﬂecting the uncertainty in the estimate of the transition rate between vacua, It has been estimated to be between 0.1 and 1 [64]. The sphaleron rate that is evaluated within the thermal volume 1/T 3,

sph 6 Γ 10− T, (1.57) B+L¡ ' must be compared to the Hubble rate

T 2 = 1.67√g∗ , (1.58) H mpl

19 where m = 1.22 10 GeV denotes the Planck mass and g∗ is the eﬀective number of P l × degrees of freedom. At very high temperatures T T , sphalerons are out of equilibrium, c as Γsph . Equilibrium is achieved at B+L¡ H Γsph = T 1013GeV. (1.59) B+L¡ H → ∼ When T falls below the EW phase transition temperature T 100 GeV, the sphaleron rate c ∼ becomes again smaller than the Hubble rate, and the baryon-number violating processes is out of equilibrium. This result provides an important constraint on any baryogenesis

mechanism which operates above Tc.

18 C and CP Violation

In the fermion sector of the SM the C and CP symmetries are violated. The latter effect is related to the mixing of the three generations (see the section 1.1.2). The quarks mixing is parameterized by the CKM matrix. The Jarlskog invariant that is shown for the description of CP violation in the SM cannot be large enough for baryogenesis. A dimensionless measure of the strength of CP violation, using the relevant temperature of the Universe, which must be at least of order 100 GeV for sphalerons to be effective. One then finds that

nb n¯b J 20 η = 10− (1.60) − 12 ≡ nγ (100 GeV) ∼

10 which is much too small to account for η 10− , thus opening the speculation for other ∼ mechanisms for baryogenesis beyond the SM [42].

Departure from Thermal Equilibrium

The last requirement in Sakharov’s criteria, the departure from thermal equilibrium, is an important ingredient to explain the baryon asymmetry within the SM of particle physics. The condition for the out-of-equilibrium decay is [65]

T α c , (1.61) mpl where the case of T 100 GeV would require extremely weak couplings α. At some c ∼ critical temperature Tc the minimum becomes meta-stable and the phase transition may proceed. Phase transitions can be classified into first order, second order or crossovers. A thermodynamic quantity that changes from one phase to another is called the order parameter. First-order phase transitions occur with a measurable changing discontinuously of the order parameter as a function of temperature. In second-order phase transitions, as the temperature is varied, the change in the order parameter slowly decreases until it is continuous at the phase transition point. The crossover is a smooth transition between the two phases. The difference between the first and the second order type of transition is determined by the behavior of the scalar potential at finite temperature, as shown in figure 1.3. In a first-order phase transition, there is a barrier between the symmetric and broken phases. Such transition have super-cooled (out of equilibrium) symmetric states when the temperature decreases and are of use for baryogenesis purposes. In a second order transition or a smooth cross-over there is no barrier between the symmetric and broken phases. Actually, when the broken phase is formed, the origin (symmetric phase) becomes a

19 Baryogenesis 31

determined by the behavior of the Higgs potential at finite temperature, as shown in figure 10. In a first order transition, the potential develops a bump which sep- arates the symmetric and broken phases, while in a second order transition or a smooth cross-over there is no bump, merely a change in sign of the curvature of maximum. The phase transition may be achieved by a thermal fluctuation for a field located the potential at H = 0. The critical temperature Tc is defined to be the tem- at the origin. perature at which the two minima are degenerate in the first order case, or the temperature at which V ′′(0) = 0 in the second order case.

T>Tc V V T>Tc

T=Tc T=Tc

H H Fig. 10. Schematic illustration of Higgs potential evolution with temperature for ﬁrst (left) and second Figure 1.3: Higgs(right) orderpotential phase transition. evolution with temperature for the ﬁrst (left) and the second

(right) order phaseA first transition. order transition proceeds by bubble nucleation (fig. 11), where inside the bubbles the Higgs VEV and particle masses are nonzero, while they are still vanishing in the exterior symmetric phase. The bubbles expand to eventually A first-ordercollide phase and transition fill all of space. proceeds If the Higgs by VEVbubblev isnucleation, large enough inside where the inside bub- the bubbles bles, sphalerons can be out of equilibrium in the interior regions, while still in the Higgs VEVequilibrium and particle outside masses of the bubbles. are nonzero, A rough while analogy they to GUT are baryogenesis still vanishing is that in the exterior symmetric phase.sphalerons The outside bubbles the bubbles expand correspond to eventually to B-violating fillY allboson of space. decays, which If the Higgs VEV are fast, while sphalerons inside the bubbles are like the B-violating inverse Y de- is large enoughcays. (i.e. The Strong latter should enough be slow; First otherwise Order they Phase will relax Transition the baryon (SFOPT))asymmetry the bubbles sphalerons canback be out to zero. of equilibrium in the interior regions. In that case the baryon number violating interactionsIn a second are out order of EWPT, equilibrium even though in the the sphalerons bubble go walls from and being a in net equi- baryon number librium to out of equilibrium, they do so in a continuous way, and uniformly can be generatedthroughout during space. the phaseTo see why transition the difference [66]. between these two situations is important, we can sketch the basic mechanism of electroweak baryogenesis, due to Cohen, Kaplan and Nelson [32]. The situation is illustrated in figure 12, which portrays a section of a bubble wall moving to the right. Because of CP-violating Condition forinteractions SFOPT in the bubble wall, we get different amounts of quantum mechanical reflection of right- and left-handed quarks (or of quarks and antiquarks). This Now, we can tryleads to to see a chiral whether asymmetry the SFOPT in the vicinity can ofoccur the wall. in the There SM. is anThe excess basic of tool for doing so is the finite temperature effective potential of the Higgs field. The high temperatures effective potential, Veff , is given by

V (φ, T ) = m2 (T )φ2 ET φ3 + λφ4, (1.62) eff ∼ φ − here

2 2 2 mφ(T ) = λvT + aT , (1.63) − 1 3 3 2 2 3/2 E = 3g + (g + g0 ) , (1.64) 12√2π 2 λ 3 1 a = + (g2 + g 2) + y2, (1.65) 2 16 0 4 t

20 where, vT is VEV at temperature T and yt is the coupling to the top quark. At the critical

temperature Tc, Veff (φ, T ) becomes

2 2 vTc Veff (φ, Tc) = λφ φ , (1.66) − √2

since this is the form that has degenerate minima at φ = 0 and φ = vTc . From Eqs. (1.62) and (1.66) one can ﬁnd that 1 m2 (T ) = λv2 ,ET = √2λv , (1.67) φ 2 Tc c Tc

therefore, the ratio vTc /Tc can be ﬁnd,

v E 3g3 Tc = = . (1.68) Tc 2√λ ∼ 16πλ The sphaleron rate Γ exp( E /T ) must be suppress at T to prevent the washout of sph ∼ − sph c the baryon asymmetry. The bigger v(T )/T is at the critical temperature is the less washout of the baryon asymmetry through the sphalerons process. Therefore, the ratio vTc /Tc can determines how strongly the sphalerons are suppressed inside the bubbles. So, if

vTc & 1, (1.69) Tc or equivalently, E 8π sph , (1.70) T & g the condition for a ﬁrst-order transition to be strong is satisﬁed. By combining Eqs. (1.68) and (1.69), we can infer a bound on the Higgs mass [67]. Numerical simulations [68, 69, 70,

71, 72, 73, 74] have shown that there is a ﬁrst order phase transition if mH < 72 GeV. This is far below the current limit m 125 GeV, then it is impossible to have a SFOPT in the H ∼ SM.

1.4 Dark matter and its constraints

In 1933 Zwicky [157], by applying Newtons laws and measuring the speeds of individual galaxies within a cluster of galaxies, could deduce the mass of the cluster. He also determined the amount of visible matter in the clusters by measuring the brightness of the galaxies that form them. Those two measurements showed that a typical giant cluster of galaxies comprises at least ten times more invisible matter than what is visible [158]. This was the ﬁrst of many

21 observations that show that there is a form of matter in the Universe that we can only detect due to its gravitational eﬀects on visible matter. Zwicky named this mysterious matter ”Dark matter”.

There are various evidence for DM from a wide range of length scales, from galaxies to the entire Universe. For reviews on DM see e.g. refs. [159, 160, 161].

The most convincing evidence comes from rotation curves of galaxy. The circular rotation velocity of stars and gas under Newtonian gravity, v(r), is given by

r GM(r) v(r) = , (1.71) r where r is the radial distance from the galactic centre and M(r) is the mass distribution, Z M(r) = 4π ρ(r)r2dr. (1.72)

If the mass distribution of the galaxy were like that of the luminous matter in the galaxy, we would expect to see this decrease in circular velocity at the outskirts of a galaxy with a density which drops approximately exponentially with radius. Instead we see at rotation curves for large radii v(r) roughly constant by a matter component distributed spherically 2 with density ρ ∝ r− . This leads to the conclusion that there exists more matter than we can see.

There is also lot of other evidence from galaxy cluster scales. This evidence comes from measuring the mass of galaxy clusters and comparing that to the amount of visible light from the galaxy. The evidence includes using X-rays to determine the temperature proﬁle of the gas in the cluster which, when combined with equations of hydro-static equilibrium, can provide estimates of the total mass.

One can also use gravitational lensing to determine the mass of a cluster by observing how much it bends light originating from behind the cluster. All these measurements, including those on galactic scales, consistently show that there is about ﬁve times more mass than can be accounted for by stars and gas [160]. There have been suggestions that instead of explaining all of these measurements by postulating DM, one should instead modify the laws of gravity [163]. This proposal is strongly disfavored by one of the most spectacular pieces of evidence in favor for DM, the Bullet Cluster [164].

The Bullet Cluster is a system consisting of two galaxy clusters that recently collided with each other. From observing the system in visible light (X-rays) show that the matter, which interacts electromagnetically, is left between the two visible galaxies. However, the weak

22 lensing measurements show that there is much matter that passed by, without interactions. This matter can be the DM [165, 166].

At the cosmological scale, we have evidence for DM from the CMB radiation. By studying the CMB angular power spectrum in combination with data from BBN, one can determine both the density of normal matter (baryons) and the density of DM. The results from the Planck Satellite [46] give a DM density of Ω h2 = 0.1199 0.0027, which is signiﬁcantly DM ± larger than the density of baryons Ω h2 = 0.02226 0.00016. We know that Ω 1 [167], b ± tot ≈ therefore DM makes up about 23% of its total energy density of the Universe, ﬁve times more than the density of SM baryons.

For a particle to be a DM candidate, it has to be massive, stable on cosmological time scales and not interact too strongly with baryons or photons. A viable particle physics model of DM needs to be able to reproduce the observed DM relic density.

Consider a stable DM particle with mass MDM , that in the early Universe is in thermal equilibrium. As DM is stable, the only way to change the number of DM particles is through pair production or annihilation. As long as the interaction rate of these processes is larger than the expansion rate given by the Hubble parameter , these interactions can H keep the particles in thermal equilibrium. As the Universe expands, the Hubble parameter and the interaction rate decreases. When the interaction rate becomes smaller than the Hubble rate, we get chemical decoupling and freeze-out of DM. After freeze-out, the number of DM particles will remain constant and the number density will decrease with the normal expansion of the Universe. To study this in detail, we consider the annihilation rate, given by

η = σv n , (1.73) h iann eq where σv is the thermally averaged annihilation cross-section and n is the number h iann eq density of DM in thermal equilibrium. The early Universe may be described in the language of thermodynamics and we may write a Boltzmann equation for the number density of DM particles during the evolution of the Universe,

n˙ + 3 n = σv (n2 n2 ), (1.74) H −h iann − eq where n is the present time number density of DM and the 3 n term includes the eﬀect H of the expanding Universe. The equilibrium number density can easily be determined from 3 standard statistical physics and it depend on temperature T as neq ∝ T , when the DM is 3/2 T/MDM relativistic and neq ∝ (MDM T ) e when it is non-relativistic one.

23 If we rewrite the Eq. (1.74) using y = n/s and x = MDM T , where s is the entropy density, we get

dy x σv s = − h iann (y2 y2 ). (1.75) dx − − eq H A general scale for a process to drop out of equilibrium is when σv . The number h i H density remains constant, which decreases with T 3 in the same way as the density for normal matter does. This is generally referred to as ‘freeze-out’.

Formally one must solve the Boltzmann equation to ﬁnd an expression for the relic density of DM. However, accounting for the measured energy density, a useful approximation exists for a thermal WIMP [168],

27 3 1 3 10− cm s− Ω h2 × . (1.76) DM ' σv h iann For masses around those of the SM an annihilation cross-section is of order

26 3 1 σv 3 10 cm s− . (1.77) h iann ∼ ×

The annihilation process, which may proceed along three way, corresponding to the SM particles-DM scattering, the DM-DM annihilation into the SM particles and pair production of DM particles via SM-SM particles annihilation. These three processes are vital to the experimental search for DM, each corresponding to one of the three canonical search tech- niques: direct detection (scattering), indirect detection (annihilation) and collider studies (pair production).

In the direct detection, one looks for recoil (energy) from collisions between DM particles and heavy nuclei, see e.g. [169]. There are several experiments to search for this phenomenon, and there exists a pronounced tension between their reported results and those looking directly for WIMPs. In general, the DM particle and the cross section for its scattering oﬀ a nucleus, σDM N , are the important parameters for the direct detection experiments, − therefore these parameters can be constrained by the direct detection experiments. The most stringent constraints on the interactions of potential WIMPs (currently from the LUX collaboration [170], which is directly searching for DM interactions via nuclear recoil) rule out the claimed annual modulation from the CoGeNT [171] and DAMA [172] collaborations.

In the indirect detection, one looks for high energy photons produced by high energy SM particles after the DM annihilates, see e.g. [173]. Among the indirect detection experiments are PAMELA, INTEGRAL, HESS and Fermi-LAT [174, 175, 176]. In the indirect detection

24 the annihilation signal is very sensitive to the density of DM, so the most promising place to look would be towards the galactic centre where the DM density is larger.

To look for DM at colliders, one would search for signals with missing energy recoiling against visible objects. Collider search for mono-jets plus missing energy can give compet- itive constraints in the low DM mass region [177, 178, 179]. In this region direct detection experiments lose sensitivity as the recoil energy becomes too small.

1.5 Content of the thesis

This thesis is organized as follow: In chapter 2, we present the structure of the cSMCS and discuss the physical conditions for a spontaneous CP violation in such model. There are three neutral Highs particles in this model. Assuming the lightest one to be the 125 GeV Higgs boson found at LHC, we calculate masses of the additional Higgs scalars and perform a numerical study of the allowed region of parameters. The scenario, according to which the SM-like Higgs particle comes mostly from the SM-like SU(2) doublet with a small modiﬁcation coming from the singlet, is in agreement with the newest Rγγ and EW Precision Test (EWPT) [1, 2]. It have been found that the Jarlskog invariant, measuring the CP mixing and a possibility of the CP violation in the scalar sector, can be enhanced as compared to the one in the SM [1, 2]. We show this model provides strong enough ﬁrst-order EW phase transition to suppress the baryon-violating sphaleron process and leads to proper description in the baryogenesis [3, 6].

In the chapter 3, we present our predictions for the production rates of the cSMCS Higgs bosons at the LHC. The scalar spectrum of the cSMCS includes three neutral Higgs particles with the lightest one considered to be the 125 GeV Higgs boson found in 2012 at the LHC. In the considered model, the SM-like Higgs boson comes mostly from the SM-like SU(2) doublet, with a small correction from the singlet. To calculate the production rates of the Higgs bosons, we use a conventional effective LO QCD framework and the unintegrated parton distribution functions (UPDF) of Kimber-Martin-Ryskin (KMR). We first compute the SM Higgs production cross-section and compare the results to the existing calculations from different frameworks as well as to the experimental data from the CMS and ATLAS collaborations. We have shown that our approach is capable of producing sound predictions in this case and we use it for calculation of the cSMCS predictions for the Higgs bosons production at the LHC. These predictions for yet undetected Higgs bosons of the cSMCS model that may provide some clues for the future discovery [4].

25 In chapter 4, we study the general structure of the Inert Doublet Model with a complex singlet (cIDMS) with focus on the properties of the Dark Matter (DM). It contains an analysis of the Higgs couplings and a comparison with the existing LHC data and EWPT. We present our study of relic density for a DM candidate of the model, which is assumed to be the lightest neutral Z2-odd scalar particle [1, 5]. The chapter 5 is devoted to exploring the Higgs sector of the Two Higgs Doublet Model

(2HDM) based on the Veltman condition. We have chosen the soft Z2 symmetry breaking version of the 2HDM with non-zero vacuum expectation values for both Higgs doublets (Mixed vacuum)and CP conserving in the scalar sector. We apply cancellation quadratic divergences in the this model in order to ﬁnd the mass of the heavy Higgs bosons. The Higgs boson mass of two SM-like scenarios (with 125 GeV h and with 125 GeV H particles) are considered. We have found that solution exists only for the SM-like h scenario [7].

Finally, the chapter 6 contains a summary and conclusion of the presented results.

26 Chapter 2

The Extended Standard Model with a Complex Singlet

As it was discussed in chapter 1.3, the SM of EW interaction is unable to account for the observed magnitude of the BAU for two reasons: ii) there is not enough CP violation from the CKM matrix to generate the baryon asymmetry [79, 80, 81], and i) the EW phase transition is not strongly ﬁrst-order and therefore, any baryon asymmetry generated during the EW phase transition would be washed out by unsuppressed baryon violating processes in the broken phase [10, 11, 12, 14, 15, 16, 17]. In order to address this important issues, various extensions of the SM are considered [21, 22, 23, 24, 25].

Here we shall assume that an additional sources of CP violation can be provided by extension of the SM with a neutral complex scalar singlet χ, which accompanies the SM- like Higgs doublet Φ, and a pair of iso-doublet VQ, VL + VR. Assuming nonzero vacuum expectation value for the complex singlet we analyze the physical conditions for spontaneous CP violation. This kind of extension of the SM was discussed in the literature with various motivations, see e.g. [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 32]. We consider the potential with a softly broken global U(1) symmetry, which we call the constrained SM+CS model (cSMCS). This choice allows to limit the number of the parameters. Moreover, it constitutes a part the further extension of the scalar sector, which allows for the reliable description of DM. This extension called cIDMS and will be discussed in chapter 4.

In the model, there are three neutral Higgs particles with mixed CP properties. Assuming the lightest one, predominately CP-even, to be 125 GeV Higgs boson found at LHC, we calculate masses of the other Higgs scalars and perform a numerical study of the allowed region of parameters. We calculate the analog of Jarlskog type invariant [57, 82, 83, 84]. The

27 scenario realized in the model according to which the SM-like Higgs particle comes mostly from the SM-like SU(2) doublet with a small modification coming from the singlet, is in agreement with the newest LHC Higgs data, in particular the strength signal Rγγ, as well as the EWPT. We propose seven benchmarks to test this model. This model provides a strong enough first-order EW phase transition via the soft breaking terms in the potential, to suppress the baryon-violating sphaleron process. In the presence of an iso-doublet VQ and a complex singlet, the Yukawa Lagrangian acquires additional terms. While, diagonalizing the quark mass matrix the whole Lagrangian will be modified with new term which is a function of the time-dependent CP violating phase [103]. The appearance of these terms leads to the generation of baryon asymmetry in the spontaneous baryogenesis scenario.

The content of this chapter is as follows. In section 2.1, we explain the difference between explicit and spontaneous CP violation. Section 2.2 contains a general presentation of the SMCS model and its constrained version (cSMCS). In particular, the section 2.2.3 describes the conditions for the spontaneous CP violation in the model. Physical states in the Higgs sector are discussed in section 2.3. The numerical results of scans over parameters of the model are collected in section 2.4. In section 2.5 the Jarlskog invariant for the scalars is discussed. In section 2.6, the agreement of cSMCS model with existing LHC measurements of the properties of the SM-like Higgs boson as well as comparison with data on S and T parameters are shown. The benchmarks are presented here as well. In section 2.7, the necessary conditions for the strong enough first-order EW phase transition in the model are verified. In section 2.8, the generation of a baryon asymmetry through the mixing of the SM quark and VQ is discussed. Finally section 2.9 contains our conclusion.

2.1 Explicit and spontaneous CP violation

As we reviewed in chapter 1.1, in the SM the CP symmetry is broken explicitly by the CKM phase. Models beyond the SM often introduce additional CP violating phases. However, CP violation does not necessarily is due to the explicit breaking of CP. CP violation may be a consequence of a spontaneous CP violation, which is realized when the Lagrangian of the theory respects the CP symmetry but the vacuum is not invariant under it. Such a case arises when the VEV of a scalar field operator exhibits physical CP violating phases (which cannot be removed from the theory by field re-definitions). Here we consider the case, where the additional source of CP violation (spontaneous) is provided by a neutral complex scalar singlet χ with a complex VEV.

28 2.2 The cSMCS: The SM plus a complex singlet

The full Lagrangian of this model is given by

= SM + + (ψ , Φ) + (V , χ), (2.1) L Lgf Lscalar LY f LY q where SM describes the pure gauge bosons terms as well the SM bosons-fermions interaction, Lgf describes the scalar sector of the model with one SU(2) doublet Φ and a neutral Lscalar complex scalar (spinless) singlet χ. (ψ , Φ) represents the Yukawa interaction of Φ with LY f SM fermions, see Appendix 7.2.1. (V , χ) describes the interaction between singlet scalar, LY q quarks and VQ and the mass of VQ. We consider a pair (left and right handed) of heavy

iso-doublet VQ, VL + VR, with VL and VR having the same transformation properties under

the gauge group of the SM, and where VL transforms in the same way as a SM quark doublet

QL. Essentially, the case of iso-singlet VQ have the same result as the case of iso-doublet

VQ, while their transformation are similar to uR or dR with respect to the SM gauge group [103]. In order to have more general expressions and have both up and down types of VQ (respectively, and ), is preferred to utilize the iso-doublet VQ. U D The (V , χ) is given by LY q (V , χ) = λ χQ V + MV V + h.c, (2.2) LY q V L R L R

where, M is the mass of VQ, which is considered to be heavy and λV is the coupling between singlet scalar, VQ and SM quarks. In general the couplings of all three generations of SM quarks to VQ should be considered, nevertheless, since only one linear combination of these will actually mix with the VQ, we consider only one generations of SM quarks (the heaviest),

QL.

This model allows for the SM-like scenario observed at the LHC, with the SM-like Higgs boson predominantly consisting of a neutral CP-even component of the Φ doublet.

We assume Φ and χ ﬁelds have VEVs equal to v and weiξ, respectively, where v, w, ξ R. ∈ We shall use the following ﬁeld decomposition:

! φ+ 1 Φ = , χ = (φ + iφ ). (2.3) 1 (φ + iφ ) √ 2 3 √2 1 4 2

Masses of the EW gauge bosons and the fermions are given by the VEV of the doublet, e.g. 2 2 2 MW = g v /4 for the W boson.

29 2.2.1 Potential

The scalar potential of the model can be written as follows

V = VD + VS + VDS, (2.4) with the pure doublet and the pure singlet parts (respectively VD and VS) and their interaction term VDS. The SM part of the potential, VD, is given by:

1 2 1 2 V = m Φ†Φ + λ Φ†Φ . (2.5) D −2 11 2

The potential for a complex singlet is equal to:

1 2 1 2 2 2 V = m χ∗χ m (χ∗ + χ ) S −2 s − 2 4 2 2 2 4 4 +λs1(χ∗χ) + λs2(χ∗χ)(χ∗ + χ ) + λs3(χ + χ∗ ) 3 3 +κ1(χ + χ∗) + κ2(χ + χ∗ ) + κ3(χ∗χ)(χ + χ∗). (2.6)

The doublet-singlet interaction terms are:

2 2 VDS = Λ1(Φ†Φ)(χ∗χ) + Λ2(Φ†Φ)(χ∗ + χ )

+κ4(Φ†Φ)(χ + χ∗). (2.7)

2 There are three quadratic (ma), six dimensionless quartic (λa, Λa) and four dimensionful parameters κi, i = 1, 2, 3, 4, describing linear (κ1), cubic (κ2, κ3) and mixed (κ4) terms, respectively. The linear term κ1 can be removed by a translation of the singlet ﬁeld, and therefore can be neglected.

To simplify the model, we apply a global U(1) symmetry

U(1) : Φ Φ, χ eiαχ (2.8) → → to reduce the number of parameters in the potential.

However, a non-zero VEV of χ would lead in such case to a spontaneous breaking of this U(1) symmetry and an appearance of massless Nambu-Goldstone scalar particles, what is not acceptable. Keeping some U(1) soft-breaking terms in the potential would solve this problem and at the same time would still lead to a reduction of the number of parameters of V. In what follows, we shall consider a potential with a soft-breaking of U(1) symmetry. 2 2 So we keep the U(1)-symmetric terms (m11, ms, λ, λs1, Λ1) and the U(1)-soft-breaking terms 2 (m4, κ2,3 and κ4). Simplifying slightly the notation by using: λs = λs1, Λ = Λ1, we get the potential in the following form

30 1 2 1 2 V = m Φ†Φ + λ Φ†Φ + Λ(Φ†Φ)(χ∗χ) −2 11 2 1 2 2 m χ∗χ + λ (χ∗χ) + κ (Φ†Φ)(χ + χ∗) −2 s s 4 1 2 2 2 3 3 m (χ∗ + χ ) + κ (χ + χ∗ ) + κ (χ + χ∗)(χ∗χ). −2 4 2 3 (2.9)

All the parameters of the potential V are real. Therefore, the potential is explicitly symmetric under the CP transformation Φ Φ†, χ χ∗. We shall call the model with this choice of → → parameters, the constraint SMCS, cSMCS.

2.2.2 Positivity conditions

In order to have a stable minimum, the parameters of the potential need to satisfy the positivity conditions. The potential should be bounded from below, i.e. should not go to negative inﬁnity for large ﬁeld values. As this behavior is dominated by the quartic terms, the cubic terms will not play a role here. Thus the following simple positivity conditions will apply λ, λ > 0, Λ > √2λλ . s − s (2.10)

2.2.3 Extremum conditions

The extremum conditions lead to the following constraints:

v( m2 + v2λ + 2√2w κ + Λw2) = 0, (2.11) − 11 1 4

w ( µ2 + v2Λ + 2w2λ ) + √2[3(w2 w2)κ 1 − 1 s 1 − 2 2 2 2 2 +(3w1 + w2)κ3] + v √2κ4 = 0, (2.12)

w [ µ2 + v2Λ + 2w2λ + 2√2w ( 3κ + κ )] = 0, (2.13) 2 − 2 s 1 − 2 3 where we use the VEV for the singlet scalar ﬁeld in the form: weiξ = w cos ξ + iw sin ξ = 2 2 w1 + iw2 and parameters µ1 and µ2 deﬁned as

µ2 = m2 + 2m2, µ2 = m2 2m2. 1 s 4 2 s − 4 31 Various extrema are possible, among them with vanishing one or two of vacuum expec-

tation parameters v, w1, w2 (discussion in the Appendix 7.2.2). Below we concentrate on the

case with v, w1 and w2 diﬀerent from zero, allowing for a vacuum violating CP.

2.2.4 The CP violating vacuum

Minding the Eq. (2.11), when neither v, w1 nor w2 vanish, an important relation can be obtained via subtracting Eq. (2.13) from the Eq. (2.12),

8m2 cos2 ξ + 6R cos ξ(1 + 2 cos 2ξ) + 2R cos ξ + R = 0, (2.14) − 4 2 3 4 where 2√2v2κ R = √2wκ ,R = √2wκ ,R = 4 cos ξ, 2 2 3 3 4 w all of which are of [mass]2 dimension. In addition we have v2 R = (m2 v2λ w2Λ). 4 w2 11 − 1 −

For a particular case, i.e. R2 = 0, the above equation transforms to, 8m2 cos2 ξ + 2R cos ξ + R = 0. (2.15) − 4 3 4 In ﬁgure 2.1 the regions allowed by Eqs. (2.11),(2.12) and (2.13) of parameters for a

vaccum with v, w1, w2 = 0 are presented. Figure 2.1(a) shows the region of parameters R3, 6 2 R4 and ξ as given by the Eq. (2.15), for a ﬁxed m4 is shown.

In ﬁgure 2.1(b) and (c) the allowed regions of parameters (R3,R4), for R2 = 0, and the

allowed region of the parameters (R2,R4), for R3 = 0 are shown, respectively. These regions 2 are in agreement with the Eq. (2.14), for a ﬁxed m4.

When v and both w1, w2 are diﬀerent from zero and κ4 = 0, the Eq. (2.14) simpliﬁes to 4m2 cos ξ + 3R (1 + 2 cos 2ξ) + R = 0, (2.16) − 4 2 3

For a particular case R2 = 0, Eq. (2.16) transforms to:

2 R3 4m4 cos ξ + R3 = 0, cos ξ = 2 − 4m4 4m2 < R < 4m2. (2.17) → − 4 3 4 2 The regions of the parameters R2, R3 and ξ allowed by Eq. (2.16), for fixed m4, are shown in figure 2.2. Figure 2.3a is a two-dimensional version of the figure 2.2. Another two- 2 dimensional plot showing the allowed regions of R3 and 4m4, for R2 equal zero (see Eq. (2.17)), is presented in figure 2.3b.

32 Figure 2.1: The spontaneously CP violation: the allowed (shaded) regions of the parameters R ,R ,R for 1 < cos ξ < 1 and 4m2 = 500 GeV2. The boarder lines correspond to the 2 3 4 − 4 cos ξ = 1 limits. (a) Plot 3d for parameters R , R and ξ based on Eq. (2.15); (b)Regions ± 3 4 for R3, R4 allowed by Eq. (2.14); (c) Regions for R2 and R4 given by Eq. (2.14).

2.3 Physical states in the Higgs sector

The neutral ﬁeld φ1 from the SU(2) doublet mix with φ2 and φ3 from the singlet. Mass

squared matrix Mmix in the basis of φ1, φ2, φ3 can be written as follows:   M11 M12 M13   Mmix =  M21 M22 M23  , (2.18) M31 M32 M33 where the Mij(i, j = 1, 2, 3) are: 1 M = ( m2 + 3v2λ + w2Λ) 11 2 − 11

M12 = v(w1Λ + √2κ4),

M13 = vw2Λ 1 1 M = m2 m2 + v2Λ + (w2 + 3w2)λ 22 − 4 − 2 s 2 2 1 s +3√2w (κ + κ ) κ v2/√2w2 1 2 3 − 4 1 M = w (2w λ + √2( 3κ + κ )) 23 2 1 s − 2 3 1 1 M = m2 m2 + v2Λ + (w2 + 3w2)λ 33 4 − 2 s 2 1 2 s +√2w ( 3κ + κ ). (2.19) 1 − 2 3 33 (a)

(b) (c)

Figure 2.2: The regions of the parameters R2, R3 and ξ as follows from Eq. (2.16), for ﬁxed 2 2 4m4 = 500 GeV , from diﬀerent perspectives.

Figure 2.3: The correlation between the parameters R , R and m2, for 1 < cos ξ < 1. The 2 3 4 − dashed lines corresponds to the cos ξ = 1 limits, not allowed for CP violation. (a) Shaded ± 2 2 regions for R2 and R3 allowed by Eq. (2.16), at 4m4 = 500 GeV ; (b) Shaded regions for 2 4m4 and R3 allowed by Eq. (2.17), for R2 = 0.

34 The extremum condition have not been applied to get Mij elements presented in Eq. (2.19). When the the extremum condition Eqs. (2.11-2.13) is applied the diagonal elements change to

2 M11 = v λ, 2 w 2 2 2 M22 = 3κ2 + κ3(1 + 2(w1 w2)/w ) √2w1 − κ v2/w2 + 2w2λ , − 4 1 s 2 M33 = 2w2λs. (2.20)

2 Diagonalization of Mmix (2.19) gives the mass-eigenstates h1, h2, h3:     h1 φ1  h  = r  φ  , rM 2 rT = diag(M 2 ,M 2 ,M 2 ),  2   2  mix h1 h2 h3 h3 φ3 (2.21)

We will consider the following mass hierarchy for the neutral Higgses Mh1 < Mh2 . Mh3 .

The rotation matrix r = r1r2r3 depends on three mixing angles (α1, α2, α3). The individual rotation matrices are given by (here and below ci = cos αi, si = sin αi):     c1 s1 0 c2 0 s2

r1 =  s c 0  , r2 =  0 1 0  ,  − 1 1    0 0 1 s 0 c − 2 2  1 0 0    r3 =  0 c3 s3  . (2.22) 0 s c − 3 3 All αi vary over an interval of length π. The full rotation matrix r depends on the mixing angles in the following manner:

r = r1r2r3  c c c s c s s c c s + s s  1 2 3 1 − 1 2 3 1 3 2 1 3 =  c s c c + s s s c s s + c s  . (2.23)  − 2 1 1 3 1 2 3 − 3 1 2 1 3  s c s c c − 2 − 2 3 2 3 The inverse of r can be used to obtain the reverse relation between hi and φi:   c1c2 c2s1 s2 1 − − r− =  c s c s s c c + s s s c s  . (2.24)  3 1 − 1 2 3 1 3 1 2 3 − 2 3  c c s + s s c s s + c s c c 1 3 2 1 3 − 3 1 2 1 3 2 3 35 1 The element (11) of both rotation matrices r and r− are equal to

1 r(11) = r(11)− = c1c2.

The Physical masses are M 2 v2λ, h1 ' 1 q M 2 = (M + M (M + M )2 + 4M 2 ). (2.25) h2,3 2 22 33 ∓ 22 33 23 Two important relations can be read from the above rotation matrices, namely:

h = c c φ + (c s c s s )φ + (c c s + s s )φ (2.26) 1 1 2 1 3 1 − 1 2 3 2 1 3 2 1 3 3 and φ = c c h c s h s h . (2.27) 1 1 2 1 − 2 1 2 − 2 3

These relations describe the composition of the SM-like Higgs boson h1 in terms of the CP-

even (φ1 and φ2) and the CP-odd (φ3) components. It signals the mixing in the model. In the section 2.4, we use these equations to perform a scanning of the relevant regions of

parameters. We shall treat h1 as the 125 GeV Higgs boson.

2.4 Allowed regions of parameters for CP violating vacuum

In what follows, we present results of a numerical analysis of the allowed regions of parameters of the cSMCS model, with the CP violating vacuum, in agreement with the positivity and

extremum conditions as well as the perturbativity conditions. For simplicity the κ4 term is neglected in the main part of analysis. We assume it is negligible, being generated at 1 one loop with strength given by 16π2 κ3Λ [85], where coupling Λ we keep small to ensure perturbativity of our calculation. Moreover, we have checked that neglecting the κ4 term does not change basic properties of the model. We assume v being bounded to the region

246 GeV < v < 247 GeV and that the mass of the lightest Higgs particle h1 lies in range

M [124.00, 127.00] GeV, (2.28) h1 ∈ in agreement with recent LHC results for the Higgs boson [211]. We take masses of two additional, heavier Higgs scalars to be

Mh3 & Mh2 > 150 GeV. (2.29)

36 Figure 2.4: The allowed region for quartic parameters and masses. a) Λ versus λs is presented.

In the light shaded region only positivity conditions were applied. b) Mh2 versus λs and c)

Mh3 versus λs.

The parameters of the Higgs sector are varied in the following ranges:

1 < Λ < 1, 0 < λ < 1, 1 < ρ < 1, 0 < ξ < π, (2.30) − s − 2,3 where we used dimensionless parameters ρ2,3 = κ2,3/w. From assumption that M 2 m2 v2λ, and M 125 GeV we estimate range of λ to h1 ≈ 11 ≈ h1 ≈ be:

0.2 < λ1 < 0.3. (2.31) In order to have an appropriate range for the parameter v we set the ranges of remaining quadratic variables as follows:

90000 GeV2 < µ2, µ2, m2 < 90000 GeV2. (2.32) − 1 2 11

The range of values of VEV for the singlet, w, was not set in the analysis - it was derived from the scan. The obtained allowed regions for quartic parameters and masses are shown

in ﬁgure 2.4. In ﬁgure 2.4a, the allowed region of Λ versus λs is shown. We got strong

constraints on the singlet self coupling λs, to be greater than 0.2, and on the doublet-singlet coupling Λ , to be below 0.2. Note, that the constraints of Λ and λ arise mainly from the | | s mass limits (2.28). The positivity condition can constrain only the region of negative Λ, as follows from (2.10). This correspond to a light shadowed region in the plot ﬁgure 2.4a.

Figures 2.4b and 2.4c show the allowed regions of masses for h2 and h3, respectively. For

higher λs larger masses are possible, respectively up to 650 and 800 GeV. Note, that larger

37 Figure 2.5: The allowed regions fo cubic parameters: a) ρ2 versus ρ3, b) ρ2 versus ξ, c) ρ3 versus ξ. values of quadratic parameters µ2 , µ2 , beyond the range given in Eq. (2.61), would lead | 1| | 2| to larger allowed masses for scalars h2, h3.

The allowed regions for the cubic parameters ρ2 and ρ3 are important from point of view of CP violation condition Eq. (2.16), they are shown in figure 2.5. Figure 2.5a shows the allowed by scan over parameters the (ρ2, ρ3) region. Note, that it reproduces roughly results presented on figure 2.3a, obtained solely from the extremum conditions Eq. (2.16). In figures

2.5b,c the allowed regions of the phase ξ versus ρ2,ρ3 are presented. In both panels there are two allowed regions, symmetric with respect to ξ π/2, with a gap around the central ∼ value ξ = π/2 (which corresponds to w 0). 1 ≈ The result of the scanning over other potential parameters are shown in figure 2.6. In the figure 2.6a the allowed region of the (w, v) plane is presented. The VEV of singlet w reaches the highest value of 800 GeV, however the most points are concentrated at low value of w (2 - 50 GeV). This is related to the fact that we limit λ m2 /v2 (Eq. (2.31)), what ≈ 11 leads to the small w according to the Eq. (2.11). Domination of small w is seen also in the figure 2.6c, where the allowed region of w as a function of Λ is shown as well as in figure2.6d, 2 where the w as a function of m11 is presented. Here, the concentration of points is observed 2 for value of m11, close to the mass square of the SM Higgs boson, as expected. The allowed regions of ξ versus Λ is shown in figure 2.6b, where a symmetry and a gap for ξ π/2 is ∼ seen in the ξ distributions, as discussed above.

The allowed regions of masses of the Higgs bosons h2 and h3 are shown in ﬁgure 2.7, once more showing the symmetry and the gap in the ξ distribution. The maximal values

38 Figure 2.6: The allowed regions: (a) the correlation between v and w (b) the correlation 2 between Λ and ξ (c) the correlation between w and Λ and (d) w versus m11.

39 Figure 2.7: Allowed regions for Higgs masses: a) Mh2 versus ξ and b) Mh3 versus ξ.

of masses, around 650 GeV (h2) and 800 (h3) GeV, can be reached for ξ around 1 and symmetrically around 2 radians (i.e. for ξ equal 1.5 0.5 radians). ±

2.5 J type-invariants

In this section, we calculate the Jarlskog type invariant for the cSMCS model. C. Jarlskog has introduced such quantity originally for description of the mixing quark sector [57]. It has been shown that the Jarlskog type invariants can be used for the scalars to ﬂag the existence of the CP-violation in the models with an extended scalar sector [82, 83, 84]. The interpretation of Jarlskog invariant in 2HDM was performed in e.g. [24, 88, 22], with a conclusion that if the Jarlskog quantity J1 is diﬀerent from zero then there is a CP violation in 2HDM.

The J1-type considered by us for the model can be deﬁned by mixing elements of the squared mass matrix [2.18] as follows:

J = M M (M M ) + M (M 2 M 2 ), (2.33) 1 12 13 22 − 33 23 13 − 12 which for vacuum state, described by the Eq. (2.16), leads to

2 2 2 2 2 2 2 J1 = 2Λ v w m4 sin ξ cos ξ = 2Λ v w1w2m4, (2.34)

2 with m4 given by Eq. (2.16). So, in order to have J1 = 0 the non-vanishing complex VEV 6 2 of a singlet is needed as well as the U(1)-violating quadratic term - m4. As follows from Eq.

40 6 Figure 2.8: J1 invariant (in GeV ) as a function of Λ in linear (a) and log (b) scale.

2 (2.16), nonzero value of m4 means non vanishing of at least one cubic term for the singlet. Further - a interaction between a doublet and singlet is necessary. It is well know that in 5 the SM the Jarlskog invariant is of the order of 10− [81]. The ﬁgure 3.7 shows the range of 6 6 the dimensionful (GeV ) invariant J1 for the considered model. By normalizing it by v , as 6 we choose v to represent temperature of the EW TEW , we get the highest value for J1/v 3 2 | | around 10− . It can be larger for larger m . | 4| In the Appendix 7.2.6, we calculate the Jarlskog type invariant for the case with κ = 0. 4 6

2.6 Comparison with data

In the considered cSMCS model we examine the SM-like scenario with the lightest neutral Higgs particle being the 125 GeV Higgs particle observed at LHC. Not only mass, but also direct couplings to fundamental particles should be close to the ones measured at the LHC. We found that this is indeed a case for our model. Below we collect main formulas and constraints from the model as coming from the LHC data on 125 GeV Higgs bosons and measurement of oblique corrections. We ﬁnish this section by presenting 7 benchmarks.

2.6.1 Properties of h1 Higgs boson in the light of LHC data

The couplings of the lightest Higgs particle (h1) to the quarks and the gauge bosons in the cSMCS model1, as compared with the corresponding couplings of the SM Higgs, are modiﬁed

(suppressed) by a factor r11 (see Appendix 7.2.1). In particular, for the Higgs boson decay

1VQ are not included in this calculation, as they are heavy and will decouple, see Appendix 7.2.1.

41 into vector bosons (V = Z,W ) we have

2 Γ(h VV ∗) = r Γ(H VV ∗). (2.35) 1 → 11 SM → Further constraints on the parameters of our model can be obtained by comparing the decay of the light Higgs boson h1 and of the SM Higgs boson into γγ. This is done using the signal strength Rγγ:

σ(gg h ) BR(h γγ) = → 1 1 → Rγγ σ(gg H ) BR(H γγ) → SM SM → Γ(h gg) BR(h γγ) = 1 → 1 → , (2.36) Γ(H gg) BR(H γγ) SM → SM → taking into account that the production of the Higgs bosons in the LHC is dominated by the gluon fusion processes and that the narrow width approximation can be applied. The Higgs h1 decay width into gluons is given by:

Γ(h gg) = r2 Γ(H gg). (2.37) 1 → 11 SM →

The main contribution in the one-loop coupling of h1 to photons is due to the W boson and top quark, and therefore in our model the corresponding amplitude and the decay rate are equal to (see Appendix 7.2.4):

A(h γγ) = r (ASM + ASM ) 1 → 11 W t Γ(h γγ) = r2 Γ(H γγ). (2.38) → 1 → 11 SM →

Since the total width of the light Higgs boson h1 is given by

Γ r2 ΓSM , (2.39) tot ≈ 11 tot the signal strengths Eq. (4.29) is equal to

r2 . (2.40) Rγγ ≈ 11 Both and are smaller than 1, by the same amount, compatible with recent LHC RVV Rγγ measurements.

Note, that the total decay width for heavier Higgses can be signiﬁcantly modiﬁed with respect to the SM, if hi (heavier) can decay into the lighter hj particles, since

2 SM X Γtot ri1Γtot + Γhi hj hj . ≈ → i=2,3;j=1,2,3;i>j

42 The partial decay width for such decay channels h h h , where i > j, is given by i → j j

g2 4M 2 !1/2 Γ(h h h ) = hihj hj 1 hj , (2.41) i j j 32πM M 2 → hi − hi

where ghihj hj is the coupling between Higgs bosons, see Appendix 7.2.5 for corresponding expressions. In the considered model the signal strength for γγ for h1 as well as for h2, h3 can only be smaller than (or equal to) 1. Below, we present our predictions for several benchmarks, all for r2 0.81 0.98, in agreement with the LHC data on the 125 GeV 11 ∼ − Higgs couplings to ZZ (see Appendix 7.1).

2.6.2 Oblique parameters S,T,U

The oblique parameters S, T and U provide an indirect probe of physics beyond the SM for theories with SU(2) U(1) gauge content. The additional particles introduce corrections L × Y to the gauge boson propagators in the SM that can be parametrized by these parameters. They can then be used to constrain models of new physics beyond the SM. These parameters are only sensitive to new physics that contributes to the oblique corrections, i.e. the vacuum polarization corrections to four-fermion scattering processes [89]. The oblique corrections can be parameterized in terms of four vacuum polarization functions: the self-energies of the photon, Z boson, and W boson, and the mixing between the photon and the Z boson induced by loop diagrams,

2 2 Πγγ(q ) = q Πγγ0 (0) + ..., (2.42) 2 2 ΠZγ(q ) = q ΠZγ0 (0) + ..., (2.43) 2 2 ΠZZ (q ) = ΠZZ (0) + q ΠZZ0 (0) + ..., (2.44) 2 2 ΠWW (q ) = ΠWW (0) + q ΠWW0 (0) + ..., (2.45)

2 where Π0 denotes the derivative of the vacuum polarization function with respect to q . The constant pieces of Πγγ and ΠZγ are zero because of the renormalization conditions. New physics contributions is encoded in δΠ(q2),

Π(q2) = ΠSM (q2) + δΠ(q2). (2.46)

43 The S, T and U parameters deﬁne as

2 2 2 2 4swcw cw sw S = ( )[δΠZZ0 (0) − δΠZγ0 (0) δΠγγ0 (0)], (2.47) α − swcw − 1 δΠWW (0) δΠZZ (0) T = ( )[ 2 2 ], (2.48) α MW − MZ 2 4sw 2 2 U = ( ) δΠ0 (0) c δΠ0 (0) 2s c δΠ0 (0) s δΠ0 (0) , (2.49) α WW − w ZZ − w w Zγ − w γγ 2 where α = e /(4π) is the fine-structure constant, sw = sin θw, cw = cos θw are the sine and cosine, respectively, of the weak mixing angle. T is sensitive to the isospin violation, i.e. it measures the difference between the new physics contributions of neutral and charged current processes at low energies, while S gives new physics contributions to neutral current processes at different energy scales. U is generally small in beyond SM models. The latest values of the oblique parameters, determined from a fit with reference mass-values of top and Higgs boson Mt,ref = 173 GeV and Mh,ref = 125 GeV are [90], see also Appendix 7.1.

S = 0.05 0.11,T = 0.09 0.13,U = 0.01 0.11. (2.50) ± ± ± The oblique parameters in cSMCS model, following the method introduced in [89], are described in the Appendix 7.2.3. In section 2.6.3, we present our benchmarks and will show they are in agreement with current data for S and T parameters.

2.6.3 Benchmarks

Here we present seven benchmarks for the considered model showing agreement with the above mentioned constraints. Properties of benchmarks are presented in table 2.1 and 2.2.

The table 2.1 shows mixing angle α1,2,3 and masses for h1,h2 and h3. The highest mass of h3 is 760 GeV while the lowest one is 179 GeV. The table contains as well the prediction of the considered model for the S and T parameters (all being in agreement with the current 6 6 data within 3σ) and J1/v invariant. The Jarlskog type invariant J1/v can be positive or 5 4 negative, with range of its (absolute) value from 3.5 10− to 9.5 10− . × × In table 2.2 the calculated for h as well as h and h are presented together with their Rγγ 1 2 3 total widths. The largest decay widths, from 7 to 17 GeV, are obtained for benchmark A3,

A4 and A5 for relatively heavy h3 (for masses around 600 GeV). Note, that only benchmark points A6 and A7 corresponds to relatively light (mass below 200 GeV) Higgs bosons h2 and h3 and only these two benchmarks arise from negative Λ, as presented in ﬁgure 2.9. We have used these benchmarks for prediction of the production of Higgs bosons h1,h2 and h3 at the LHC, see chapter 3.

44 6 B α1 α2 α3 Mh1 Mh2 Mh3 S T J1/v 4 A1 -0.047 -0.053 1.294 124.64 652.375 759.984 -0.072 -0.094 -2.2 10− × 4 A2 -0.048 0.084 0.084 124.26 512.511 712.407 -0.001 -0.039 7.2 10− × 4 A3 0.078 0.297 0.364 124.27 582.895 650.531 0.003 -0.046 4.5 10− × 4 A4 0.006 -0.276 0.188 125.86 466.439 568.059 -0.013 -0.169 -9.5 10− × 6 A5 0.062 -0.436 0.808 125.21 303.545 582.496 0.002 -0.409 5.0 10− × 5 A6 -0.210 0.358 0.056 124.92 181.032 188.82 0.003 -0.010 -4.0 10− × 5 A7 -0.205 0.403 0.057 125.01 175.45 178.52 0.002 -0.020 -3.5 10− × Table 2.1: Benchmark points A1 A7, masses are given in GeV. −

h1 h2 h3 h1 h2 h3 B Rγγ Rγγ Rγγ Γtot Γtot Γtot A1 0.98 0.0021 0.0028 0.0042 0.304 0.781 A2 0.98 0.0021 0.0070 0.0042 0.145 1.31 A3 0.98 0.0055 0.085 0.0042 0.566 12.24 5 A4 0.92 3.3 10− 0.074 0.0043 0.001 7.08 × A5 0.81 0.0029 0.17 0.0043 0.002 17.51 A6 0.82 0.19 0.11 0.0043 0.119 0.163 A7 0.81 0.18 0.15 0.0043 0.871 0.083

Table 2.2: Values of Rγγ and Γtot for benchmark points A1 A7. The Signal strength of the +0.27 − +0.25 SM Higgs boson into γγ from ATLAS [95] (1.14 0.25) and CMS [96] (1.11 0.23). The total − − widths are given in GeV (see Appendix 7.1).

2.7 The electroweak phase transition

At very high temperatures, far above the EW scale, the EW gauge symmetry is unbroken with no baryon number violation. The Universe cools down and expands and near the EW scale bubbles of broken EW symmetry appear and subsequently expand in the surrounding unbroken phase. The EW baryon asymmetry can be realized if this change of phase proceeds by a ﬁrst-order phase transition. If the phase transition is strongly ﬁrst-order, the baryon violating processes are out of equilibrium in the bubble walls and a net baryon number can be

45 1.0x10-3

J 0.0 A5 1 A6 A7

A4 -1.0x10-3 -0.2 -0.1 0.0 0.1 0.2 Λ

6 Figure 2.9: J1 invariant (divided by v ) for the benchmarks points. generated during the phase transition. Phase transition is strong enough when [104, 105, 106], v Tc 1, (2.51) Tc ≥ where Tc corresponds to the critical temperature, see discussion in section 1.3. To study the EW phase transition in the present model, we express the complex scalar

χ in terms of its real and imaginary parts, χ = (φ2 + iφ3)/√2. For the potential at zero temperature we have

2 1 2 1 2 µ1 2 V (T ) = m Φ†Φ + λ Φ†Φ φ 0 −2 11 2 1 − 4 2 2 µ2 2 1 2 2 φ + Λ(Φ†Φ)(φ + φ ) − 4 3 2 2 3 1 2 2 2 1 3 2 + λs(φ + φ ) + κ2(φ 3φ2φ ) 4 2 3 √2 2 − 3

1 3 2 + κ3(φ + φ2φ ) + √2κ4(Φ†Φ)φ2. (2.52) √2 2 3 The one-loop thermal corrections to the eﬀective potential at ﬁnite temperature T are (see reference [106] for review),

4 2 X niT m ∆V = I i , (2.53) thermal 2π2 B,F T 2 i where Z ∞ 2 h √x2+yi IB,F (y) = dx x ln 1 e− , (2.54) 0 ∓

46 the minus and plus sign corresponds to the bosons and the fermions, respectively. In Eq.

(2.53), mi is the field-dependent mass and ni is the number of degrees of freedom, see Appendix 7.3.2. Since the barrier is produced at tree-level, it is sufficient to include the high temperature expansion in the one-loop thermal potential (i.e. only keeping T 2 terms) [107, 108]. Therefore, the one-loop thermal potential using the high temperature approximation is given by, 1 1 µ2 µ2 V (T ) = m2 Φ Φ + λ(Φ Φ) + 1 φ2 + 2 φ2 2 11 † 2 † 4 2 4 3 1 1 Λ(Φ Φ)(φ2 + φ2) + λ (φ2 + φ2)2 2 † 2 3 4 s 2 3 1 3 2 1 3 2 +κ2 (φ 3φ2φ ) + κ3 (φ + φ2φ ) √2 2 − 3 √2 2 3 T 2 +√2κ (Φ Φ)φ + κ φ , (2.55) 4 † 2 34 3 2 where, 2m2 + m2 + 2m2 T 2 m2 = m2 + (3λ + Λ + W Z t ) , 11 − 11 2v2 3 1 1 T 2 µ2 = µ2 + (Λ + 2λ ) , 2 1 −2 1 s 3 1 1 T 2 µ2 = µ2 + (Λ + 2λ ) , 2 2 −2 2 s 3 κ34 = √2(κ3 + κ4). (2.56) The extremum conditions of the effective potential Eq. (2.53) at temperature T , with respect to the fields φ1, φ2 and φ3 are

2 2 2 v(m11 + λv + Λw + 2√2κ4w1) = 0, (2.57)

w (µ2 + Λv2 + 2λ w2) + √2 3κ (w2 w2) 1 1 s 2 1 − 2 2 + κ (3w2 + w2) + κ v2 + κ T 2 = 0, (2.58) 3 1 2 4 3 34

w [µ2 + Λv2 + 2λ w2 + 2√2( 3κ + κ )w ] = 0. (2.59) 2 2 s − 2 3 1 The solution of the Eqs. (2.57),(2.58) and (2.59) at extreme temperatures is as follows,

κ34 v = 0, w1 − , w2 0. (2.60) ≈ 2λs + Λ ≈

At extreme temperatures the scalar component φ3 and VQ decouple from the model because their contribution to the finite temperature effective potential are Boltzmann suppressed and therefore, the potential is similar to the SM plus a real singlet [109]. Now, we will scan over the parameter space of the model under the condition for strong first order EWPT,

47 Veff (vTc ,Tc) = Veff (0,Tc) (i.e. two degenerate minima at critical temperature, Tc),

v /T 1. Tc c ≥

We consider Tc to be smaller than 250 GeV and vTc below its zero-temperature value v0 = 246 GeV. In our scanning the following regions for parameters space of the model fulﬁlling positivity and unitarity conditions is considered, namely:

Λ [ 0.23, 0.23], λ [0, 1], ρ [ 1, 1], ξ [0, π], ∈ − s ∈ 2,3,4 ∈ − ∈ m2 , µ2 , µ2 [ 90000, 90000]GeV2, (2.61) 11 1 2 ∈ − and we take λ1 in the range, see Eq. (2.31):

0.2 < λ1 < 0.3. (2.62)

The masses of Higgs scalars Mh2 and Mh3 are taken to be

Mh3 & Mh2 > 150 GeV. (2.63) In section 2, we have shown that these ranges of parameters are in agreement with LHC data and measurements of the oblique parameters. The results of our scan are shown in the ﬁgure

2.10. In ﬁgure 2.10(a) the allowed region of vTc /Tc as a function of Tc is shown. Within the

interval 100 < Tc < 200 the ratio vTc /Tc ratio can reach 2.5. The figure 2.10(b) shows that 3 v /T 1 is possible for ρ > 10− . The similar results are obtained for ρ and ρ . Tc c ≥ | 3| 2 4 We see that the strongly enough first-ordered EW phase transition is possible in our model. Since the out of equilibrium condition can be achieved for strong enough first-order phase transition, in the bubble walls, we conclude that a successful BAU in our model is possible [106].

2.8 Baryogenesis

In this section we describe the baryon asymmetry resulting from a mixing of the SM quarks and heavy VQ [109, 103]. We follow reference [103], with iso-doublet VQ.

The mass terms in the presence of the complex singlet are (see Eq. (2.1)):

(V , χ) = λ χQ V + MV V + h.c. (2.64) LY q V L R L R In general the couplings of all three generations of SM quarks to VQ should be considered, nevertheless, since only one linear combination of these will actually mix with the VQ, we

48 Figure 2.10: The allowed regions of critical temperature T , v and ρ for strongly ﬁrst- c Tc | 3| order phase transition. (a) (T ,v /T ) and (b) ( ρ , v /T ). The scatter points are selected c Tc c | 3| Tc c to satisfy the criterion, (v /T ) 1 (see text for details). Tc c ≥ consider only one (the heaviest) generation QL. In order to estimate the resulting baryon asymmetry we will follow the spontaneous baryogenesis scenario [110]. This mechanism works when there is an interaction of the form

= ∂ ξJ µ , (2.65) L µ B

µ where JB is the baryon current. Therefore, to generate baryon asymmetry, the phase of the singlet VEV should be time-dependent, otherwise, such constant phase can be easily rotated away with the redeﬁning the VL and VR [103]. On diagonalizing the mass terms, we obtain mass eigenstates Q and V ,

 0      QL a b QL         =     , (2.66) 0 V b∗ a∗ V L − L where,

" # 1/2 λ w2 − a = 1 + V , M " # 1/2 λ w λ w2 − b = V 1 + V e iξ. (2.67) M M −

49 Diagonalizing the quark mass matrix results in some non-diagonal kinetic terms. In addition, couple of time-dependent terms appear in the Lagrangian (see Appendix Eq.2.67), namely

µ µ QLiγ ∂µQL + V Liγ ∂µVL µ µ Q iγ ∂ Q0 + V iγ ∂ V 0 + ∆ + const. → 0L µ L 0L µ L Lk (2.68)

Since the CP violation disappears for a constant phase, in calculation of the baryon asymmetry only the following kinetic term needs to be considered

2 2 λV w 0 0 ∆ = ξ˙(Q γ Q0 V γ V 0 ), (2.69) Lk − M 2 0L L − 0L L with ξ˙ being the time derivation of the phase. During the weak phase transition, CP violation in the scalar potential will produce a non-zero spatial average for the time derivative of phase, which splits particle-antiparticle energy levels inside the domain walls produced during the weak phase transition. Such term increases the baryon density of the Universe. Conventionally, the amount of BAU is calculated via the following relation, Z Γ (T ) n = N sph µ dt, (2.70) B − f 2T B

where Nf is the number of ﬂavors in the model. The sphaleron rate, Γsph, is deﬁned as 4 Γsph = K(αW T ) in the symmetric phase and K is the numerical factor, that it has been

estimated to be between 0.1 and 1 [64]. The chemical potential, µB, for the third generation is as follows [103], 5 λ2 w2 µ = V ξ.˙ (2.71) B −6 M 2 We assume that the mass parameter M is much larger than the temperature, i.e. M T . ≥ Thus the sphaleron ﬂuctuations cannot produce VQ pairs and we get the number density of

baryons nB at the temperature T as follows 5Kα4 λ2 w2 n = W V δξT 3, (2.72) B 2 M 2 where δξ is the total change of the phase ξ. The BAU is determined via the ratio of the baryon number to the entropy [42]. The entropy density deﬁnes as

2π2 s = g T 3, (2.73) 45 ∗ therefore, n 225Kα4 λ2 w2 B = W V δξ, (2.74) 2 2 s 4π g∗ M

50 2 where the SU(2) gauge coupling α = 3.4 10− and g∗ 100 is the eﬀective number W × ∼ of degrees of freedom in the thermal equilibrium. The observations from WMAP gives the

following value for nB/s ratio [45, 46]

nB 11 = 8.7 0.3 10− . (2.75) s ± × From Eq. (2.74) and Eq. (2.75) we get,

2 2 λV w 3 K δξ = 1.14 0.3 10− . (2.76) M 2 ± × The numerical analysis of the equations (2.74) and (2.75) have been performed via scanning the involving parameters in the following ranges,

M [0.3, 13]TeV, ∈ λ [0, 1], V ∈ w [2, 400]GeV, ∈ δξ [0, π], (2.77) ∈ with numerical factor K = 1. The range of the parameter w is obtained from our previous analyze presented in section 2.4 (see ﬁgure 2.6). Figure 2.11 illustrates the parameter space

allowing the generation of observed BAU for the nB/s ratio within 2σ. The result were obtained from scanning in the ranges given by Eq. (2.77) with the central value of Eq. (2.75). In ﬁgure 2.11(a), the distribution of the parameters (δξ, w) is shown. The parameter space for the (w, M) points is shown in ﬁgure 2.11(b). Since M and w are independent parameters, their correlation is a direct consequence of the constraint (2.76). Based on these results, we conclude that our model provides successful BAU.

2.9 Conclusion and outlook

In this chapter we have presented the cSMCS - an extension of the SM containing a complex singlet with a non-zero complex VEV and a pair of heavy iso-doublet VQ, which allows for the spontaneous CP violation. We considered the potential with a softly broken global U(1) symmetry. Within our model diﬀerent vacua can be realized, here we have focused on the case with the CP violating vacuum. We have derived a simple condition for existence of such vacuum, and found that at least one cubic (or mixing) term for χ is needed in order to have spontaneous CP violation. In this model there are three neutral Higgs particles. The model can easily accommodate the SM-like Higgs, with mass around 125 GeV, in agreement with

51 Figure 2.11: The allowed region for δξ, w and M in the Eq. (2.75) of the given ranges, i.e. Eq. (2.77), for acceptable BAU within 2σ. (a) is the correlation of δξ and w. (b) is the correlation of w versus M.

LHC data and measurements of the oblique parameters. In the present work the possibility of the ﬁrst-order EW phase transition for the cSMCS model is investigated. This model provides a strong enough ﬁrst-order EW phase transition via the soft breaking terms (κ2,

κ3 and/or κ4) in the potential, to suppress the baryon-violating sphaleron process. The parameter space of the model for the valid regions of BAU is scanned, concluding that the enlargement of the cSMCS model successfully predict an acceptable value for BAU.

We provide seven benchmarks, in agreement with collider data, for future tests of the model, see chapter 3.

52 Chapter 3

A phenomenological study on the production of Higgs bosons in the cSMCS model at the LHC

Throughout the years numerous theoretical and phenomenological attempts have been made, trying to explore diﬀerent aspects of the production of the Higgs particles at the LHC, within the SM, e.g. the references [111, 112, 113, 114, 115, 116, 117, 118, 119]. Here, we study

the production of the Higgs bosons h1, h2, h3 in the cSMCS model at the LHC, using kt- factorization framework. As we have shown in the chapter 2, the cSMCS model contains three neutral Higgs particles which the lightest is the h1 =125 GeV Higgs boson found at the LHC and the other two Higgs scalars, hi, i = 2, 3 are taken to have masses

Mh3 & Mh2 > 150 GeV.

The main contribution to the cross-section for the Higgs bosons production at the LHC,

P + P H + X, 1 2 → gives the so-called gluon-gluon fusion sub-process, i.e.

g∗(k ) + g∗(k ) H(p), (3.1) 1 2 → see the ﬁgure 3.1 part (a). Also, the Higgs boson production accompanied with a single jet or double jets can be traced back to the weak-boson fusion processes (ﬁgure 3.1 part (b))

and g∗ + g∗ H + g, g∗ + q∗ H + q and q∗ +q ¯∗ H + g sub-processes (parts (c), (d) and → → → (e) of the ﬁgure 3.1, respectively), which are expected to give roughly one tenth of the total

53 Higgs production rate. It has been shown that one can replace such complicated calculation by using a higher-order correction factor (i.e. the K-factor) [111]. In principle, one has to include the contributions of all quark ﬂavors in such diagrams. However, since the SM Higgs boson coupling to the top quark is considerably stronger compared to the other quarks, we consider only top-quark loops in our calculation.

푞′ 푔 푞 푡 푊±/푍0 퐻

(a) (b)

Figure 3.1: The main contributing sub-processes in the total cross-section for the production of the Higgs bosons at the LHC.

In the collinear factorization framework, the total cross-section for the production of a Higgs boson can be written as the partonic cross-section for the involving sub-process

(ˆσgg H ), times the probability of appearing that particular partonic conﬁguration at the top → of the evolution ladder of the individual hadrons, i.e.

Z 1 Z 1 dx1 dx2 2 2 σP +P H+X = x1g(x1, µ1) x2g(x2, µ2) → 0 x1 0 x2

2 2 2 2 σˆgg H (x1, k1,t = 0, µ1; x2, k2,t = 0, µ2). (3.2) × →

2 The single-scaled (gluonic) parton distribution functions (PDF), g(xi, µi ), are the solutions of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [120, 121, 122, 123]. These functions parametrize the probability of ﬁnding a gluon, emitting from the ith hadron and carrying the fraction xi of its longitudinal momentum. parameters µi are the ultra-violet cutoﬀs, related to the virtuality of the exchanged gluon during the inelastic

54 scattering. In the Eq. (3.2) ki,t are the transverse momenta of the incoming gluons. Ne- glecting the transverse momentum contributions of the incoming partons can seriously lower the precision of the calculations, predominantly for the event with the high center-of-mass energy and the small-x regions [124, 125, 126, 127, 111]. Knowing this, have brought up the necessity of introducing transverse momentum dependent parton distribution functions (TMD PDF), notably the Ciafaloni-Catani-Fiorani-Marchesini (CCFM) evolution equation [128, 129, 130, 131, 132], and the Balitski-Fadin-Kuraev-Lipatov (BFKL) evolution equation [133, 134, 135, 136, 137]. Other approach based on the unintegrated parton distribution

functions (UPDF) with kt-factorization are the leading order (LO) Kimber-Martin-Ryskin (KMR) and next-to-leading order (NLO) Martin-Ryskin-Watt (MRW) formalisms [124, 125]. Recently, it has been shown that these UPDF, specially in the KMR formalism, provide successful descriptions of the existing high energy experimental data [138, 139, 140, 141, 142].

In this chapter, we calculate the total cross-section for production of the Higgs particles in

the cSMCS. First, we calculate the total cross-section for the production of HSM particle and compared the results with the existing SM predictions from other theoretical analysis and also with the experimental data from the CMS and the ATLAS collaborations [143, 144, 145].

Afterwards, we present our prediction for the Higgs particles h1, h2, h3 production, going through the benchmarks of the cSMCS (see the section 2.6.3), which are in agreement with the latest measurements of the EWPT (i.e. the S and T parameters) and the signal strength, h1 . Rγγ

3.1 Calculation of the Higgs production cross-section

Assuming that the gluons entering the g∗ + g∗ H sub-process have some non-negligible → transverse momenta, the total cross-section for the Higgs particle production, using the

deﬁnition of the kt-factorization [146], is given by

Z µ2 2 2 dkt 2 2 a(x, µ ) = 2 fa(x, kt , µ ), (3.3) kt

where a(x, µ2) represents the solutions of the DGLAP evolution equations for both quarks 2 2 2 2 and qluons, i.e. xq(x, µ ) and xg(x, µ ) respectively. Here, the fa(x, kt , µ ) are the corresponding UPDF of KMR formalism (for more description of the structure and the kinematics of the UPDF see the references [124, 125, 140]). Thus the Eq. (3.2) can be rewritten as

55 follows:

Z 1 Z 1 Z 2 Z 2 dx1 dx2 ∞ dk1,t ∞ dk2,t σP +P H+X = 2 2 → 0 x1 0 x2 0 k1,t 0 k2,t f (x , k2 , µ2)f (x , k2 , µ2) × g 1 1,t 1 g 2 2,t 2 2 2 2 2 σˆgg H (x1, k1,t, µ1; x2, k2,t, µ2). (3.4) × →

The partonic cross-section,σ ˆgg H , is deﬁned as →

dφgg H 2 dσˆgg H = → (g∗(k1) + g∗(k2) H(p)) , (3.5) → Fgg H |M → | → where ki and p respectively represent the 4-momenta of the incoming gluons and the produced

Higgs boson. dφgg H and Fgg H are the corresponding particle phase space and the ﬂux → → factor, 3 d p (4) dφgg H = δ (k1 + k2 p) , (3.6) → 2E −

Fgg H = x1x2s, (3.7) → with s being the center of mass energy squared,

s = (p + p )2 = 2p p , 1 2 1 · 2 in the inﬁnite momentum frame (where one can safely neglect the masses of the incoming hadrons in comparison with their momenta (pi mi)). The dφgg H can be expressed in → terms of the transverse momenta of the produced Higgs boson pt, its rapidity yH , and the azimuthal angles of its emission, ϕ,

d3p π dϕ = dp2 dy . (3.8) 2E 2 t H 2π

In the Eq. (3.5), is the matrix element of the g∗(k ) + g∗(k ) H(p) sub-process, equal M 1 2 → to (see the Appendix 7.4):

α2 (µ2) G 2 = S F τ 2 D(τ) 2 (m2 + p2)2 cos2ϕ. |M| 288π2 √2 | | H t (3.9)

In a high-energy inelastic collision at the LHC, one can consider the following kinematics in the center-of-mass frame √s p = (1, 0, 0, 1), i 2 ± 2 2 ki = xiPi + ki, , ki, = ki,t, i = 1, 2 , (3.10) ⊥ ⊥ − 56 and express the law of the transverse momentum conservation for the g∗(k )+g∗(k ) H(p) 1 2 → sub-process as:

k1, + k2, = p , (3.11) ⊥ ⊥ ⊥ 2 2 with p = pt being the transverse momentum of the produced Higgs boson. The longitudi- ⊥ − nal fractions x can be expressed by the transverse mass of the Higgs boson, m2 m2 +p2, i H,t ≡ H t its rapidity and the parameter s,

mH,t x = e+yH , 1 √s mH,t y x = e− H . (3.12) 2 √s

Putting the above formulas together, we derive the master equation for the production of the Higgs bosons: Z 2 2 GF dk1,t dk2,t dϕ 2 σP +P H+X = 2 2 dyH cos ϕ → √2 k1,t k2,t 2π f (x , k2 , µ2) f (x , k2 , µ2) × g 1 1,t g 2 2,t 2 2 2 2 αS(µ ) τ D(τ) 2 22 | 2| mH + pt . (3.13) × 144π x1x2smH 2 2 To determine the density functions of the incoming gluons, fg(xi, ki,t, µ ). We utilize the KMR formalism and obtain

2 Z zmax 2 2 2 2 αS(kt ) fg(x, kt , µ ) = Tg(kt , µ ) dz 2π x X x x [P (LO)(z) q , k2 × gq z z t q x x +P (LO)(z) g , k2 ]. (3.14) gg z z t

The variable zmax = µ/(µ+kt) describes the angular ordering constraint, as a consequence of 2 2 the color coherence eﬀect of successive gluonic emissions [147]. Ta(kt , µ ) is the probability of survival, which limits the parton emissions between the scales kt and µ. It factors over the virtual contributions from the gluonic LO DGLAP equation and can be deﬁned as:

2 Z µ α (k2) dk2 Z zmax T (k2, µ2) = exp[ S dz g t 2 − 2 2π k kt 0 P (LO)(z) + n P (LO)(z)], (3.15) × gg f gq (LO) with nf being the number of active quark ﬂavors. Pab (z) are the LO splitting functions, parameterizing the probability of a parton with the longitudinal momentum fraction x to be

emitted from a parent parton with the fraction x0, z = x/x0 [142, 148].

57 In the following section, we will introduce some of the numerical methods that have been used in the calculation of the master Eq. (3.13) to predict the total cross-section for the production of the Higgs bosons of the cSMCS, with the lightest being the SM-like Higgs boson.

3.2 Numerical analysis

We use the UPDF of KMR, Eq. (3.14), to solve numerically the master Eq. (3.13), utilizing the VEGAS algorithm in the Monte-Carlo integration [149]. The required PDF for the prepa- ration of these UPDF are provided in the form of libraries, e.g. the MMHT2014 libraries, the reference [150], where the single-scaled solutions of the DGLAP evolution equations have

been ﬁtted to the experimental data on the F2 structure function from e-p deep inelastic scattering and the high-energy hadron-hadron scatterings. We chose the hard-scale of the UPDF as the transverse mass of the produced Higgs boson, i.e.

2 2 1/2 µ = (mH + pt ) . (3.16)

One should note that the upper and the lower boundaries of the transverse momentum integrations in the Eq. (3.13) are respectively and zero. Nevertheless, since the KMR ∞ UPDF rapidly converge to zero in the kt > µ domain, it is safe to introduce an upper bound for these integrations in the following form

k = µ 4(m2 + p2 )1/2. (3.17) t,max max ≡ H t,max Further domain has no inﬂuence on our results. On the other hand, it is important to note

that the UPDF of the kt-factorization are being deﬁned only in the QCD perturbative regime,

i.e. for kt > µ0 with µ0 = 1 GeV, as the minimum scale for which the DGLAP evolution of the integrated PDF is valid. We have to deﬁne our treatment of these distribution functions

in the non-perturbative region, kt < µ0. A natural choice to by-pass this obstacle is to fulﬁll the requirement that 2 2 2 lim fg(xi, ki,t, µ ) ki,t. k2 0 ∼ i,t→ So, for the non-perturbative region, we choose

2 2 2 2 ki,t 2 2 2 fg(xi, ki,t < µ0, µ ) = 2 xig(xi, µ0)Tg(µ0, µ ). (3.18) µ0 Also, we set the boundaries of the rapidity integration in accordance with the speciﬁcations of the detectors (i.e. y < 2.5 for the CMS report [143] and y < 2.4 for ATLAS reports | H | | H | 58 [144, 145], excluding the 1.37 < y < 1.52 region for the later). Otherwise, we choose to | H | integrate over the y < 10 domain. According to the Eq. (3.12), and the 0 < x < 1 | H | i constraint, further rapidity domain will have no inﬂuence on our result.

At this point, we must mention that while calculating the total production rate for the SM Higgs boson production in the collinear approximation, the higher order QCD corrections to the LO g∗ + g∗ H sub-process are significant. These correction are either kinematic in → nature (e.g. corrections from the (b), (c), (d) and (e) diagrams in the figure 3.1) or arise from real parton emissions or virtual loop corrections. It is however customary to compensate for these neglected contributions by the means of introducing an additional factor into the main calculations, called the K-factor, which is defined as the ratio of the corrected results to the LO results. It has been suggested that introducing a K-factor as

πα (µ2) K = exp C S c , (3.19) A 2

2 1/3 with CA = 3 and µc = (mH pt ) can absorb the main part of these higher order corrections [111].

Finally, we are ready to calculate the cross-section for the production of the SM Higgs boson and the cSMCS Higgs bosons, using the master Eq. (3.13) in the kt-factorization framework. To switch from the SM to the cSMCS, we change the mass of the considered Higgs boson and replace the couplings of SM Higgs boson with quarks,mf /v, with its corresponding cSMCS couplings, i.e. ri1mf /v. We use the benchmarks presented in the table 2.1 for the mass of the considered Higgs boson.

3.3 Results and discussions

Before discussing the cSMCS predictions, it is necessary to prove that our framework can in fact produce reliable results. We test our approach by calculating the cross-section for the production of the SM Higgs boson HSM . The figure 3.2 presents the differential cross- section for the production of the SM Higgs boson HSM (dσH /dpt) versus the transverse momentum of the produced particle (pt). Parts (a) and (b) illustrate our results for the LHC with the center-of-mass energy E = 8 TeV and the rapidity regions y < 2.4 and CM | H | y < 2.5, respectively. The main results are being presented by solid black curves while the | H | blue stripped patterns mark the uncertainty bounds (which are determined by manipulating the hard-scale µ by a factor of 2). Part (c) shows the results for the center-of-mass energy E = 13 TeV and rapidity region y < 2.4 (excluding the 1.37 < y < 1.52 region). The CM | H | | H | 59 results are compared with CMS data at 8 TeV in the figure 3.2(a) and with ATLAS data at 8 TeV and 13 TeV in the figure 3.2(b) and 3.2(c), respectively. The data points are the results of measurements of the CMS and the ATLAS collaborations, references [143, 144, 145]. A similar comparison is presented in the figure 3.3, regarding the rapidity contribution of the

SM Higgs boson production. i.e. dσH /dyH versus yH . We conclude that our framework, at least within its uncertainty bounds, can give an acceptable description of the SM Higgs particle.

1 1 1

10 10 10

P+P H +X P+P H +X P+P H +X

SM SM SM

Uncertainty Uncertainty Uncertainty

CMS ATLAS ATLAS

E =8 TeV E =8 TeV E =13 TeV

SM SM SM

0 0 0 M =125 GeV M =125 GeV M =125 GeV

10 H 10 H 10 H [pb/GeV] [pb/GeV] [pb/GeV] t t t /dp /dp /dp H H H d d d

-1 -1 -1

10 10 10

-2 -2 -2

10 10 10

0 50 100 150 200 0 50 100 150 200 0 50 100 150 200

p [GeV] p [GeV] p [GeV]

t t t

(a) (b) (c)

Figure 2 Figure 3.2: Diﬀerential cross-section for the production of the SM Higgs boson as a function of the transverse momentum of the SM Higgs boson at the LHC. The calculations have been

performed for the center-of-mass energies Ecm = 8 TeV and 13 TeV. The black solid curves illustrate the main prediction while the blue stripped patterns show the uncertainty bound for the results. The uncertainty bounds are determined via manipulating the hard-scale µ by a factor of 2. The results have been compared with the experimental data from the CMS (a) and the ATLAS (b and c) collaborations [143, 144, 145]. To prepare the KMR UPDF, we have utilized the PDF of MMHT2014.

Figure 3.4, illustrates our predictions for diﬀerential cross-section for the production

of the SM Higgs boson at the LHC for ECM = 14 TeV, as functions of its pt. Part (a) depicts the general behavior of this production rate, exclusively within the SM. Part (b) outlines an comparison between our calculations and the similar results from NLL+LO (next-to-leading logarithmic re-summation plus LO calculations) and NLL+NLO (next-to- leading logarithmic re-summation plus NLO calculations) analysis performed in the collinear factorization, in [113]. The collinear results show a slightly higher peak, compared to the KMR framework. Otherwise, the general behavior of these frameworks are identical. In

60 P+P H +X P+P H +X P+P H +X

SM SM SM

60 60 60

Uncertainty Uncertainty Uncertainty

CMS ATLAS ATLAS

50 50 50 E =8 TeV E =8 TeV E =13 TeV

SM SM

M =125 GeV M =125 GeV M =125 GeV

H H

40 40 40 [pb] [pb] [pb] H H H /dy /dy /dy

30 30 30 H H H d d d

20 20 20

10 10 10

0 0 0

0.0 0.8 1.6 2.4 0.0 0.8 1.6 2.4 0.0 0.8 1.6 2.4

y y y

H H H

(a) (b) (c)

Figure 3 Figure 3.3: Diﬀerential cross-section of the production of the SM Higgs boson at the LHC as a function of the rapidity of the produced particle. other details are as in the ﬁgure 3.2.

the part (c), we demonstrate the diﬀerential cross-section of the production of the SM-like

Higgs boson h1 from the cSMCS. The curves A1 through A7 correspond to the cSMCS benchmarks, presented in the table 2.1. These computations are presented with respect to the SM uncertainty bounds. Also, the ﬁgure 3.5 presents a similar comparison regarding the

yH contribution of the Higgs boson production for the SM HSM and the cSMCS h1.

Furthermore, we have calculated the total rate of production of the SM Higgs boson

HSM at the LHC (σH ) as a function of the center-of-mass energy of the hadronic collision

(ECM ) and compared the results with the existing experimental data from CMS and ATLAS [143, 144, 145], see the ﬁgure 3.6. Our predictions seem to be realistic. Furthermore, we

have computed for ECM = 14 TeV the mass distribution of the Higgs boson HSM , what is presented as the ﬁgure 3.7. The peculiar pattern which is seen in the cross-section is originated from the imaginary part of D(τ), the Eq. (7.31), which becomes non-zero at

mH = 2mt.

At this point, after proving the correctness of our approach in describing the SM Higgs

HSM production at the LHC, we present our predictions regarding the expected production

rate for h1 and for the heavier Higgs boson h2 and h3 in the cSMCS model. The ﬁgures 3.8 and 3.9, present a comparison of the diﬀerential cross-sections of the production of the cSMCS

Higgs bosons h1, h2 and h3 as a function of the transverse momentum at ECM =14 TeV, in accordance with the A1-A7 benchmarks of the cSMCS (table 2.1). The calculations have been performed using the UPDF of KMR, with ECM = 14 TeV, illustrating the transverse momentum and rapidity distributions of these diﬀerential cross-sections. Parts (a), (b)

61 1

3 3 10

SM-Uncertainty P+P H +X P+P H +X (KMR) h

1 SM SM

A1 Uncertainty Uncertainty

NLL+LO E =14 TeV

0 SM NLL+NLO

10 M =125 GeV

H A4

E =14 TeV

A5 2 2

M =125 GeV A6

-1 [pb/GeV] [pb/GeV] [pb/GeV]

t 10 t t /dp /dp /dp H H H d d d

1 1

-2

-3

0 0 10

80 160 240 320 20 40 60 80 100 10 20 30 40

p [GeV] p [GeV] p [GeV]

t t t

(a) (b) (c)

Figure 4 Figure 3.4: Diﬀerential cross-section of the production of the Higgs boson at the LHC as a function of the transverse momentum at ECM = 14 TeV. Part (a) illustrates the main results and the corresponding uncertainty bounds for the SM Higgs boson HSM . Part (b) presents a comparison between our results for HSM (the solid black curve and its blue strip- patterned uncertainty bounds) and the results of similar calculations within the collinear framework, i.e. the NLL+LO (red dashed curve) and the NLL+NLO (dotted green curve). The collinear results are from the reference [113]. Part (c) presents the predictions of the various benchmarks of cSMCS for the lightest SM-like Higgs boson h1.

60 10

SM-Uncertainty P+P H +X h

1 SM

A1 Uncertainty

E =14 TeV

M =125 GeV

H A4

A5 40 10

[pb] A7 [pb] H H

30 /dy /dy H H d d

20 10

-1

0 10

0 1 2 3 4 5 0 1 2 3 4 5

y y

H H

(a) (b)

Figure 5

Figure 3.5: Diﬀerential cross-section of the production of the Higgs boson at the LHC HSM

(a) and the cSMCS h1 (b) as a function of the rapidity of the produced particle with ECM = 14 TeV. Other details are as in the parts (a) and (c) of the ﬁgure 3.4, respectively.

and (c) correspond to the SM-like h1 (see ﬁgure 3.4(c)), h2 and h3 Higgs bosons in the cSMCS model, respectively. It is apparent that the behavior of these predictions is rather

62 3

P+P H +X

Uncertainty

1 [pb]

10 H

CMS

ATLAS

M =125 GeV

-1

6 12 18

E [TeV]

Figure 6

Figure 3.6: Inclusive total cross-section of the production of the SM Higgs boson HSM at the LHC as a function of the central-mass energy of proton-proton collision, at the LHC. The solid black curve illustrates the main results while the uncertainty bounds (blue stripped pattern) have been produced via manipulating the hard-scale of the UPDF, µ, by a factor of 2. The experimental data are from the CMS (black diamonds) and the ATLAS (white circles) collaborations [143, 144, 145].

P+P H +X

Uncertainty

E =14 TeV

0 [pb]

10 H

-1

-2

-3

250 500 750 1000

m [GeV]

Figure 7

Figure 3.7: Inclusive total cross-section of the production of the Higgs bosons HSM as a function of the mass at ECM = 14 TeV. The solid black curve illustrates the main results while the uncertainty bounds (blue stripped pattern) have been produced via manipulating the hard-scale of the UPDF, µ, by a factor of 2.

diverse, covering different kinematic regions. The differential cross-section for the h2 and h3 are generally smaller than the differential cross-section for h1. The upper and the lower 1 5 curves for h2 from A6 and A4 benchmarks with a spread of 10− pb/GeV to 10− pb/GeV,

63 1 3 respectively and for h3 from A7 and A1 benchmarks with a spread of 10− pb/GeV to 10− pb/GeV, respectively. In the ﬁgure 3.9, the general behavior of the curves is similar to the

ﬁgure 3.8 with a single diﬀerence: the upper curve in the h3 case belongs to the benchmark 3 7 3 5 A5. The h2 and the h3 curves have a spread of 10− pb to 10− pb and 10− pb to 10− pb, respectively.

We believe that results presented in the ﬁgures 3.8 and 3.9 are reliable estimations for the Higgs boson signals within the cSMCS and will be particularly useful in the on-going experimental research regarding light and heavy Higgs bosons at the LHC.

1 0 1 1 0 0 1 0 0

h 1 A 1 h 2 A 1 h 3 A 1 A 2 A 2 A 2 E C M = 1 4 T e V E C M = 1 4 T e V E C M = 1 4 T e V A 3 1 0 - 1 A 3 1 0 - 1 A 3

0 A 4 A 4 A 4 1 0 A 5 A 5 A 5

] ] - 2 ] - 2

V A 6 V 1 0 A 6 V 1 0 A 6 e A 7 e A 7 e A 7 G G G / / / b b b p p p

[ - 1 [ - 3 [ - 3

t 1 0 t 1 0 t 1 0 p p p d d d / / / 1 2 3 h h h σ σ σ d d 1 0 - 4 d 1 0 - 4 1 0 - 2 1 0 - 5 1 0 - 5

1 0 - 3 1 0 - 6 1 0 - 6 8 0 1 6 0 2 4 0 3 2 0 8 0 1 6 0 2 4 0 3 2 0 8 0 1 6 0 2 4 0 3 2 0

p t [ G e V ] p t [ G e V ] p t [ G e V ] ( a ) ( b ) ( c )

F i g u r e 8 Figure 3.8: Diﬀerential cross-section for the production of the cSMCS Higgs bosons at the LHC as a function of the transverse momentum of the produced Higgs bosons at

ECM =14 TeV. Parts (a), (b) and (c) present the benchmark predictions of the cSMCS

Higgs bosons, h1, h2 and h3 in accordance with the benchmarks given in the table 2.1

3.4 Conclusions

In this Chapter we have calculated the production rates for the SM Higgs bosons and the Higgs bosons from the cSMCS, using an eﬀective LO partonic matrix element and the UPDF of the KMR formalism. The calculations for the SM Higgs boson have been compared with the existing experimental data of the CMS and the ATLAS collaborations, showing that our computations, within the given uncertainty bounds, present an acceptable platform to describe the Higgs production at the LHC. Afterwards, we have presented our predictions regarding the distribution of the transverse momentum and the rapidity of the produced

SM-like h1 and heavy Higgs bosons h2 and h3, from the cSMCS at the LHC. Detecting

64 1 0 2 1 0 - 2 1 0 - 2

h 1 A 1 h 2 A 1 h 3 A 1 A 2 A 2 A 2 E = 1 4 T e V - 3 E = 1 4 T e V - 3 E = 1 4 T e V C M A 3 1 0 C M A 3 1 0 C M A 3 A 4 A 4 A 4 A 5 1 0 - 4 A 5 1 0 - 4 A 5 1 0 1 A 6 A 6 A 6 ] ] ]

b A 7 b A 7 b A 7

p p - 5 p - 5 [ [ [

1 0 1 0 H H H y y y d d d / / / 1 2 2

h h - 6 h - 6

σ σ 1 0 σ 1 0 d d d 1 0 0 1 0 - 7 1 0 - 7

1 0 - 8 1 0 - 8

1 0 - 1 1 0 - 9 1 0 - 9 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5

y H y H y H ( a ) ( b ) ( c )

F i g u r e 9 Figure 3.9: Diﬀerential cross-section for the production of the cSMCS Higgs bosons at the

LHC as a function of the rapidity of the produced Higgs particles at ECM = 14 TeV. Other details are as in the ﬁgure 3.8. heavier Higgs bosons, if happen, will open the doors for further exploration of these ideas. These predictions may provide some clues regarding the dynamics of the next discovery.

65 66 Chapter 4

IDMS: Inert Doublet Model with a complex singlet

The gravitational eﬀects of DM have been observed in galaxies, clusters of galaxies, the large- scale structure of the Universe and the CMB Radiation. Such observations indicate that DM accounts for 85% of the matter density in the Universe and for 23% of its total energy density [36]. DM should be neutral, stable, weakly interacting and consistent with the observed large-scale structure of the Universe. For reviews on DM see e.g. refs. [159, 160, 161].

One of the simplest models for scalar DM that is the Inert Doublet Model (IDM), a

version of a Two Higgs Doublet Model with an exact Z2 symmetry [162].

We extended the cSMCS model model by a Z2-odd SU(2) doublet with zero vacuum expectation value (Inert Doublet). This model has an exact Z2 symmetry and in addition to providing a possible new source for CP violation and proper description for the baryogenesis can provide the correct relic density of DM, while fulﬁlling direct and indirect DM detection limits, and simultaneously agreeing with the LHC results (see e.g. [151, 152, 153, 154, 155, 156]).

The content of this chapter is as follows. Section 4.1 contains the presentation of the general model, in particular its scalar potential. In section 4.2 we present in detail a constrained version of our model, the mass eigenstates in the neutral and charged charged sectors and study the parameter space of the model. Section 4.3.3 contains an analysis of Higgs couplings and a comparison with LHC data. In section 4.4 we present our study of relic density for a DM candidate of the model, which is assumed to be the lightest neutral Z2-odd scalar state. Conclusions are presented in section 4.5, where we also discuss possible implications

67 for neutrino physics. Detailed formulas, benchmark points and values related to the LHC and DM analysis are presented in the Appendix 7.5.

4.1 The IDMS: The IDM plus a complex singlet

We shall consider a Z2-symmetric model that contains a SM-like Higgs doublet Φ1, which is involved in a generation of the masses of gauge bosons and fermions, as in the SM. There is also an inert scalar doublet Φ2, which is odd under a Z2 symmetry. This Φ2 doublet has VEV= 0 and can provide a stable DM candidate. Then, we have the neutral complex singlet χ with hypercharge Y = 0 and a non-zero complex VEV and a pair of heavy iso-doublet

VQ, VL + VR. The singlet χ can play several roles in models with two doublets and a singlet, leading to diﬀerent scenarios. CP violation can be explicit, provided by the singlet interaction terms, or spontaneous, if χ C. Here we shall take χ to be even under a Z2 transformation h i ∈ deﬁned as:

Z :Φ Φ , Φ Φ , χ χ, SM ﬁelds SM ﬁelds, VQ VQ. (4.1) 2 1 → 1 2 → − 2 → → → We shall consider the case when the CP symmetry can be violated by a non-zero complex χ . h i The full Lagrangian of the model looks as follows:

= SM + + (ψ , Φ ) + (V , χ), = T V, (4.2) L Lgf Lscalar LY f 1 LY q Lscalar − where SM describes boson-fermion interaction as in the SM, describes the scalar Lgf Lscalar sector of the model. Terms (ψ , Φ ) and (V , χ) represent the Yukawa interaction of LY f 1 LY q Φ1 with SM fermions and the interaction of singlet scalar with VQ, respectively. The kinetic term in has the standard form: Lscalar µ µ T = (DµΦ1)† (D Φ1) + (DµΦ2)† (D Φ2) + ∂χ∂χ∗, (4.3) with Dµ being a covariant derivative for an SU(2) doublet.

We take the Yukawa interaction in the form of the Model I in the 2HDM1, where only

Φ1 couples to fermions. 1In 2HDM lagrangian, four different types of Yukawa interactions arise. In type I, all the quarks couple to a single scalar doublet, here it is Φ1. In type II the down-type right-handed quarks dR couple to one of the first doublet, and the up-type right-handed quarks to the other doublet. The names Model III and Model IV were used for the flipped and lepton-specific models, respectively.

68 In our model only Z2-even ﬁelds Φ1 and χ acquire vacuum expectation values, which we denote by v and weiξ, respectively, where v, w, ξ R. We shall use the following ﬁeld ∈ decomposition:

! ! φ+ φ+ Φ = 1 , Φ = 2 , (4.4) 1 1 (φ + iφ ) 2 1 (φ + iφ ) √2 1 6 √2 4 5 χ = 1 (φ + iφ ). (4.5) √2 2 3

Thus, the Z2 symmetry (4.1) is not violated spontaneously. Also, U(1)EM is not broken, and there is no mixing between the neutral and charged components. Masses of gauge bosons 2 2 2 and fermions are given by the VEV of the ﬁrst doublet as in the SM, e.g MW = g v /4 for the W boson.

The full scalar potential of the model can be written as

V = VIDM + VS + VDS, (4.6)

where we have separated the pure doublet and the pure singlet parts (respectively VIDM and

VS) and their interaction term (VDS). The IDM part of the potential, VIDM , is given by:

1 2 2 1 2 2 V = (m Φ†Φ + m Φ†Φ ) + (λ (Φ†Φ ) + λ (Φ†Φ ) ) + λ (Φ†Φ )(Φ†Φ ) IDM − 2 11 1 1 22 2 2 2 1 1 1 2 2 2 3 1 1 2 2 (4.7) λ 2 2 † † 5 † † +λ4(Φ1Φ2)(Φ2Φ1) + 2 ((Φ1Φ2) +(Φ2Φ1) ). The general singlet part of the potential is equal to (see Eq. (2.6) ):

2 2 m3 m4 2 2 2 2 2 4 4 VS = χ∗χ (χ∗ + χ ) + λs1(χ∗χ) + λs2(χ∗χ)(χ∗ + χ ) + λs3(χ + χ∗ ) − 2 − 2 (4.8) 3 3 +κ1(χ + χ∗) + κ2(χ + χ∗ ) + κ3(χ(χ∗χ) + χ∗(χ∗χ)).

The doublet-singlet interaction terms are:

2 2 2 2 VDS = Λ1(Φ1†Φ1)(χ∗χ) + Λ2(Φ2†Φ2)(χ∗χ) + Λ3(Φ1†Φ1)(χ∗ + χ ) + Λ4(Φ2†Φ2)(χ∗ + χ ) (4.9) +κ4(Φ1†Φ1)(χ + χ∗) + κ5(Φ2†Φ2)(χ + χ∗).

We assume that all parameters of V (4.6) are real, and it is not diﬃcult to see that the ? potential is explicitly invariant under the standard CP transformation Φ Φ† , χ χ . 1,2 → 1,2 →

Since V is Z2-symmetric and the chosen vacuum state (4.4,4.5) will not spontaneously break this symmetry, the problem of cosmological domain walls will not arise in this model.

69 In total, there are four quadratic parameters, twelve dimensionless quartic parameters and

ﬁve dimensionful parameters κ1,2,3,4,5. The linear term κ1 can be removed by a translation of the singlet ﬁeld, and we will omit it below.

One could reduce this general model by invoking additional symmetries besides the im- posed Z2 one (see e.g. [183, 184, 185, 186, 187, 188, 189, 190] for various symmetry assign- ments). In particular, to simplify the model one can apply a global U(1) symmetry, as we discuss below. Similarly, had we chosen to assign a Z2-odd quantum number also to χ (or if singlet was odd under an additional Z20 symmetry), it would have also resulted in a variant of the model with a simpliﬁed potential, where all terms with an odd number of ﬁeld χ would be absent. Obviously, in those cases having a Z2 (or Z20 ) symmetric vacuum state would require χ = 0, and thus there would be no additional CP violation in the model. h i

4.2 The constrained IDMS: cIDMS

Similarly as for the cSMCS model presented in chapter 2, we will reduce the most general IDMS potential (4.6-4.9) by imposing a global U(1) symmetry:

U(1) : Φ Φ , Φ Φ , χ eiαχ. (4.10) 1 → 1 2 → 2 → However, a non-zero VEV, χ would lead to a spontaneous breaking of this continuous h i symmetry and appearance of massless Nambu-Goldstone scalar particles, which are not phenomenologically viable. Keeping some U(1)-soft-breaking terms in the potential would solve this problem and at the same time would still lead to a reduction of the number of parameters in V .

The parameters of the IDMS potential can be divided into the following groups:

2 2 2 1. U(1)-symmetric terms: m11, m22, m3, λ1,2,3,4,5, λs1, Λ1,2,

2 2 2. U(1)-soft-breaking terms : m4, ρ2,3, ρ4,5,

3. U(1)-hard-breaking terms λs2, λs3, Λ3,4.

In what follows we shall consider a potential with soft-breaking of the U(1) symmetry 2 by the singlet cubic terms ρ2,3 and quadratic term m4 only, neglecting the remaining ones

(ρ4,5). We recall that Φ1 is the SM-like Higgs doublet responsible for the EW symmetry

2 Recall that ρ1 can be removed from (4.6) by translation of χ.

70 breaking and for providing masses of gauge bosons and fermions. In addition, we want to use it as a portal for DM interactions with the visible sector, as in the IDM. We shall assume

therefore that there is no direct coupling of Φ2 to χ, thus setting the U(1)-invariant term

Λ2 = 0. The ﬁeld χ shall then interact with the DM particles only through mixing with the

neutral component of Φ1. We are therefore left with the following U(1)-symmetric terms 2 2 2 2 (m11, m22, m3, λ1 5, λs1, Λ1) and U(1)-soft-breaking terms (m4, ρ2,3). − We shall call our model, the model with this choice of parameters, cIDMS. The cIDMS potential is then given by:

2 2 1 h 2 2 i 1 V = m Φ†Φ + m Φ†Φ + λ Φ†Φ + λ Φ†Φ − 2 11 1 1 22 2 2 2 1 1 1 2 2 2

2 2 λ † † † † 5 † † +λ3 Φ1Φ1 Φ2Φ2 + λ4 Φ1Φ2 Φ2Φ1 + 2 Φ1Φ2 + Φ2Φ1 (4.11)

2 m3 2 χ∗χ + λ (χ∗χ) + Λ (Φ†Φ )(χ∗χ) − 2 s1 1 1 1 2 m4 2 2 3 3 (χ∗ + χ ) + κ (χ + χ∗ ) + κ [χ(χ∗χ) + χ∗(χ∗χ)]. − 2 2 3

4.2.1 Parameter choice

Once the potential (4.6) is restricted only to U(1)-symmetric or U(1)-soft-breaking terms, no more terms will be generated when we move beyond tree-level. For our choice of parameters, the cIDMS, we assume that some of U(1)-symmetric or U(1)-soft-breaking terms are set manually to zero. One may ask these terms will remain zero, or if they will be generated

at loop level. Indeed, it turns out that some terms we neglected, namely κ4 and Λ2 are 1 generated already at the 1-loop level, with their β functions being proportional to 16π2 and product of Λ1 and κ3 and, respectively, a combination of λ4, κ3 [191]. This shows that our parameter choice is not protected against loop corrections, which was expected, as those terms are allowed by the symmetry we chose to consider. However,

it is important to notice that loop contribution for κ4 depend on the parameter Λ1, i.e. the

mixing parameter between Φ1 and χ. In our analysis we chose scenarios where this parameter is small, leading to the Higgs particle being SM-like, which is a favoured interpretation of current LHC data.3

3 2 2 2 The linear term, with β function 1/16π (m κ3 +m (3κ2 +κ3)), even if removed by translation of ﬁelds ∝ 3 4 at tree-level, appears when we include loop corrections. The resulting tadpole diagram can be interpreted

as the shift in vacuum energy. If κ1 is kept non-zero at tree-level, one can remove it consistently at every loop level [192]. In any case, this term is not relevant for the presented work.

71 One can notice also that if κ3 is equal to zero, then both κ4 and Λ2 remain zero also at loop level.

The potential of Eq. (4.11) has in addition the mass term m22 for second doublet and

the terms of interaction between doublets with the couplings λ2,3,4,5 in compare with the potential of the cSMCS model (see Eq. (2.68)).

4.2.2 Positivity conditions

In order to have a stable minimum, the parameters of the potential need to satisfy positivity conditions. Namely, the potential should be bounded from below, i.e. should not go to negative infinity for large field values. This behavior is dominated by the quartic terms and the cubic terms will not play a role here. Thus the following conditions will apply to a variety of models that will differ only by their cubic interactions.

We use the method of [193], which uses the concept of co-positivity for a matrix build of coeﬃcients in the ﬁeld directions. For the cIDMS, the positivity conditions read:

λ , λ , λ 0, λ¯ = λ + θ[ λ + λ ](λ λ ) + √λ λ > 0, 1 2 s1 ≥ 12 3 − 4 | 5| 4 − | 5| 1 2 ¯ λ1S = Λ1 + √2λ1λs1 > 0, (4.12) q p 1 √λ λ λ + [λ + θ[ λ + λ ](λ λ )]√λ + Λ λ2 + λ¯ λ¯ λ¯ > 0, 2 1 2 s1 3 − 4 | 5| 4 − | 5| s1 1 2 12 1S 2S ¯ where λ2S = √2λ2λs1 > 0.

4.2.3 Extremum conditions

It is useful to re-express dimensionful parameters κ2,3 in terms of the dimensionless parameters ρ (we consider them being of order (1)) as: 2,3 O

κ2,3 = wρ2,3, (4.13) with w being an absolute value of the singlet VEV.

The minimization conditions lead to the following constraints for three quadratic param-

72 eters from V (4.11):

2 2 2 m11 = w Λ1 + v λ1, (4.14) 2 2 2 2 w m = v Λ1 + 2w λs1 + ( 3ρ2 + 3ρ3 + 2ρ3 cos 2ξ), (4.15) 3 √2 cos ξ − 2 2 w m = (3ρ2 + ρ3 + 6ρ2 cos 2ξ). (4.16) 4 2√2 cos ξ

2 The m22 parameter is not determined by the extremum conditions.

2 The squared-mass matrix Mij, for i, j = 1, ...6, is given by:

2 2 ∂ V Mij = , (4.17) ∂φiφj Φ = Φ ,χ= χ i h ii h i

with φi being the respective fields from the decomposition (4.4,4.5). This definition along 2 + with the normalization defined in (4.4,4.5) gives the proper mass terms of Mϕϕ ϕ− for the 2 Mϕ 2 charged scalar fields, and 2 ϕ for the neutral scalar fields.

4.2.4 Vacuum stability

The tree-level positivity conditions (4.12), which ensure the existence of a stable global minimum, correspond to λ > 0 in the SM. It is well known, that radiative corrections coming from the top quark contribution can lead to negative values of the Higgs self-coupling, resulting in the instability of the SM vacuum for larger energy scales. Full analysis of the stability of the cIDMS potential beyond tree-level is beyond the scope of this thesis. However, it has been shown in reference [154] that for the IDM the contributions from additional scalar states will in general leads to the relaxation of the stability bound at high energies and allow the IDM to be valid up to the Planck scale. Since cIDMS contains two more scalar states, this condition should hold here as well.

73 4.2.5 Mass eigenstates

The neutral sector

The form of the neutral part of the squared-mass matrix (4.17) for φi, (i = 1, ..., 6) allows us to identify the physical states4 and their properties:

 2  Mmix(3 3) 0(3 3) × ×  M 2 0 0  M 2 =  H  (4.18)  2   0(3 3) 0 M 0   × A  0 0 0

As there is no mixing between four Z2-even ﬁelds φ1,2,3,6, and two Z2-odd ﬁelds φ4,5, we can divide the particle content of the model into two separate sectors: the Z2-even sector, called the Higgs sector, and the Z2-odd sector, called the inert (or Dark) sector. Below we list the particle content of the neutral sector:

1. The Goldstone ﬁeld, GZ = φ6, is a purely imaginary part of the ﬁrst doublet Φ1.

2. There is a mixing between the singlet χ and the real neutral ﬁelds of Φ1 (namely φ1, φ2

and φ3) resulting in three neutral scalars h1, h2, h3. Masses of the these Higgs particles

depend only on the following parameters of the potential: λ1, Λ1, ρ2,3, λs1 (similar to cSMCS model).

3. In the inert sector the DM candidate is stable and it is the lighter of the two neutral

components of Φ2 (φ4 or φ5), which we identify as the scalar particles H and A. Masses of those particles are just like in the IDM: 1 M 2 = ( m2 + v2λ ),H = φ , (4.19) H 2 − 22 345 4 2 1 2 2 M = ( m + v λ− ),A = φ , (4.20) A 2 − 22 345 5

where λ = λ + λ + λ , λ− = λ + λ λ . 345 3 4 5 345 3 4 − 5 M 2 M 2 λ = H − A . (4.21) 5 v2

If λ5 < 0 then H, as a neutral lighter state, is our DM candidate. Since Z2 symmetry

is exact in our model, the Z2-odd particles have limited gauge and scalar interactions (they interact in pairs only) and they do not couple to fermions. Masses of inert

4The Physical states in the Higgs sector in cIDMS is identical to that in the cSMCS model

74 2 particles (also charged scalars) depend only on λ3,4,5 and m22. These parameters do

not inﬂuence masses of the Higgs particles from the Z2-even sector. In that sense, the masses of particles from the Higgs and the inert sectors can be studied separately. On this level, the only connection between parameters from these two sectors is through

the positivity constraints. As in the IDM, λ2 does not inﬂuence the mass sector and

it appears only as a quartic coupling between the Z2-odd inert particles.

The charged sector

The Z2-odd charged scalar H± comes solely from the second doublet, as in the IDM; its mass is given by 2 1 2 2 M ± = ( m + v λ ). (4.22) H 2 − 22 3

Notice, that the mass relations for the Z2-odd sector from the IDM hold, namely 2 2 2 2 v (λ4 + λ5) 2 2 v (λ4 λ5) M = M ± + ,M = M ± + − . (4.23) H H 2 A H 2 The neutral particle H is a DM candidate, therefore λ4 + λ5 < 0, resulting in MH < MH± .

If we allow an additional mixing between Φ2 and χ through a non-zero Λ2,4 and ρ5 then the squared-mass formulas are modiﬁed as M 2 M 2 + ∆, with ∆ = 1 w2(Λ + H,A,H± → H,A,H± 2 2 2Λ4 cos 2ξ + 2√2ρ5 cos ξ). Still, the IDM relations (4.21) and (4.23) hold.

2 In the inert sector, three quartic parameters, λ3,4,5, and one quadratic parameter m22, are relevant. The remaining quartic parameter, λ2, appears only in the quartic interaction of Z2- odd particles and is therefore not constrained by the analysis of the mass spectrum. However, we expect that – as in the IDM – combined unitarity, perturbativity and global minimum conditions may provide constraints for this, otherwise practically unlimited, parameter [194].

The masses of Z2-odd scalars, and therefore parameters of the potential given by relations (4.21) and (4.23), are already constrained by experimental and theoretical results.

4.3 Scanning and experimental constraints

Every model of new physics should be theoretically self-consistent and stay in agreement with the available experimental data. This poses various constraints on the parameter space of a cIDMS. Among the experimental constrains are the LEP experiment, the oblique parameters S and T, the bound from the LHC and the constraints from the measurements of the DM relic density.

75 4.3.1 LEP bounds

We performed the scanning, use inert parameters changing in the range allowed by the perturbativity constraints, with H being the DM candidate:

0 < λ < 1, 1 < λ < 1, 1 < λ < 0, (4.24) 2 − 3,4 − 5 with v = 246 GeV.

The results obtained at LEP provide several sources of bounds on new physics.

1. The LEP studies of invisible decays of Z and W ± gauge bosons require that there is

no decay of W ± or Z into inert particles, which gives the following limits [195, 196]:

MH± + MH,A > MW ± ,MH + MA > MZ , 2MH± > MZ . (4.25)

2. Searches for charginos and neutralinos at LEP have been translated into limits of region of masses in the IDM [196] excluding

M M > 8 GeV if M < 80 GeV M < 100 GeV. (4.26) A − H H ∧ A We shall adopt the same limit for inert particles in the studied cIDMS.

2 3. Note that, as in the IDM, the value of MH± provides limits for m22, which is not 2 2 constrained by the extremum conditions. Demanding that MH± > 0 results in m22 < λ v2, which for range of 1 λ 1 reduces to m2 < v2. This constraint is modiﬁed 3 − ≤ 3 ≤ 22 by taking account of the ”model-independent” limit from LEP for the charged scalar mass [197]: 2 4 2 M ± > 70 90 GeV m 5 10 GeV (4.27) H − ⇒ 22 . ·

2 Figure 4.1 shows the correlation between the charged-scalar mass and m22. Large 2 values of M ± correspond to large values of m . H − 22

4. Mass splittings between the Z2-odd particles are given by combinations of λ4 and λ5, which are constrained by the perturbativity conditions. If we demand that λ < 1 | 3,4,5| then in the heavy mass regime all particles will have similar masses, as they are all driven to high scales by the value of m2 (4.23). This is visible in ﬁgure 4.2. Notice − 22 that mass splitting of the order of 200 GeV is allowed only for the smaller masses.

76 2 Figure 4.1: Charged scalar mass MH± as a function of m22.

(a) (MA,MH ) (b) (MH± ,MH )

Figure 4.2: (a) Relation between MH and MA. (b) Relation between MH and MH± . Both correlations for random scanning with λ < 1 and m2 < 106 GeV2. | 3,4,5| | 22|

4.3.2 Oblique parameters S,T,U

As was mentioned in the section 2.6.2, EWPT provides strong constraints for New Physics beyond the SM. In particular, additional particles may introduce important radiative corrections to gauge boson propagators. These corrections can be parameterized by the oblique parameters S, T and U. The value of these parameters will be inﬂuenced both by the presence of extra (heavy) Higgses present in the cIDMS and by inert particles H±, H and A. In our work we have checked the compatibility of our benchmark points with the 3σ bounds on S and T , following the method described in [89]. For detailed formulas see Appendix 7.5.2. Speciﬁc values for given sets of parameters of the cIDMS model are presented in table 7.4, in Appendix 7.5.4. In general, we took the IDM results as the guidance points for our analysis, and found that indeed the cIDMS represents the same behaviour: additional heavy particles, including the heavy Higgses, can be accommodated in the model without violating EWPT constraints.

77 4.3.3 LHC constraints on Higgs parameters in the cIDMS

Higgs signal strength in the cIDMS

Further constraints on the parameters of our model (cIDMS) can be obtained by comparing

the light Higgs signal (h1), and the one arising from the SM, with the LHC results (see Appendix 7.1). This is done by using the following signal strength (see section 2.6.1):

σ(gg h ) BR(h XX) = → 1 1 → , (4.28) RXX σ(gg φ ) BR(φ XX) → SM SM → for X = γ, Z, ..., assuming the gluon fusion is the dominant Higgs production channel at the LHC and the narrow-width approximation. The expression for reduces to: RXX Γ(h gg) BR(h XX) = 1 → 1 → . (4.29) RXX Γ(φ gg) BR(φ XX) SM → SM →

In this model the Higgs (h1) decay width into gluons and vector bosons (V = Z,W ) are 1 similar to cSMCS model and suppressed by a factor r11 (where r11 is the (11) element of r− deﬁned by (2.24).

The one-loop coupling of h1 to photons receives contributions mainly from the W boson 5 and top quark , as well as the charged scalar H± from the inert sector, so the amplitude can be written as (see Appendix 7.5.1):

SM SM A(h γγ) = r (A + A ) + A ± , (4.30) 1 → 11 W t H and similar expression for the amplitude describing h Zγ (see Appendix 7.5.1). A 1 → Therefore, the decay widths into two photons and into a photon plus a Z boson, are given, respectively, by

Γ(h γγ) = r2 1 + η 2Γ(φ γγ), (4.31) 1 → 11| 1| SM → Γ(h Zγ) = r2 1 + η 2Γ(φ Zγ), (4.32) 1 → 11| 2| SM → where g + − v A ± g + − v ± η = h1H H H , η = h1H H AH . (4.33) 1 2r M 2 ASM + ASM 2 2r M 2 SM + SM 11 H± W t 11 H± AW At

+ − The triple coupling λh1H H is given by

+ − gh1H H = vλ3r11, (4.34) 5VQ are not included in this calculation, as they are heavy and will decouple.

78 meaning it is also modiﬁed with respect to the IDM by a factor of r11. In the total width of the SM Higgs boson we can neglect the contributions coming from the Higgs decay into Zγ and γγ.6 The total Higgs decay width in the cIDMS can be

signiﬁcantly modiﬁed with respect to the SM if h1 can decay invisibly into inert particles. The partial decay width for the invisible channels h ϕϕ, where ϕ = A, H, is: 1 → g2 4M 2 1/2 Γ = Γ(h ϕϕ) = h1ϕϕ 1 ϕ , (4.35) inv 1 32πM M 2 → h1 − h1 with

gh1AA = λ345− vr11 and gh1HH = λ345vr11. Therefore, in regions of masses where Higgs-invisible decays could take place, the total width of the Higgs boson in the cIDMS is given by

Γ r2 ΓSM + Γ . (4.36) tot ≈ 11 tot inv Finally, the signal strengths from Eq. (4.29) can be written as follows,

2 1 2 2 1 2 2 1 = r ζ− , = r 1 + η ζ− , = r 1 + η ζ− , (4.37) RZZ 11 Rγγ 11| 1| RZγ 11| 2| where ζ is deﬁned as Γinv ζ 1 + 2 SM . (4.38) ≡ r11Γtot

For the cIDMS case r11 = c1c2, where c1 = cos α1 and c2 = cos α2 are deﬁned by the rotation angles in the scalar sector, Eq. (2.23), and thus

2 2 1 2 2 2 1 2 2 2 1 = c c ζ− , = c c 1 + η ζ− , = c c 1 + η ζ− . (4.39) RZZ 1 2 Rγγ 1 2| 1| RZγ 1 2| 2| Notice that there is a limit on , i.e. 1. It is not possible to enhance this RZZ RZZ ≤ decay with respect to the SM. and can be bigger than 1 if there is a constructive Rγγ RZγ interference between the SM and the cIDMS contributions.

Numerical analysis of the Higgs signal strenghts

We scan over parameter space in ranges:

0.2 λ 0.3, 1 Λ , λ , ρ 1, 0 ξ π, ≤ 1 ≤ − ≤ 1 3,4 2,3 ≤ ≤ ≤ 0 < λ < 1, 0 < λ < 1, 1 < λ < 0, (4.40) s1 2 − 5 106( GeV)2 < m2 < 5 104( GeV)2. − 22 · 6Bear in mind that this approximation is established in order to obtain some analytical expressions for

the corresponding ratios, Rγγ , RZγ and RZZ whose results will guide our DM analysis.

79 with v = 246 GeV and w = 300 GeV.

From ﬁgure 4.3, it is clear the ratios , and can present deviations from the Rγγ RZγ RZZ SM value up to 20%. Figure 4.3a shows the correlation between and , while ﬁgure Rγγ RZγ 4.3b correspond to and R . Rγγ ZZ

(a) (Rγγ ,RZγ ) (b) (Rγγ ,RZZ ) Figure 4.3: (a) Correlation between and . (b) Correlation between and . Rγγ RZγ Rγγ RZZ

If < 1 then both and are correlated with , and . Rγγ RZγ RZZ Rγγ Rγγ ∼ RZγ Rγγ ∼ RZZ Notice that there is a possibility of enhancement of both and . This is in agreement Rγγ RZγ with the IDM, where a correlation between enhancement in γγ and Zγ channels exists [217]. 2 Note that the upper limit for MH± comes from the lower limit for m22 from set (4.40).

and as functions of M ± are shown in figure 4.4a and figure 4.4b, respectively. Rγγ RZγ H For smaller masses of the charged scalar there is a possibility of enhancement of both Rγγ and . For heavier M ± the maximum values tend to the SM value, however deviation up RZγ H to 20 %, i.e. 0.8, is possible. Note that the situation is similar to the one from the Rγγ,Zγ ≈ IDM, where significant enhancement, e.g. = 1.2, was possible only if M ± 150 GeV, Rγγ H . and for heavier masses 1 [217]. Rγγ →

A similar result is presented in figure 4.6, which depicts γγ as function of the dimen- 2 R 2 sionful parameter m22. Significant enhancement is possible only for small values of m22 , 2 | | which correspond to small values of MH± . For large negative values of m22, i.e. heavy masses of all Z -odd scalars, the preferred value of is close to the SM value. Then the heavy 2 Rγγ particles effectively decouple from the SM sector and their influence on the SM observables is minimal, as expected. This effect is also visible in the IDM.

80 (a) (MH± ,Rγγ ) (b) (MH± ,RZγ )

Figure 4.4: (a) as function of M ± . (b) as function of M ± . Rγγ H RZγ H

(a) (MH ,Rγγ ) (b) (MH ,RZγ ) Figure 4.5: (a) as function of M . (b) as function of M . Rγγ H RZγ H

Figure 4.6: as function of m2 . Rγγ 22

Comment on invisible Higgs decays at the LHC

Measurements of invisible decays of the SM-like Higgs at the LHC set very strong constraints on Higgs-portal type of DM models (see e.g. [198] and detailed use of constraints in [155] for the IDM, or [199] for the 3HDM). In general, a DM candidate with mass below approximately

81 53 GeV annihilating mainly into b¯b through the Higgs exchange cannot be in agreement

with the LHC limits and relic density constraints. The remaining region, 53 GeV . MH . 62.5 GeV, corresponds to the Higgs-resonance, and the tree-level behavior is roughly the same in all Higgs-portal-type DM models. In principle, calculations in this region require loop corrections both for the annihilation cross-section, and the scattering cross-section, which is

beyond the scope of this work. Therefore, in our analysis we will focus on MH > Mh1 /2,

and comment on the region MH < Mh1 /2 in section 4.4 for completeness. The partial decay width of Higgs into invisible particles, for example a DM candidate from the cIDMS, is given by (4.35), and therefore depends on the DM candidate’s mass and its coupling to the Higgs. The cIDMS acts here as a standard Higgs-portal type of DM model and we obtain results known already for the IDM. Figure 4.7 shows the permitted range of parameter λ , as a function of mass of M , assuming that Br(h inv) is smaller than 345 H 1 → 0.37 (which is the value from ATLAS, denoted by dashed line [200]) and 0.20 (which is the value coming from global ﬁt analysis, solid line [201]).7

0.06

0.04

0.02

345 0.00 λ

-0.02

-0.04

-0.06 0 10 20 30 40 50 60

MH [GeV] Br(h→inv)=0.37 Br(h→inv)=0.20

Figure 4.7: Constraints for λ345 from measurements of Higgs invisible decays branching ratio, with the assumption that only h HH channel is open. Solid line: Br(h inv) = 0.20, 1 → → dashed line: Br(h inv) = 0.37. →

If we demand that Br(h inv) < 0.37 allowed region of DM-Higgs coupling is λ 0.02 1 → | 345| . for mass M below 30 GeV. For Br(h inv) < 0.20 we obtain λ 0.015. This H ∼ 1 → | 345| . limit will be combined with the relic density measurements in section 4.4 and it will provide

7 This can be treated as a limit for DM-Higgs coupling in the cIDMS, as g = c1c2λ345, with c1c2 HHh1 ≈ 0.99 for all considered SM-like scenarios.

82 strong constrain, comparable with the one obtained from DM direct detection searches, for

low DM mass region. In figure 4.8a we see that for a 20% deviation of Rγγ from (below) the SM model value, the invisible branching ratio is actually Br(h inv) < 0.20. On the 1 → other hand, figure 4.8b shows that when the invisible channels are open, the dimensionless parameter λ should be small (as mentioned above) in order to get an invisible branching | 345| ratio below 20%. In both figures the horizontal line at Br(h inv) = 0.20 should be 1 → understood as a reference point, so that all the points above it are ruled out by current experiment results.

(a) (b) Figure 4.8: (a) Br(h inv) as a function of . (b) Br(h inv) as a function of λ . 1 → Rγγ 1 → 345 In both panels, all the points above Br(h inv) = 0.2 are ruled out by current experiment 1 → results.

Further constraints for the DM candidate H come obviously from astrophysical measurements of DM relic density, and direct and indirect detection. Those will be discussed in section 4.4.

4.3.4 DM constraints

We present results and limits from the dedicated DM experiments used in this thesis.

1. We expect the relic density of DM to be in agreement with Planck data [46]:

Ω h2 = 0.1199 0.0027, (4.41) DM ± which leads to the 3σ bound:

2 0.1118 < ΩDM h < 0.128. (4.42)

83 If a DM candidate fulﬁlls this requirement, then it constitutes 100 % of DM in the 2 Universe. A DM candidate with ΩDM h smaller than the observed value is allowed, however in this case one needs to extend the model to have more DM candidates to complement the missing relic density. Regions of the parameter space corresponding 2 to value of ΩDM h larger than the Planck upper limit are excluded. In this work 2 calculation of ΩDM h was performed with an aid of micrOMEGAs 3.5 [180]. In these calculations all (co)annihilation channels are included, with states with up to two virtual gauge bosons allowed.

2. The strongest constraints for light DM annihilating into bb or ττ from indirect detection experiments are provided by the measurements of the gamma-ray ﬂux from Dwarf Spheroidal Galaxies by the Fermi-LAT satellite, ruling out the canonical cross-section 26 3 σv 3 10− cm /s for M 25 40 GeV [174, 175]. h i ≈ × DM . − For the heavier DM candidates PAMELA and Fermi-LAT experiments provide similar 25 3 limits of σv 10− cm /s for M = 200 GeV in the bb, ττ or WW channels h i ≈ DM [181]. H.E.S.S. measurements of signal coming from the Galactic Center set limits of 25 24 3 σv 10− 10− cm /s for masses up to TeV scale [176]. h i ≈ − 3. Current strongest upper limit on scattering cross section of DM particles on nuclei

σDM N is provided by the LUX experiment [182]: −

46 2 σDM N < 7.6 10− cm for MDM = 33 GeV. (4.43) − ×

4.4 Dark Matter in the cIDMS

In this section we will discuss properties of DM in the model. Because we can treat the cIDMS as an extension of the IDM, we will start with the brief description of DM phenomenology of the later. In both models H is a DM candidate if λ5 < 0. In the IDM the DM annihilation channels that are dominant for the DM relic density are HH h ff¯ for M M and → → H . W HH WW and HH h WW for M M . If the mass splittings M M or → → → H & W A − H M ± M are small then the coannihilation channels HA(H±) Z(W ±) ff 0 also play H − H → → an important role.

The regions of masses and couplings that correspond to the proper relic density have been studied in many papers (see e.g. [152, 153, 195, 202, 203, 204, 205]). In general, there are four regions of DM mass where the measured relic density can be reproduced: light DM particles with mass below 10 GeV, medium mass regime of 50 80 GeV with two − 84 distinctive regions: with or without coannihilation of H with the neutral Z2-odd particle A, medium mass region 80 150 GeV with very large mass splittings, and heavy DM of mass − larger than roughly 550 GeV, where all inert particles have almost degenerate masses and so coannihilation processes between all inert particles are crucial. These regions are further constrained or excluded (as it is the case with the low DM mass region) by direct and indirect detection experiments, and by the LHC data (see e.g. [155, 154, 156] for recent results).

Adding the singlet ﬁeld χ changes this picture, although certain properties of the IDM persist. In our model there is no direct coupling between the inert doublet Φ2 and the singlet χ, and the only interaction is through mixing of χ with the ﬁrst doublet Φ1. This means, that the inert scalars interaction with gauge bosons is like in the IDM, while the inert scalars-Higgs boson interaction changes with respect to the IDM. The IDM Higgs particle h corresponds in our case to φ1, so h φ1, where φ1 = β1h1 + β2h2 + β3h3 is given by the → P3 2 mixing parameters in (2.27, with β1 = c1c2 ), and obviously i=1 βi = 1. (The IDM case corresponds to β 0). The important processes for the cIDMS are: 2,3 →

HH h ff,¯ HH h WW (ZZ), (4.44) → i → → i → HH W W, (4.45) → HA(H±) Z(W ±) ff 0, (4.46) → →

IDM IDM IDM with couplings g = β g , g ¯ = β g , with g being the respective couplings hiHH i hHH hiff i hff¯ hXX of h to HH and ff¯ in the IDM. Following sum rules hold:

3 3 X 2 IDM 2 2 X IDM 2 g = (g ) = λ , g ¯ = (g ) . (4.47) hiHH hHH 345 hiff hff¯ i=1 i=1

Since both ghiHH and ghiff¯ have an extra βi coeﬃcient with respect to the IDM, the rate 2 for Higgs-mediated processes (4.44) will change by βi . If we are to consider an IDM-like case with β β then we could expect to reproduce results for the IDM. However, the 2,3 1 interference between diagrams may be in principle important, and as our analysis shows, they do inﬂuence the results.

4.4.1 Benchmarks

In this section we discuss properties of DM for chosen benchmarks in agreement with constraints from LHC/LEP:

85 A1: Mh1 = 124.83 GeV,Mh2 = 194.46 GeV,Mh3 = 239.99 GeV, (4.48)

A2: Mh1 = 124.85 GeV,Mh2 = 288.16 GeV,Mh3 = 572.25 GeV, (4.49)

A3: Mh1 = 125.36 GeV,Mh2 = 149.89 GeV,Mh3 = 473.95 GeV. (4.50)

By choosing values of Mh1,h2,h3 we determine parameters from the Higgs sector: λ1, λs1, Λ1, ρ2,

ρ3 and ξ. The corresponding values of parameters of the potential for each benchmark are presented in Appendix 7.5.3.

The above values were chosen to illustrate diﬀerent possible scenarios:

For A1 all Higgs particles are relatively light, although only one, the SM-like Higgs h1,

is lighter than 2MW . Case A2 is similar to A1; the important diﬀerence is the value of Mh3 , which is signif- icantly heavier, and of the order of 500 GeV or 1 TeV, respectively.

In scenario A3 there are two Higgs particles that have mass below 2MW : h1 (the

SM-like Higgs) and h2.

We treat 2MW as the distinguishing value because two Higgs particles of masses smaller than 2MW inﬂuence the DM phenomenology by introducing another resonance region in the medium DM mass regime.

Below we shall discuss properties of DM for the listed benchmark points. In our analysis we focus on three diﬀerent mass regions8:

± 1. light DM mass: 50 GeV < MH < Mh1 /2 with MA = MH + 50 GeV,MH = MH + 55 GeV,

± 2. medium DM mass: Mh1 /2 < MH < MW with MA = MH + 50 GeV,MH = MH + 55 GeV,

3. heavy DM mass: MH & 500 GeV with MA = MH± = MH + 1 GeV, which are based on studies of the IDM. These mass splittings are in agreement with all collider constraints, including the EWPT limits, for all studied benchmark points (see table 7.4 in Appendix 7.5.4 for exact values).

8 Very light DM particle from the IDM with MH . 10 GeV is excluded by combined relic density and Higgs-invisible decay limits from the LHC [155].

86 We are not going to address the possibility of accidental cancellations in region MW < M < 160 200 GeV [204], leaving it for the future work. Note however, that this region H − could in principle be modiﬁed with respect to the IDM in benchmark A2.

4.4.2 Light DM

In this work we define the light DM region as 50 GeV < MH < 62 GeV. As mentioned in section 4.3.3, the SM-like Higgs particle can decay invisibly into a HH pair (or also into AA, if we allow MA < Mh1 /2). Measurements of invisible decays strongly constrain the value of the DM-Higgs coupling, which in case of cIDMS is c1c2λ345. The results presented in this section were obtained for benchmark A1. Other benchmarks were also tested and they provide no noticeable change in the results. In all considered benchmarks β = c c 1 and the main annihilation channel of DM particles is HH 1 1 2 ≈ → h b¯b, regardless of the values of M and M . 1 → h2 h3 2 In the figure 4.9 the relation between ΩDM h and MH is presented, for a few chosen values of λ . As discussed before, λ 0.015 0.02 is the boundary value which is in 345 | 345| ∼ − agreement with LHC limits for Br(h inv). From figure 4.9 one can see that this value → gives the proper relic density for masses of the order of 53 GeV, which is a result that had been previously obtained for One- and Two-Inert Doublet Models [155, 199]. This value of the coupling for masses below 53 GeV results in a relic density well above the Planck limits, which leads to overclosing of the Universe. For these smaller masses, to obtain a proper relic density, one needs to enhance the DM annihilation by taking a bigger value of coupling ( λ 0.05, 0.07 ), which at the same time will lead to the enhanced Higgs invisible decays | 345 ∼ | and this is not in agreement with the LHC results. For masses bigger than 53 GeV coupling corresponding to the proper relic abundance gets smaller ( λ 0.002), fitting into LHC | 345| ∼ constraints.

As discussed in section 4.3.3, if the Higgs can decay invisibly, its total decay width is strongly affected with respect to the SM, and therefore it is not possible to obtain enhancement in the Higgs di-photon decay channel, i.e. < 1, see figure 4.5. This was confirmed Rγγ by a direct check we performed, and the detailed values are presented in the Appendix 7.5.4 in table 7.5. The maximum allowed value of for parameters which are in agreement Rγγ both with the relic density constraints, and with the LHC invisible branching ratio limits, is between 0.85 0.91 for benchmarks A1-A2. It is interesting to note, that for bench- Rγγ ≈ − mark A3, i.e. the one with two relatively light Higgs particles, the results are different, here differs from the SM value by more than 20%. This is an important difference, because Rγγ 87 for light DM particles calculation of relic density does not depend on the chosen benchmark.

Similar situation happens with values of , which are close to the SM value for bench- RZγ marks A1-A2 (depending on the values of parameters one can obtain both an enhancement or a suppression with respect to = 1), however for benchmark A3 this channel is sup- RZγ pressed by more than 20 %.

2 ΩDMh

0.6 λ345=0.002

0.5 λ345=0.015

0.4 λ345=0.05

0.3 λ345=0.07 0.2 Planck+3σ 0.1 Planck-3σ MH[GeV] 52 54 56 58 60 62

2 Figure 4.9: Values of DM relic density (ΩDM h ) with respect to DM mass (MH ) for chosen values of λ345 parameter, for benchmark A1. Horizontal lines represent 3σ Planck bounds, region above is excluded, in region below additional DM candidate is needed to complement missing DM relic density. Calculations done for MA = MH + 50 GeV,MH± = MH + 55 GeV, however exact values of those parameters do not inﬂuence the output, as the coannihilation eﬀects are surpressed.

4.4.3 Medium DM

In this section we focus on the medium mass region from the cIDMS, i.e. masses of DM candidate between M /2 62 GeV and M 83 GeV. h1 ≈ W ≈

Figures 4.10a-4.10c show the behaviour of relic density with respect to λ345 for masses of DM candidate changing between Mh1 /2 and MW , for chosen cIDMS benchmark points A1-A2 (figure 4.10a) and A3 (figure 4.10b). The results for the IDM are well known in the literature; we have included them for comparison in figure 4.10c. There is a near-resonance region, M M /2, symmetric around λ 0. Larger DM masses correspond to greater H ∼ h 345 ≈ annihilation into gauge bosons, causing asymmetry with respect to λ345 = 0. Also, the increased annihilation rate leads to the lowered relic density.

This behaviour is repeated by benchmark points A1-A2 of cIDMS, where both additional

Higgs particles are heavier than 2MW . However, one can see that the presence of these additional states is non-negligible. It is important to stress that even for β β , i.e. the 2,3 1 case that was supposed to be close to the IDM, the impact of three Higgs states on the value

88 of relic density is signiﬁcant. In general, the annihilation of DM particles is enhanced and therefore the relic density for a given mass is lower with respect to DM candidate from the IDM. This means, that in the cIDMS for the masses of DM candidate bigger than 79 GeV relic density is below the Planck limit, while for the IDM masses of up to 83 GeV can be in agreement with the measured value.

A new phenomena with respect to the IDM can happen if one of the extra Higgs bosons

is lighter than 2MW , which is the case for benchmark A3. As the mass of DM candidate gets closer to this h2-resonance, i.e. MDM & 70 GeV, the eﬀective annihilation cross-section increases, resulting in the relic density below the observed value. Clearly, the annihilation

rate is enhanced and dominated by the Higgs-type exchange through h2 (note the symmetric distribution around λ345 = 0), in contrast to the previously discussed cases, whereas for the heavier masses the annihilation into gauge bosons is starting to dominate, therefore pushing the good region towards negative values of λ345.

Ω h2 h2 M = 64 GeV DM ΩDM H 0.6 M = 64 GeV 0.6 H M = 66 GeV H 0.5 M = 66 GeV 0.5 H MH = 68 GeV MH = 68 GeV 0.4 0.4 MH = 70 GeV M = 69 GeV 0.3 H 0.3 MH = 72 GeV MH = 70 GeV 0.2 MH = 74 GeV 0.2 MH = 72 GeV 0.1 MH = 76 GeV 0.1 λ345 MH = 73 GeV λ345 -0.20 -0.15 -0.10 -0.05 0.05 0.10 MH = 77 GeV -0.04 -0.02 0.02 0.04 (a) A1-A2 (b) A3

2 ΩDMh MH = 63 GeV 0.6 MH = 66 GeV 0.5 MH = 69 GeV 0.4 MH = 72 GeV

0.3 MH = 75 GeV

0.2 MH = 77 GeV

0.1 MH = 80 GeV λ345 -0.25 -0.20 -0.15 -0.10 -0.05 0.05 MH = 82 GeV

2 Figure 4.10: Relation between DM relic density ΩDM h and λ345 for chosen values of MH for (a) benchmark A2, (b) benchmark A3, (c) the IDM. Horizontal lines represent Planck limits for Ω h2 = 0.1199 3σ, region above is excluded. Calculations done for M = DM ± A MH + 50 GeV,MH± = MH + 55 GeV, however exact values of those parameters do not inﬂuence the output, as the coannihilation eﬀects are surpressed.

The diﬀerence between benchmarks is even more striking if one studies good regions

of relic density in the plane (MH , λ345), as presented in ﬁgure 4.11. For cases A1-A2 the

89 behaviour follows that of the IDM, with the corresponding couplings being slightly smaller. Nevertheless, the scenario is repeated and one can clearly see the shift towards negative values of λ345. In case of benchmark A3 the situation is completely diﬀerent; not only the mass range is signiﬁcantly reduced with respect to the previous cases and the IDM, but also the values of coupling are much smaller, concentrated symmetrically around zero.

Relic density constraints (PLANCK) Case A2 Case A3 0.05 0.05 Excluded Excluded

0 0

345 -0.05 345 -0.05 λ λ

-0.1 -0.1

-0.15 -0.15 64 66 68 70 72 74 76 78 64 66 68 70 72 74 76 78

MH [GeV] MH [GeV]

Figure 4.11: Relic density constraints on the mass of the DM candidate and its coupling to SM Higgs boson, with the white and gray regions representing too low (not excluded, but an additional DM candidate needs to be added to the model) and too high (excluded) relic abundance, respectively. Red and blue regions corresponds to relic density in agreement with Planck measurements for benchmark A2 and A3, respectively.

The cIDMS, as other scalar DM models, can be strongly constrained by results of direct detection experiments. The current strongest limits come from LUX experiment, and are presented in ﬁgure 4.12. There are also results of calculation of DM-nucleus scattering cross-section, σDM,N for the benchmark points discussed in this section. Red regions denote benchmarks A1-A2, while blue regions correspond to benchmark A3. The diﬀerence between those two groups is clear. In case of benchmark A3, the coupling is usually much smaller than in cases A1-A2, therefore the resulting cross-section will be also smaller9, falling well below the current experimental limits.

LHC analysis provides us with further constraints for the studied region. For benchmarks A1-A3 values of and are within the ATLAS & CMS experimental uncertainties, Rγγ RZγ 9 Recall that the DM scattering oﬀ nuclei is mediated by the Higgs particles, h1, h2, h3, therefore the strength of this scattering will directly depend on the value of DM-Higgs couplings.

90 2 σDM,N[cm ] 10-43 10-44 A3 -45 10 A1 10-46 10-47 LUX 10-48 MH[GeV] 65 70 75 80 Figure 4.12: Direct detection constraints for considered benchmarks (A1-A2: red, A3: blue). All points are in agreement with relic density measurements and collider constraints. Black line: upper LUX limit.

with the preferred value of and below 1. The value of these signal strengths depends Rγγ RZγ on the exact values of parameters and an enhancement is possible, but not automatic. All values are listed in table 7.6 in Appendix 7.5.4.

Case A3 diﬀers from the other two benchmarks because of the presence of an extra light Higgs particle. For points that have good relic density, allowed values of are close to Rγγ 0.75, with also below 1, namely 0.79 (see the table 7.6 in Appendix Rγγ ≈ RZγ RZγ ∼ 7.5.4).

Recall however, that in contrast with the low DM mass region, here the diﬀerence between two groups of benchmarks is visible already for calculations of DM relic density.

4.4.4 Heavy DM

In the heavy mass regime all inert particles have similar masses, because of perturbativity 2 limits for self-couplings λi. Those masses are driven by the value of m22, which can reach large negative values. Therefore, the mass splittings given by combination of λ4,5 are small. In this analysis we choose them to be:

MA = MH± = MH + 1 GeV. (4.51)

2 Figure 4.13 presents the relation between relic density ΩDM h and DM-Higgs coupling

λ345 for benchmarks A1, for ﬁxed values of DM mass.

91 2 ΩDMh 0.20 MH = 700 GeV

0.18 MH = 675 GeV

MH = 650 GeV 0.16 MH = 625 GeV 0.14 MH = 600 GeV

0.12 MH = 575 GeV

λ345 MH = 550 GeV -0.4 -0.2 0.2 0.4 Figure 4.13: Heavy DM candidate: relation between relic density and DM-Higgs coupling

λ345 for benchmarks A1 for chosen values of MH . Results for A2 and A3 are equivalent to A1. Horizontal lines denote 3σ Planck limits.

It is interesting to note, that this region of masses is more similar to the low DM mass region, than to the medium mass region. Although all benchmarks result in the very similar values of Ω h2, just like for the light DM, there is a diﬀerence when it comes to and DM Rγγ . Again, for cases A1-A2 the preferred value of is bigger, this time tending towards RZγ Rγγ the close neighborhood of 1. For case A3 resulting values are smaller, close to = 0.8. Rγγ Detailed values are presented in table 7.7 in Appendix 7.5.4.

4.5 Conclusions and Outlook

In this work we have studied the cIDMS – an extension of the SM, namely a Z2 symmetric

Two-Higgs Doublet Model with a complex singlet. This model, apart from having a Z2-odd scalar doublet, which may provide a good DM candidate, contains a complex singlet with a non-zero complex VEV, which can bring additional sources of CP violation. This is a feature that is missing in the IDM.

Within the considered model different scenarios can be realized. We have focused on the case where the SM-like Higgs particle, existence of which has been confirmed by the ATLAS and CMS experiments at the LHC, comes predominantly from the first, SM-like doublet, with a small modification coming from the singlet. In addition to the SM-like Higgs there are two other neutral Higgs particles, and their presence can strongly influence Higgs and DM phenomenology.

92 We constrain our model by comparing the properties of the light Higgs particle (h1) from the cIDMS with the one arising from the SM. LHC results provide limits for the Higgs-decay signal strengths, in particular h γγ. There are correlations and . 1 → Rγγ ∼ RZγ Rγγ ∼ RZZ The maximum value for h ZZ signal strength is 1. For smaller masses of the charged 1 → scalar there is a possibility of enhancement of both and . For heavier M ± the Rγγ RZγ H maximum values tend to the SM value. and can be bigger than 1 if there is Rγγ RZγ constructive interference between the SM and the cIDMS contributions. Notice, that this enhancement is possible simultaneously as in the IDM, i.e. there is a correlation between enhancement in γγ and Zγ channels.

The cIDMS can provide a good DM candidate, which is in agreement with the current experimental results. The low DM mass region, which we define as masses of H below Mh1 /2, reproduces behavior of known Higgs-portal DM models, like the IDM. For MH . 53 GeV it is not possible to fulfill LHC constraints for the Higgs invisible decay branching ratio and relic density measurements at the same time. For 53 GeV . MH . 63 GeV we are in the resonance region of enhanced annihilation with very small coupling λ345 corresponding to proper relic density. This region is in agreement with collider and DM direct detection constraints, however we expect the loop corrections to play an important role here. It is important to stress that, while DM phenomenology does not depend on the chosen benchmark point (A1-A2), there is a difference when it comes to the LHC observables. Values of for Rγγ benchmark A3 are smaller than in all other cases, being always below 1.

For heavier DM mass, our studies show that the annihilation cross-section is enhanced with respect to the IDM and therefore relic density in the cIDMS is usually lower than for the corresponding point in the IDM. This is the case both in the medium and the heavy DM mass region.

The most striking change with respect to the IDM arises in the relic density analysis with the possibility of having an additional resonance region if the mass of one of additional

Higgs particles is smaller than 2MW . For our chosen benchmark points it happens in case A3. Corresponding DM-Higgs couplings, and therefore the resulting DM-nucleus scattering- cross-section constrained by results of direct detection experiments, are much smaller for A3 than for other benchmark points. This point, however, results in the much smaller values of and . These values are on the edge of 20 % diﬀerence with respect to the SM value, Rγγ RZγ and – while not being yet excluded by the experiments within current experimental errors, they are not favored. For other studied benchmark points, both relic density calculations, and the LHC observables, do not depend very strongly on the exact values of masses of Higgs particles. Preferred values of are of the order of 0.95. Rγγ 93 In the heavy mass region all inert particles are heavier than the particles from the SM sector and the impact on the Higgs phenomenology can be minimal. For example, this is the region where is the closest to the SM value. Rγγ

94 Chapter 5

Implication of Quadratic Divergences Cancellation in the Two Higgs Doublet Model

The SM describes the physics of elementary particles with a very good precision [206]. Experiments have conﬁrmed its predictions with remarkable precision. One of the most precise aspects of the model is associated with the Higgs sector. However, the SM is not a complete perfect model, since it is unable to provide adequate explanations for many questions in particle and astrophysics. One of the problems of the SM is the naturalness of the Higgs mass. From experimental data we know that the Higgs boson mass (125 GeV) is of the order of the EW scale, yet from the naturalness perspective this mass should be much larger than the EW scale. This is because of the large radiative corrections to the Higgs mass which implies an unnatural tuning between the tree-level Higgs mass and the radiative corrections. These radiative corrections diverge, showing a quadratic sensitivity to the largest scale in the theory[207]. Solutions to this hierarchy problem imply new physics beyond the SM, which must be able to compensate these large corrections to the Higgs boson mass. This goal can be obtained with the presence of new symmetries and particles. The quadratic divergences were studied within the SM by Veltman [37]. He suggested that the radiative corrections to the scalar mass vanish (or are kept at a manageable level). This is popularly known as the Veltman condition.

In this chapter, we apply Veltman condition in the 2HDM (Model II of the Yukawa interaction) to predict masses of the additional neutral scalars in two possible scenarios, with the SM like-h and the SM like-H bosons. In section 5.1, we introduce the 2HDM with

95 nonzero vacuum expectation for both doublets (Mixed Model), present formulae for masses of Higgs bosons as well as the basic couplings for the neutral Higgs particles and deﬁne two SM-like scenarios. The conditions for cancellation of quadratic divergences for the 2HDM are presented in section 5.2. In sections 5.3, we discuss the properties of the approximate (analytical) solution of these conditions. The numerical solutions for two considered scenarios are presented in section 5.4. In section 5.5 the experimental constraints are described which are fulﬁlled by the benchmarks presented in section 5.6. Finally, section 5.7 is devoted to the summary and conclusions.

5.1 Mixed Model with a soft Z2 symmetry breaking

The Higgs sector of the 2HDM consists of two SU(2) scalar doublets Φ1 and Φ2. The 2HDM 2 2 2 potential depends on quadratic and quartic parameters, respectively m11, m22, m12 and λi (i= 1..., 5), from which ﬁve Higgs bosons masses come up after the spontaneous symmetry breakdown (SSB). The most general SU(2) U(1) invariant Higgs potential for two doublets + 0 × Φ = (φ , Φ )†, with a soft Z symmetry (Φ Φ , Φ Φ ) violation is given by: 1,2 1,2 1,2 2 1 → 1 2 → − 2 λ λ V = 1 (Φ+Φ )2 + 2 (Φ+Φ )2 + λ Φ+Φ Φ+Φ + λ Φ+Φ Φ+Φ 2 1 1 2 2 2 3 1 1 2 2 4 1 2 2 1 1 m2 m2 m2 + ( λ (Φ+Φ )2 + h.c.) 11 Φ+Φ 22 Φ+Φ ( 12 (Φ+Φ ) + h.c.). (5.1) 2 5 1 2 − 2 1 1 − 2 2 2 − 2 1 2 All parameters are assumed to be real, so that CP is conserved in the model.

In order to have a stable minimum, the parameters of the potential need to satisfy the positivity conditions leading to the the potential bounded from below. This behavior is governed by the quadratic terms, for them positivity conditions read as follows: p p λ > 0, λ > 0, λ > λ λ , λ < λ + λ + λ λ . (5.2) 1 2 3 − 1 2 | 5| 3 4 1 2 The Mixed Model is based on the vacuum with nonzero VEV for both doublets, respectively < φ 0 >= υ1 = 0 and < φ 0 >= υ2 = 0, with υ2 = υ2 + υ2. The minimization conditions 1 √2 6 2 √2 6 1 2 are as follows:

m2 = υ2λ + υ2(λ 2ν), (5.3) 11 1 1 2 345 − m2 = υ2λ + υ2(λ 2ν), (5.4) 22 2 2 1 345 − where λ λ + λ + λ , ν m2 /(2υ υ ). It is well known, that such minimum is at the 345 ≡ 3 4 5 ≡ 12 1 2 same a global minimum, i.e. vacuum [39].

96 There are ﬁve Higgs particles, with masses as follows:

2 1 2 M ± = (ν (λ + λ ))υ , (5.5) H − 2 4 5 M 2 = (ν λ )υ2. (5.6) A − 5 Other two mass squared M 2 being the eigenvalues of the matrix 2 h,H M " # cos2 βλ + sin2 βν (λ ν) cos β sin β 2 = 1 345 − υ2, (5.7) M (λ ν) cos β sin β sin2 βλ + cos2 βν 345 − 2

where tan β = υ2/υ1. This matrix written in the terms of the mass squared masss for physical particles M 2 , with M M , and the mixing angle α is given by h,H H ≥ h " # M 2 sin2 α + M 2 cos2 α (M 2 M 2) sin α cos α 2 = h H H − h . (5.8) M (M 2 M 2) sin α cos α M 2 sin2α + M 2cos2α H − h H h

We concentrate on the Model II for Yukawa interaction, where the down-type quarks,

with mass denoted mD, couple only to the ﬁrst scalar doublet and the up-type quarks (mass 1 mU ) couple only to the second Higgs doublet .

The ratio of the coupling constant (gi) of the neutral Higgs boson to the corresponding SM SM coupling gi , called the relative couplings

gi χi = SM , (5.9) gi are summarized in the table 5.1 (see e.g. reference [216]). One sees that all basic couplings

χV (W and Z) χu(up-type quarks ) χd(down-type quarks ) h sin(β α) sin(β α) + 1 cos(β α) sin(β α) tan β cos(β α) − − tan β − − − − H cos(β α) cos(β α) 1 sin(β α) cos(β α) + tan β sin(β α) − − − tan β − − − A 0 iγ cot β iγ tan β − 5 − 5 Table 5.1: Tree-level couplings of the neutral Higgs bosons to gauge bosons and fermions in 2HDM (II).

can be represented by the couplings to the gauge boson V, χ = sin(β α) (cos(β α)) for V − − h(H) and the tan β parameter.

So, we consider two cases which deﬁne our SM-like scenarios:

1In the analysis leptons, which in the Model II couple as the down-type quarks, are neglected.

97 M 125 GeV, sin(β α) +1 (β α = π/2) (SM-like h scenario) • h ∼ − ∼ −

M 125 GeV, cos(β α) +1 (β = α) • H ∼ − ∼ cos(β α) 1(β α = π) (SM-like H scenario). − ∼ − − ±

In both cases we identify a SM-like Higgs boson with the 125 GeV Higgs particle observed at LHC. Therefore, in the SM-like h scenario the neutral Higgs partner (H) can only be heavier, while in the SM-like H scenario - the partner particle h can only be lighter than ∼ 125 GeV.

In this analysis, we apply the positivity conditions and use the limit on λ < 4π as a perturbativity condition. We keep the mixing angles α in the range: π/2 < α < π/2 and − 0 < β < π/2.

5.2 Cancellation of the quadratic divergences

The SM is not a stable theory at the Planck scale (hierarchy problem). This is because of the large radiative corrections to the Higgs boson mass. These corrections diverge quadratically with the largest scale in the theory. This leads to unnatural cancellation between the tree- level Higgs boson mass and the radiative corrections. This is known as a ﬁne tuning problem of the SM [207]. The most important one-loop contributions come from the diagrams, involving the gauge bosons, top quark and the Higgs boson loops, see ﬁgure 5.1.

The one-loop Veltman’s condition in the SM model is given by

2M 2 + M 2 + M 2 = 4Σ N m2 , (5.10) W Z HSM f f f

where the sum is over all SM fermions with Nf its number of color degrees of freedom, i.e. N = 3 for quarks and N = 1 for leptons. The Eq. (5.10) is satisﬁed for M 310 GeV, f f HSM ∼ while the the current limit for the Higgs mass reads, M 125 GeV. Therefore, cancella- HSM ∼ tion of the quadratically divergent loop contributions within SM requires an unrealistic value of the Higgs boson mass. Additional contributions to the quadratic divergences at one-loop in the 2HDM may allow for a proper mass for the SM Higgs boson [208, 209, 210]

The cancellation of the quadratic divergences at one-loop applied to 2HDM with Model

98 18 Hierarchy Problem

t H H

W,Z,γ H

H H H H

Figure 2.1: The most signiﬁcant quadratically divergent contributions to the Higgs mass in the Standard Figure 5.1: TheModel. most signiﬁcant quadratically divergent loop contributions from fermion, W/Z gauge bosons and Higgs bosons to the Higgs boson mass.

2 3 2 2 2 2 δ m = (3g + g0 + 8λ 8λ )Λ , (2.1) q 64π2 − t where g, g , λ and λ are the SU(2) U(1) gauge couplings, the quartic Higgs coupling and II for Yukawa interaction0 leadst to a set× ofY two conditions [210], namely: the top Yukawa coupling respectively. In terms of masses 2 2 2 12 2 6M + 3M + υ 3(3λ + 2λ + λ ) = m , (5.11) W δ Zm2 = (21m2 + m32 + m24 4m2)Λ2 ,2 D (2.2) q 16π2v2 W Z h − t cos β

2 22 2 2 2 2 22 2 2 2 2 2122 2 where m = g v /4, m = (g + g0 )v /4, m = 2λv and m = λ v /2, with v = 246 GeV. 6WM W + 3MZ Z + υ (3λ2 +h 2λ3 + λ4)t = t 2 mU . (5.12) These radiative corrections to the Higgs vacuum expectation valuesin(vev)β tend to destabilize the electroweak scale. The requirement of no ﬁne-tuning between the tree-level and the one- Among fermions we include only the dominate top and bottom quarks contributions (mD loop contribution to m2 sets an upper bound on Λ. E.g for a Higgs mass m = 115 200 GeV, → m , m m ). Expressing λ’s parameters by masses and the massh parameter− m , we have b U t and imposing that one-loop contributions are not bigger than 10 times the value of m2 1, 12 → h ! 2 δ1 δquadm2 A11 A12 M = 10 Λ < 2 3 TeVh, , (2.3) (5.13) m2 ≤ ⇒ ∼ − 2 δ2 A21 A22 MH 2 where we have implicitly used v2 = m and m2 = 2m2. With this argument, new physics where − λ h should appear to modify the ultraviolet behaviour of the SM. This is known as the ”Big 2 2 12Hierarcmb hy” problem2 [10], since2 other fundamen2 tal scales2 (MP lanckm, 12MGUT ) are larger than2 δ1 = 6M 3M 2M ± M + [1 + 3 tan β], (5.14) costhis2uppβ er−boundWon −Λ. Z − H − A 2 sin β cos β 1Obviously, if one is stricter about the maximum acceptable size of δ m2, then Λ2 decreases in the same 2 q 2 12propmortion.t 2 2 2 2 m12 2 δ = 6M 3M 2M ± M + [1 + 3 cot β], (5.15) 2 sin2 β − W − Z − H − A 2 sin β cos β and 3sin2α 2 sin α cos α A = , (5.16) 11 cos2β − sin β cos β 3cos2α 2 sin α cos α A = + , (5.17) 12 cos2β sin β cos β 3cos2α 2 sin α cos α A = , (5.18) 21 sin2β − sin β cos β 3sin2α 2 sin α cos α A = + . (5.19) 22 sin2β sin β cos β In the following we solve the set of equations (5.13) expressing the condition for cancellation of the quadratic divergences to derive masses of the partner Higgs particles.

99 5.3 Approximate solution of the cancellation conditions

It is useful to look ﬁrst at the approximate solution, which can be obtained analytically.

In the 2HDM model with the soft Z2 symmetry breaking we have for the strict SM-like (alignment) scenarios, with sin(β α) = 1 or cos(β α) = 1: − −

1 M 2 SM like h : λ λ = (tan2 β )( H ν), (5.20) − 1 − 2 − tan2 β v2 − 1 M 2 SM like H : λ λ = (tan2 β )( h ν). (5.21) − + 1 − 2 − tan2 β v2 −

These formula follow directly from equations (5.7) and (5.8). The diﬀerence of λ1 and λ2 is given in terms of tan β, ν and the mass of the neutral partner for the SM-like h or H particle, mining respectively the H or the h boson. From diﬀerence of equations (5.11) and (5.12) we found that

4 m2 m2 4m2 m2/m2 λ λ = ( b t ) = b (1 t b )(1 + tan2 β). (5.22) 1 − 2 v2 cos2 β − sin2 β υ2 − tan2 β

Combining the equations (5.20), (5.21) and (5.22), we obtain the following expressions for masses squared of the partner of the SM-like h or H Higgs particle,

2 tan2 β mt m2 M 2 = 4m2 − b + νv2. (5.23) b tan2 β 1 −

Obviously, the above prediction for the mass of the partner Higgs particle has been obtained without additional constraints. We have plotted M versus tan β, as given by Eq. (5.23) for

m12 = 0 and 100 GeV, in the ﬁgure 5.2.

100 1 0 4

1 0 3

2 2 ) m 1 2 = 0 ( G e V ) V

e 2 2 2 2 m = 1 0 0 ( G e V ) G 1 2

( 1 0 M < 1 2 7 G e V M

1 0 1

1 0 0 0 . 1 1 1 0 1 0 0 t a n β

Figure 5.2: The mass of the partner of the SM-like h(H) Higgs particle versus tan β based 2 2 2 2 on Eq. (5.23) for m12 = 0 and m12 = 100 GeV .

In the ﬁgure the mass limit 127 GeV for the SM-like particle is used. The hachured ∼ area (i.e. all masses less than 127 GeV) is the allowed region for Mh in the SM-like H scenario and the white area (i.e. all masses higher than 127 GeV) is allowed for MH in the

SM-like h scenario. Let us look at the m12 = 0 = ν case. It is clear that for the SM-like h, solutions exist only for tan β 1 and the mass of H should be larger than M = 2m . ≤ H0 t The solutions for the SM-like H exist for large tan β (tan β m /m 43) with mass of + ≥ t b ≈ h below Mh0 = 2mb. For positive ν, in the intermediate tan β new regions open up e.g. for 2 2 2 2 m12 = 100 GeV . For negative ν(m12), all curves are lying below the reference ν = 0 curves.

5.4 Solving the cancellation conditions

Here we present the results of numerical solutions of Eqs. (5.13) for the SM-like scenarios, as described above. We apply the positivity and the perturbativity constraints on parameters of the model. We have performed three scans for three considered SM-like scenarios (SM-like h, SM-like H+ and SM-like H-) with the mass window of the SM-like Higgs 124-127 GeV and the relative coupling to gauge bosons χV between 0.90 and 1.00, in agreement with the newest LHC data, which are presented in the Appendix 7.1.

We assume v being bounded to the region 246 GeV < v < 247 GeV and using following

101 regions of the parameters of the model:

M < M 1000 GeV,M ± [360, 800] GeV (5.24) h H ≤ H ∈ m2 [ 4002, 4002] GeV2,M [130, 700] GeV, 12 ∈ − A ∈

Note, that very recently the new lower bound on MH± has been derived, much higher than the used by us in the scan 360 GeV [101], namely: M ± > 570 800 GeV [87]. In the H − calculation we use mt = 172.44 GeV, mb = 4.18 GeV, MW = 80.38 GeV, MZ = 91.18 GeV [212, 213]. Solutions of the Eqs. (5.13)) were found by using Mathematica and independently by a C + + program, written by us.

Performing our scanning we found no solution for the SM-like H scenario in both cases cos(β α) 1, for mass of the charged Higgs boson larger than 360 GeV [101]. − ∼ ±

For SM-like h scenario there are solutions only for positive m12, in the region 200 - 400

GeV. Figure 5.3 shows the correlation between m12 and Mh (panel (a)) and the correlation between m12 and MH (panel (b)). In both panels, the ﬁrst region from the left (lower m12 region) is obtained for large tan β (above 40), while the right one corresponds to low tan β

(below 5). Figure 5.3(c) shows the correlation of tanβ with MH . In the low tan β the lower limit for MH is 500 GeV.

(a) (b) (c)

Figure 5.3: The correlation between m12 and Mh (a), m12 and MH (b), tanβ and MH (c).

Below, we will look closer to the obtained results, by confronting the obtained predictions for observables with experimental data. We propose 5 benchmarks for the SM-like h scenario, which will be discussed in section 5.6, in agreement with theoretical constraints (positivity and perturbativity) and experimental constraints, mainly coming from the measurements of SM-like Higgs boson. In addition, properties of the partner particles (H) is checked, which is important for future search. In such calculations the 2HDMC program was used [215].

102 5.5 Experimental constraints

We apply experimental limits from LHC for the SM-like Higgs particle h as described in the Appendix 7.1. As we already mentioned we solve the cancellation conditions by scanning

over Mh and the mixing parameters (α, β), keeping the values of mass and coupling to the gauge bosons (i.e. sin(β α)) within the experimental bounds. We confront the resulting − solutions with existing data for the 125 GeV Higgs boson, in particular the experimental data on Higgs boson couplings (χV , χt and χb) and Higgs signal strength (RZZ , Rγγ and

RZγ) from ATLAS [95] and CMS [96], as well as the combined ATLAS+CMS results [100], collected in table 7.2. Also the total Higgs decay width measured at the LHC is an important constraint, see the Appendix 7.1.

We also keep in mind other existing limits on additional Higgs particles which appear in 2HDM, as the lower mass limit of the H+ taken to be 360 GeV (based on the earlier analysis [101]). We also check if the obtained solutions are in agreement with the oblique parameters S, T , U constraints, being sensitive to presence of extra (heavy) Higgses that are contained in the 2HDM.

Below, the following short notation will be used: tβ, sβ α and cβ α for tan β, sin(β α) − − − and cos(β α), respectively. −

5.6 Benchmarks

Results from scan lead to ﬁve benchmark points h1-h5, presented in table 5.2. The result of the scanning shows that for the SM-like h scenario, solutions in agreement with existing data only exist for small tanβ (0.45 -1.07). Values of observables for the SM-like Higgs particle h for ﬁve benchmark points h1-h5 are presented in table 5.3. The large tanβ solutions,

(above 42) exist, however they lead to too large Rγγ, (above 2). For a convenience we add to the both tables experimental data (with 1 σ accuracy from the fit assuming χ 1 and | v| ≤ B 0.) SM ≥ Benchmarks correspond to solutions with masses M 505 827 GeV and M H ∼ − A ∼ 270 650 GeV and M ± 375 646 GeV. The newest result of the reference [87] with lower − H ∼ − bound on M ± 570 800 GeV can limit our benchmarks to (h3, h4) only. There is a small H ∼ − tension, at the 2 σ, for all benchmarks for the coupling hb¯b with the newest combined LHC result [100], which has surprisingly small uncertainty 0.16 (before the individual results were +0.24 +0.26 ATLAS 0.61 0.26 [95] and CMS 0.49 0.19 [96], in perfect agreement with all our benchmarks). − − 103 Also, benchmarks (h2,h3) correspond to slightly too small mass of the h in the light of the new combined CMS and ATLAS value of 125.09 0.24 GeV [214]. ± We compare our benchmarks to the experimental data on 2HDM (II) on the plot tan β versus cos(β α), see figure 5.7. Our benchmark h4 is very close to the best fit point found − by the ATLAS. We would like to point out that our benchmarks results from cancellation of the quadratic divergences and no fitting procedure has been performed. Note, that the h4 benchmark corresponds to heavy and degenerate A and H+ bosons, with mass 650 GeV, ∼ while H is even heavier with mass 830 GeV. ∼ Table 5.2: SM-like h for sin(β α) 1, Higgs bosons masses (in GeV) are shown for various − ∼ values of angles α and β. The experimental data for Mh and sβ α are from [214] and [100], − respectively.

2 B mark α tβ sβ α cβ α Mh MH MA MH± m12 − − exp - - 1.00 (0.92-1.00) - 125.09 0.24 - - - - ± h1 -1.24627 0.451897 0.995014 -0.0997376 124.426 573.832 444.16 454.424 (281.0690)2 h2 -1.10678 0.481736 0.999886 0.0150858 124.082 505.298 266.59 375.488 (191.9640)2 h3 -1.00657 0.507350 0.995518 0.0945748 124.242 736.961 567.37 598.392 (352.9160)2 h4 -0.96384 0.589698 0.997252 0.0740784 125.252 826.947 650.08 645.560 (421.8410)2 h5 -0.94625 1.077030 0.980477 -0.1966340 125.771 605.931 448.48 438.628 (309.6770)2

Table 5.3: SM-like h, sin(β α) 1. The h relative couplings, decays rates and S, T and − ∼ tot h h h h Γh U. The experimental data for χt , χb ,Rγγ and RZγ from [100], for totSM from [99] and for Γh oblique parameters from [90] are presented.

tot h h h h Γh B point χt χb Rγγ RZγ totSM S T U Γh +0.23 +0.19 +14 exp 1.43 0.22 0.57 0.16 1.14 0.18 < 9 10 10/4.1 0.05 0.11 0.09 0.13 0.01 0.11 − ± − − ± ± ± h1 0.77 1.04 1.03 0.98 0.96 -0.00 -0.02 -0.00 h2 1.03 0.99 1.09 0.97 0.96 -0.00 -0.24 -0.00 h3 1.17 0.94 1.05 0.98 0.95 -0.00 -0.07 -0.00 h4 1.12 0.95 1.11 1.08 0.97 -0.00 0.01 -0.00 h5 0.79 1.19 1.31 1.35 0.77 0.01 0.01 -0.00

It is worth to look for the properties of the heavy neutral Higgs boson H,the partner of the SM-like h bosons. The corresponding observables are given in table 5.4, here the relative couplings are calculated in respect to couplings the would-be SM Higgs boson with same mass

104 as H. The coupling to t quark is negative, what is easy to understand looking at the table 5.1. Its absolute value χH is enhanced, as compare to the SM value, and the corresponding | V | ratio varies from 1.1 to 2.30. One observes a huge enhancement in Rγγ, it is from 50 to 153 times larger the SM one, at the same time Zγ decay channel looks modest (0.31-1.44). Also the total width is similar to the one predicted by the SM - the corresponding ratio varies from 0.3 to 1.09. For a possible search for such particle the γγ channel would be the best. + The ZZ channel is hopeless, but the H decays to ZA and H W −, govern by sin(β α) − coupling, may be useful. Note, that all heavy Higgs bosons have masses below 850 GeV, in the energy range being currently probed by the LHC.

Table 5.4: SM-like h, sin(β α) 1. The partner H bosons masses, relative couplings and − ∼ decays rates are given.

tot H H H H ΓH B point MH χt χb Rγγ RZγ totSM ΓH h1 573.832 -2.30 0.34 69.14 0.62 0.88 h2 505.298 -2.06 0.49 14.88 0.31 1.09 h3 736.961 -1.86 0.59 152.71 1.21 0.78 h4 826.947 -1.61 0.66 49.95 1.44 0.53 h5 605.931 -1.10 0.85 72.79 0.79 0.28

5.7 Summary and Conclusion

In this chapter, we have investigated the cancellation of the quadratic divergences in the 2HDM, applying positivity conditions and perturbativity constraint. We have chosen the

soft Z2 symmetry breaking version of the 2HDM with non-zero vacuum expectation values for both Higgs doublets (Mixed Model) and considered two SM-like scenarios, with 125 GeV h and 125 GeV H. We applied the condition for cancellation of quadratic divergences for the Model II Yukawa interaction, in order to derive masses and couplings of the additional (i.e. not the SM-like) Higgs particles in the Model.

We have performed a three set of scanning based on the existing theoretical constraints and the LHC limits on the mass of the SM-like Higgs particle and the its coupling to gauge bosons, and the lower limits on the mass of H+, for SM-like h(H ) scenarios. We have ± found that the solutions exist only for the SM-like h scenario, for positive m12. Afterwards, we have chosen 5 benchmarks in agreements with experimental data for the 125 GeV Higgs

105 particle from the LHC. We have also checked that our benchmark points are in agreement with the oblique parameters S, T and U, at the 3 σ.

We compare our benchmarks to the experimental constraints for 2HDM (II) on the tan β versus cos(β α) plane. All benchmarks are close to or within the allowed 95% CL region, − especially our benchmark h4 is very close to the best ﬁt point found by the ATLAS. We would like to point out that our benchmarks results from cancellation of the quadratic divergences and no ﬁtting procedure has been performed. Note, that the h4 benchmark corresponds to heavy and degenerate A and H+bosons, with mass 650 GeV, while H is even heavier with ∼ mass 830 GeV. ∼ We have analyzed also the properties of the heavy neutral Higgs boson H, partner of the SM-like h bosons. The corresponding the relative couplings are calculated in respect to the would-be SM Higgs boson with same mass as H. The coupling to t quark is negative, and its absolute value is enhanced as compare to the SM value, up to 2.3 times, due to negligible H χV and small tan β. There is a huge enhancement in Rγγ, it is from 50 to 153 larger than the SM one, at the same time Zγ decay channel looks modest (0.31-1.44). Also the total width is similar to the one predicted by the SM - the corresponding ratio varies from 0.3 to 1.09. For a possible search for such particle the γγ channel would be the best. The ZZ + channel is hopeless, but the H decays to ZA and H W −, govern by sin(β α) coupling, − may be useful. Note, that all heavy Higgs bosons have masses below 850 GeV, in the energy range being probed by the LHC.

106 Figure 5.4: Regions of the 2HDM Model II excluded by to the measured rates of Higgs boson production and decays by ATLAS [102] and our benchmarks points h1-h5. The white allowed region contains the best ﬁt point denoted by a cross. Solid (short dashed) lines correspond to the observed (expected) 95 % limits, the long-dashed vertical line presents a SM prediction. 107 108 Chapter 6

Summary and Conclusion

Baryogenesis, the creation of the baryon asymmetry in the universe, is a long-standing problem in cosmology. Sakharov formulated his well-known conditions for baryogenesis: baryon number violation B, C and CP symmetry must be violated, and departure from thermal equilibrium should exist. Among many particle physics scenarios that have been proposed in the past decades, the EW baryogenesis is one of the most interesting. In this scenario the baryon number violation is caused by the anomaly that relates a change in baryon number (B + L) to a change in Chern-Simons number NCS of the EW gauge ﬁelds:

∆(B + L) = 2nF NCS.

The out-of-equilibrium conditions can be provided by an EW phase transition. This phase transition was supposed to be caused by the lowering temperature of the universe and to be suﬃciently out of equilibrium it had to be of ﬁrst order. However subsequent works have shown that for the experimentally allowed range of the Higgs mass, the EW phase transition is only a crossover. It is widely believed that a crossover transition is too close to equilibrium for the creation of the asymmetry. Furthermore, the CP violating phase in CKM matrix has been found to be much too small.

In this thesis, we have been investigated the properties of cSMCS - an extension of the SM containing a complex singlet with a non-zero complex VEV and a pair of heavy iso- doublet VQ, which allows for the spontaneous CP violation. We considered the potential with a softly broken global U(1) symmetry. Within our model diﬀerent vacua can be realized, however, we have focused on the case with the CP violating vacuum. We have derived a simple condition for the existence of such vacuum and found that at least one cubic term for χ is needed in order to have spontaneous CP violation. In the model, there are three neutral

109 Higgs particles, that the lightest is the SM-like Higgs, with the mass around 125 GeV, in agreement with LHC data and measurements of the oblique parameters. This model leads to a strong enough ﬁrst-order EW phase transition via the soft U(1) symmetry breaking terms

(κ2, κ3 and/or κ4) in the potential suppressing the baryon-violating sphaleron process. The parameter space of the model for the valid regions of BAU is scanned, concluding that the enlargement of the cSMCS model successfully predicts an acceptable value for BAU.

The possibility to test the Higgs sector of this model has been studied by us for the

LHC. The production rates for the SM Higgs bosons HSM and the Higgs bosons from the

cSMCS (h1, h2 and h3) using an eﬀective LO partonic matrix element and the UPDF of the KMR formalism have been calculated. The calculations for the SM Higgs boson have been compared with the existing experimental data of the CMS and the ATLAS collaborations, showing that our computations, within the given uncertainty bounds, present an acceptable platform to describe the Higgs bosons production at the LHC. Afterward, we have presented our predictions regarding the distribution of the transverse momentum and the rapidity of

the produced SM-like h1 and heavy Higgs bosons h2 and h3, from the cSMCS at the LHC. Detecting heavier Higgs bosons, if happen, will open the doors for further exploration of these ideas. These predictions may provide some clues regarding the dynamics of the next discovery.

We extended the cSMCS model by an Inert Doublet, which leads to the Inert Doublet Model plus a complex singlet (cIDMS). In this model, we have focused on the properties of DM, that is one of the essential ingredients of our Universe. The DM particle should be neutral, stable and weakly interacting and leads to the observed large-scale structure of the Universe. We have found that this model can provide a good DM candidate, which is in agreement with the current experimental results. In our analysis we have focused on

three diﬀerent mass regions, light DM mass (50 GeV < MH < Mh1 /2), medium DM mass

( Mh1 /2 < MH < MW ) and heavy DM mass (MH & 500 GeV). The low DM mass region,

which we deﬁne as masses of H below Mh1 /2, reproduces properties of known Higgs-portal DM models, like the IDM. For MH . 53 GeV it is not possible to fulﬁll LHC constraints for the Higgs invisible decay branching ratio and relic density measurements at the same time.

For 53 GeV . MH . 63 GeV we are in the resonance region of enhanced annihilation with very small coupling λ345 corresponding to proper relic density. This region is in agreement with collider and DM direct detection constraints. For heavier DM mass, the mere presence of heavier Higgs particles changes the annihilation rate of DM particles. Our studies show that the annihilation cross-section is enhanced with respect to the IDM and therefore relic density in the cIDMS is usually lower than for the corresponding point in the IDM. This is the case both in medium and heavy DM mass region.

110 Finally, we have investigated the 2HDM (II) using the cancellation of the quadratic

divergences conditions. For this, we have chosen soft Z2 symmetry breaking versions of the 2HDM with non-zero vacuum expectation values for both Higgs doublets (Mixed Model) and considered SM-like scenarios (with 125 GeV h and 125 GeV H) applying the condition for cancellation of quadratic divergences. We have performed a scanning based on the existing theoretical constraints and the LHC limits of the mass of the Higgs particle ( 125 GeV ∼ ) and its coupling to gauge boson and the lower limits (360 GeV ) on the mass of H+, for SM-like h(H ) scenarios. We have found that the solutions exist only for the SM-like ± h scenario. Afterward, we have chosen 5 benchmarks with properties of the light Higgs particle h in agreement with the LHC data. Moreover, we have checked the compatibility of our benchmark points with the 3σ bound on the oblique parameters S, T and U. We compare our benchmarks to the experimental constraints for 2HDM (II) on the tan β versus cos(β α) plane. All benchmarks are close to or within the allowed 95% CL region. We have − also analyzed the properties of the heavy neutral Higgs boson H, partner of the SM-like h bosons. The corresponding relative couplings are calculated with respect to the would-be SM Higgs boson with the same mass as H. The coupling to t quark is negative, and its absolute value is enhanced as compare to the SM value, up to 2.3 times, due to negligible H χV and small tan β. There is a huge enhancement in Rγγ, it is from 50 to 153 larger than the SM one, at the same time Zγ decay channel looks modest (0.31-1.44). Also, the total width is similar to the one predicted by the SM - the corresponding ratio varies from 0.3 to 1.09. For a possible search for such particle, the γγ channel would be the best. The ZZ + channel is hopeless, but the H decays to ZA and H W −, govern by sin(β α) coupling, − may be useful. Note, that all heavy Higgs bosons have masses below 850 GeV, in the energy range being probed by the LHC.

111 112 Chapter 7

Appendix

7.1 Appendix A: Experimental constraints on the Higgs sector

Here we collect the LHC data on the Higgs boson relevant for our analysis. All of them point towards to the SM-like scenario.

A. Properties of the SM-like Higgs boson

Mass of the SM-like Higgs boson observed at LHC has been measure using the

γγ and ZZ∗ channels. It is equal to [214]

M = 125.09 0.24 GeV. exp ±

Total decay width is constrained by the oﬀ-shell production - its upper limit is [96, 95], Γtot H < 5.4. Γtot H |SM +14 The limit 10 10 MeV to be compared to the SM value 4.1 MeV was presented by − CMS [99]. Signal strengths: The global signal strength obtained in the recent combined CMS and ATLAS analysis [100] is equal +0.11 µ = 1.09 0.10. − 113 Higgs Signal Strength ATLAS CMS ATLAS+CMS +0.40 +0.32 +0.26 ZZ 1.52 0.34 1.04 0.29 1.29 0.23 +0− .27 −+0.25 −+0.19 γγ 1.14 0.25 1.11 0.23 1.14 0.18 − − − Zγ RZγ < 11 RZγ < 9.5 -

Table 7.1: The signal strength of the SM Higgs boson into ZZ, γγ and Zγ from ATLAS [95],[97] and from CMS [96][98], as well as from the combined analysis ATLAS+CMS [100].

Some of the individual signal strengths for the 125 GeV Higgs are given in the table 7.1. Some of relative couplings of the 125 GeV Higgs boson obtained as a ratio of the observed coupling to the one predicted for the SM Higgs boson with the same mass are given in the table 7.2

Higgs coupling ATLAS CMS ATLAS+CMS

χZ 1 -1 1.00 (-0.97,-0.94)U(0.86,1.00) (-0.97,-0.94)U(0.86,1.00) (0.92-1.00) +0.35 +0.42 +0.23 χt 1.31 0.33 1.45 0.32 1.43 0.22 +0− .24 +0− .26 − χb 0.61 0.26 0.49 0.19 0.57 0.16 | | − − ±

Table 7.2: χZ , χt and χb from ATLAS [95] and from CMS [96], as well as the combined analysis ATLAS+CMS [100].

B. The EWPT provides strong constraints for New Physics beyond the SM. In particular, additional particles may introduce important radiative corrections to the gauge boson propagators. These corrections can be parameterized by the oblique parameters S, T and U. The T parameter is sensitive to the isospin violation, i.e. it measures the diﬀerence between the new physics contributions of neutral and charged current processes at low energies, while the S parameter gives new physics contributions to neutral current processes at diﬀerent energy scales. The U parameter is generally small in new physics models. The latest values of the oblique parameters, with reference mass-values of top and

Higgs boson Mt,ref = 173 GeV and Mh,ref = 125 GeV, are shown in from the reference [90]: S = 0.05 0.11,T = 0.09 0.13,U = 0.01 0.11 exp ± exp ± exp ± . The latest values of the oblique parameters, with reference mass-values of top and

114 Higgs boson Mt,ref = 173 GeV and Mh,ref = 125 GeV, [90] gives the 3σ bounds:

0.28 < S < 0.38, 0.30 < T < 0.48, 0.32 < U < 0.34. − − −

7.2 Appendix B

In the cSMCS, the SM is extended by a complex singlet with a non-zero complex VEV and a pair of heavy iso-doublet VQ. The couplings of scalars with fermions and gauge bosons are presented in the Appendix 7.2.1. In Appendix 7.2.2, we present the possible minima of the potential. Additionally, the formula of the oblique parameters for the cSMCS model are presented in the Appendix 7.2.3. The decays width of h γγ in the cSMCS model and → Higgs trilinear couplings have been shown in Appendices 7.2.4 and 7.2.5, respectively. In the Appendix 7.2.6, we have calculated the Jarlskog-type invariant for the case with κ = 0. 4 6

7.2.1 Coupling of scalars with gauge bosons and fermions

We can represent rotation of the fields φ , i = 1 4, from the doublet Φ and the singlet χ i − as follow  G0  ! φ + iφ  h  1 4  1  = P   , (7.1) φ2 + iφ3  h2  h3 where the 2 4 matrix P is equal to × ! i r r r P = 11 21 31 , (7.2) 0 r12 + ir13 r22 + ir23 r32 + ir33 and h , i = 1 3, are the physical field. VQ are very heavy and will decouple. Therefore, i − the coupling h with fermions is suppressed by r . contains the Yukawa interactions 1 11 LY between the fermions and the Higgs fields hi and it is given by:

X mf = ff(r h + r h + r h ) (7.3) LY − v 11 1 21 2 31 3 f

The fermions obtain mass via SSB.

The kinetic term in has aq standard form: Lscalar µ Tk = (DµΦ)† (D Φ) + ∂χ∂χ∗, (7.4)

115 with Dµ being a covariant derivative for an SU(2) doublet and can be deﬁned as

1 α α D = ∂ ig τ W ig0YB , (7.5) µ µ − 2 µ − µ the covariant derivative of the neutral singlet χ is identical with their ordinary derivative. We have 1 1 1 T = ∂ h ∂µh + ∂ h ∂µh + ∂ h ∂µh k 2 µ 1 1 2 µ 2 2 2 µ 3 3 M 2 + M 2 W +W µ + z Z Zµ w µ − 2 µ 2 + µ Mz µ + g(MwWµ W − + ZµZ )[r11h1 + r21h2 + r31h3] 2cw 2 2 g + µ g µ 2 2 + ( W W − + Z Z )[r h h + r h h µ 2 µ 11 1 1 21 2 2 4 8cw 2 + r31h3h3 + r11r21h1h2 + r11r31h3h2 + r21r31h2h3] + gauge cubic/quartic terms. (7.6) p The quadratic terms give masses to the W and Z bosons: M = 1 v g ,M = 1 v g2 + g 2. W 2 | | Z 2 0 Since the neutral singlet ﬁeld carries no hypercharge, its VEV does not contribute to the masses of the gauge bosons. Note, that the couplings Zhihj do not appear in this model. The charged current part of the Lagrangian remain similar to the SM one is given by:

5 5 g µ 1 γ CKM µ 1 γ + C = uiγ − M dj + νiγ − ei W L −√2 2 ij 2 µ +h.c. (7.7)

7.2.2 Possible minima

1. v = w = 0 The EW symmetry conserving extremum is realized when both doublets and singlet have a zero VEV. For this vacuum state the fermions and gauge bosons are massless, therefore this case is not the true vacuum state.

2. v = 0 and w = 0 In this case there are three diﬀerent solutions 6

v = 0 and w1 = 0

v = 0 and w2 = 0 v = 0, w = 0 and w = 0 1 6 2 6 116 In all these cases, similar to the previous case, the EW symmetry extremum is realized. There is no coupling between singlet and the SM fermions and therefore for this vacuum state the fermions are massless, the same holds for the gauge bosons.

3. v = 0 and w = 0 6 In the case when only the doublet acquires a VEV, singlet do not mix with the SM-like Higgs and therefore singlet maybe the DM candidate. Moreover, with this vacuum state, there is no spontaneous CP violation.

4. v = 0 and w = 0 6 6 In this vacuum state, spontaneous CP violation can be realized what is discussed in section 2.2.4. However, DM candidate does not exist (as follows from discussion in chapter 4.

5. v = 0, w = 0 and w = 0 6 1 6 2 For this choice of vacua, the real component of singlet mixes with the SM-like Higgs boson and the imaginary part of singlet would behave as a DM. In this vacuum state, there is no spontaneous CP violation.

6. v = 0, w = 0 and w = 0 6 1 2 6 For this choice of vacua, the imaginary component of singlet mixes with the SM-like Higgs boson and the real part of singlet would behave as a DM. There is no spontaneous CP violation in this choice of vacua as well.

7.2.3 Oblique parameters for cSMCS

To study the contributions to oblique parameters in the cSMCS, we use the method presented in [89]. S and T parameters in the cSMCS are given by:

117 g2 T = (r r r r )2F (M 2 ,M 2 ) 2 2 12 23 13 22 h1 h2 64π MW αem − − (r r r r )2F (M 2 ,M 2 ) − 12 33 − 13 32 h1 h3 (r r r r )2F (M 2 ,M 2 ) − 22 33 − 32 32 h2 h3 +3(r )2(F (M 2 ,M 2 ) F (M 2 ,M 2 )) 11 Z h1 − W h1 3(F (M 2 ,M 2 ) F (M 2 ,M 2 )) − Z href − W href +3(r )2(F (M 2 ,M 2 ) F (M 2 ,M 2 )) 21 Z h2 W h2 − +3(r )2(F (M 2 ,M 2 ) F (M 2 ,M 2 )) (7.8) 31 Z h3 − W h3 and

g2 S = (r r r r )2G(M 2 ,M 2 ,M 2 ) 2 2 12 23 13 22 h1 h2 Z 384π Cw − +(r r r r )2G(M 2 ,M 2 ,M 2 ) 12 13 − 13 32 h1 h3 Z +(r r r r )2G(M 2 ,M 2 ,M 2 ) 22 33 − 32 32 h2 h3 Z 2 2 2 2 2 +(r11) Gb(M ,M ) Gb(M ,M ) h1 Z − href Z +(r )2G(M 2 ,M 2 ) + (r )2G(M 2 ,M 2 ) 21 b h2 Z 31 b h3 Z 2 2 2 +log(Mh1 ) log(Mhref ) + log(Mh2 ) − 2 +log(Mh3 ) , (7.9) where the following functions have been used

1 M 2M 2 M 2 F (M 2,M 2) = (M 2 + M 2) 1 2 log( 1 ), (7.10) 1 2 2 1 2 − M 2 M 2 M 2 1 − 2 2 2 16 5(m1 + m2) 2(m1 m2) G(m1, m2, m3) = − + −2 3 m3 − m3 3 m2 + m2 m2 m2 + 1 2 1 − 2 m3 m1 m2 − m3 − 3 (m1 m2) m1 rf(t, r) + − 2 log + 3 , (7.11) 3m3 m2 m3 The function f is given by  t √r √r ln − r > 0,  | t+√r | f(t, r) = 0 r = 0, (7.12)  √ r  2√ r arctan − r < 0, − t 118 with the arguments deﬁned as

t m + m m , r m2 2m (m + m ) + (m m )2. (7.13) ≡ 1 2 − 3 ≡ 3 − 3 1 2 1 − 2

Finally, Gb(m1, m2) can be written as follows

2 79 m1 m1 m1 Gb(m1, m2) = − + 9 2 2 + 10 + 18 3 m2 − m2 − m2 2 3 m1 m1 m1 + m2 m1 6 2 + 3 9 log − m2 m2 − m1 m2 m2 2 − 2 m1 m1 f(m1, m1 4m1m2) +(12 4 + 2 ) − . − m2 m2 m2 (7.14)

7.2.4 Decays h γγ in the cSMCS →

The couplings of the lightest Higgs particle (h1) to the quarks and the gauge bosons in the cSMCS model, as compared with the corresponding couplings of the SM Higgs, are modiﬁed (suppressed) by a factor r . Therefor, the decay width Γ(h γγ) is given by [217, 218], 11 → Γ(h γγ) = r2 Γ(φ γγ). (7.15) 1 → 11 SM →

Then the ratio Rγγ turns out, = r2 (7.16) Rγγ 11 where the form factors for this decay are

4 4M 2 4M 2 ASM = A t ,ASM = A W , (7.17) t 3 1/2 M 2 W 1 M 2 h1 h1 where

A (τ) = 2τ [1 + (1 τ)f(τ)] , 1/2 − A (τ) = [2 + 3τ + 3τ(2 τ)f(τ)] , (7.18) 1 − − A (τ) = τ [1 τf(τ)] , 0 − − and   arcsin2(1/√τ) for τ 1 f(τ) = 2 ≥ (7.19) 1 1+√1 τ log − iπ for τ < 1  4 1 √1 τ − − − −

119 7.2.5 Higgs trilinear couplings

For coupling among Higgs bosons we have, 1 h g = r2 (Λr v + r ( 3√2κ + √2κ h2h1h1 2 13 21 22 − 2 3 2 +2λsw1) + 6λsr23w2) + r12(Λr21v

+3r22(√2(κ2 + κ3) + 2λsw1) + 2λsr23w2) 2 +r11(3λr21v + Λ(r23w2 + r22w1))

+2Λr11(r13(r23v + r21w2)

+r12(r22v + r21w1)) + 2r12r13(r23( 3√2κ2 i − +√2κ3 + 2λsw1) + 2λsr22w2) . (7.20)

The g coupling can be obtained from the above expression by substitution r r , h3h1h1 2j → 3j and for g by substitution r r and then r r . h3h2h2 2j → 3j 1j → 2j

7.2.6 J-invariant for κ = 0 4 6

The J1 can be deﬁned by mixing elements of the squared mass matrix Eq.2.18). This parameters for the case κ = 0 is given by 4 6 J = v2w 1 2 × h w2Λ2(1/√2) κ + 3κ (1 + 2(w2 w2)/w2) κ v2/w2 3 2 1 − 2 − 4 κ Λ(w2/w ) √2λ w + κ 3κ (3 + 2(w2 w2)/w2) − 4 1 s 1 3 − 2 1 − 2 2 2 + κ4v /w i + 4κ2 λ w + (κ 3κ )/√2 . (7.21) 4 s 1 3 − 2 2 2 Note, that even for Λ = κ2 = κ3 = 0 the J1 is not vanishing: J1 = 4v w2w1κ4λs.

7.3 Appendix C

The one-loop thermal corrections to the eﬀective potential at ﬁnite temperature T are (see reference [106] for review),

4 2 X niT m ∆V = I i , thermal 2π2 B,F T 2 i

120 where Z ∞ 2 h √x2+yi IB,F (y) = dx x ln 1 e− , (7.22) 0 ∓ the minus and plus sign corresponds to the bosons and the fermions, respectively. In this

Appendix, the evaluation of this integral, the ﬁeld-dependent mass mi and the number of degrees of freedom ni are given.

7.3.1 Evaluation of the integral

The integral Eq. (2.54) is evaluated as follow,

∂ 1 Z x2 1 I (y) = ∞ dx , (7.23) ∂y B,F 2 (x2 + y)1/2 exp((x2 + y)1/2) 1 0 −

Z 4 ∞ 2 x π IB,F (y) y=0 = dxx ln(1 e− ) = , (7.24) | 0 − −45

∂ 1 Z x π2 I (y) = ∞ dx = . (7.25) ∂y B,F |y=0 2 ex 1 12 0 −

7.3.2 The ﬁeld-dependent mass mi

In the cSMCS model the ﬁeld-dependent masses mi of gauge bosons, Goldestone boson, mφ1 ,

mφ2 and mφ3 , which are used in Eq. (2.53) are given by

g2φ2 φ2 M 2 = 1 ,M 2 = (g2 + g 2) 1 , W 4 Z 0 4 2 2 2 2 mG = λφ1 + Λ(φ2 + φ3) + 2√2κ4φ2), m2 = 3λφ2 + Λ(φ2 + φ2) + 2√2κ φ ), φ1 1 2 3 4 2 1 m2 = 3λ φ2 + λ φ2 + 3√2(κ + κ )φ + Λφ2, φ2 s 2 s 3 2 3 2 2 1 1 m2 = 3λ φ2 + λ φ2 + √2( 3κ + κ )φ + Λφ2. φ3 s 3 s 2 − 2 3 2 2 1 (7.26)

ni is the number of degrees of freedom as,

nW = 6, nZ = 3, nG = 3, nφ,φ2,φ3 = 1, nt = 12. (7.27)

121 7.4 Appendix D: LO Higgs Production Matrix Ele- ment

Using the Feynman rules for the loop-diagram (a) of the ﬁgure 3.1 (and its permutation), one can write the corresponding matrix element as:

Z 4 2 ∞ d q (g∗(k ) + g∗(k ) H(p)) = 2πα (µ ) 1 2 S 4 M → 0 (2π) 1/2 GF γ.(q + k2) + mt mt T r × √2 (q + k )2 m2 2 − t γ.(q) + m [γ µ(λ , k )ta] t γ ν(λ , k )tb × µ a 1 1 q2 m2 ν b 2 2 − t γ.(q k ) + m − 1 t + [k k ] . (7.28) × (q k )2 m2 1 ↔ 2 − 1 − t

In the Eq. (7.28), αS and GF are respectively the running coupling constant of the strong interaction and the Fermi’s constant. q is the 4-momenta of the exchanged particle in the µ top-quark loop. a (λi, ki) are the polarization vectors of the incoming gluons. To calculate 2, one has to multiply the expression (7.28) by its complex conjugate, do the traces and |M| perform the integration. Additionally, since the incoming gluons are virtual, one has to take into account the so-called non-sense polarization (see the references [140, 219]) through the following identity: kµ kν X µ ν i,t i,t (λ, k )∗ (λ, k ) = . (7.29) i i k2 λ i,t µ λi, ki,t and a are the spin state, the transverse momenta and the color index of the incoming gluons, respectively.

Hence, after rather lengthy calculations, one obtains:

α2 (µ2) G 2 = S F τ 2 D(τ) 2 (m2 + p2)2 cos2ϕ, |M| 288π2 √2 | | H t (7.30)

2 2 where τ = 4mt /mH and

" 2# 3 1 τ 1 + √1 τ D(τ < 1) = 1 + − ln − iπ , 2 2 1 √1 τ − − − 3 1 D(τ 1) = 1 + (1 τ)arcsin2 . (7.31) ≥ 2 − √τ

122 It is easy to conﬁrm that at the limit of large m , i.e. as τ , t → ∞ 1 lim τD(τ) = 1 + . (7.32) τ τ →∞ O Afterwards, neglecting the transverse momentum of the produced boson, p 0, the Eq. t → (3.9) returns to its conventional from in the collinear approximation [220].

7.5 Appendix E

In the cIDMS, the SM has being extended by a Z2-odd SU(2) doublet with zero vacuum expectation value (Inert Doublet), a complex singlet with a non-zero complex VEV and a pair of heavy iso-doublet VQ, V + V . The decays width of h γγ and h Zγ in this L R 1 → → model have been shown in the Appendix 7.5.1. The formula of the oblique parameters for the cIDMS model are present in the Appendix 7.5.2. We have used these relations in our calculations. Our benchmark points for DM analysis are shown in the Appendix 7.5.3.

7.5.1 Decays h γγ and h Zγ in cIDMS → → The decay width, Γ(h γγ), in the IDMS model is given by [217, 218], → Γ(h γγ) = r2 1 + η 2Γ(φ γγ). (7.33) → 11| 1| SM →

Then the ratio Rγγ turns out, = r2 1 + η 2, (7.34) Rγγ 11| 1| where + − ± gh1H H v AH η1 = 2 SM SM . (7.35) 2r11MH± AW + At The form factors for this decay are,

2 4MH± A ± = A , H 0 M 2 h1 4 4M 2 ASM = A t , (7.36) t 3 1/2 M 2 h1 4M 2 ASM = A W , W 1 M 2 h1

123 where,

A (τ) = 2τ [1 + (1 τ)f(τ)] , 1/2 − A (τ) = [2 + 3τ + 3τ(2 τ)f(τ)] , (7.37) 1 − − A (τ) = τ [1 τf(τ)] , 0 − − and   arcsin2(1/√τ) for τ 1 f(τ) = 2 ≥ (7.38) 1 1+√1 τ log − iπ for τ < 1.  4 1 √1 τ − − − −

The decay width, Γ(h Zγ), in the IDMS model is given by, → Γ(h Zγ) = r2 1 + η 2Γ(φ Zγ) (7.39) → 11| 2| SM → and the ratio for this process turns out,

= r2 1 + η 2, (7.40) RZγ 11| 2| where + − ± gh1H H v H η2 = SMA SM , (7.41) 2r M ± + 11 H AW At 2 2 2 (1 2 sin θW ) 4MH± 4MH± ± H = − I1 2 , 2 , A − cos θW Mh MZ 8 2 2 2 SM (1 3 sin θW ) h 4Mt 4Mt t = 2 − A1/2 2 , 2 , A cos θW Mh MZ 2 2 SM h 4MW 4MW W = A1 2 , 2 , (7.42) A Mh MZ with

Ah (τ, λ) = I (τ, λ) I (τ, λ), 1/2 1 − 2 sin2 θ 2 sin2 θ 2 Ah(τ, λ) = cos θ 4 3 W I (τ, λ) + 1 + W 5 + I (τ, λ) , 1 W 2 2 2 1 − cos θW τ cos θW − τ 2 2 2 τλ τ λ τ λ 1 1 I (τ, λ) = + [f(τ) f(λ)] + g(τ − ) g(λ− ) , 1 2(τ λ) 2(τ λ)2 − (τ λ)2 − − − − τλ I (τ, λ) = [f(τ) f(λ)] , (7.43) 2 −2(τ λ) − − and  q  1 1 arcsin √τ for τ 1  τ − ≤ g(τ) = 1 (7.44) √1 1+√1 1/τ  − τ log − iπ if τ > 1.  2 1 √1 1/τ − − −

124 7.5.2 Oblique parameters for cIDMS

To study contributions to oblique parameters in the cIDMS, we use the method presented

in [89]. There are 6 neutral ﬁelds (including a Goldstone boson) coming from Φ1,Φ2 and χ,

related to the physical ﬁelds h1 3,H,A through: −

 G0   h   0   1  ϕ1 + iG    H   H + iA  = V   , (7.45)    A    ϕ2 + iϕ3    h2  h3

The 3 6 rotation matrix V is given by ×

  i r11 0 0 r21 r31   V =  0 0 1 i 0 0  , (7.46) 0 r12 + ir13 0 0 r22 + ir23 r32 + ir33

where rij are the elements of the inverse rotation matrix deﬁned in chapter 2(Eq.2.3).

Charged sector contains only a pair of charged scalars H± from doublet Φ2. S and T parameters in the cIDMS are given by:

g2 T = 2 2 64π MW αem × 2 2 2 2 2 2 F (M ± ,M ) + F (M ± ,M ) F (M ,M ) H H H A − H A (r r r r )2F (M 2 ,M 2 ) (r r r r )2F (M 2 ,M 2 ) − 12 23 − 13 22 h1 h2 − 12 33 − 13 32 h1 h3 (r r r r )2F (M 2 ,M 2 ) + 3(r )2(F (M 2 ,M 2 ) F (M 2 ,M 2 )) − 22 33 − 32 32 h2 h3 11 Z h1 − W h1 2 2 2 2 2 2 2 2 2 +3(r11) (F (M ,M ) F (M ,M )) 3(F (M ,M ) F (M ,M )) Z h1 − W h1 − Z href − W href +3(r )2(F (M 2 ,M 2 ) F (M 2 ,M 2 )) 21 Z h2 W h2 − +3(r )2(F (M 2 ,M 2 ) F (M 2 ,M 2 )) , (7.47) 31 Z h3 − W h3

125 and g2 S = 2 2 384π Cw × 2 2 2 2 2 2 2 2 (2s 1) G(M ± ,M ± ,M ) + G(M ,M ,M ) w − H H Z H A Z +(r r r r )2G(M 2 ,M 2 ,M 2 ) 12 23 − 13 22 h1 h2 Z +(r r r r )2G(M 2 ,M 2 ,M 2 ) 12 13 − 13 32 h1 h3 Z 2 2 2 2 2 2 2 +(r22r33 r32r32) G(M ,M ,M ) + (r11) Gb(M ,M ) − h2 h3 Z h1 Z 2 2 2 2 2 2 2 2 Gb(M ,M ) + (r21) Gb(M ,M ) + (r31) Gb(M ,M ) − href Z h2 Z h3 Z 2 2 2 2 2log(MH± ) + log(MA) + log(MH ) + log(Mh1 ) − log(M )2 + log(M )2 + log(M )2 , (7.48) − href h2 h3

where used functions are deﬁned in Appendix 7.2.3.

7.5.3 Benchmarks

We propose three benchmark points to be used in DM analysis1. Chosen values of masses of Higgs particles and corresponding parameters are listed in table 7.3. We also present rotation 2 matrices rAi for each benchmark. These matrices diagonalize the scalar mass matrix, Mmix in the following way, M 2 = r M 2 rT = diag(M 2 ,M 2 ,M 2 ). (7.49) f Ai mix Ai h1 h2 h3

 0.999465 0.00682726 0.0319988 

rA1 =  0.0324672 0.328031 0.944109  (7.50)  −  0.0040509 0.944642 0.328077 − −  0.987153 0.0555822 0.149795 

rA2 =  0.159095 0.255572 0.95361  (7.51)  −  0.0147203 0.965191 0.261131 −  0.90504 0.0113276 0.425176  − − rA3 =  0.424229 0.0477451 0.904295  (7.52)  −  0.0305436 0.998795 0.0384057 − − − Benchmarks A1, A2 and A3 are presented in table 7.3. 1In tables in appendices 7.5.3 and 7.5.4, we are listing parameters with a larger precision to allow the reader to reproduce our results.

126 Mh1 Mh2 Mh3 A1) 124.838 194.459 239.994 A2) 124.852 288.161 572.235 A3) 125.364 149.889 473.953

λ1 λs1 Λ1 ρ2 ρ3 ξ A1) 0.2579 0.2241 -0.0100 0.0881 0.1835 1.4681 A2) 0.2869 0.8894 -0.1563 0.6892 0.6617 0.8997 A3) 0.2830 0.6990 0.0928 0.3478 0.2900 0.4266

Table 7.3: We present the masses of the scalars (in GeV) in the upper table. In the lower table, the values of dimensionless parameters from the scalar potential are presented.

7.5.4 S,T and , for the benchmarks of the cIDMS Rγγ RZγ Table 7.4 presents values of oblique parameters S and T for benchmarks A1, A2 and A3. The 3σ bounds are:

0.28 < S < 0.38, 0.30 < T < 0.48, 0.32 < U < 0.34. (7.53) − − − Table 7.5, 7.6 and 7.7 contain values of and for diﬀerent DM mass, for benchmarks Rγγ RZγ A1-A3. All those points are in agreement with collider and DM constraints.

Mh1 Mh2 Mh3 δA δ MH S T 3σ ± 50 55 50 0.0025 0.0050 Yes A1) 124.838 194.459 239.994 50 55 75 0.0024 0.0051 Yes 1 1+ 600 -0.0078 0.0000 Yes 50 55 50 0.0029 -0.0378 Yes A2) 124.852 288.161 572.235 50 55 75 0.0028 -0.0377 Yes 1 1+ 600 -0.0075 -0.0418 Yes 50 55 50 0.0027 -0.1968 Yes A3) 125.364 149.889 473.953 50 55 75 0.0026 -0.1967 Yes 1 1+ 600 -0.0077 -0.2019 Yes

Table 7.4: Values of oblique parameters S and T for benchmark points A1 A3 and chosen − masses of inert scalars. All studied cases are in agreement with EWPT constraints. Also, we use δA, = MA,H± MH . ± −

127 Benchmark A1 Benchmark A2 M (GeV) λ M (GeV) λ H | 345| Rγγ RZγ H | 345| Rγγ RZγ 50 0.015 0.8770 0.9365 50 0.015 0.8556 0.9136 53 0.015 0.8826 0.9405 53 0.015 0.8610 0.9175 56 0.015 0.8886 0.9449 56 0.015 0.8668 0.9218 59 0.015 0.8952 0.9503 59 0.015 0.8733 0.9270 50 0.002 0.9014 0.9596 50 0.002 0.8793 0.9361 53 0.002 0.9045 0.9611 53 0.002 0.8823 0.9375 56 0.002 0.9073 0.9624 56 0.002 0.8851 0.9388 59 0.002 0.9100 0.9636 59 0.002 0.8877 0.9400 50 0.001 0.9020 0.9601 50 0.001 0.8799 0.9366 53 0.001 0.9050 0.9615 53 0.001 0.8829 0.9379 56 0.001 0.9078 0.9627 56 0.001 0.8856 0.9392 59 0.001 0.9104 0.9639 59 0.001 0.8882 0.9403 Benchmark A3 M (GeV) λ H | 345| Rγγ RZγ 50 0.015 0.7192 0.7679 53 0.015 0.7238 0.7712 56 0.015 0.7287 0.7748 59 0.015 0.7341 0.7792 50 0.002 0.7391 0.7870 53 0.002 0.7417 0.7881 56 0.002 0.7440 0.7892 59 0.002 0.7463 0.7902 50 0.001 0.7396 0.7872 53 0.001 0.7421 0.7884 56 0.001 0.7445 0.7894 59 0.001 0.7466 0.7904

Table 7.5: Low DM mass region: values of and for chosen values of M and λ Rγγ RZγ H 345 for MA = MH + 50 GeV,MH± = MH + 55 GeV. Points listed above correspond to DM relic density in agreement with Planck results. Values of and do not depend on the sign Rγγ RZγ of λ345.

128 Benchmark A1 Benchmark A2 M (GeV) λ M (GeV) λ H 345 Rγγ RZγ H 345 Rγγ RZγ 64 0.0125 0.9116 0.9646 64 0.0125 0.8893 0.9410 66 0.019 0.9116 0.9646 66 0.019 0.8893 0.9410 68 0.02 0.9130 0.9665 68 0.02 0.8906 0.9416 70 0.018 0.9149 0.9660 70 0.018 0.8925 0.9424 72 -0.097 0.9401 0.9764 72 -0.097 0.9179 0.9525 74 -0.039 0.9295 0.9719 74 -0.039 0.9067 0.9481 76 -0.116 0.9458 0.9783 76 -0.116 0.9227 0.9544 77 -0.123 0.9474 0.9800 77 -0.123 0.9242 0.9550 78 -0.136 0.9501 0.9800 78 -0.136 0.9268 0.9560 Benchmark A3 M (GeV) λ H 345 Rγγ RZγ 64 -0.02 0.7542 0.7936 66 -0.017 0.7546 0.7938 68 0.006 0.7513 0.7925 69 0.004 0.7523 0.7929 70 -0.003 0.7540 0.7937

Table 7.6: Medium DM mass region: values of and for chosen values of M and Rγγ RZγ H λ345 for MA = MH + 50 GeV,MH± = MH + 55 GeV. Points listed above correspond to DM relic density in agreement with Planck results.

129 Benchmark A1 Benchmark A2 M (GeV) λ M (GeV) λ H 345 Rγγ RZγ H 345 Rγγ RZγ 550 0 0.9986 0.9989 550 0 0.9741 0.9743 575 0.2 0.9967 0.9981 575 0.2 0.9723 0.9737 575 -0.2 1.0005 0.9995 575 -0.2 0.9760 0.9750 600 0.23 0.9966 0.9981 600 0.23 0.9722 0.9736 600 -0.23 1.0006 0.9995 600 -0.23 0.9761 0.9751 625 0.25 0.9966 0.99981 625 0.25 0.9722 0.9736 625 -0.25 1.0006 0.9995 625 -0.25 0.9761 0.9751 650 0.28 0.9966 0.9980 650 0.28 0.9722 0.9736 650 -0.28 1.0007 0.9996 650 -0.28 0.9762 0.9751 675 0.3 0.9966 0.9981 675 0.3 0.9722 0.9736 675 -0.3 1.0007 0.9996 675 -0.3 0.9762 0.9751 700 0.33 0.9965 0.9980 700 0.33 0.9721 0.9736 700 -0.33 1.0007 0.9996 700 -0.33 0.9762 0.9751 Benchmark A3 M (GeV) λ H 345 Rγγ RZγ 550 0 0.8188 0.8190 575 0.2 0.8173 0.8184 575 -0.2 0.8203 0.8196 600 0.23 0.8172 0.8184 600 -0.23 0.8204 0.8196 625 0.25 0.8172 0.8184 625 -0.25 0.8205 0.8196 650 0.28 0.8172 0.8184 650 -0.28 0.8205 0.8196 675 0.3 0.8172 0.8184 675 -0.3 0.8205 0.8196 700 0.33 0.81714 0.8196 700 -0.33 0.82057 0.81964

Table 7.7: Heavy DM mass region: values of and for chosen values of M and Rγγ RZγ H λ345 for MA = MH± = MH + 1 GeV. Points listed above correspond to DM relic density in agreement with Planck results.

130 7.6 Appendix F: Oblique parameters for 2HDM

For the 2HDM (mixed), using formulas from [89] adapted for a two-doublet case, the following expressions for S and T parameters are obtained:

2 2 2 g 2 MA MH± T = M ± (1 log ) (7.54) 2 2 H 2 2 2 64π Mwαem × − MH± MA MA 2 −2 2 2 2 2 MA MA MH± MH± + sin (β α)MH ( 2 2 log 2 2 2 log 2 ) − MA MH MH − MH± MH MH −2 2 2− 2 2 2 MA MA MH± MH± + cos (β α)Mh ( 2 2 log 2 2 2 log 2 ) − MA Mh Mh − MH± Mh Mh − − +3 cos2(β α)(F (M 2 ,M 2 ) + F (M 2 ,M 2) F (M 2 ,M 2 ) F (M 2 ,M 2)) , − Z H Z h − W H − W h

g2s2 S = w 2 96π αem × 2 2 2 2 2 (2s 1) G(M ± ,M ± ,M ) w − H H Z + sin2(β α)G(M 2 ,M 2 ,M 2 ) − H A Z 2 2 2 2 2 2 2 2 + cos (β α)(G(Mh ,MA,MZ ) + Gb(Mh ,MZ ) + Gb(MH ,MZ ) − 2 2 2 2log(M ± ) + log(M ) + log(M ) , (7.55) − H A H where used functions are deﬁned in Appendix 7.2.3.

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