Investigating higgsino dark matter in the semi-constrained NMSSM

Dark matter (DM) is a crucial yet elusive component of the Universe, accounting for 26.8% of its total mass and energy. Despite its invisibility and lack of direct detection, the presence of DM is strongly supported by observational evidence such as galaxy rotation curves, gravitational lensing, and anisotropies in the cosmic microwave background (CMB) [1−3]. Among the proposed DM candidates, weakly interacting massive particles (WIMPs) are of particular interest. They are consistent with the predictions of the Lambda Cold Dark Matter ($ \Lambda $CDM) model and have masses ranging from the MeV to TeV scales [4−7]. The theoretical framework suggests that WIMPs were once in thermal equilibrium in the early universe and that their current abundance can be explained by the freeze-out mechanism [8, 9]. This mechanism is crucial not only for understanding WIMP relic densities, but also for explaining other processes in the early universe, such as big bang nucleosynthesis and recombination [10]. However, the continued lack of direct detection has led to broader investigations of WIMP properties focusing on their potential annihilation channels and various detection methods.

The role of R-parity in supersymmetry (SUSY) is crucial because it ensures the stability of the lightest supersymmetric particle (LSP), making it a potential DM candidate. In the Minimal Supersymmetric Standard Model (MSSM), the LSP is usually a neutralino, which can be a bino ($ \tilde{B} $), wino ($ \tilde{W} $), or higgsino ($ \tilde{H}_u $ or $ \tilde{H}_d $) [11−14]. These four types of neutralino as LSPs have received a lot of attention [15−34]. For bino-dominated LSPs, the main problem is the risk of excessive DM. To avoid this by increasing the annihilation rate in the early universe, the preferred strategies include coannihilation mechanisms with wino, slepton, or stop [35, 36]. If the LSP is not the only source of DM, a higgsino-dominated LSP in the 100−300 GeV mass range could still contribute to DM, given that the DM-nucleon scattering cross-section is significantly reduced owing to the lower relic density [37, 38]. For a bino-higgsino mixed LSP, the efficient scattering with nucleons makes it difficult to meet the current limits of direct detection. Overall, the lack of definitive results from DM detection experiments tends to constrain the parameter space, favoring a bino-dominated LSP.

In the Next-to-Minimal Supersymmetric Standard Model (NMSSM), which extends the MSSM by adding a singlet, the requirement for the 125 GeV Higgs mass puts less pressure on the stops, thus relaxing the tight constraints from DM experiments [39]. This modification introduces an additional singlino in the neutralino sector, drawing significant attention to the DM sector of the NMSSM [40−64]. Unlike in the MSSM, a bino-dominated DM candidate can coannihilate with a singlino-dominated neutralino, resulting in the measured abundance. In the NMSSM, DM is likely to reach the observed abundance primarily through two mechanisms: coannihilation with higgsino-dominated electroweakinos (EWkinos), or through resonant annihilations facilitated by singlet-dominated CP-even or CP-odd Higgs bosons [65]. In the General NMSSM, singlino-like DM could achieve the observed relic abundance through different channels, such as $ \chi^0_1\chi^0_1 \to h h $, with the theory significantly favoring singlino-dominated DM over bino-like DM over a wider range of parameters [40].

In previous studies of ours [66−68], we focused on the funnel annihilations of light DM within the semi-constrained NMSSM (scNMSSM). The scNMSSM, also known as the NMSSM with non-universal Higgs mass, allows the Higgs sector to deviate from universality at the GUT scale. Specifically, the soft masses $ m_{H_u}^2 $, $ m_{H_d}^2 $, and $ m_S^2 $ can differ from $ M_0^2 + \mu^2 $. Additionally, the trilinear couplings $ A_{\lambda} $ and $ A_{\kappa} $ can be independent from $ A_0 $. We found that the light DM in the scNMSSM is typically singlino-dominated, with four possible funnel annihilation mechanisms allowing the singlino-dominated LSP to acquire the right relic density [67, 68]. Furthermore, we found that highly singlino-dominated DM can reach sufficient relic density. However, both the higgsino asymmetry and spin-dependent cross-section remain small. Conversely, when there is a sizeable higgsino component in light DM, the relic density is always low [66]. Moreover, DM dominated by higgsino can potentially be detected through direct detection, which may enable probing significant mass splittings [69], and its annihilation signals are expected to be observed by the upcoming Cherenkov Telescope Array (CTA) [70], exploiting the unique sensitivity of the CTA to TeV-scale DM. This sensitivity could surpass that of all other existing or planned detectors, establishing CTA as a critical instrument for indirect DM searches. Therefore, in this study, we focused on higgsino-dominated DM in the scNMSSM, aiming to determine whether it has the correct DM relic density and to study its detection and annihilation processes. We examined the relic density of DM to ensure it agrees with cosmological observations, and analyzed the potential for direct and indirect detections based on current experimental limits. Additionally, we explored future collider detection prospects and specific annihilation channels that could affect the detectability of higgsino-dominated LSP.

This paper is organized as follows. In Sec. II, we provide a brief overview of the neutralino and chargino part of the scNMSSM. In Sec. III, we discuss our calculation methodology and provide a comprehensive analysis of the obtained results. In Sec. IV, we conclude by summarizing the main results and their implications for future studies.

The NMSSM extends the MSSM by adding a singlet superfield $ \hat{S} $. The superpotential is modified to

where $ W_{\mathrm{MSSM}}^{\mu=0} $ represents the MSSM superpotential without the $ \mu $-term. The vacuum expectation value (VEV) of the singlet scalar, $ v_s $, generates the $ \mu $-term dynamically,

The NMSSM also includes specific soft SUSY breaking terms:

where $ H_u $ and $ H_d $ are the Higgs doublet scalars, $ A_{\lambda} $ and $ A_{\kappa} $ are trilinear couplings, and $ m_S $ is the singlet scalar soft mass.

In the scNMSSM, the Higgs sector can deviate from universality at the GUT scale, allowing for the soft masses $ m_{H_u}^2 $, $ m_{H_d}^2 $, and $ m_S^2 $ to differ from $ M_0^2 + \mu^2 $. The trilinear couplings $ A_{\lambda} $ and $ A_{\kappa} $ can also can also be independent from $ A_0 $. Thus, the scNMSSM parameter space is defined by nine parameters:

where $ M_{0} $ and $ M_{1/2} $ are the universal sfermion and gaugino masses, and $ A_{0} $ is the universal trilinear coupling in the sfermion sector.

The electroweakino sector within the NMSSM is predicted to include five neutralinos and two pairs of charginos. The five neutralinos are composed of the following eigenstates: $ \tilde{B} $, $ \tilde{W}_{3} $, $ \tilde{H}_{d}^{0} $, $ \tilde{H}_{u}^{0} $, and $ \tilde{S} $. The mass matrix is

where $ s_\beta $, $ c_\beta $, $ s_W $, and $ c_W $ represent $ \sin \beta $, $ \cos \beta $, $ \sin \theta_W $, and $ \cos \theta_W $, respectively. $ M_1 $ and $ M_2 $ are the bino and wino masses, respectively, evolved from $ M_{1/2} $ through RGEs to the SUSY scale. The higgsino mass is $ \mu $, and the singlino mass is $ 2 \kappa v_{s} $. Diagonalizing this matrix, one can obtain the five neutralino mass eigenstates, $\tilde{\chi}^{0}_i\; (i=1,,2,...,5)$. The lightest neutralino is usually the LSP in the NMSSM, making it a perfect DM candidate.

The two pairs of charginos are composed of the eigenstates $ \tilde{H}^{\pm} $ and $ \tilde{W}^{\pm} $. The mass matrix is

where $ M_2 $ is the wino mass, as mentioned above, and the charginos can be the mixture of charged higgsinos and winos. Diagonalizing this matrix, we obtain two physical states in the chargino sector, typically denoted as $ \tilde{\chi}^{\pm}_i\; (i=1,2) $. The lighter chargino, $ \tilde{\chi}^{\pm}_1 $, is dominated by either charged higgsinos or winos, depending on whether $ \mu $ or $ M_2 $takes a smaller value, respectively.

The NMSSM Higgs sector includes three CP-even Higgs bosons ($ H_i $, $ i=1,2,3 $), two CP-odd Higgs bosons ($ A_i $, $ i=1,2 $), and a pair of charged Higgs bosons $ H^{\pm} $. Some of the CP-even Higgs bosons are the SM-like Higgs $ H_{\rm{SM}} $, another doublet-dominated $ H_D $, and the singlet-dominated $ H_S $. Similarly, the two CP-odd Higgs bosons are the doublet-dominated $ A_D $ and the singlet-dominated$ A_S $.

The coupling of the lightest neutralino $ \tilde{\chi}^{0}_1 $ to the Higgs basis states is given by [39]

where the values of $ N_{ij} $ are obtained from the diagonalization of the neutralino mass matrix, and $ C_{X \tilde{\chi}^{0}_1 \tilde{\chi}^{0}_1} $ denotes the couplings of the Higgs basis states ($ H_{\rm{SM}}/H_D/H_S/A_D/A_S $) to the LSP $ \tilde{\chi}^0_1 $.

In the scNMSSM, the gaugino masses are unified at the GUT scale, resulting in the following mass ratios at the SUSY scale: $ M_1 : M_2 : M_3 = 1 : 2 : 6 $. Therefore, the mass of the wino ($ M_2 $) is larger than that of the bino ($ M_1 $), i.e., $ M_2 > M_1 $. A higgsino-dominated LSP is quantified by $ |N_{13}|^2 + |N_{14}|^2 > 0.5 $ and has a mass close to $ \mu $. This condition is met when both $ M_1 $ and $ 2 \kappa v_{s} $ are larger than $ \mu $. In this case, the NLSP (next-to-lightest supersymmetric particle) is another higgsino-dominated neutralino $ \tilde{\chi}^{0}_2 $ or a higgsino-dominated chargino $ \tilde{\chi}^{\pm}_1 $. Given that both are higgsino-dominated, the mass of the NLSP is also close to $ \mu $. Therefore, to achieve the correct relic density, the LSP can annihilate not only through funnel annihilation but also through coannihilation with the NLSP or other sparticles featuring similar masses.

When examining DM characteristics, the DM relic density of the LSP $ \tilde{\chi}^{0}_1 $ is critical. It is calculated by solving the Boltzmann equation for the number density n of the LSP:

where H denotes the Hubble rate, representing the expansion rate of the universe; $ n_{\text{eq}} $ is the equilibrium number density; and $ \langle v\sigma_{\text{eff}} \rangle $ is the thermally averaged effective cross section for annihilation and coannihilation processes.

The relic density $ \Omega h^2 $ is then calculated from the equilibrium solutions of this equation. It can be expressed through the following simplified formula [71]:

where $ m_{\tilde{\chi}^{0}_1} $ is the mass of the LSP, $ n_0 $ is the current number density of the DM, and $ \rho_c = {3H_0^2}/{8\pi G_N} $ represents the critical density of the Universe. In this study, we used the $\textsf{micrOMEGAs}$ package to calculate the DM relic density and scattering cross-section of DM with nucleons.

In this study, we focused on the relic density and annihilation channels of higgsino-dominated DM in the scNMSSM, considering a wide mass range from GeV to TeV. We also explored the implications of direct and indirect detection experiments, as well as collider searches, on the properties and viability of higgsino-dominated DM. In this study, the LSP must be higgsino-dominated, which is defined by

This is an additional selection to ensure a higgsino-dominated LSP which does not result from experimental or theoretical constraints. Therefore, we set the relevant parameter space in the scNMSSM as follows:

This parameter space scan is consistent with a previous study of ours [72]. We used the$\textsf{NMSSMTools-6.0.2}$ package [73−76] to conduct the scans and calculate relevant properties. Both theoretical and experimental constraints were considered. Theoretical constraints include vacuum stability and absence of Landau poles [73, 74]. Experimental constraints include flavor constraints [77−80], a global fit of LHC Higgs data [81−85], DM relic density [86, 87], and direct and indirect DM searches [88−91]. Constraints on SUSY searches at the LHC and LEP were implemented using the $\textsf{SModelS-v2.2.1}$ package [92−98], while constraints on searches for additional Higgs bosons and exotic decays of the SM-like Higgs were applied using the $\textsf{HiggsBounds-5.5.0}$ package [99−104].

For the samples satisfying the above theoretical and experimental constraints, we observed the following properties. The mass range of the lightest neutralino varies from 100 GeV to 4 TeV and is predominantly higgsino-dominated. The mass of the lightest neutralino, predominantly higgsino-dominated, ranges from 100 GeV to 4 TeV. Similarly, the mass of the lightest chargino, $ \tilde{\chi}_{1}^{\pm} $, also spans from 100 GeV to 4 TeV, mirroring that of the lightest neutralino. This similarity arises because both the lightest neutralino and lightest chargino are higgsino-dominated, resulting in their masses being approximately equal to $ \mu $. Given that $ \mu < M_1 $, an upper limit is set on their masses. In the Higgs sector, $ h_1 $ is identified as the 125 GeV SM-like Higgs; $ h_2 $ and $ h_3 $ are classified as heavy CP-even Higgs bosons, while $ a_1 $ and $ a_2 $ are heavy CP-odd Higgs bosons.

Figure 1 shows the DM relic density and higgsino component of the LSP in the scNMSSM. Note the following key points from this figure:

Figure 1.  (color online) Surviving samples are shown in the planes of $ \kappa $ versus $ \lambda $ (left and middle) and $ \mu $ versus $ M_{1/2} $ (right). From left to right, the colors represent $ \mu $, DM relic density $ \Omega h^2 $, and higgsino component in the LSP $ |N_{13}|^2 + |N_{14}|^2 $, respectively. Samples with larger values of $ \mu $ are plotted on top of those with smaller values.

● In the left and middle panels, the surviving samples are shown on the $ \kappa $ versus $ \lambda $ plane. Regarding the higgsino-dominated LSP, for which the higgsino is the lightest neutralino and the singlino has a larger mass, the following relationship exists:

Therefore, all surviving samples are in the region $ |\kappa/\lambda| > 1/2 $.

● In the left panel, the color indicates $ \mu $. Note that $ \mu \gtrsim 3 $ TeV is restricted to a small region. This is because, as the higgsino mass increases, the DM relic density becomes excessively large. The coannihilation rate slows down, necessitating funnel annihilation processes to reduce the DM density.

● In the middle panel, the color indicates the DM relic density $ \Omega h^2 $. Note that in the region with larger $ \mu $, which corresponds to a heavy higgsino LSP, there are few samples with the correct relic density. This is because heavy higgsino LSPs require both coannihilation and funnel annihilation to achieve the correct relic density. Samples with $ \mu \approx 1000 $ GeV appear to have the right DM relic density, which is likely a primary result of coannihilation. In contrast, samples with $ \mu \lesssim 1000 $ GeV do not have the correct relic density.

● In the right panel, the surviving samples are displayed on the $ \mu $ versus $ M_{1/2} $ plane, with the color indicating the higgsino component in the LSP, $ |N_{13}|^2 + |N_{14}|^2 $. Note that almost all the samples are highly higgsino-dominated, with $ |N_{13}|^2 + |N_{14}|^2 \gtrsim 0.9 $. Additionally, $ M_{1/2} $ constrains the upper bound of the higgsino LSP. Given that $ \mu < M_1 $ and the gaugino masses satisfy the ratio $ M_1 : M_2 : M_3 = 1 : 2 : 6 $, the bino mass $ M_1 $ is proportional to $ M_{1/2} $.

In Fig. 2, the surviving samples are displayed on the planes of the light CP-odd Higgs boson mass $ m_{a_1} $ versus LSP mass $ m_{\tilde{\chi}^{0}_1} $. Colors represent relevant mass degeneracies (upper, lower left, and lower middle panels) and DM relic density $ \Omega h^2 $ (lower right panels). The color of the Higgs-LSP mass degeneracies $ 1/(|m_{H/A}/m_{\tilde{\chi}_1^{0}}-2|+1) $ approaching one indicates that Higgs masses tend to be $ 2 m_{\tilde{\chi}_{1}^{0}} $. Similarly, the stau-LSP mass degeneracies $ m_{\tilde{\tau}}/m_{\tilde{\chi}_1^{\pm}} $ approaching one indicate that stau masses tend to be $ m_{\tilde{\chi}_{1}^{0}} $. Colors closer to zero indicate highly mass degeneracies. The mass degeneracies of different annihilation mechanisms are based on the following properties [105, 106]:

Figure 2.  (color online) The surviving samples are displayed on the planes of the light CP-odd Higgs boson mass $ m_{a_1} $ versus the LSP mass $ m_{\tilde{\chi}^{0}_1} $. Colors represent relevant Higgs-LSP mass degeneracies (upper panels and lower left panel), stau-LSP mass degeneracies (lower middle panel) and DM relic density $ \Omega h^2 $ (lower right panel). The color of the Higgs-LSP mass degeneracies $ 1/(|m_{H/A}/m_{\tilde{\chi}_1^{0}}-2|+1) $ approaching one indicates that Higgs masses tend to be $ 2 m_{\tilde{\chi}_{1}^{0}} $. Similarly, the stau-LSP mass degeneracies $ m_{\tilde{\tau}}/m_{\tilde{\chi}_1^{\pm}} $ approaching one indicate that stau masses tend to be $ m_{\tilde{\chi}_{1}^{0}} $. Colors closer to zero indicate larger mass degeneracies. In the upper panels, lower left, and middle panels, samples with smaller mass degeneracies are plotted on top of those with smaller values, while in the lower right panels, samples with larger DM relic density $ \Omega h^2 $ are plotted on top.

Note that the coannihilation mechanism requires the NLSP mass to be very close to the LSP mass. For instance, chargino $ \tilde{\chi}_{1}^{\pm} $, stau $ \tilde{\tau}_{1} $, and stop $ \tilde{t}_{1} $ coannihilation occur when their masses are very close to the LSP mass, leading to the corresponding coannihilation. Specifically, for the higgsino-LSP scenario in the scNMSSM, all samples satisfy $ \tilde{\chi}_{1}^{\pm} $ coannihilation, with a small fraction satisfying $ \tilde{\tau}_{1} $ coannihilation, and no samples satisfying $ \tilde{t}_{1} $ coannihilation.

The funnel annihilation mechanism requires the CP-even or CP-odd Higgs $ H/A $ mass to be close to twice the LSP mass. For surviving samples, $ h_1 $ is the 125 GeV SM-like Higgs, and the minimum LSP mass is larger than 100 GeV. Therefore, a higgsino-dominated LSP does not undergo $ h_1 $ funnel annihilation. Only $ h_2 $, $ h_3 $, $ a_1 $, and $ a_2 $ funnel annihilation mechanisms are present. In the scNMSSM framework, our analysis shows that there are samples with a higgsino-dominated LSP that achieve the correct DM relic density with no need for funnel annihilation. However, for heavier higgsino-dominated LSP, funnel annihilation is necessary to achieve the correct DM relic density.

In Fig. 2, we used $ 1/(|m_{H/A}/m_{\tilde{\chi}_1^{0}}-2|+1) $ to represent the Higgs-LSP mass degeneracy. This expression approximates to one when the condition for the A/H funnel mechanism is satisfied, i.e., $ m_{\tilde{\chi}_1^{0}} \approx 2m_{H/A} $, and it approaches zero in the opposite case. Note also the following aspects in this figure:

● For the surviving samples, all samples satisfy the $ \tilde{\chi}_{1}^{\pm} $ coannihilation condition. This is because the masses of both the higgsino-dominated LSP and the chargino $ \tilde{\chi}^{\pm}_1 $ are approximately equal to $ \mu $.

● In the upper left panel, $ h_2 $ funnel samples seem to occupy two distinct regions on the $ m_{a_1} $ versus $ m_{\tilde{\chi}^{0}_1} $ panel. One region is concentrated where $ m_{\tilde{\chi}^{0}_1} \lesssim 2400\; {\rm{GeV}} $ and $ m_{a_1} \lesssim 8000 \;{\rm{GeV}} $; the other region is concentrated around $ m_{a_1} = 2m_{\tilde{\chi}^{0}_1} $.

● In the upper middle panel, $ h_3 $ funnel samples are relatively few and seem to be distributed in the region where $ m_{a_1} < 2m_{\tilde{\chi}^{0}_1} $. In the upper right panel, $ a_1 $ funnel samples seem to be distributed around the line $m_{a_1} = 2m_{\tilde{\chi}^{0}_1}$. In the lower left panel, $ a_2 $ funnel samples are relatively more abundant and seem to be distributed in the region where $ m_{a_1} < 2m_{\tilde{\chi}^{0}_1} $.

● In the lower middle panel, the distribution of samples satisfying $ \tilde{\tau}_1 $ coannihilation is analyzed. These samples are dispersed throughout the plane, indicating that $ \tilde{\tau}_1 $ coannihilation occurs across all examined regions.

● In the lower right panel, there seems to be a distinct region where the higgsino-dominated LSP mass is less than 1300 GeV. In this region, the DM relic density is insufficient, but it increases with mass until it reaches around 1300 GeV, where the DM relic density becomes sufficient. This is because all samples satisfy the $ \tilde{\chi}_{1}^{\pm} $ coannihilation condition. When the LSP mass is low, the coannihilation process is more efficient, leading to easier annihilation of DM and hence to an insufficient DM relic density. As the LSP mass increases, coannihilation decreases, causing the DM relic density to rise. When the LSP mass exceeds 1300 GeV, $ \tilde{\chi}_{1}^{\pm} $ coannihilation becomes less efficient, leading to an excess of DM. At this point, additional funnel annihilation mechanisms are needed to assist in DM annihilation.

In Fig. 3, the surviving samples are displayed on the planes of the light CP-odd Higgs boson mass $ m_{a_1} $ versus LSP mass $ m_{\tilde{\chi}^{0}_1} $ (left), $ \kappa $ versus $ \lambda $ (middle), and $ \mu $ versus $ M_{1/2} $ (right). Each color represents a different DM annihilation mechanism. Given that all surviving samples have a higgsino-dominated LSP, and the light chargino $ \tilde{\chi}^{\pm}_1 $ is also higgsino-dominated, each sample inherently satisfies the $ \tilde{\chi}_1^{\pm} $ annihilation criterion. Therefore, except for the points representing only $ \tilde{\chi}_1^{\pm} $ coannihilation (purple), all other points correspond to hybrid annihilation mechanisms. We have categorized the samples into six classes based on the different DM annihilation mechanisms they satisfy:

Figure 3.  (color online) Surviving samples are shown in the planes of the light CP-odd Higgs boson mass $ m_{a_1} $ versus the LSP mass $ m_{\tilde{\chi}^{0}_1} $ (left), $ \kappa $ versus $ \lambda $ (middle), and $ \mu $ versus $ M_{1/2} $ (right). Each color represents a different DM annihilation mechanism. All samples satisfy the $ \tilde{\chi}_1^{\pm} $ annihilation criteria and are classified according to the DM annihilation mechanisms they meet as follows: only the $ \tilde{\chi}_1^{\pm} $ coannihilation (purple), singlet-dominated CP-even Higgs funnel annihilation (blue), singlet-dominated CP-odd Higgs funnel annihilation (dark green), doublet-dominated Higgs funnel annihilation (light green), doublet-dominated Higgs funnel annihilation with $ \tilde{\tau}_1 $ coannihilation (pink), and $ \tilde{\tau}_1 $ coannihilation (magenta).

● $ \tilde{\chi}_1^{\pm} $ coann. (only): solely the $ \tilde{\chi}_1^{\pm} $ coannihilation;

● $ H_S $ funnel ($ \tilde{\chi}_1^{\pm} $ coann. hybrid): singlet-dominated CP-even Higgs funnel annihilation combined with $ \tilde{\chi}_1^{\pm} $ coannihilation;

● $ A_S $ funnel ($ \tilde{\chi}_1^{\pm} $ coann. hybrid): singlet-dominated CP-odd Higgs funnel annihilation combined with $ \tilde{\chi}_1^{\pm} $ coannihilation;

● $ H_D/A_D $ funnel ($ \tilde{\chi}_1^{\pm} $ coann. hybrid): doublet-dominated Higgs funnel annihilation combined with $ \tilde{\chi}_1^{\pm} $ coannihilation;

● $ H_D/A_D $ funnel - $ \tilde{\tau}_1 $ coann. ($ \tilde{\chi}_1^{\pm} $ coann. hybrid): doublet-dominated Higgs funnel annihilation combined with $ \tilde{\tau}_1 $ coannihilation and $ \tilde{\chi}_1^{\pm} $ coannihilation;

● $ \tilde{\tau}_1 $ coann. ($ \tilde{\chi}_1^{\pm} $ coann. hybrid): $ \tilde{\tau}_1 $ coannihilation combined with $ \tilde{\chi}_1^{\pm} $ coannihilation.

The classification of these samples is based on the criteria described by Eqs. (16)–(19). When comparing Fig. 3 with Figs. 1 and 2, the following conclusions can be drawn:

● In the left panel, the blue samples, which represent only $ \tilde{\chi}_1^{\pm} $ coannihilation, correspond to an LSP mass below 1300 GeV. This annihilation channel constrains the upper mass limit of the LSP owing to the DM relic density requirements. As shown in the lower right panel of Fig. 2, the DM relic density increases with larger LSP masses. This is because the efficiency of $ \tilde{\chi}_1^{\pm} $ coannihilation decreases as the mass increases. Consequently, larger LSP masses, if solely relying on $ \tilde{\chi}_1^{\pm} $ coannihilation, would result in a DM relic density that exceeds the experimental upper limits.

● In the left panel, for samples with an LSP mass below 1300 GeV, a fraction also falls into the category of $ \tilde{\tau}_1 $ coannihilation combined with $ \tilde{\chi}_1^{\pm} $ coannihilation. It is important to note that the primary mechanism for these parameter points remains $ \tilde{\chi}_1^{\pm} $ coannihilation. For LSP masses above 1300 GeV, achieving the correct DM relic density requires the use of Higgs funnel annihilation mechanisms. Depending on the proportion of singlet versus doublet components in the Higgs, their distribution varies. Blue samples, satisfying the conditions for $ H_S $ funnel annihilation, typically have LSP masses below 2500 GeV. In contrast, samples satisfying the conditions for $ A_S $(dark green) and $ H_D/A_D $(light green) funnel annihilations can span any LSP mass range. Pink samples, which involve funnel annihilation combined with $ \tilde{\tau}_1 $ coannihilation, also cover any LSP mass. However, the main contributors to the DM relic density in these samples are still $ \tilde{\chi}_1^{\pm} $ coannihilation and Higgs funnel annihilation.

● In the middle panel, combined with the middle panel of Fig. 1, we observe that in the region with a large $ \mu $ value, corresponding to a heavy higgsino-dominated LSP, $ A_S $ (dark green) and $ H_D/A_D $ (light green) funnel annihilations are possible. Additionally, there is a small presence of Higgs funnel annihilation combined with $ \tilde{\tau}_1 $ coannihilation (pink). Only a few samples in this region reach the correct DM relic density.

● In the left panel, within the region where $ \mu $ is relatively low, we observe different probabilities of Higgs funnel annihilation. For instance, in regions where $ \mu $ is less than 600 GeV, $ A_S $ funnel annihilation (dark green) is more likely to occur. When $ \mu $ decreases below 200 GeV, $ H_S $ funnel annihilation (blue) becomes more common.

In Fig. 4, surviving samples are displayed on the planes of the rescaled spin-independent (SI) DM-nucleon cross-section $ \sigma_{\rm{SI}} \times \Omega/\Omega_0 $ (left), rescaled spin-dependent (SD) DM-nucleon cross-section $ \sigma_{\rm{SD}} \times \Omega/\Omega_0 $ (middle), and rescaled thermally averaged DM annihilation cross-section times velocity $ \langle \sigma v \rangle \times (\Omega / \Omega_0)^2 $ (right) versus LSP mass $ m_{\tilde{\chi}^{0}_1} $, where $ \Omega_0 = 0.1187 $. The colors indicate the DM relic density $ \Omega h^2 $ (upper panels), and different DM annihilation mechanisms (lower panels). We used the rescaled SI/SD DM-nucleon cross-section and rescaled DM pair annihilation rate because the LSP may not constitute all of the DM. We hypothesize that the higgsino does not necessarily account for all of the observed DM relic density. This consideration allows for the higgsino to function either as the sole DM component or as part of a broader DM composition, possibly through an extension of the model or alternative freeze-out/freeze-in mechanisms. The following conclusions can be drawn from this figure:

Figure 4.  (color online) Surviving samples are displayed across the planes of the rescaled spin-independent DM-nucleon cross-section $ \sigma_{\rm{SI}} \times \Omega/\Omega_0 $ (left), rescaled spin-dependent DM-nucleon cross-section $ \sigma_{\rm{SD}} \times \Omega/\Omega_0 $ (middle), and rescaled thermally averaged DM annihilation cross-section times velocity $ \langle \sigma v \rangle \times (\Omega / \Omega_0)^2 $ (right) versus LSP mass $ m_{\tilde{\chi}^{0}_1} $, where $ \Omega_0 = 0.1187 $. In the upper panels, colors indicate the DM relic density $ \Omega h^2 $. In the lower panels, colors represent different DM annihilation mechanisms, as in Fig. 3. Samples with higher DM relic densities $ \Omega h^2 $ are plotted above those with lower values in the upper panels. In the left panels, the orange, blue, green, and black solid curves indicate the spin-independent (SI) DM-nucleon cross-section detection limits of LUX (2017) [107], XENON1T (2018) [108], PandaX-4T (2021) [90], and LZ (2022) [91], respectively. The black dotted, dashed, and dot-dashed curves indicate the future detection limits of XENONnT (20t.y) [109], LUX-ZEPLIN (LZ) projected [110], and PandaX-xT (200t.y) [111], respectively. The neutrino floor [112] is indicated by the orange shaded region. In the middle panels, the orange, cyan, blue, green, and black solid curves indicate the spin-dependent (SD) DM-nucleon cross-section detection limits of LUX (2017) [113], XENON1T (2019) [114], XENONnT (2023) [115], PandaX-4T (2022) [89], and LZ (2022) [91], respectively. The black dotted, dashed, and dot-dashed curves indicate the future detection limits of XENONnT (20t.y) [109], LZ projected [110], and PandaX-xT (200t.y) [111], respectively. In the right panels, the orange and green solid curves indicate the upper limits on the pair-annihilation rate from the Milky Way dSphs based on six years of Fermi Large Area Telescope data [116] and 112 hours of observations of the Galactic Center with HESS [117], respectively. The black dashed curves indicate the projected sensitivity from the CTA on the pair-annihilation rate [118].

● In the upper left and middle panels, samples more easily escape SD DM-nucleon cross-section constraints, whereas the SI DM-nucleon cross-section constraint is significantly stricter. This is because the higgsino-dominated LSP has a very small higgsino asymmetry, denoted as $ |N_{13}^2 - N_{14}^2| $. Given that the SD cross-section is proportional to the square of this asymmetry [66], it remains small, enabling samples to easily escape experimental SD constraints.

● Concerning the upper left panels, future experiments such as XENONnT (20t.y) and LUX-ZEPLIN (LZ) (1000 day) are expected to cover all samples with the correct DM relic density, while the upcoming PandaX-xT (200t.y) will cover nearly all samples, approaching the neutrino floor. In the lower left panels, XENONnT (20t.y) and LZ (1000 days) are expected to cover nearly all samples with $ \tilde{\tau}_1 $ coannihilation, and the upcoming PandaX-xT (200t.y) will cover different Higgs funnel annihilations. Only samples with insufficient DM relic density are likely to escape future experimental constraints.

● Regarding the middle panels, future SD DM-nucleon cross-section constraints from XENONnT (20t.y), LZ (1000 days), and PandaX-xT (200t.y) will only cover a small fraction of samples with insufficient DM relic density, including parts of the $ \tilde{\chi}_1^{\pm} $ coannihilation (only) and $ H_S $ and $ A_S $ funnel annihilation samples. With respect to the right panels, future DM annihilation rate constraints from the CTA projections will include some samples with sufficient DM relic density, including parts of the $ A_S $ and $ H_D/A_D $ funnel annihilation samples. The CTA will be the world's most sensitive gamma-ray observatory, covering photon energies from tens of GeV to over 300 TeV with very high angular resolution [119]. It is particularly sensitive to the `thermal' annihilation cross-section, which is crucial for explaining the observed DM abundance in standard WIMP models.

In Fig. 5, surviving samples are displayed on the planes of the mass difference between NLSP and LSP, $ \Delta m ({\rm{NLSP}}, {\rm{LSP}}) $, versus the NLSP mass $ m_{\rm{NLSP}} $. Here, higgsino-dominated charginos and next-to-lightest neutralinos, serving as NLSP, have nearly equal masses, $ m_{\rm{NLSP}} = m_{\tilde{\chi}2^{0}} \approx m{\tilde{\chi}_1^{\pm}} $, indicating a compressed EWkino spectrum. Detection limits for higgsino-dominated electroweak processes by future detectors are derived from Fig. 8.10 in Ref. [120]. For HL-LHC, HE-LHC, and FCC-hh, the constraints are based on a search involving two soft leptons in the production of $ \tilde{\chi}_2^{0} \tilde{\chi}_1^{\pm} $, $ \tilde{\chi}_2^{0} \tilde{\chi}_1^{0} $, and $ \tilde{\chi}_1^{\pm} \tilde{\chi}_1^{\pm} $ [121]. For the International Linear Collider (ILC), the constraints come from searches for $ \tilde{\chi}_1^{\pm} $ through disappearing track analyses in a higgsino-dominated LSP scenario [122]. For a very close mass difference $ \Delta m ({\rm{NLSP}}, {\rm{LSP}}) < $ 1 GeV, monojet searches have been considered at the FCC-hh, HE-LHC, and HL-LHC [123, 124]. Several conclusions can be drawn from this figure.

Figure 5.  (color online) Surviving samples are displayed in the planes of mass difference between NLSP and LSP, $ \Delta m ({\rm{NLSP}}, {\rm{LSP}}) $, versus NLSP mass $ m_{\rm{NLSP}} $, where higgsino-dominated charginos and next-to-lightest neutralinos as NLSP have nearly identical masses, $ m_{\tilde{\chi}_2^{0}}\approx m_{\tilde{\chi}_1^{\pm}} $. In the left panel, colors represent the DM relic density $ \Omega h^2 $, with samples having larger values plotted above those with smaller values. In the right panel, colors represent various DM annihilation mechanisms. Detection limits for higgsino-dominated electroweak processes by future detectors are taken from Fig. 8.10 in Ref. [120]. The blue and red curves represent detection limits at the HL-LHC with $ 3 {\; {\rm{ab}}}^{-1} $ at 14 TeV using soft leptons; red and magenta dotted curves depict the HE-LHC with $ 15 {\; {\rm{ab}}}^{-1} $ at 27 TeV and the FCC-hh with $ 30 {\; {\rm{ab}}}^{-1} $ at 100 TeV [121], respectively. The light green solid and dot-dashed curves refer to the ILC at $ 0.5 {\; {\rm{ab}}}^{-1} $and 500 GeV, and $ 1 {\; {\rm{ab}}}^{-1} $and 1000 GeV [122], respectively. The gray, light blue, and green dot-dashed curves refer to the Compact Linear Collider (CLIC) at $ 1 {\; {\rm{ab}}}^{-1} $and 380 GeV, $ 2.5 {\; {\rm{ab}}}^{-1} $and 1500 GeV, and $ 5 {\; {\rm{ab}}}^{-1} $and 3000 GeV [120], respectively. The black, blue, and light green slash-hatched regions indicate mono-jet searches at the FCC-hh, HE-LHC, and HL-LHC [123, 124], respectively.

● Regarding the left panel, samples with insufficient DM relic density and NLSP mass below 350 GeV can be probed at HL-LHC for mass splittings $ \Delta m ({\rm{NLSP}}, {\rm{LSP}}) $ between 1.6 and 50 GeV. The NLSP detection limit at HE-LHC is approximately1.5 times higher than that at HL-LHC, while FCC-hh projections show NLSP detection limits up to 1 TeV, with sensitivity dependent on mass splittings $ \Delta m ({\rm{NLSP}}, {\rm{LSP}}) $. The sensitivity of lepton colliders is independent of $ \Delta m ({\rm{NLSP}}, {\rm{LSP}}) $.$ {\rm{ILC}}_{500} $ and $ {\rm{ILC}}_{1000} $ can detect NLSP masses up to 250 GeV and 500 GeV, respectively, while $ {\rm{CLIC}}_{1500} $ and $ {\rm{CLIC}}_{3000} $ can reach 650 GeV and 1300 GeV. Monojet searches at hadron colliders can extend the search for small $ \Delta m ({\rm{NLSP}}, {\rm{LSP}}) $ scenarios. For $ \Delta m ({\rm{NLSP}}, {\rm{LSP}}) < 1 \text{GeV} $, the higgsino monojet search could reach an NLSP mass of 200 GeV, 490 GeV, and 1400 GeV at the 14 TeV HL-LHC, 27 TeV HE-LHC, and 100 TeV FCC-hh colliders, respectively.

● Concerning the right panel, samples with the Higgs funnel annihilation mechanism are difficult to probe with hadron colliders and require lepton colliders. The lepton colliders $ {\rm{ILC}}_{500} $ and $ {\rm{ILC}}_{1000} $ can cover some samples with $ H_S $ and $ A_S $ funnel annihilation mechanisms, while $ {\rm{CLIC}}_{3000} $ can cover some samples with $ H_D/A_D $ funnel annihilation mechanisms, as well as some samples with $ \tilde{\tau}_{1} $ coannihilation.

Table 1 presents five benchmark points detailing the DM sector, where the relative contributions of channels to $ \Omega h^2 $ represent the percentage contributions of different DM annihilation channels to the DM relic density. Points 1 and 2 belong to the sole $ \tilde{\chi}_1^{\pm} $ coannihilation; note that a smaller LSP mass leads to an insufficient DM relic density in these cases. Points 3 and 4 are associated with Higgs funnel annihilation combined with $ \tilde{\chi}_1^{\pm} $ coannihilation, showing that $ \tilde{\chi}_1^{\pm} $ coannihilation still predominantly contributes to the DM relic density. Point 5 belongs to the $ \tilde{\tau}_1 $ coannihilation combined with $ \tilde{\chi}_1^{\pm} $ coannihilation, where $ \tilde{\chi}_1^{\pm} $ coannihilation remains the main contributor and $ \tilde{\tau}_1 $ coannihilation contributes only approximately 1.7%.

	P1	P2	P3	P4	P5
$ \lambda $	$ 1.64 \times 10^{-4} $	$ 2.52 \times 10^{-3} $	$ 3.98 \times 10^{-1} $	$ 9.58 \times 10^{-2} $	$ 1.58 \times 10^{-1} $
$ \kappa $	0.26	0.21	0.54	−0.19	0.54
$ \tan\beta $	5.7	10.5	7.1	26.1	30.6
$ \mu $/GeV	302	1103	1348	3800	823
$ M_0 $/GeV	6820	9303	4917	6156	218
$ M_{12} $/GeV	8024	9964	6983	9141	6765
$ A_0 $/GeV	−5023	−2643	6675	−3336	−2497
$ A_\lambda $/GeV	9900	−4528	7246	7035	−4788
$ A_\kappa $/GeV	−3427	−8386	−6337	3064	−7811
$ m_{h_1}$/GeV	125	126	124	126	126
$ m_{h_2}$/GeV	28555	31225	2803	5590	4044
$ m_{h_3} $/GeV	939301	183648	7865	14031	4815
$ m_{a_1} $/GeV	28556	31225	4184	5589	4044
$ m_{a_2} $/GeV	65007	46134	7865	7751	4915
$ \tilde{\chi}^{0}_1 $/GeV	316	1145	1385	3864	849
$ \tilde{\chi}^{0}_2 $/GeV	318	1146	1389	3868	851
$ \tilde{\chi}^{0}_3 $/GeV	3728	4689	3228	4288	3110
$ \tilde{\chi}^{0}_4 $/GeV	6767	8445	3784	7681	5607
$ \tilde{\chi}^{0}_5 $/GeV	1002573	193954	5842	14999	5753
$ \tilde{\chi}^{\pm}_1 $/GeV	317	1145	1387	3867	850
$ \tilde{\chi}^{\pm}_2 $/GeV	6767	8445	5842	7681	5607
$ \tilde{t}_1 $/GeV	9214	12772	9658	12007	8338
$ \tilde{\tau}_1 $/GeV	7332	9991	5365	6118	857
$ \Omega h^2 $	0.01	0.12	0.07	0.12	0.09
$\langle \sigma v \rangle /({\rm{cm} }^3 {\rm{s} }^{-1})$	$ 9.67 \times 10^{-26} $	$ 7.78 \times 10^{-27} $	$ 5.25 \times 10^{-27} $	$ 3.07 \times 10^{-26} $	$ 1.41 \times 10^{-26} $
$\sigma_{\rm{SI} } /{\rm{cm} }^2$	$ 4.35 \times 10^{-47} $	$ 2.71 \times 10^{-47} $	$ 2.42 \times 10^{-46} $	$ 4.12 \times 10^{-46} $	$ 6.18 \times 10^{-47} $
$\sigma_{\rm{SD} } /{\rm{cm} }^2$	$ 1.71 \times 10^{-43} $	$ 1.20 \times 10^{-44} $	$ 8.26 \times 10^{-46} $	$ 2.02 \times 10^{-44} $	$ 4.62 \times 10^{-44} $
Relative Contributions of Channels to $\Omega h^2$	$\tilde{\chi}_1^+ \tilde{\chi}^0_1 \to \text{SM 36.2}$% $\tilde{\chi}_1^+ \tilde{\chi}^0_2 \to \text{SM 23.5}$% $\tilde{\chi}_1^+ \tilde{\chi}_1^- \to \text{SM 19.7}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_2 \to \text{SM 15.2}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_1 \to \text{SM 4.0}$% $\tilde{\chi}^0_2 \tilde{\chi}^0_2 \to \text{SM 1.4}$%	$\tilde{\chi}_1^+ \tilde{\chi}^0_1 \to \text{SM 31.7}$% $\tilde{\chi}_1^+ \tilde{\chi}^0_2 \to \text{SM 27.4}$% $\tilde{\chi}_1^+ \tilde{\chi}_1^- \to \text{SM 20.1}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_2 \to \text{SM 15.7}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_1 \to \text{SM 2.9}$% $\tilde{\chi}^0_2 \tilde{\chi}^0_2 \to \text{SM 2.0}$%	$\tilde{\chi}_1^+ \tilde{\chi}_1^- \to \text{SM 38.5}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_1 \to \text{SM 19.6}$% $\tilde{\chi}^0_2 \tilde{\chi}^0_2 \to \text{SM 13.5}$% $\tilde{\chi}_1^+ \tilde{\chi}^0_1 \to \text{SM 12.7}$% $\tilde{\chi}_1^+ \tilde{\chi}^0_2 \to \text{SM 9.9}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_2 \to \text{SM 6.0}$%	$\tilde{\chi}_1^+ \tilde{\chi}_1^- \to \text{SM 45.0}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_1 \to \text{SM 28.2}$% $\tilde{\chi}^0_2 \tilde{\chi}^0_2 \to \text{SM 19.9}$% $\tilde{\chi}_1^+ \tilde{\chi}^0_1 \to \text{SM 2.2}$% $\tilde{\chi}_1^+ \tilde{\chi}^0_2 \to \text{SM 1.8}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_2 \to \text{SM 0.8}$%	$\tilde{\chi}_1^+ \tilde{\chi}^0_1 \to \text{SM 32.6}$% $\tilde{\chi}_1^+ \tilde{\chi}^0_2 \to \text{SM 24.8}$% $\tilde{\chi}_1^+ \tilde{\chi}_1^- \to \text{SM 19.5}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_2 \to \text{SM 15.3}$% $\tilde{\chi}^0_1 \tilde{\chi}^0_1 \to \text{SM 3.8}$% $\tilde{\chi}^0_2 \tilde{\chi}^0_2 \to \text{SM 1.7}$% $\tilde{\chi}_1^+ \tilde{\tau}_1 \to \text{SM 0.8}$% $\tilde{\chi}^0_1 \tilde{\tau}_1 \to \text{SM 0.4}$% $\tilde{\chi}^0_2 \tilde{\tau}_1 \to \text{SM 0.3}$% $\tilde{\tau}_1 \bar{\tilde{\tau}}_1 \to \text{SM 0.1}$% $\tilde{\tau}_1 \tilde{\tau}_1 \to \text{SM 0.1}$%

Table 1.  Five benchmark points for surviving samples, where the relative contributions of channels to $ \Omega h^2 $ represent the percentage contributions of different DM annihilation channels to the DM relic density.

In this study, we explored the higgsino-dominated DM relic density, annihilation mechanism, and detectability within the scNMSSM, covering a mass range from 100 GeV to over 4 TeV. We started by scanning the parameter space with $\textsf{NMSSMTools}$, taking into account both theoretical and experimental constraints, such as vacuum stability, Higgs boson data, SUSY searches at the LHC, DM relic density, and direct and indirect DM searches. To ensure that the higgsino components are predominant in the LSP, we imposed the condition $ |N_{13}|^2 + |N_{14}|^2 > 0.5 $, confirming that all samples contain higgsino-dominated LSPs suitable as DM candidates. We also confirmed that $ h_1 $ is consistent with the 125 GeV SM-like Higgs, and the LSP mass is between 100 GeV and 4 TeV.

Furthermore, we computed the DM relic density and explored various annihilation mechanisms. We assessed the potential for both direct and indirect detections using existing experimental limits. We considered the prospects for detection at future colliders. In this study, we also investigated the characteristics and annihilation mechanisms of the lightest supersymmetric particle (LSP) and the next-to-lightest supersymmetric particle (NLSP) within a higgsino-dominated framework. Our findings can be summarized as follows:

● Characterization of LSP and NLSP: The mass of the higgsino-dominated LSP is closely related to the $ \mu $ parameter, and the NLSP is typically either another higgsino-dominated neutralino, $ \tilde{\chi}^{0}_2 $, or a higgsino-dominated chargino, $ \tilde{\chi}^{\pm}_1 $. Given their higgsino domination, the NLSP mass similarly approximates the $ \mu $ value.

● Annihilation mechanisms: The primary annihilation mechanism for the higgsino-dominated LSP involves coannihilation with its NLSP counterpart,$ \tilde{\chi}^{\pm}_1 $. This study further identified several annihilation pathways, including Higgs funnel annihilation mechanisms and $ \tilde{\tau}_1 $ coannihilation processes. We categorized the particle samples into three main categories based on their respective DM annihilation mechanisms:

– Sole $ \tilde{\chi}_1^{\pm} $ coannihilation;

– Higgs funnel annihilation mechanisms involving $ H_S $, $ A_S $, and $ H_D/A_D $, combined with $ \tilde{\chi}_1^{\pm} $ coannihilation hybrids;

– $ \tilde{\tau}_1 $ coannihilation, which may or may not include Higgs funnel annihilation mechanisms, alongside $ \tilde{\chi}_1^{\pm} $ coannihilation.

• DM relic density and detection: All examined samples satisfy the $ \tilde{\chi}_1^{\pm} $ coannihilation criteria. The efficiency of the coannihilation mechanism decreases with increasing LSP mass, affecting the DM relic density. Particularly, samples with a larger $ \mu $ (heavier LSP) exhibit lower probabilities of achieving the correct relic density without additional funnel annihilation mechanisms. The higgsino asymmetry ($ |N{13}^2 - N_{14}^2| $) significantly influences the spin-dependent (SD) DM-nucleon cross-section, enabling many samples to evade experimental SD constraints.

• Future experimental prospects: Upcoming experiments, such as XENONnT, LZ (1000 days), and PandaX-xT, aim to cover nearly all samples that reach the correct DM relic density ($ \mu \gtrsim 1300 $ GeV) through SI DM direct detection. In parallel, the Cherenkov Telescope Array (CTA) is expected to further scrutinize DM annihilation rates for indirect detection, particularly through $ A_S $ and $ H_D/A_D $ funnel mechanisms. Additionally, the detection capabilities at lepton colliders surpass those at hadron colliders. The future ILC and CLIC are expected to detect samples with higher mass NLSPs effectively. Notably, the CLIC operating at 3000 GeV could probe all samples with insufficient DM relic density ($ \mu \lesssim 1300 $ GeV).