The triplet-like dark matter is mainly composed of ∆p component with the composition (OD32)2> 2/3. With such large triplet fraction, one expects that the corresponding coupling is different from those of doublet-like dark matter. The relic density of triplet-like dark matter is shown in the left panel of Fig. 6.6. Unlike the doublet case, the suitable relic density consistent with the PLANCK data occurs almost everywhere in the dark matter mass especially for the mass above 10 GeV.
For dark matter mass between 1 to 10 GeV, the dominant annihilation cross section is given by the ¯ccand ¯τ τ final states. The full diagrams of this process are given by Fig. 6.2.
However, only the s-channel Higgs exchange is relevant for this mass range especially the one mediated by h1 and h2. This comes from the dark matter couplings with those two Higgses which are given by
λDD∗hi = i−λH∆vOH1i− λΦ∆vΦOH2i+ 2λ∆v∆OH3i (OD32)2. (6.7) One expects that for i equals to 1, the first term on the right hand side would be dominant and unsuppressed by the Higgs mixing matrix. Thanks to the large value of vΦand v∆ compared to v, the second and third term could have comparable value with the first term. The same argument also hold for i equals to 2 which describes the h2exchange. Thus, as long as the mh2 is not too far from mh1, these two contribution need to be taken into account. The h3 mediator on the other hand, is sub-dominant due to its heavy mass. As before, the small value of c quark and τ lepton Yukawa couplings will make the corresponding annihilation cross section becomes small such that they enhance the relic density in this low mass regime.
As a side note, the right hand side of Eq. (6.7) consists of three different terms that can be cancelled or enhanced each other depending on the relative signs of these terms. Furthermore, the interference between h1, h2 and another possible mediator in general, could make the resulting cross section to have spreading values as one can see in Fig. 6.6.
Similar to the inert doublet-like case, the ¯bb channel dominates the annihilation cross section when the dark matter mass is larger than 10 GeV. This will make the cross section become larger and reduce the relic density. As a result, the observed relic abundance from PLANCK data can be satisfied in this range. The next final state to be considered is the W+W− which opens at the dark matter mass around 40 GeV. This channel occurs via p-channel contact interaction, s-p-channel Higgses and gauge bosons exchange, and the t-p-channel charged Higgs exchange as depicted in Fig. 6.5 with the relevant replacement WL+ → W+. The p-channel contact interaction is suppressed by (OD22)2. The charged Higgs exchange is sub-dominant due to its heavy mass as will be explained later when we discuss coannihilation.
100 101 102 103 (right) planes. The gray area on the left has no coannihilation or resonance. The gray area on the right is excluded by PLANCK data at 2σ . Some orange squares are above the XENON1T limit due to ISV cancellation at the nucleus level.
For the neutral gauge boson mediator, the dominant amplitude is given by the Z′exchange and its proportional to dark matter coupling with the Z′which is given by
gDDµ ∗Z′= i
which is proportional toOi jG gauge boson mixing matrix. Note that the second and third terms on the right hand side come from Goldstone-like contribution which can give non-negligible role in Z′exchange. In our scan, the dominantO2iGcomponent is given byO23G which corresponds to Z′′ exchange. However, due to its larger mass, this channel would be sub-dominant. The SM Z exchange on the other hand, is suppressed byO21G. In non-relativistic approximation, this will contribute to the p-wave annihilation cross section.
Another contribution comes from the three Higgses exchange with the first two Higgses h1 and h2dominating the cross section. Thanks to both cancellation (enhancement) in the coupling in Eq. (6.7) and also the destructive (constructive) interference between h1 and h2 diagrams, the corresponding cross section will be small (large). In the non-relativistic approximation, these Higgses exchange will lead to the velocity independent or s-wave annihilation cross section. Taking into account the similar enhancement (cancellation) in Z′ mediator coupling of Eq. (6.8), these three amplitudes control the dark matter relic abundance.
The next final state is ZZ pair that occurs via p-channel contact interaction, t and u-channel of D, e∆,W′exchange, and the s-channel of three Higgses interaction. This channel is also controlled by the h1 and h2exchange while the other contributions from the t- and u-channels are suppressed not only by the masses of D, e∆,W′but also by the mixing matrix elements. For the D mediator, the amplitude is proportional to the (O21G)2and hence reduces the associated amplitude. The suppression of e∆ is originated from the vanishing small values of OD22 and also ofO21G. On the other hand, the W′exchange contributes sub-dominantly due to its dependence on (OD22)2. The p-channel suffers from large (O21G)2suppression. Moving further, the h1h1final state is dominated by four point contact interaction, s-channel h1and h2exchange, t- and u-channel D exchange as given in Fig (6.3). As expected the p-channel gives the most important contribution while the other three contributions are comparable to each other. For the p-channel interaction, the relevant coupling is
λ3DD∗h1h1 = i
λH∆(OD32)2(OH11)2 , (6.9) while another relevant coupling for h1h1h1which accounts for h1exchange is given by
λh1h1h1= i6λHv(OH11)3− 3λH∆v∆(OH11)2OH31+ 3λHΦvΦ(OH11)2OH21 , (6.10) and the corresponding h1h1h2coupling is written as
λh1h1h2 = i
λHΦvΦ(OH11)2OH22− λH∆v∆(OH11)2OH32 . (6.11) From the left panel of Fig. 6.6, one can see a very clear h1resonance near mD≈ 63 GeV.
As in typical resonance effect, the corresponding relic density will be too small. However, the observed relic density from the PLANCK data can still be accommodated in triplet dark matter case thanks to the suitable adjustment via cancellation among different terms in Eq. (6.7) for i equals to 1.
When the dark matter mass is above 100 GeV, the longitudinal components of the W and Zboson start to dominate the annihilation cross section. The SM Higgs boson h1final state also gives important contribution. On average, in this mass regime, the most dominating final states are the WL+WL− (≥ 50%), h1h1(∼ 25%) and ZLZL(∼ 20%). As before, the diagrams relevant to these three final states are described in Figs. 6.5, 6.3, 6.4 for WL+WL−, h1h1, and ZLZL respectively. There is an exact cancellation between the p-channel contact interaction, t-channel as well as u-channel D exchange in ZLZL final state. This will leave the s-channel Higgses exchange to control the corresponding annihilation cross section. In the case of WL+WL− the situation is quite different as there is a big mass difference between D and
charged Higgs. However, the energy dependence of the amplitude is still vanishing small via similar cancellation between p-channel contact interaction and the t-channel charged Higgs exchange. This is realized by taking the limit of s >> mD, mH−. In this limit, Eq. (6.2) is
where we have taken t =−2s in this high energy limit. Since we only scan for the dark matter mass below 10 TeV, we can not see this cancellation as in doublet-like dark matter.
As a result, the points in high energy regime are spreading everywhere and do not have the tendency of having larger relic abundance. This can be seen from the left panel of Fig. 6.6.
Another mechanism that comes into play in high energy dark matter mass is given by the appearance of new resonances and coannihilations. At mD> 100 GeV, there are a lot points that satisfy 2mD≈ mh2. This will generate the h2resonance as depicted by the red points in the left panel of Fig. 6.6. This resonance contribution occurs via the s-channel h2exchange that appears in any dominant final states mentioned before. As opposed to the SM Higgs resonance which located around 63 GeV, the h2resonance spreads in a wide range of the dark matter mass. This is due to the fact that mh2 can take any value greater than mh1 and less than mh3. In the case of coannihilation, one needs to remember that the triplet-like dark matter is characterized by the ∆pcomponent. This fraction is dominated by the (3,3) component of the mixing matrix in Eq. (3.19). Therefore, the mass splitting between D and charged Higgs H± is no longer described by Eq. (6.1). In fact, from our scan result, the charged Higgs mass is always more than twice heavier than the D mass. In addition, the mass splitting between D and e∆ is also large. This is due to the choice of large v∆ that makes the (3,3) component always smaller than (2,2) component of dark scalar mixing matrix in Eq. (3.19).
Thus, the triplet-like dark matter provides the natural dark matter candidate in G2HDM model. Based on these two mass splittings, there is no coannihilation between triplet -like dark matter and either H± or e∆. Interestingly, at above 400 GeV, there is coannihilation effect coming from W′and heavy fermion fH as depicted by the gold square and green points in the left panel of Fig. 6.6. These two coannihilations are important when the annihilation cross section becomes ineffective. This conclusion can be drawn based on the location of these coannihilation points which are situated above the 2σ red line.
The important channel for triplet-like dark matter-nucleon interaction is given by the t-channel hiand Ziexchange. The heavy fermion exchange is not relevant due to its heavy mass suppression in the propagator. In particular, the dominant contribution to the spin independent cross section is given by the h1, Z and Z′mediator. Out of these three dominant
contributions, the h1gives the most significant impact on the cross section. The upper bound of this Higgs exchange can even reach 10−41cm2 which is 5 order of magnitude above the current XENON1T limit for the dark matter mass less than 500 GeV. This is due to the big range of the λDD∗h1 coupling given in Eq. (6.7). Above 500 GeV, the h1, Z and Z′ contributions become comparable to each other. The typical cross section in this mass range is below 10−45cm2which can escape the current XENON1T data. Unlike the doublet-like case, the Z boson exchange is no longer characterized by the SM coupling as one can see from Eq. (6.8). From this equation, we see that the Ziexchange is controlled by the SU (2)H gauge coupling as well asO2iGcomponents. This will make the SM Z boson exchange become much smaller compared to the similar exchange in the doublet-like case. Further comment needs to be made regarding the comparable contribution between Z and Z′. As one naively expects, the Z exchange is more important than the Z′one due to the heavier mass of the later contribution. This comparable contribution can be understood from the mixing matrix given in Eq. (3.36) especially the lower 3x3 mixing part. Thanks to the large value of vΦcompared to the SM VEV, the (2,2) component will strongly mix with the (3,3) component while the mixing between the (1,1) components are quite small. Thus, one expects thatO21G will be much smaller thanO22G. In addition, as long as the Z′mass is not much heavier than the Z mass, the resulting cross sections between these two contributions would be comparable as observed in our case.
The ISV effect is still observed even though it becomes milder than the doublet-like case thanks to the h1contribution. The dark matter-neutron cross section is slightly larger than the proton one. This can be understood from the value of| fn/ fp| ∼ O(1). Moreover, there is a variation of the sign between fp and fn for each points in our scan. In some cases, fpwill have the same sign with fnwhile in other case they have opposite signs. Thus, the maximal cancellation fp/ fn≈ −0.7 can not be realized in this case. Due to the Z and Z′exchange, the dark matter-nucleon and the anti-dark matter-nucleon cross sections are different. As mentioned before, we take the average value between these two contributions.
The result of our calculation for direct detection is shown in the right panel of Fig. 6.6.
After calculating the averaged cross section in the nucleus level, we project the result on the dark matter-neutron cross section. The blue points denote the allowed region that satisfy the SGSC constraints as well as 2σ relic density. The orange region describes the points that survive the SGSC, relic density, and XENON1T constraints. Note that there are few orange points located above the XENON1T exclusion line. This is due to the ISV effect that weakens the XENON1T exclusion limit. From this figure, the allowed dark matter mass which satisfies all the constraints is located above 400 GeV.
101 102 103
Fig. 6.7 The annihilation cross section at the present universe (left) and DM-neutron elastic scattering cross section (right) for f∆> 2/3 case. In the left panel, the annihilation final state is classified to be three main types: W+W− (blue), b¯b (green), and h1h1(orange). However, the exclusion by ID is marked by unfilled and light colors. In the right panel, the region allowed by SGSC+RD+ID+DD constraints is marked by filled dark blue squares. However, the region excluded by SGSC+RD+ID and SGSC+RD+DD is marked in orange crosses and light blue squares. Projected sensitivities from the CTA experiment for the W+W−and b¯bfinal states are also shown.
The indirect dark matter search constrains the allowed thermally averaged cross section in the recent universe. Since the temperature of the current universe is very low ∼ 2.7 K, the p-wave cross section is extremely suppressed compared to the s-wave part. In addition, there is no coannihilation is present time as the heavier Z2 particles are expected to have decayed into the lightest one. Thus, it is sufficient to consider the dark matter annihilation cross section alone. The corresponding Feynman diagrams of indirect detection are similar to the relic density diagrams. The indirect detection (ID) constraints for triplet-like DM is given in Fig. 6.7. All the points in this figure satisfy both the SGSC and 2σ relic density constraints. The left panel shows the present time of dark matter annihilation as a function of the dark matter mass. We only present the most dominant final states which are given by ¯bb and W+W− pairs. As the dark matter mass get less than∼ 70 GeV, the dark matter pair dominantly annihilates into ¯bb. The allowed region coming from this annihilation is depicted by the green triangle points while the excluded ones are represented by the orange triangle points. At the region near SM Higgs resonance, it is shown that the shape of the thermally averaged cross section forms a dip. This seems to contradict the typical cross section which has the spiky peak in the appearance of the resonance. This happens
because of the 2σ relic density constraints imposed on this region. Near the resonance, the corresponding annihilation cross section would have a very large value and hence suppressed the relic density. In order to satisfy the observed relic density, the λDD∗h1 coupling needs to be extremely fine tuned to the small value such that the associated cross section becomes smaller. Thus, this fine tuned coupling will make the recent dark matter annihilation cross section to have smaller value. Above 100 GeV, the most dominant contribution is given by W+W− final state. The ID allowed region in this final states are shown by the blue points while the ID exclusion region is represented by the red square area. Note that the current DM ID sensitivity can only apply strongly for the DM mass located between 10 GeV and few hundred GeV. Furthermore, the future CTA sensitivity [59] described by the red (purple) line for the W+W− (¯bb) final states might be able to put the limit in the TeV region. In the right panel of Fig. 6.7 we compare the exclusion limit given by recent XENON1T data (blue squares) and Fermi gamma-ray constraints (orange crosses). One can see the XENON1T exclusion (unfilled squares) power is much stronger than Fermi gamma-ray exclusion. The blue filled region describes the area that passes all the constraints discussed so far: SGSC, 2σ PLANCK relic density, XENON1T data and Fermi gamma-ray exclusion.