Numerical Methodology

In this section, we will explain our method to collect the data in studying the dark matter phenomenology of G2HDM model. We perform random scan by using Fortran code to get the sample points consistent with the constraints and conditions from various theoretical and phenomenological arguments. We first do the scan on the scalar sector by inputing λ_H, λ_Φ, λ_∆, λ_HΦ, λ_H∆, λ_Φ∆, λ_HΦ^′ , λ_H^′, M_H∆, M_Φ∆, and v_∆ as our input parameter. The value of standard model VEV v is fixed to 246 GeV. Furthermore, unlike in [26] where the value of v_Φwas fixed, we will also scan v_Φin the range between 20 TeV to 100 TeV. As v_Φ provides the highest mass scale in the scan, this parameter sets the typical energy scale of G2HDM model.

In order to get the vacuum expectation value (VEV), we use these scalar parameters to minimize the scalar potential according to equations (3.14), (3.15) and (3.16). These conditions will make the parameters µ_H, µ_Φ, and µ_∆ that appear in the scalar potential expressible in terms of other scalar parameters. Next, we enter these 12 parameters in order to satisfy unitarity condition. This is done by considering the 2→ 2 scalar-scalar scattering amplitude in the high energy limit such that we only need to consider four point contact interaction which is free from center of mass energy suppression. For instance, the corresponding scalar particles in the initial and final states of these scatterings in the diagonal channels can be (√^hh amplitudes of these diagonal channels as well as other off-diagonal channels are all directly related to the scalar quartic couplings and they can be found in [26].

Next, we numerically diagonalize the scalar mixing mass matrices given in Eqs. (3.17) and (3.19). The output of this diagonalization procedure are the masses of the physical scalar Higgs bosons m_h1, m_h2, m_h3 and the corresponding mixing angles components O^H_{i j} as well as the mass of the physical dark scalars m_Dand m

∆e with their corresponding mixing angles components O^D_{i j}. The resulting mass is ordered in ascending manner m_h1 < m_h2< m_h3 and m_D< m

e∆. We also calculate the charged Higgs mass m_H± and heavy fermion mass m_fH. As for the observed standard model higgs, we identify this by m_h1and constraint it to have the value 125.09± 0.24 GeV. Based on this, we further calculate the branching ratio of the Higgs decay into γγ and ττ.

The points that satisfy all the theoretical and phenomenological constraints will be passed to calculate the mass spectrum in the gauge boson sector. In this sector we take additional scan parameter as our input namely g_Hand g_X. However, we fix the Stueckelberg mass M_X to 2 TeV. We then do the numerical diagonalization on the gauge boson mixing matrix in Eq. (3.36). The outcome of this diagonalization are three neutral gauge boson masses in ascending order similar to those of scalar Higgs mass m_Z < m_Z′< m_Z′′ and the gauge boson

Fig. 5.7 The typical diagram used to study perturbative unitarity in scalar-scalar scattering.

The S_i, S_j(S_k, S_l) denote the scalar particles in the initial (final) states respectively. These scalars can be one of the following pairs (√^hh

2, ^G√^0G0

2 , H₂^0∗H₂⁰, H⁺H⁻, ^φ√^2φ2 2 , ^G

0 HG⁰_H

√2 , G_H^pG^m_H,

δ₃δ₃

√2, ∆_p∆_m) as described in the text.

mixing angles components O^G_{i j}. The lightest gauge boson is identified as the SM Z boson which has the value within 3σ accuracy as 91.1876± 0.0021 GeV. The W^′gauge boson is a member of the Z2odd family and has the potential to be a viable dark matter candidate. In order to avoid this, we demand that W^′is always heavier than D. This is realized by imposing the minimum value of g_Hminas

g_Hmin= 2m_D q

v²+ v²

Φ+ 4v²

∆

. (5.12)

We also do a special scan to get the inert doublet-like dark matter candidate f_H0∗

2 > 0.67.

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

λ_HΦ^′ < M_Φ∆

2v_∆ . (5.13)

This condition will make the scan range of λ_HΦ^′ became very limited.

We collect about 5 million points that satisfy all the conditions in scalar and gauge sector constraints (SGSC) and further pass these points to MicrOMEGAS [45] to calculate relic density, DM-nucleon cross section and annihilation cross section at present time.

Parameter Doublet-like Triplet-like Goldstone-like λ_H [0.12, 2.75] [0.12, 2.75] [0.12, 2.75]

λ_Φ [10⁻⁴, 4.25] [10⁻⁴, 4.25] [10⁻⁴, 4.25]

λ_∆ [10⁻⁴, 5.2] [10⁻⁴, 5.2] [10⁻⁴, 5.2]

λ_HΦ [−6.2, 4.3] [−6.2, 4.3] [−6.2, 4.3]

λ_H∆ [−4.0, 10.5] [−4.0, 10.5] [−4.0, 10.5]

λ_Φ∆ [−5.5, 15.0] [−5.5, 15.0] [−5.5, 15.0]

λ_HΦ^′ [−1.0, 18.0] [−1.0, 18.0] [−1.0, 18.0]

λ_H^′ [−8√

2π, 8√

2π] [−8√

2π, 8√

2π] [−8√

2π, 8√ 2π]

M_H∆/GeV [0.0, 15000] [0.0, 5000.0] [0.0, 5000.0]

M_Φ∆/GeV [0.0, 5.0] [−50.0, 50.0] [0.0, 700]

v_∆/TeV [0.5, 2.0] [0.5, 20.0] [14.0, 20.0]

v_Φ/TeV [20, 100] [20, 100] [20, 28.0]

g_H [see text, 0.1] [see text, 0.1] [see text, 0.1]

g_X [10⁻⁸, 1.0] [10⁻⁸, 1.0] [10⁻⁸, 1.0]

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

Finally, the annihilation cross section for each DM composition is passed to LikeDM [51]

for the calculation of indirect detection likelihood. Due to the different underlying physical consideration, each dark matter case will have different phenomenology. Therefore, we further analyze the resulting points calculated by MicrOMEGAS. From here, one can determine the relevant couplings and parameters which are sensitive to the dark matter composition in doublet-, triplet- or Goldstone-like case. The complete parameter set that we scanned in this study for each case is summarized in Table 5.1.

Chapter 6

Dark Matter in G2HDM: Numerical Results

Having explained our methodology and strategy in the previous chapter, we further discuss in detail the numerical results of complex scalar dark matter study in this chapter. We start our discussion by imposing the relic density requirement and collect the surviving parameters to face the direct search constraint. Next, the viable parameters space are fed against the indirect search experimental data. We will discuss the phenomenology of each dominant component of the dark matter candidate starting from the inert doublet-like DM, SU (2)_H triplet-like DM, and Goldstone boson-like DM in different sections.

6.1 Inert Doublet-like DM

The doublet-like dark matter is characterized by the dominant H₂⁰ component with the composition (O^D₂₂)²> 2/3. In G2HDM model, the doublet-like dark matter is equal to the inert dark matter candidate in IHDM model in the limit of the mass degeneracy between the scalar (S) and pseudo-scalar (A) part of the inert scalar. This doublet-like dark matter is the main motivation behind the construction of G2HDM as already mentioned before. Therefore, in this part of result session, we will give a detail discussion in this doublet dark matter candidate. We show the relic density as a function of the DM mass in the left panel of Fig. 6.1.

Similar to the IHDM model, the cosmological interesting relic abundance Ω_Dh²∼ 0.1 occurs in three different mass regions [56, 57]: (i) the low DM mass between 10 GeV to 40 GeV, (ii) the intermediate DM mass between 150 GeV to 500 GeV, and (iii) the heavy DM mass greater than 500 GeV. The underlying physics behind these three regimes are different.

10⁰ 10¹ 10² 10³

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

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

Fig. 6.2 Feynman diagrams of dark matter annihilation into ¯τ τ , ¯ccand ¯bb final states.

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

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

In the intermediate mass range between 150 GeV to 500 GeV, the dominant annihilation cross sections are given by the W⁺W⁻, h₁h₁, ZZ and ¯tt final states. The W⁺W⁻ final state occurs via four point contact interaction (p-channel), s-channel of three Higgses and three neutral gauge bosons exchange, and t-channel of charged Higgs exchange. If one takes the non-relativistic approximation, the scalars exchange diagrams lead to the s-wave annihilation cross section while the gauge bosons exchange diagrams will contribute to the p-wave part. One expects that due to their heavy masses, the h₃and Z^′′ exchange will be sub-dominant compared to the other mediators. Next, the h₁h₁ final state is mediated via p-channel contact interaction, three Higgses exchange in the s-channel, D, e∆,W^′exchange in the t and u-channel. In this case, one expects that the most dominating diagram is given by the p-channel interaction as there is no propagator and hence no mass suppression. The next dominant amplitudes are given by h₁and h₂ exchange, as well as D and e∆ exchange.

The contributions from the W^′and h₃exchange are small due to their large mass suppression.

It is interesting that the heavier dark scalar e∆ gives comparable contribution with that of D in the t/u-channel. This happens because of the mass splitting between them is small. We will discuss this further in the coannihilation case below. The same argument also holds in the case of ZZ final state which has exactly similar diagrams as the h₁h₁final state. Finally, for the ¯tt final state, the diagram is similar to those of low mass dark matter case. The heavy fermion contribution is suppressed due to its typical TeV mass scale. All of these contributions will make the relic density to have small value in this intermediate mass range.

Unfortunately, one can not obtain the points that satisfy the PLANCK data as can be seen from the left panel of Fig. 6.1. This happens because the doublet-like dark matter is the member of the usual SU (2)_L doublet. Its interactions with the SM particles such as h₁and Z are controlled the SM couplings. Therefore, one would expect that these kind of interactions are unsuppressed by any mixing angles and hence make the corresponding annihilation cross sections larger.

Before moving further to higher dark matter mass > 500 GeV, let us discuss about the coannihilation in doublet-like dark matter. As one can see from the left panel of Fig. 6.1.

The relevant contribution of coannihilation is given by the charged Higgs. The mass splitting

between charged Higgs and doublet dark matter is approximately given by m²_H±− m²D≈ −1

2λ_H^′v². (6.1)

This tells us that the initial abundance of charged Higgs H^±compared to the D abundance is determined by the λ_H^′ parameter. This coannihilation contribution starts to become important when the dark matter mass is larger than 100 GeV. The possible coannihilation cross section in this case is the annihilation between D and H^± as well as H^± with themselves. In this study, we learn that it is the second interaction which gives us the dominant coannihilation contribution. This is due to the fact that the interaction of H^± between themselves also occur via the electromagnetic interaction. As we already know that the electromagnetic interaction is stronger than the typical WIMP dark matter interaction. As an additional remark, we mention that the doublet dark matter solution which is characterized by the (2,2) component of mixing matrix in Eq. (3.19) is very hard to find and one needs to do a dedicated scan in order to get the desired points. This lies on the fact that the (3,3) element of the mass matrix in Eq. (3.19) is smaller than the (2,2) element due to the presence of the large v_∆in the denominator of the (3,3) element. This will make the (3,3) element always smaller than the (2,2) element and hence the natural DM candidate in this model behaves like the triplet

∆_pcomponent. If one insists to have doublet-like DM, one needs to go to particular direction in the parameter space. However, by doing this, the mass splitting between D and e∆ will always be less than 10 % as the argument inside the square root of Eq.( 3.21) is very small.

With this setup in mind, one expects that the coannihilation between D and e∆ will appear everywhere in the dark matter mass. However this is not the case. One needs to remember that the coannihilation cross section is controlled by the interaction between the dark matter and coannihilating particles. We found that the coannihilation with e∆ is very small compared to the charged Higgs case. This is the reason we do not plot the e∆ coannihilation in the left panel of Fig. 6.1.

Finally, in the heavy mass region m_D> 500 GeV, the annihilation cross sections are dominated by the SM Higgs h₁h₁(Fig. 6.3) as well as the longitudinal components of the gauge bosons Z_LZ_L (Fig. 6.4) and W_L⁺W_L⁻ (Fig. 6.5). For the SM Higgs final state, the important contribution is given by four point like interaction, three Higgses exchange in s-channel, t and u channel exchange of D and e∆ as shown in Fig. 6.3. The W^′exchange is suppressed by its heavy mass and hence its contribution is sub-dominant. The typical amplitudes for the longitudinal gauge bosons scattering scale with the energy. Thus, in order to preserve the unitarity, one needs to cancel this energy dependence. For the Z_LZ_L final state the relevant diagram is described in Fig. 6.4. There is an exact cancellation in the

Fig. 6.3 Relevant diagrams for h₁h₁final state. For the doublet case, both D and e∆ exchange in t- and u-channels are important. For the triplet-like and Goldstone-like dark matter, only Dexchange is relevant.

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

energy dependence between four points contact interaction with the t and u-channels. The contribution of W^′ exchange is suppressed and therefore not relevant for the annihilation cross section. On the other hand, the diagram with e∆ mediator is suppressed by the mixing angle (O^D₃₂)². Thus, the only relevant diagrams for Z_LZ_L final state are given by the ones with Higgses exchange.

The amplitudes that scale with the center-of-mass energy also observed in the W_L⁺W_L⁻ final state. The corresponding diagrams for this process are given in Fig. 6.5. The cancellation

Fig. 6.5 Relevant diagrams for W_L⁺W_L⁻ final state.

occurs between the p-channel contact interaction and t-channel charged Higgs exchange.

However, unlike the Z_LZ_L case, the cancelation in this case is not exact. The sum of these two amplitudes is given by

i(Mp+MH⁻)_L= ie²(O^D₂₂)² 2m_W²s²_W ×

"

(s− 2mW² )

2 +(t− m²D)² (t− m²_H−)

#

, (6.2)

where t = m²_D+ m_W² −₂^s. If m_D= m_H−, the second term on the right hand side is equal to t− m²Dand the exact cancelation can be realized. Thus, the cancellation depends on the mass splitting between D and charged Higgs H⁻ which is controlled by λ_H^′ as in Eq. (6.1). As one goes to higher dark matter mass, this mass splitting become smaller and the cancelation can be realized. The next relevant diagram is given by the Z_imediator. In the non-relativistic limit, these amplitudes lead to the p-wave annihilation cross section which proportionals to the square of the dark matter velocity v². In addition, in the case of Z^′ (Z^′′) exchange,

these two diagrams are further suppressed by the off-diagonal elements of gauge boson mixing matrixO₁₂^G (O₁₃^G) leaving only the SM Z boson exchange as the leading order diagram.

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

λ_DD∗hi= i−2λHvO^H_1i− λHΦv_ΦO^H_2i+ λ_H∆v_∆O^H_3i (O^D₂₂)², (6.3) where the index i runs from 1 to 3. The coupling λ_DD∗hiis a function of the scalar parameters λ s which are bounded and can not be arbitrarily large. As a result, when dark matter mass getting bigger the annihilation cross section becomes smaller due to the dark matter mass suppression in the denominator. Therefore, the resulting relic density will be bigger in the high mass scale as one can see from the left side of Fig. 6.1. This is consistent with the unitarity constraint.

The next experimental constraint to be considered is the direct detection experiment given by the XENON1T 2018 [47] and CRESST-III 2017 [58] data. In the right panel of Fig. 6.1, the grey points represent all the points we scan that satisfy SGSC constraints.

One sees that for the dark matter mass larger than 10 GeV, the doublet-like dark matter is excluded by the XENON1T and CRESST-III experiments. There are points that escape these two experimental constraints which are situated below 10 GeV, however in this regime, the observed relic abundance will be too large compared to the PLANCK observation and therefore can not be a suitable dark matter candidate.

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

g_ZDD^µ = i gc_W

2 +g^′s_W 2



(O^D₂₂)²O11^G(p_D∗− pD)^µ (6.4)

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

proportional toO₁₁^G ∼ 1. This can be seen from third component of SU (2)_L, SU (2)_H, and generator of U (1)_X group respectively. The first term on the right hand side of this equation is the SM coupling while the second and third terms are suppressed by off diagonal component of the gauge boson mixing matrixO^G. In the limit of SM case (g_H= g_X = 0), the ratio between neutron and proton coupling to Z is and Q = 0. In this case, dark matter interacts differently with proton and neutron and hence the resulting cross section will violate isospin conservation. Since the inert doublet-like dark matter is a member of SU (2)L doublet, the dominant contribution comes from the SM Z exchange.

One notes that there is a difference between the dark matter-neutron cross section and dark matter-proton cross section. This is the effect of the ISV mentioned before. This can be easily seen from the Eq. (6.5). At quark level, the SM Z boson couples differently with up-type and down-type quark and the difference follows from the different values of

One notes that there is a difference between the dark matter-neutron cross section and dark matter-proton cross section. This is the effect of the ISV mentioned before. This can be easily seen from the Eq. (6.5). At quark level, the SM Z boson couples differently with up-type and down-type quark and the difference follows from the different values of