fGp> 2/3
Fig. 4.1 The relation between the ratio v∆/vΦ and parameter fGp for all the scanned DM parameter space after SGSC.
to specify the dominant component is that we want to have large coverage of the parameter space for our scan. If we increase this value, for example 0.95, it is difficult to find points in the parameter space of our scan.
In the mass matrix 4.15, the v∆/vΦ ratio control the composition of Goldstone boson in DM. However, when the ratio close to 1, the Z′ has large mixing with ZSM, which is disfavor by electroweak precision test(EWPT). Also, to get the value of fGp
larger than ∼ 0.8 will resulting tachyonic DM mass. Thus, the value 2/3 is fair choice.
In general, GpH, H20
∗ and ∆p are complex fields, thus the scalar DM candidate are also complex scalar.
4.2 Numerical Analysis
Now we want to test the G2HDM for experiment of DM searches, if we found points in parameter space that satisfied DM constraints, it will indicates the G2HDM can be a BSM candidate for DM. In this thesis we will illustrate our scalar DM candidate in the search of relic density and direct detection, both of them are relative strict constraints.
4.2.1 Constraints and Range of Parameters
The constraints of scalar sector parameters were studied in [32], where they examined the vacuum stability(VS) of the scalar potential, the perturbative unitarity(PU) for
38 CHAPTER 4. DARK MATTER IN G2HDM scattering amplitudes, and the Higgs physics (HP) of Higgs diphoton and τ+τ− decay in LHC. The parameters for our DM study are all constrained by VS+PU+HP.
By the gauge sector study in ref.[31] for the Z′ constraints, we set our scan of vΦ
start from 20TeV, and set the upper limit to 100GeV by considering the energy scale of future collider. In addition, we required the SM-like Z1 is the lightest among Zi, and the mass of Z1 within its 3σ of measured value 91.1876 ± 0.0021GeV. We will mark the scalar and gauge sector constraints as SGSC in the follows.
Since we choose Z2-odd scalars D as our DM in this thesis, there will be several constraints arise from mass spectrum. The mass of DM have to be lighter than all the other Z2-odd, mH± > mD, mW′(p,m) > mD, and mfH > mD. We will implement these conditions in our scan. By eq.(4.24), the mass condition of mW′(p,m), it imposing the lower bound of gH
gH min = 2mD
qv2+ vΦ2 + v∆2
(4.34) We want to eliminate fermion coannihilation as much as possible to simplify our analysis. However, the Yukawa couplings are proportional to fermions. Meanwhile, we want the new Yukawa couplings to be reasonably small in order to minimize their effects on perturbative unitarity and renormalization group running. Thus, we set the mfH = 1.2mD, and as suggest by the LHC SUSY colored particles search in [33], we set the lower limit of heavy fermions to 1.5TeV, thus we have
mfH = max [1.5TeV, 1.2mD] (4.35) And as showed in sec.4.1.6, we found the new fermions acquired mass solely from Φ2. By the condition in eq.(4.35), the constraints on Yukawa couplings are
yfH = max
Before going to calculate the DM properties for relic density and direct detection, we randomly scan all the parameters in the range list in Table.4.3 for the model to test for varies of theoretical and phenomenological constraints, and passed those points satisfied constraints to DM search. The input parameters in scalar sector are λH, λΦ, λ∆, λHΦ, λH∆, λΦ∆, λ′HΦ, λ′H, MH∆, MΦ∆, v∆, and vΦ. We scanned vΦ in the range between 20TeV to 100TeV, except for Goldstone-like DM, as we discussed in
4.2. NUMERICAL ANALYSIS 39 Parameter Doublet-like Triplet-like Goldstone-like
λH [0.12, 2.75] [0.12, 2.75] [0.12, 2.75]
λΦ [10−4, 4.25] [10−4, 4.25] [10−4, 4.25]
λ∆ [10−4, 5.2] [10−4, 5.2] [10−4, 5.2]
λHΦ [−6.2, 4.3] [−6.2, 4.3] [−6.2, 4.3]
λH∆ [−4.0, 10.5] [−4.0, 10.5] [−4.0, 10.5]
λΦ∆ [−5.5, 15.0] [−5.5, 15.0] [−5.5, 15.0]
λ′HΦ [−1.0, 18.0] [−1.0, 18.0] [−1.0, 18.0]
λ′H [−8√
2π, 8√
2π] [−8√
2π, 8√
2π] [−8√
2π, 8√ 2π]
gH [See text, 0.1] [See text, 0.1] [See text, 0.1]
gX [10−8, 1.0] [10−8, 1.0] [10−8, 1.0]
MH∆/GeV [0.0, 15000] [0.0, 5000.0] [0.0, 5000.0]
MΦ∆/GeV [0.0, 5.0] [−50.0, 50.0] [0.0, 700]
v∆/TeV [0.5, 2.0] [0.5, 20.0] [14.0, 20.0]
vΦ/TeV [20, 100] [20, 100] [20, 28.0]
Table 4.3 Parameter ranges used in the scans mentioned in the text. MX is fixed at 2 TeV in this work and MY is set to be zero throughout the scan.
previous section, the allowed fraction fGp of Goldstone-like DM are in a narrow range to realize this scenario. Since vPhiis the largest masss scale among all the parameters, it represents the energy scale of G2HDM.
For the gauge sector, we have gH, gX, MX, and MY as free parameters. As we mentioned in sec.4.1.5, we set MY = 0. We implement the heavy Stueckelberg mass scenario to set the MX = 2TeV as discussed in [31]
4.2.2 Relic Density
As we discussed in section 3.1, one need to solve Boltzmann equation for the relic density. Furthermore, when the mass difference between DM and other Z2-odd particles ( ˜∆, H±, W′(p,m), and new fermions ) are sufficiently small (usually smaller than 10%
of DM mass) the number density are not suffer to the Boltzmann suppression before DM decoupling, since the DM can coannihilation with these particles. The resonance of SM Z and Higgs boson can play important roles for the doublet-like DM, when 2mD ≈ mZ,H, while there no Z resonance for triplet-like and Goldstone-like DM, since both of them are SU(2)L singlet.
Since mW′± and mfH contains the parameters from gauge and fermion sector, hence the coannihilation to DM can happen in wide range of parameter space. However, we
40 CHAPTER 4. DARK MATTER IN G2HDM already choose a mass condition for heavy fermion in eq.(4.35), thus the coannihilation with heavy fermion is in a narrow range in our scan.
The scalar ˜∆ is in the same mass matrix with D, the mass splitting is controlled by the square root term in eq.(4.16). The coannihilation of doublet-like DM with charged Higgs happened when
m2H±− m2D ≈ −1
2λ′Hv2 (4.37)
In the approximation when DM mass is dominated by the (2, 2) element in eq.(4.15).
m2H±− m2D ≈ −1
2λ′Hv2 (4.38)
We use the latest result Ωh2 = 0.120 ± 0.001 from the PLANCK collaboration[17]
at 2σ to compare our calculation of relic density, and we notated this constraint as RD.
4.2.3 Direct Detection
The dominant channel for direct detection are showed in Fig.4.2. In general, the DM interactions with proton and neutron can be different, this is so-called isospin violation(ISV) DM, which come from the difference of couplings in the DM interaction with up/down type quark. For instance, the coupling strength of DM interaction with quark mediated by Zi boson is characterized by the quantum number assignment for SU(2)L× U(1)Y × SU(2)H × U(1)X
gqqZV¯ i ∝ i 2
g cW
T3−2Qqs2WOG1i+ gHT3′O2iG+ gXXOG3i
(4.39)
Thus we can expect the difference of couplings for up and down quarks. Thus, the ratio of effective couplings in eq.(3.7) can be different from 1. In particular, for the xenon target, the maximum cancellation between proton and neutron happened when the ratio fn/fp ≈ −0.7. In G2HDM, for the scalar DM with non-negligible doublet composition, we can find the evident of ISV when the interaction mediated by Zi. The most stringent limit of direct detection for DM is from the null result of XENON1T[18]
for mass ⪆ 5 GeV, and CRESST-III [19] for mass below 5 GeV to sub- GeV. We notated these constraints for direct detection as "DD". In the calcualation of micrOMEGAs [34]
we only consider the isotope of xenon with mass number A = 131 and proton number Z = 54. However, the data published by XENON1T assumed the cross section of DM are isospin conserved, hence the effective coupling fn equal to fp. For compared our calculation of ISV DM with the experiment data, we then calculated the cross section
4.2. NUMERICAL ANALYSIS 41 in nucleus level by using eq.(3.7), and transform the experimental data to nucleus level by
where the µ2p and m2A are DM-proton and DM-nucleus reduced mass respectively.
Since our DM is complex scalar, we also have another peculiar property. In general, for complex scalar DM is the DM-nucleon interaction and antiDM-nucleon DM are different. In the limit of heavy mediate, we can integrate them to obtain effective interactions between DM and nucleon. The spin independent interaction for complex scalar DM can be written into two terms of effective operators, one for odd operator another for even operator
LD = 2λN,eMDDD∗ψ¯NψN + iλN,o(∂µDD∗− D∂µD∗) ¯ψNγµψN (4.41) where ψN denote the nucleon field operator, λN,e denote the coupling of even operator, and λN,o denote the coupling of odd operator. The effective coupling of DM/antiDM with the nucleon is given by
λN = λN,e± λN,o
2 (4.42)
where the plus and minus sign stands for DM-nucleon and antiDM-nucleon interaction respectively. The even operator stays the same after exchanges of D with D∗, while the odd operator change sign after this exchanges. The numerical value of the effective couplings fp and fn for D are not the same as those for D∗, hence the cross section of σDN and σD∗N are different in general. We denoted the cross section in eq.(3.7) of DM with nucleus as σDN and antiDM with nucleus as σD∗N.
Fig. 4.2 The dominant Feynman diagrams of direct detection with Z gauge boson and Higgs boson as mediator. The diagram with heavy fermions mediator can be ignored for its insignificant contribution.
42 CHAPTER 4. DARK MATTER IN G2HDM In addition, the couplings of doublet DM to Zi are
gD
Where Di with i = 1, 2, 3 notated for Goldstone-like, doublet-like, and triplet-like respectively. For the doublet-like DM, the element of mixing matrix O22D is not suppressed. Hence, by eq.(4.39) and eq.(4.43), we found the cross section of doublet-like DM in direct detection has large contribution from Zi bosons. Since the SM coupling constant g and g′ are typically larger than gH and gX, so the interaction will be dominated by Z1 exchange. This will cause the significant ISV effect, where the DM-neutron cross section is three orders of magnitude larger than the DM-proton cross section. The ISV effect is also observed in triplet-like DM, but triplet-like DM only interact with SU(2)H gauge boson, and the O32D mixing in dark sector will further reduced the couplings of DM to Z1 results in mildly ISV effect, moreover, the cross section that contribute by Z1 can comparable with Z2, Z3, and SM Higgs h1.