From previous sections, we have learned that the doublet-like DM scenario cannot fulfill the DM constraints and that Goldstone boson-like DM requires ISV cancellations close to maximal to pass the XENON1T limit. Therefore, in this section we are going to discuss the allowed G2HDM parameter space based on the triplet-like DM.
In Fig. 6.11, we present the λiallowed region of SGSC constraints (green region) and SGSC+RD+DD constraints (red scatter points). Comparing the green region with the red scatter points, one can easily see that λΦ, λH∆, and λΦ∆ are mostly constrained by RD+DD
10−3
Fig. 6.11 A summary plot for the scalar parameter space allowed by the SGSC constraints (green region) and SGSC+RD+DD constraints (red scatter points). The numbers written in the first block of each column are the 1D allowed range of the parameter denoted in the horizontal axis after SGSC+RD+DD cut.
constraints. As discussed in previous section, the dark matter relic abundance is mainly controlled by three dominant contributions coming from W+W−, h1h1, and ZZ final states.
Furthermore, we already see that only dark matter mass larger than 400 GeV can satisfy all the DM constraints. The dominant contribution of the gauge boson final state in this case is given by its longitudinal component. In this range, those three dominant annihilation cross sections are determined by λDD∗h1 and λDD∗h2 originated from s-channel h1and h2exchange.
Thus, to understand which parameters are sensitive to the dark matter phenomenological
constraints in general, it is sufficient to look at these two couplings. One can see from Eq. (6.7) that there are three dominant terms that contribute to the DM-DM-Higgs couplings, λH∆vOH11, λΦ∆vΦOH22, and λ∆v∆OH33. Clearly, λH∆, and λΦ∆ are restricted by the allowed Higgs coupling sizes. In addition, the p-channel in the h1h1final state also depends strongly on λH∆. However, λ∆and v∆are not constrained because the cross section is suppressed by the heavy mediator mh3 and the condition v∆< vΦ. Next, due to a rather loose requirement on the triplet-like dark matter OD32 > 2/3 one may expect the contribution from another dark matter component. For triplet case, another dominating contribution comes from Goldstone-like part and can be as large as (OD12)2≈ 1/3 while the doublet component is strongly suppressed. Therefore, one needs also to consider the first lines of both Eq. (6.13) and Eq. (6.14) to account this impurity effect. Due to the large value of vΦand v∆, the first line of Eq. (6.14) will put constraint on the λΦ∆ and λΦ respectively. Moreover, from the first line of Eq. (6.13), one can see the additional constraint on λΦ∆ and λΦ as this coupling directly depend on these two parameters. In addition, there is also two additional terms λHΦ and λHΦ′ that only appear in the first two terms of the first line in Eq. (6.13). These two terms are not suppressed by any mixing angle and only account for the impurity effect.
As a consequence, one expects that the λHΦ and λHΦ′ will be mildly constrained. As an additional remark, there is also a dependence on the λ s in the components of the orthogonal mixing matrices ODi j and OHi j. This will induce indirect constraints on the λ s inside these two matrices even though it is very difficult to see this dependence unlike the λ s appear explicitly in the relevant couplings.
Next, we project the allowed G2HDM parameter space to the two VEVs vΦ and v∆, two mass scales MH∆ and MΦ∆, and two new gauge couplings gH and gX. Importantly, only gauge coupling gH and the VEV vΦcan be further constrained. Interestingly, we found such a exclusion comes from the lower allowed DM mass. The allowed DM mass values range from hundreds of GeV to TeV. This range is reflected in gH since the minimal value we choose for gH is given by Eq. (5.12) and depends directly on the DM mass. In sum, a good scalar DM candidate in G2HDM requires gH> 7.09× 10−3and vΦ> 22.7 TeV.
0
Fig. 6.12 A summary plot table of the parameter space of the two VEVs vΦand v∆, two mass scales MH∆ and MΦ∆, and two new gauge couplings gH and gX. The color scheme is the same as Fig. 6.11.
Chapter 7 Summary
The G2HDM is a novel two Higgs doublet model with a DM candidate arises naturally without imposing any ad hoc discrete symmetry. After SU (2)H symmetry breaking, one can find three potential DM candidates: the lightest new dark scalar, heavy neutrino, and the SU(2)H gauge boson W′(p,m). Though these three candidates are all interesting, we focused this paper on the most popular one, the new scalar DM, which is also well discussed in the inert doublet Higgs DM model. Different to the inert doublet Higgs DM model, the mixing betweenZ -odd scalars adds a touch of complexity since DM in G2HDM not only comes from the inert doublet but may also be Goldstone boson-like and SU (2)H triplet-like.
We took the dominant composition ( fj> 2/3) as a criteria to classify them but the mixture between them can be simply inferred. In this paper, we have discussed these three types individually with two assumptions: that all the new non-SM heavy fermion are heavy enough to have mostly negligible contributions and that DM shall be thermally produced before the freeze-out temperature. We have comprehensively shown their detectability and exclusions by the current SGSC and DM constraints (mainly RD+DD).
For the inert doublet-like DM, we found some interesting features. First, the main difference between the inert doublet DM in IHDM and G2HDM is that in IHDM there is a mass splitting between scalar and pseudoscalar while they are completely degenerated in G2HDM. As long as the mass splitting in IHDM remains larger than the exchanged energy between DM and nucleon in the direct detection experiment, the interaction mediated by the Zgauge boson remains suppressed. Since this mass splitting does not exist in G2HDM, such interactions are unsuppressed and bring the spin independent cross section up to∼ 10−38cm2, which is significantly above the XENON1T 95% C.L. limit for masses above∼10 GeV and above the CRESST-III result down to 2 GeV.
On the other hand, at the mass region mD≲ 10 GeV, DM is over abundant because of off-shell annihilation channels. Hence, we confirmed that the inert doublet-like DM can be completely excluded by RD+DD constraints.
Next, a SU (2)H triplet-like scalar DM was discussed. Since the composition fH2 has to be tiny in order to avoid the tension with DM DD, the triplet-like DM can mostly mix with the Goldstone boson GHp,m. There is no Z-resonance region in the triplet-like DM for DM annihilation and the parameter space is more or less consistent with Higgs portal DM.
However, DD is still the most stringent constraint comparing with ID and collider constraints.
The allowed DM mass mD by SGSC+RD+DD is required to be heavier than 300 GeV.
Despite weaker constraints from ID and collider constraints, it might be possible to detect the heavy DM mass region by the future CTA and 100 TeV colliders even if a DM signal is not found at direct detection experiments before hitting the neutrino floor.
We explored the Goldstone boson-like DM but we found it is not possible to find a pure Goldstone boson-like DM. The non-tachyonic DM condition and EWPT constraints prohibit the composition fGP > 0.75 unless one would like to move to a more fine-tuned region where v∆/vΦ ≫ 1 and v∆≫ 20 TeV. This will cause the Goldstone-like dark matter to receive a significant component coming from the triplet. Furthermore, this triplet component would typically reduce the coupling strength λGDD∗h1 and λGDD∗h2 relevant to determine the relic abundance. As a result, the annihilation cross section is smaller than the triplet-like case, resulting in larger relic density and less points within the PLANCK relic density measurement.
Furthermore, XENON1T measurement excludes almost all the points with appropriate relic density, except for those close to maximal cancellation due to isospin violation ( fn/ fp≈ 0.7) where direct detection sensitivity is notably reduced. Therefore, a small region with mass above 100 GeV can pass all the DM constraints applied in this work.
Finally, we presented the impact of DM constraints on the G2HDM parameter space.
Based on the triplet-like DM case, we found λΦ, λH∆, λΦ∆, gH, and vΦ are significantly constrained by DM constraints, mainly RD+DD. Interestingly, the lower limit gH> 7.09× 10−3 for vΦ< 100 TeV is within reach for the future linear or circular lepton-antilepton machines and 100 TeV hadron colliders.
Appendix A
Relevant Couplings
In the following, we list the relevant couplings contribute to the dark matter analysis in various processes discussed in the text. We use conventional notation g and g′ to denote the Standard Model SU (2)L and U (1)Y coupling. The cW and sW denote the usual cosine and sine of the Weinberg angle. In addition, for the scalar-scalar-gauge vertex, we adopt the convention that all momentum are incoming as shown in the vertex below.
Fig. A.1 The DD∗W+W− coupling for p-channel interaction.
A.1 Dominant Couplings for Dark Matter
Fig. A.2 The dominant DD∗Ziand DD∗hicouplings for the inert doublet-like dark matter.
Fig. A.3 The dominant DD∗Ziand DD∗hicouplings for the SU (2)H triplet-like dark matter.
Fig. A.4 The dominant DD∗Ziand DD∗hicouplings for the SU (2)H Goldstone boson-like dark matter.
Appendix B
Benchmark Points for Monojet
Although in our study we do not impose collider search as our global dark matter constraint, we take 10 benchmark points to study dark matter production at the collider. These 10 points are located just below the XENON1T exclusion line.
TableB.110benchmarkpointsforthemono-jetoftheSU(2)Htriplet-likeDM.
BenchmarkpointABCDEFGHIJ
λH0.2480.3110.7101.6210.2120.5350.5750.1820.3640.141 λΦ3.5150.6202.4432.6380.9241.4501.6551.9310.4420.930 λ∆0.7353.1294.0070.2791.2740.7720.4721.8214.0820.045 λHΦ−0.964−0.303−1.975−0.652−0.297−0.8201.594−0.177−0.4140.127 λH∆0.340−0.0431.464−0.939−0.313−0.9130.5370.169−1.3581.069 λΦ∆0.4292.5250.7061.2982.0780.9361.0563.1101.2381.067 λ ′HΦ1.2139.7895.1815.0696.0402.9723.8863.8221.9090.910 λ ′H−21.179−1.930−1.2051.454−3.071−1.627−3.097−12.631−2.870−5.374MH∆(GeV)178.7742.82760.38684.841.83×10 31.60×10 32.42×10 33.26×10 33.81×10 32.52×10 3
MΦ∆(GeV)9.78×10 −20.8155.4458.81813.39116.02523.5436.01719.2025.523v∆(100GeV)125.4753.9469.0748.2647.3336.2137.3922.9532.389.99vΦ(1TeV)62.1029.4030.2524.5032.0026.6127.1963.3846.5859.93gH×10 26.8762.7516.4985.3786.0668.1398.4208.4518.7479.376 gX6.26×10 −75.20×10 −51.66×10 −84.70×10 −72.90×10 −23.60×10 −77.24×10 −73.58×10 −23.31×10 −23.62×10 −6
MX(GeV)2000mD(GeV)94.83192.82468.96565.00890.98921.351123.491633.271815.832238.29 mh2(TeV)150.7853.48194.2224.4523.2041.3531.1129.4383.7221.18mh3(10TeV)164.6835.01666.9056.3444.0945.3649.52124.6244.0281.72mZ′(GeV)2000.00404.36982.74658.80863.191083.111144.831415.401389.162000mZ′′(GeV)2135.152000.002000.002000.002248.862000.002000.003784.112933.292809.32
Ωh 2×101.1931.1801.1861.1881.2031.2031.1921.2071.2041.195 σ SI×10 −46(cm 2)1.0584.0656.6229.64712.27512.70615.68816.81313.45517.866 Treelvl.(pb)1.55×10 −52.38×10 −48.02×10 −96.08×10 −51.49×10 −57.80×10 −69.51×10 −118.07×10 −112.46×10 −76.47×10 −11
Looplvl.(pb)1.77×10 −62.73×10 −84.36×10 −63.69×10 −91.49×10 −84.44×10 −107.11×10 −116.49×10 −129.62×10 −131.32×10 −13
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