SU(2) H Goldstone Boson-like DM

It is easy to note that the (1,1) and (3,3) elements of Eq. (3.19) have a see-saw behaviour controlled by the value of v_∆. The (2,2) element remains almost unaffected thanks to the term proportional to v²_Φ. Goldstone boson-like DM is characterized by a large value in the element (1,1) of Eq. (3.19) when compared to the (1,3) element,−MΦ∆v_Φ/2. The size of the (1,2) element is not relevant since the (2,2) element remains much larger. The difference in size between the (1,1) and (1,3) elements is best measured by taking the ratio between them which is roughly approximated by 2v_∆/v_Φ. In other words, the v_∆/v_Φratio controls the Goldstone boson composition of the DM mass eigenstate. This is illustrated in Fig. 6.8. Note that when the ratio v_∆/v_Φ grows close to 1, which means v_Φ∼ 20 TeV in our scan range, EWPT disfavour the presence of light Z^′states and larger mixing with the SM Z. Values of

f_G^p larger than∼ 0.8 are accessible only through negative MΦ∆ resulting in tachyonic DM masses.

Thus, in the case of Goldstone boson-like DM, there will be a mixture coming from the pure triplet component, Goldstone-doublet and triplet-doublet components. There is a change in the dominant DM annihilation cross section in Goldstone boson-like DM. The most dominant channel is still given by the W_L⁺W_L⁻ final state which contributes around 50%.

The next important process is given by Z^′Z^′final state. In both two cases, the s-channel h1

0.0 0.2 0.4 0.6 0.8 1.0

v∆/vΦ

0.0 0.2 0.4 0.6 0.8 1.0

fGP

After SGSC

Tachyonic DM

EWPTdisfavoured

Fig. 6.8 Correlation between the ratio v_∆/v_Φ and the mixing parameter f_GP after applying the constraints from the scalar and gauge sectors.

and h₂ exchange affect the whole amplitude. However, unlike in previous two scenarios where doublet or triplet component fully controls the corresponding dominant couplings (λ_DD∗h1 and λ_DD∗h2), there is negative effect coming from the "impurity" contributions in the λ_GDD∗h1 and λ_GDD∗h2. The expression of λ_GDD∗h1 is given as

λ_GDD∗h1 = i(λHΦ+ λ_HΦ^′ )vO^H₁₁− λΦ∆v_∆O^H₃₁+ 2λΦv_ΦO^H₂₁ (O^D₁₂)² (6.13) + iM_H∆O^D₁₂+ λ_HΦ^′ v_ΦO^D₃₂ O^D₂₂O^H₁₁

+ i

λ_H∆vO^H₁₁− 2λ∆v_∆O^H₃₁+ λΦ∆v_ΦO^H₂₁ (O^D₃₂)²,

where the first term inside the bracket on the right hand side denotes the pure Goldstone contribution, the second term describes the mixing between Goldstone-doublet and doublet-triplet, and the last term represents the pure triplet contribution. The last two lines denote the

"impurity" terms. In the case of λ_GDD∗h2, the coupling is given by

λ_GDD∗h2 = i−λΦ∆v_∆O^H₃₂+ 2λΦv_ΦO^H₂₂ (O^D₁₂)² (6.14) + i−2λ∆v_∆O^H₃₂+ λ_Φ∆v_ΦO^H₂₂ (O^D₃₂)²,

where the first (second) line denotes Goldstone (triplet) contribution.

As a result, these dominant couplings will have smaller value and make the corresponding cross section decreased. This can be easily seen from the left panel of Fig. 6.9 where there are a lot of points reside above the observed relic density band. In addition, the presence of

10⁰ 10¹ 10² 10³

Fig. 6.9 Goldstone boson-like SGSC allowed regions projected on the (mD, ΩDh²) (left) and (m_D, σ_n^SI) (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 . The orange squares above the XENON1T limit present the ISV cancellation at the nucleus level.

new dominant Z^′Z^′ final state stems from the fact that the coupling λ_GDD∗Z is suppressed by the off-diagonal rotation matrix elements O₂₁^G while λ_GDD∗Z^′ contains the somewhat largerO₂₂^G. One expects smallO₂₁^G due to the SM-likeness of Z in this model. Let’s write λ_GDD∗Z(′)= g_GDD∗Z(′)(p_D∗− pD)^µ. The expression of g_GDD∗Z is given by

g_GDD∗Z =−i

2g_H(O^D₁₂)²+ 2(O^D₃₂)²

O21^G− ic_We

2s_W(O^D₂₂)²O11^G, (6.15) where the first (second) term on the right hand side coming from Goldstone (triplet) com-ponent which is suppressed byO₂₁^G while the last term suppressed by (O^D₂₂)² denotes the doublet contribution. Furthermore, the coupling of g_GDD∗Z^′ is written as

g_GDD∗Z^′=−i

2g_H(O^D₁₂)²+ 2(O^D₃₂)²

O₂₂^G, (6.16)

where only Goldstone and triplet component are present in this case. In contrast to the λ_GDD∗h1 and λ_GDD∗h2, the existence of non-Goldstone contribution gives an enhancement to both of the λ_GDD∗Z and λ_GDD∗Z^′ couplings. In addition, the p-channel contact interaction between D and Z^′also contributes significantly to the Z^′Z^′final state. This contact coupling

is expressed as

where one can also see the enhancement effect coming from the triplet contribution. Thus in the case of Z^′Z^′final state, not only h₁and h₂s-channel exchange is important, but also the t-channel of D exchange which is controlled by λ_GDD∗Z^′ as well as p-channel contact interaction even though the h₁and h₂exchange control the range of the cross section thanks to a big range in the corresponding couplings in Eq. (6.13) and Eq. (6.14).

Coannihilation in this case is very similar to the triplet-like DM case. The most relevant coannihilations happen with W^′and heavy fermions for large masses and large relic density.

Coannihilation with W^′starts close to DM mass of 300 GeV and mostly above relic density of 10⁻¹. As before, this is where the usual DD^∗annihilation channels become smaller leaving more room for coannihilations that, otherwise, would be negligible. For the case of heavy fermions, coannihilation happens for DM masses above 1 TeV and mostly for the upper bound of relic density, where DD^∗coannihilation is even more suppressed than for the W^′ case.

As one can see in Fig. 6.9, the DM-neutron cross section is between the orders of magnitude 10⁻⁴⁶and 10⁻⁴²cm². The most dominant contribution comes from h₁exchange peaking for DM masses of order 30 GeV. The next dominant channel is given by Z and Z^′ bosons exchange. The contributions from these two gauge bosons result in the base DM-neutron cross section that sits just below 10⁻⁴⁵cm². The interference between h₁, Z, and Z^′ exchange makes the spin independent cross section varies in a wide range. Furthermore, the Goldstone-like dark matter provides us some interesting result regarding the ISV. All the DM compositions discussed have some amount of ISV. But Goldstone boson-like DM is the one where f_n/ f_pis closer to the maximally cancelling value of∼ −0.7. In our analysis, most of the times the D^∗-nucleus cross section is larger than that of D-nucleus, resulting in an average DM-nucleus cross section dominated by D^∗-nucleus. For the cases where f_n/ f_p≈ −0.7 for D^∗-nucleus this would result in noticeable ISV cancellation. In the right panel of Fig. 6.9, the orange points pass the RD constraint from PLANCK and the DD limit set by XENON1T at nucleus level thanks to ISV cancellation.

In the ID side, there is no relevant constraints for this Goldstone boson-like case. Most of the points with relic density in agreement with PLANCK have a very low annihilation cross section in the present and are far beyond the reach of current observations. For DM masses below 100 GeV, the annihilation is dominated by b¯b final state with 90% of the total cross

10²

Fig. 6.10 The present time total annihilation cross section by dominant annihilation channel (left) and the DM-neutron elastic scattering cross section (right) for f_GP > 2/3. Some blue filled squares above the XENON1T limit are due to the ISV cancellation at the nucleus level.

Projected sensitivities from the CTA experiment for the W⁺W⁻ and b¯b final states are also shown.

section in average. For DM mass above the mass of the W^±, the W⁺W⁻ final state dominates completely with an average of 50% of the total cross section, However, W⁺W⁻ final state may compose the total cross section almost completely for some points while it may goes as low as 17% for others. Unlike triplet-like DM, ID alone does not further constrain the points allowed by PLANCK. Fig. 6.10 shows the final result for this section including ID. The right panel shows the zoomed in region of points allowed by the RD constraint. As mentioned before, ISV cancellation ( f_n/ f_p≈ −0.7) reduces the sensitivity of the XENON1T result and some points pass all the constraints (SGSC+RD+ID+DD) even though they are above the direct detection limit at nucleon level.