Current bounds and future prospects of light neutralino dark matter in the NMSSM

Unlike its minimal counterpart, the next to minimal supersymmetric Standard Model (NMSSM) allows the possibility that the lightest neutralino could have a mass as small as ∼ 1 GeV while still providing a significant component of relic dark matter (DM). Such a neutralino can provide an invisible decay mode to the Higgs as well. Further, the observed SM-like Higgs boson ( H 125 ) could also have an invisible branching fraction as high as ∼ 19% . Led by these facts, we first delineate the region of parameter space of the NMSSM with a light neutralino ( M ˜ χ 01 < 62 . 5 GeV) that yields a thermal neutralino relic density smaller than the measured relic density of cold dark matter, and is also compatible with constraints from collider searches, searches for dark matter, and from flavor physics. We then examine the prospects for probing the NMSSM with a light neutralino via direct DM detection searches, via invisible Higgs boson width experiments at future e þ e − colliders, via searches for a light singlet Higgs boson in 2 b 2 μ , 2 b 2 τ and 2 μ 2 τ channels and via pair production of winos or doublet Higgsinos at the high luminosity LHC and its proposed energy upgrade. For this last-mentioned electroweakino search, we perform a detailed analysis to map out the projected reach in the 3 l þ =E T channel, assuming that chargino decays to W ˜ χ 01 and the neutralino(s) decay to Z or H 125 þ ˜ χ 01 . We find that the HL-LHC can discover SUSY in just part of the parameter space in each of these channels, which together can probe almost the entire parameter space. The HE-LHC probes essentially the entire region with Higgsinos (winos) lighter than 1 TeV (2 TeV) independently of how the neutralinos decay, and leads to significantly larger signal rates.


I. INTRODUCTION
The LHC Run-II has ushered in a new era in terms of energy, luminosity and discovery potential. At the end of the Run-II with about 140 fb −1 of data collected at 13 TeV, the status of the Higgs as the last frontier to be conquered as concerns the Standard Model has been firmly established [1]. As yet, there is no unambiguous evidence for any physics beyond the Standard Model (BSM). In case of the primary candidate for BSM physics viz TeV scale supersymmetry (SUSY), large swaths of parameter space have been ruled out. However the LHC 13 TeV still holds promise of further exploring the BSM landscape including SUSY. In particular, the electroweak sector of SUSY is still largely unconstrained. A better coverage of this sector is expected with larger integrated luminosities, as production of electroweakinos not only suffers from weak production cross sections but limits from the current searches can also be evaded as an intricate combination of parameters can lead to compressed spectra, see for example [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Compressed Higgsinos are expected in models of natural SUSY [17,18]. Experiments at the LHC have also begun to probe such scenarios [19][20][21]. Moreover, the electroweak sector contains one of the most promising dark matter (DM) candidate, the lightest neutralino. It is imperative that we continue to probe the complementarity between LHC searches for SUSY and astrophysical observations as far as supersymmetric dark matter is concerned [22,23]. A study of complementarity between future direct detection experiments and future collider searches for the case of a light neutralino DM (M˜χ0 1 ≤ 62.5 GeV), both thermal and nonthermal, in the MSSM framework was carried out in [24]. Results in [24] indicated that the MSSM parameter space with a light neutralino DM and with a correct or underabundant relic density could be entirely probed via the future dark matter experiments.
This then naturally motivates us to examine the extent to which the considerations of [24] would be altered in the simplest extension of the MSSM, the next-to-minimal supersymmetric Standard Model (NMSSM) [25,26]. As in [24] we focus on M˜χ0 1 ≤ 62.5 GeV, that can potentially contribute to invisible decays of the Higgs boson discovered at CERN [27][28][29][30], hereafter referred to as H 125 . In the MSSM with heavy sfermions, the current limit for light neutralinos after taking into account various cosmological, astrophysics and collider constraints lies around 30 GeV [24,[31][32][33][34][35][36][37]. This arises mainly from a combination of the direct detection constraint which requires a weak coupling of the DM to the Higgs and of the relic density constraint which requires that the neutralino annihilates efficiently through a mediator that is nearly on-shell. Within the NMSSM, much lighter neutralinos, as light as 1 GeV, can satisfy all current constraints [38][39][40][41][42][43][44][45]. This is because additional scalars present in the NMSSM may provide an efficient annihilation mechanism for light neutralinos in the early universe [46][47][48]. Morever, the singlet component of these new scalars entails that they can evade detection at the LHC even if their masses are below that of the standard model-like Higgs [49][50][51][52][53][54][55][56]. In this article, assuming a standard cosmological scenario, we perform a comprehensive exploration of the light neutralino as a thermal cold dark matter candidate in the NMSSM by imposing current low energy, collider and astrophysical constraints. These include flavor physics constraints, LEP bounds, recent results from LHC measurements of the Higgs sector including searches for light Higgses [51,[57][58][59] and measurements of the Higgs signal strengths, direct chargino/neutralino searches at the LHC [60][61][62] as well as limits from direct detection experiments in spin-independent neutralino-nucleon scattering (Xenon-1T [63]) and from indirect detection of DM in the photon channel (Fermi-LAT [64]). We concentrate on the electroweakino sector and assume that squarks and sleptons are heavy and do not play an important role in DM or collider observables. After having established the region in parameter space compatible with current experimental results, we assess the impact of future searches. We reemphasize that for the (thermally produced) neutralino LSP (lightest SUSY particle) to not produce too much dark matter in the early universe, we require a light spin-zero particle close to twice the LSP mass. Correspondingly, we choose parameter space points which also feature M A 1 and M H 1 below 122 GeV. We focus on three specific directions that we find more promising for discovery: (1) multiton direct detection experiments, Specifically, we assess the reach of direct electroweakino searches in the WZ mediated and WH 125 mediated 3l þ = E T search channel, at HL-LHC as well as HE-LHC. The impact of the projected search limits from direct light Higgs searches in the 2b2τ, 2b2μ, 2μ2τ and 4μ channels at the future upgrades of LHC are also examined [65]. Finally we emphasize the complementarity of future measurements of an invisible branching ratio for the Higgs, searches for Higgs and electroweakino at colliders, and direct and indirect detection experiments for discovering or probing the light neutralino in the NMSSM. The rest of the paper is organized as follows. In Sec. II, we present the NMSSM framework and discuss the parameter space of interest in Sec. III. The relevant constraints and their implications are described in Sec. IV. The characteristic features of the allowed parameter space region obtained after imposing the current constraints are discussed in Sec. V. In Sec. VI, we investigate the future reach of Xenon-nT (through spin-independent WIMPnucleon interactions), the future prospects for ILC and CEPC (through Higgs invisible width measurements) and the scope of direct light Higgs searches at future runs of LHC, and study their implications on the allowed parameter space. In Sec. VII, we explore the reach via searches for direct production of electroweakinos in the 3l þ = E T final state at the HL-LHC and the HE-LHC. We conclude in Sec. VIII.

II. THE NMSSM FRAMEWORK
The NMSSM Higgs sector consists of two doublet Higgs superfields,Ĥ u andĤ d , and a Higgs singlet superfieldŜ. The scale invariant NMSSM superpotential has the form [66] where W MSSM (μ ¼ 0) refers to the MSSM superpotential without the μ-term, while λ and κ are dimensionless parameters, andĤ u ·Ĥ d ¼Ĥ þ The soft SUSY breaking terms which contain the Higgs scalar fields are where A κ and A λ are the trilinear soft-breaking parameters, while m H d , m H u and m S are the soft breaking Higgs masses. An effective μ-term with μ ¼ λv s is generated when S develops a vacuum expectation value. The F-and D-terms also contribute to the Higgs scalar potential, and are given by where, g 1 and g 2 are the Uð1Þ Y and SUð2Þ L gauge couplings of the SM. Expanding the scalar potential, V soft þ V F þ V D , around the real neutral vacuum expectation values (vevs), v u , v d and v s of H u , H d and S, respectively, gives the physical Higgs states. Following the notation of [67], and H I d , and a second pseudoscalar Higgs is obtained from A S . At tree-level, the symmetric squared mass matrix (M 2 s ) of the neutral scalar Higgs bosons in the Higgs interaction basis fH NSM ; H SM ; H S g 1 is given by [66][67][68] Here, β is defined as tan −1 v u v d , m Z represents the Z boson mass and v ¼ the elements of the symmetric 2 × 2 pseudoscalar Higgs squared mass matrix (M 2 p ) in the interaction basis fA NSM ; A S g 2 can be written as follows [66][67][68]: 0 The Higgs mass eigenstates are defined in terms of the Higgs interaction basis where, s ik and p jk are obtained by diagonalizing the corresponding mass squared matrices. In addition to the 3 CP-even neutral Higgs states, H 1 , H 2 , H 3 , one of which is identified with H 125 (H 1 being the lightest and H 3 being the heaviest), and, 2 CP-odd neutral Higgs bosons, A 1 , A 2 (A 1 being the lighter one), the NMSSM Higgs spectrum also contains two charged Higgs boson. The tree-level mass of the charged Higgs bosons is given by where, M W represents the mass of W boson.
It can be observed from Eq. (6)-(11) that the tree-level Higgs sector of the NMSSM can be parametrized by 6 input parameters: Compared with the MSSM, the electroweakino sector of the NMSSM is phenomenologically richer and contains 5 neutralinos and 2 charginos. The neutralino mass matrix, in the basis of fB;W 3 ;H 0 d ;H 0 u ;Sg (B: bino,W 3 : neutral wino, H 0 d andH 0 u : neutral Higgsinos,S: singlino) is given by (following the notation of [67]) where, M 1 is the bino mass parameter, M 2 is the wino mass parameter, m Z is the mass of the Z boson and θ W is the Weinberg angle. The neutralino mass eigenstates are given bỹ where, the N ij 's are obtained by diagonalizing the neutralino mass matrix in Eq. (13). It follows from Eq. (13) that the neutralino sector at tree level is governed by the following input parameters: It must be noted that the phenomenology of the electroweakino and Higgs sectors is modified from that in the MSSM due to additional parameters λ; κ; A λ ; A κ .
Standard big bang cosmology with thermally produced neutralinos with M˜χ0 1 ≲ 62.5 GeV as the cold dark matter results in too large a density of cold dark matter, unless resonance enhancements of annihilation are operative in the early universe. The Z boson and the SM-like Higgs boson enables the LSP neutralino to generate a correct or underabundant relic density in the M˜χ0 1 ∼ M Z =2 and M˜χ0 1 ∼ M H 125 =2 regions. At lower LSP masses (≲M Z =2), efficient annihilation can be achieved via exchange of a light scalar Higgs (H 1 ) or a pseudoscalar Higgs (A 1 ), with mass, M A 1 =H 1 ∼ 2M˜χ0 1 .

III. SCANNING THE NMSSM PARAMETER SPACE
The first step of our analysis is to delineate the NMSSM parameter space region which is compatible with current constraints from collider and dark matter searches. A detailed discussion of constraints is found in Sec. IV. Note that a lightχ 0 1 is completely consistent with all the current data in the NMSSM unlike in the MSSM and we intend to focus on the case of a light neutralino LSP with mass M˜χ0 1 < 62.5 GeV. We find that there is a significant region with M˜χ0 1 < 62.5 GeV where the invisible decay of H 125 into aχ 0 1 pair is kinematically allowed. We zero in on this region and also explore prospects for invisible Higgs boson searches at future facilities.
The main focus of this study involves the exploration of the Higgs and electroweakino sectors of the NMSSM parameter space. The relevant input parameters which captures the physics of these two sectors are: λ, κ, A λ , A κ , tan β, μ, M 1 , M 2 [from Eq. (12) and (15)], M 3 : the gluino mass parameter, A t , A t , A τ : the stop, sbottom and stau trilinear coupling, and M U 3 R , M D 3 R , M Q 3 L : the mass of the third generation squarks. The first and second generation squark masses, and the slepton masses are fixed at 3 TeV.
The scenario with λ, κ ≈ 0 is referred to as the effective MSSM [25]. From Eq. (1), it becomes clear that the singlet superfieldŜ does not interact with the MSSM Higgs superfields,Ĥ u andĤ d , when λ → 0. In the limit of effective MSSM, the singlet scalar, the singlet pseudoscalar and the singlinolike neutralino does not couple with the MSSM sector. Thus, it is not possible to distinguish between the λ, κ → 0 limit (with λhsi fixed) of the NMSSM and the MSSM scenario. It is however possible for the singlinolike neutralino to be the LSP with mass ∼2κhsi. In order to allow the interaction between the singlet sector and the MSSM sector, it is essential to have a nonzero λ. For the case of a singlinolike LSP and λ ≪ 1, the MSSM-like NLSPs would undergo cascade decay into singlinolike LSP þ SM particles with a long lifetime which could also lead to potential LLP (long-lived particles) signatures, the study of which is beyond the scope of this work. Therefore, in this article, we focus on the regions of parameter space where λ and κ are not ≈0 or ≪ 1.
The NMSSMTools-5.3.1 [46,[69][70][71][72][73] package is used to generate the particle spectrum, and to compute the couplings and branching fraction of the Higgses and the branching ratios of the SUSY particles. The presence of a large number of input parameters makes it difficult to find parameter space points in the M˜χ0 1 ≲ 62.5 GeV region and it becomes essential to choose an optimized scan range for the input parameters. Initially, a random scan of the parameter space is performed using the NMSSMTools-5.3.1 package, for a wide range of input parameters. In the next step, the parameter space points, thus generated, are checked against the theoretical, collider and astrophysical constraints, implemented within NMSSMTools-5.3.1. Parameter space points are then chosen from the previous step to be used as seeds for the Markov Chain Monte Carlo (mcmc) scanning technique implemented in the NMSSMTools-5.3.1 package. The parameter space points generated from the mcmc scan are distributed over the following range of input parameters: 0.01 < λ < 0.7; 10 −5 < κ < 0.05; 3 < tan β < 40 100 GeV < μ < 1 TeV; 1.5 TeV < M 3 < 10 TeV 2 TeV < A λ < 10.5 TeV; −150 GeV < A κ < 100 GeV The collider and astrophysical observables that constrain the points generated from this scan are detailed in the next section.

IV. CONSTRAINTS
As we have just mentioned, our demand that the singlino is lighter than 62.5 GeV, along with the chosen range of input parameters, leads to H 1 and A 1 lighter than 125 GeV, so that H 2 then plays the role of the observed SM-like Higgs boson. The SM-like Higgs boson is required to lie within the mass range allowed by the measurements at the LHC and its couplings must be compatible with those measured at the LHC. The NMSSM parameter space considered in our study is further constrained by the low-energy flavor physics limits, LEP limits, searches of directly produced light Higgs bosons and measurements of the Higgs boson signal strengths at the LHC, gluino searches, direct searches of electroweakinos in the 3l þ = E T and the l þ l − þ = E T final state as well as from direct and indirect DM searches. These constraints are discussed in more detail below.
(i) Mass of SM-like Higgs boson: The combined measurement by the ATLAS and CMS collaborations has determined the Higgs boson mass to lie within 124.4-125.8 GeV (3σ) [1]. Adopting a conservative approach, we require H 2 (we will henceforth also refer to this as H 125 ) to be within the interval 122-128 GeV, to allow for theoretical uncertainties in the Higgs boson mass computation [74][75][76]. (ii) Limits from LEP: Measurements at LEP have excluded a chargino with mass below M˜χAE 1 ≲ 103.5 GeV [77]. This constraint, together with our choice M 1 ¼ 2 TeV [see Eq. (16)] requires that the LSP below 62.5 GeV is dominantly singlino, with only small Higgsino, wino and bino admixture. We also impose the upper limit on the production cross section ofχ 0 1χ 0 GeV as well as upper limits on e þ e − → ZH j and e þ e − → A i H j processes in various final states. The NMSSMTools-5.3.1 framework was used to implement these constraints. (iii) Upper bound on relic density: Results from the PLANCK Collaboration have put the DM relic density at Ω obs: DM h 2 ¼ 0.120 AE 0.001 [78], and assuming a 2σ window, the relic density can fall within an interval of 0.118-0.122. micrOMEGAs [79,80] is used to compute the relic density of the LSP neutralino in the standard cosmological scenario, and adopting a conservative approach, we only exclude over-abundant DM, that is we require Ω obs: This constraint would not apply if DM production entails nonstandard cosmology [81][82][83]. In this work, the scenario of nonstandard cosmology has not been considered. (iv) Flavor physics constraints: Constraints from the measurement of flavor physics observables offer sensitive probes of new physics scenarios. We impose the flavor physics constraints through bounds on the branching fraction of the rare decay modes B → X s γ, B s → μ þ μ − and B þ → τ þ ν τ . Recent measurements obtain BrðB → X s γÞ ¼ ð3.32 AE 0.16Þ × 10 −4 [84], BrðB s → μ þ μ − Þ ¼ ð3.0 AE 0.6 þ0.3 −0.2 Þ × 10 −9 [85] and BrðB þ → τ þ ν τ Þ ¼ ð1.06 AE 0.19Þ × 10 −4 [84]. In the current study, we use micrOMEGAs-5.0.6 [79,86,87] to compute these branching fractions and we allow 2σ experimental uncertainty. Note that we do not use the NMSSMTools-5.3.1 framework to implement the aforementioned Bphysics constraints, rather, we impose the more recent bounds on the flavor physics observables discussed in this section. Additionally, the constraints on ϒð1sÞ → H=Aγ, ΔM s and ΔM d are also imposed using the  The total invisible branching fraction of the SM like Higgs can also be probed by directly searching for the invisibly decaying Higgs boson through its production in association with a vector boson [89,90] and jets [91] or vector boson fusion [92]. We have imposed upper limits obtained from such studies. The ATLAS Collaboration has also used LHC Run-II data (L ∼ 140 fb −1 ) to probe the invisibly decaying Higgs, produced via VBF mode, and have set an upper limit of 13% [93]. We must note that Ref. [93] is a preprint and the published result from the ATLAS collaboration used LHC Run-I data, considered Higgs production via WH 125 , ZH 125 and VBF modes, and have set an upper limit of 25% [94]. Similar search by the CMS Collaboration, using the entire Run-I data and 35.9 fb −1 of the Run-II data, have derived an upper limit at 19% [95]. The prospects of probing the invisibly decaying Higgs boson at the future LHC has also been studied, see for example Refs. [65,96]. In the context of our analysis, the invisible decay modes of the SM-like Higgs (H 125 ) are: 1 . We impose an upper limit of 13% on the sum of these branching fractions. We note that the Higgs signal strength constraints also impose an indirect upper limit on the invisible branching fraction of the Higgs boson. The indirect limit can be comparable or, at times, even stronger than the direct upper limit, viz see Ref. [24]. In our case, we find the indirect upper limit to be nearly comparable with the latest direct upper limit from ATLAS [93] and our results do not change upon increasing the direct upper limit on the invisible branching fraction of the Higgs boson from 13% to the published limit of 19%. , and, therefore, the assumption of Brðχ 0 2 → χ 0 1 ZÞ ∼ 100% does not always hold true. As a result, the wino exclusion limit derived in [62] cannot be directly applied to the parameter space of interest. The limits derived in [62] have been translated to the case of Higgsino-like NLSP's (M 2 > μ) in [44], where the direct production of mass-degenerate Higgsino-likeχ 0 2χ1 AE andχ 0 3χ1 AE , is considered. This translation procedure enables recasting of the electroweakino search limits derived within a simplified model framework to any generic parameter space point. This translation scheme is implemented in the NMSSMTools-5.3.1 package, and has been used to evaluate the impact of direct electroweakino search limits in the 3l þ = E T final state, performed using LHC ffiffi ffi s p ¼ 13 TeV dataset collected at ∼36 fb −1 [62], on the parameter space under study.
The ATLAS Collaboration has also probed the electroweakino sector via direct chargino pair-production in the WW-mediated opposite-sign dilepton + = E T final state within a simplified model framework with winolikeχ 1 AE [101]. This search was performed using the LHC Run-II data collected with L ∼ 139 fb −1 . The limits obtained from this search exclude a winolike chargino up to M˜χAE 1 ∼ 400 GeV for a binolikeχ 0 1 with mass up to M˜χ0 1 ∼ 90 GeV assuming 100% branching ratio in theχ AE 1 → W AEχ0 1 channel. This condition holds true for the parameter space considered in this study since the only possible two body decay mode forχ 1 AE is into a W AEχ0 1 pair. Therefore these search limits can directly be translated in the current analysis by excluding dominantly winolike charginos (wino-content iñ χ 1 AE ≳ 90%) with mass M˜χ 1 AE ≲ 400 GeV. (x) Direct detection constraints: The spin-independent (SI) and spin-dependent (SD) DM-nucleon scattering cross sections form the basis of DM direct detection experiments. We impose the latest upper limits on spin-independent WIMP-nucleon cross section (σ SI ; Xenon-1T [63]), the spin-dependent WIMP-neutron cross section (σ SD−neutron ; Xenon-1T [102]), and the spin-dependent WIMP-proton cross section (σ SD−proton ; PICO-60 [103]) derived at 90% C.L.. Among the points allowed by the latest upper limits on σ SI , a small fraction (≲4%) of points are excluded by the current upper limits on σ SD−proton and σ SD−neutron . These points which are exclusively excluded by the current upper limits on the spindependent WIMP-neutron/proton cross sections feature a large λð≳0.1Þ and a relatively large Higgsino component isχ 0 1 . We illustrate in Fig. 1 the impact of current constraints in the M 2 -μ plane. The grey points are excluded by either the LEP limits, LHC signal strength constraints, B physics constraints or direct light Higgs searches and sparticle searches at the LHC. In particular, the points generated in the large M 2 region are excluded by constraints on the Higgs sector including the requirement of having a Higgs around 125 GeV as well as the constraints on the Higgs signal strengths and from the LEP searches for light Higgses. Indeed in this region, the one-loop corrections to the Higgs masses from charginos can be large [66], thus impacting both the mass spectrum of the scalars and their singlet/doublet content. Note also that the Higgs sector depends strongly on the third generation squark sector, hence the impact of the Higgs constraints could be relaxed in a more general framework where parameters of the squark sector are allowed to vary, and not kept fixed as we have in Eq. (16). We also indicate those points (green) that are excluded by the latest upper limits from the direct detection experiments and those (orange) that are excluded by the current direct electroweakino searches limits discussed in Sec. IV. The blue colored points represent the set of currently allowed parameter space points with light neutralinos that we will use to study the impact of current indirect detection constraints and the reach of future experiments. thermally averaged DM annihilation cross section times the relative velocity between the DM candidates (hσvi) inχ 0 1χ 0 1 → bb; τ þ τ − channels [64]. In the current study, we use micrOMEGAs-5.0.6 to compute hσvi. Recall that the annihilation rate scales as the square of the local dark matter density. Assuming thermal DM, this then is scaled by a factor ξ ¼ Ω thermal χ 0 1 =0.12. This then means that the detection rate will scale by ξ 2 . After this scaling, we find that only a few points are constrained by current FERMI-LAT limits, see Fig. 2. Moreover values of ξ 2 hσvi can lie several orders of magnitude below current upper limits as well as below the typical value expected, 3 × 10 −26 cm 3 = sec, even for points with ξ close to unity.
The very small values of ξ 2 hσvi that we found in our scenarios can be explained as follows. As already noted, the relic density constraint requires that DM annihilation be enhanced by s-channel exchange of a boson (Z or scalar/pseudoscalar) near a resonance. For low dark matter masses below ∼10 GeV where one may also have to worry about CMB constraints (from DM annihilation injecting electromagnetic energy during the era of matterradiation decoupling), the enhancement is due to very narrow A 1 =H 1 resonances which have very small couplings and tiny values of Γ=M A 1 =H 1 , ∼10 −9 -10 −7 , resulting in a very strong velocity dependence of the annihilation cross section. Typically the thermal energy of the DM in the early universe will allow for the required enhancement of the cross section while for the lower DM galactic velocities, there is no such enhancement. For the same reason, CMB constraints which involve even smaller velocities, will also not apply [104], even though the thermal relic density may be close to its observed value. The only exception is when the DM mass is very near M A 1 =H 1 =2 or even slightly above, then the full resonant enhancement can occur at small velocities v ∼ 10 −3 c while for the higher velocities in the early universe v ∼ 0.3c only the tail of the resonance will contribute. The fact that DM annihilation cross section in the galaxy can be enhanced as compared with the one in the early universe-the so-called Breit-Wigner enhancement-was discussed in [40,105]. It requires a very narrow resonance, as is the case here for the light scalar/pseudoscalar here, as well as a fine-tuned mass difference M˜χ0 In our scan we find a few points which feature such a large cross section and are thus excluded by FermiLAT. Following the same line of argument, we also expect only a few fine-tuned points to be subject to CMB constraints. The parameter space points from Fig. 2 which are allowed by all constraints discussed above will be referred to as the allowed parameter space points in the remainder of this paper.

V. THE THERMAL NEUTRALINO
Having discussed the relevant experimental constraints, we will move on to discuss the characteristic features of the NMSSM parameter space under study. At this point, we reiterate that we remain confined to the region where the right) channels has been shown in the y-axis against M˜χ0 1 in the x-axis. The parameter space points shown here are allowed by all the current constraints discussed till now in the present section. The black dashed line represents the 95% C.L. upper limits on the hσvi from search for excess γ-ray emission in dwarf galaxies by the FERMI-LAT Collaboration [64]. The orange colored points are excluded by the current indirect limits, while the blue colored points are still allowed. decay of SM-like Higgs boson into a pair of LSP neutralinos is kinematically feasible, and, correspondingly, we impose an upper bound on the mass ofχ 0 1 , M˜χ0 1 ≤ 62.5 GeV. In addition, the parameter space is also subjected to the collider and astrophysical constraints discussed in Sec. IV.
In the MSSM parameter space considered in our previous work [24], it was observed that the imposition of the relic density constraint resulted in a lower bound (M˜χ0 1 ≳ 34 GeV) on the mass of the LSP neutralino. The lower bound on M˜χ0 1 was implied by the presence of only the Z boson and the Higgs boson as mediators for efficient χ 0 1χ 0 1 annihilation. Correspondingly, the MSSM parameter space points were confined to the Z funnel (∼M Z =2) and Higgs-funnel (M H 125 =2) regions only. The NMSSM framework, on the other hand, features additional light Higgses (H 1 and A 1 ) with mass below M Z , which can potentially mediate the annihilation of lighter LSP neutralinos with mass below the Z funnel region. Correspondingly, we obtain allowed parameter space points with M˜χ0 1 as low as ∼1 GeV, which are consistent with the upper bound on relic abundance.
In order to emphasize upon the mass correlations between the LSP neutralino and light Higgs boson states, we plot the allowed parameter space points from Fig. 2 in the M˜χ0 1 -M A 1 plane shown in Fig. 3. The color palette in Fig. 3 represents the mass of lightest scalar Higgs boson, M H 1 . It is evident from Fig. 3 that the allowed points below the Z funnel region are mostly populated along the line M A 1 ∼ 2M˜χ0 1 , those that lie below this line actually satisfy the condition M H 1 ∼ 2M˜χ0 1 . Accord with the relic density constraint is then attained by efficient annihilation via the A 1 =H 1 resonance. We also exhibit the currently allowed parameter space points in the M˜χ0 1 -M˜χ 1 AE plane in Fig. 4. The color palette in Fig. 4 corresponds to the singlino content of the LSP.
The LEP limit on the chargino mass (M˜χ 1 AE > 103.5 GeV) implies a lower limit on the doublet Higgsino and wino mass parameters and restricts them to values roughly above ∼100 GeV. Since our region of interest concerns M˜χ0 1 ≤ M H SM =2, theχ 0 1 has to be either binolike or singlinolike. However, a bino or singlino dominatedχ 0 1 can only satisfy the relic density upper bound if it undergoes coannihilation or pair annihilates via a resonance at roughly twice its mass. Within the NMSSM parameter space considered in the present analysis, co-annihilation is not feasible since there is a large mass splitting between the LSP and all other sparticles. Thusχ 0 1χ 0 1 annihilation through an intermediate resonance remains the only feasible option with either a light scalar and pseudoscalar Higgs states at mass ∼2M˜χ0 1 . In the present analysis, M 1 has been fixed at ∼2 TeV. Therefore, the possibility of obtaining a binodominatedχ 0 1 is eliminated. Moreover, a bino-dominatedχ 0 1 requires a nonzero doublet Higgsino admixture in order to couple with a Higgs state and since the chargino mass limits constrains the amount of Higgsino fraction inχ 0 1 , the relic density limit disfavors a binolike LSP neutralino below the Z funnel region. 3 The singlinolike neutralino, on the other The possibility of Z funnel annihilation of a binolike LSP also occurs in the MSSM and has been examined in detail in Ref. [24]. Since our goal was to study new possibilities in the MSSM, we have taken the bino to be very heavy in our study.
hand, can couple with either a singlet scalar or pseudoscalar Higgs with the coupling being proportional to κN 2 15 . Therefore the only possibility to obtain a LSP neutralino below ≲34 GeV that satisfies the relic density constraint, is to have a singlinolikeχ 0 1 and at least one singlet Higgs states at roughly twice its mass.
A study of the composition of light pseudoscalar and scalar Higgses within the allowed parameter space points show that A 1 and H 1 are dominantly singlet in nature, 4 with singlet fraction ≳90%. Similarly, in accordance with the previous discussion, the LSP neutralinos are also found to be dominantly singlino in nature, as illustrated in Fig. 4.

VI. PROSPECTS AT FUTURE EXPERIMENTS
This section is subdivided into two parts. In the first part, we study the projected reach of Xenon-nT (via bounds on σ SD−neutron ), PICO-250 (via bounds on σ SD−proton ) and at the FCC, ILC and CEPC (through the measurement of the invisible width of H 125 ). The second part investigates the scope of direct light Higgs boson searches in the 2b2μ, 2b2τ and 2μ2τ channels at the future upgrades of LHC. This is, of course, in addition to continuing searches for signals from direct production of superpartners at the LHC, some of which are discussed in Sec. VII.

A. Reach of Xenon-nT, FCC-hh and the future electron-positron colliders
The projected reach of Xenon-nT [106] in probing SI WIMP-nucleon cross sections extends upto a factor of ∼50 beyond the current limits from Xenon-1T in the DM mass range of ∼15 GeV to 62.5 GeV. Correspondingly, a considerable region of the currently allowed parameter space is expected to lie within the future reach of Xenon-nT. We show the projected reach of Xenon-nT (represented as blue dashed line) in the ξσ SI -M˜χ0 1 plane in Fig. 5 (left). The points shown in Fig. 5 (left) correspond to the currently allowed parameter space points. A significant fraction of those points fall within the projected reach of Xenon-nT while another large fraction remains out of reach. Moreover, an important fraction of these points, especially at low masses, lie below the coherent neutrino scattering floor and thus will remain out of reach of even larger detectors. In the same figure, the black points represent those parameter space points for which the invisible branching fraction of the SM-like Higgs boson, BrðH 125 → invisibleÞ < 0.24%, and thereby, are outside the projected reach of even the CEPC invisible Higgs boson branching fraction measurements [107]. The color palette represents the value of BrðH 125 → invisibleÞ for those allowed parameter space points which have BrðH 125 → invisibleÞ > 0.24% and hence fall within the CEPC's projected capability of Higgs invisible branching fraction measurements. The invisible branching fraction has been  [108], FCC-ee [109], ILC [110], CEPC [107] and FCC-hh [111]. The orange colored points fall below the coherent neutrino scattering floor, while the green colored points are outside Xenon-nT's projected reach and above the coherent neutrino scattering floor. The blue colored points are within the projected reach of Xenon-nT. computed by adding the contributions from the following decay modes: An important observation to be made from Fig. 5 is that CEPC will be able to probe a small fraction of parameter space points in the M˜χ0 1 ≲ 10 GeV region which may be forever outside the reach of DM detectors unless directional detection technologies become available.
The complementarity between direct detection experiments and Higgs boson invisible width measurements at the future electron-positron colliders is further explored in Fig. 5 (right), where the allowed parameter space points are displayed in the BrðH 125 → invisibleÞ-M˜χ0 1 plane together with the future reach of HL-LHC (≳2.8%) [108], FCC-ee (≳0.63%) [109], ILC (≳0.4%) [110], CEPC (≳0.24%) [107] and FCC-hh (≳0.01%) [111]. The color code illustrates whether points are within (blue) or outside (orange and green) the projected reach of Xenon-nT (via σ SI measurements). The orange colored points also lie below the coherent neutrino scattering floor. It can be observed from Fig. 5 that a majority of the points in the M˜χ0 1 ≳ 10 GeV region would be accessible via invisible Higgs boson branching fraction measurements at future e þ e − colliders, except when the invisible decay of H 125 is kinematically suppressed.
In the NMSSM, the spin-independentχ 0 1 -nucleon interaction is mainly mediated through the CP-even Higgs bosons and the squarks. Since we have set the squark mass to a large value (M U 3 , the main contribution to σ SI comes from the t-channel exchange of H 1 and H 125 between theχ 0 1 and the light flavored quarks. The respective amplitudes from the exchange of H 1 and H 125 can also undergo destructive interference for some specific choice of input parameters with substantial finetuning [112][113][114][115][116]. Such scenarios, also referred to as blind spots, can escape the current constraints on σ SI and also be compatible with the relic density limits. In NMSSM with singlinolikeχ 0 1 LSP, these blind spots can arise when M˜χ0 1 =ðμ sin βÞ ∼ 1 [117]. In this work, although we do not probe the blind spot scenarios exclusively, however, we do obtain parameter points with small σ SI values (which fall below the projected reach of Xenon-nT), many among which even fall below the neutrino scattering floor [orange colored points in Fig. 5 (right)]. We obtain such points over the entire range of M˜χ0 1 ≲ M H 125 =2 region as illustrated in Fig. 5. Spin-dependent measurements at the future experiments can be potential probes for these points in the M˜χ0 1 ≳ 15 GeV region. The projected reach of PICO-250 [118] and Xenon-nT [119] in probing SD WIMP-proton and WIMP-neutron cross sections extend up to a factor of ∼75 and ∼7, respectively, beyond the current limits from PICO-60 [103] and Xenon-1T [102] in the DM mass range of ∼15 GeV to 62.5 GeV. We illustrate the future reach of PICO-250 and Xenon-nT in probing the currently allowed parameter space points in the ξσ SD−proton -M˜χ0 1 FIG. 6. Left: the currently allowed parameter space points are shown in the ξσ SD−neutron -M˜χ0 1 plane. The solid blue and dashed blue lines represent the current upper limits on σ SD−neutron from Xenon-1T [102] and the projected upper limits on σ SD−proton from Xenon-nT [119], respectively. Right: allowed parameter space points are shown in the ξσ SD−neutron -M˜χ0 1 plane. The solid blue and dashed blue lines represent the current upper limits on σ SD−proton from PICO-60 [103] and the projected upper limits on σ SD−proton from PICO-250 [118], respectively. The color code is similar to that followed in Fig. 5 (left). and ξσ SD−neutron -M˜χ0 1 plane in Fig. 6 (left) and (right), respectively. Here as well, a significant region of currently allowed parameter space points fall within the projected reach of Xenon-nT as well as PICO-250 while another considerable fraction remains out of reach. In Fig. 7 we again illustrate the currently allowed parameter space points in the ξσ SI -M˜χ0 1 plane. The grey and orange points fall below the coherent neutrino scattering floor [they correspond to the orange colored points in Fig. 5 (right)] and will remain outside the projected coverage of any future σ SI measurement experiment. The orange points, however, fall within the projected reach of Xenon-nT [119] and PICO-250 [118] via σ SD−neutron and σ SD−proton measurements, respectively. Thus, it can be observed from Fig. 7 that spin-dependent measurements can provide coverage of parameter space points which lie even below the coherent neutrino scattering floor in the M˜χ0 1 ≳ 15 GeV region.

B. Direct searches for the light Higgs bosons at LHC luminosity and energy upgrades
The projected sensitivity of HL-LHC and HE-LHC in probing light A 1 via direct searches in the H 125 → A 1 A 1 =H 1 H 1 → 2b2μ channel has been discussed in [65], where the corresponding 95% C.L. projection contours on σ H 125 =σ H SM × BrðH 125 → A 1 A 1 =H 1 H 1 → 2b2μÞ have also been derived, however, under the assumption that σ H 125 =σ H SM ¼ 1. We have translated these projection limits on to the currently allowed NMSSM parameter space of our interest and the projected reach has been shown in Fig. 8, where, the horizontal and vertical axes represent M A 1 and BrðH 125 → A 1 A 1 → 2b2μÞ, respectively. The orange colored points in Fig. 8 are within the projected reach of HL-LHC at 95% C.L. while the red colored points fall within the projected reach of HE-LHC at 95% C.L. The blue colored parameter space points would be undetectable at HE-LHC via direct light Higgs searches in the 2b2μ final state. These projections show that the discovery potential of light Higgs bosons produced via direct decays of H 125 is not very strong. Indeed no points are observable with 5σ sensitivity. We note though that we have made no attempt to optimize the analysis (which was based on an ATLAS analysis for ∼36 fb −1 ) for either the increased luminosity or increased energy of the HE-LHC. Our conclusion should be viewed with caution.
Projected limits from direct light Higgs searches in the 2b2μ channel, at the HL-LHC and the HE-LHC, has the potential to probe the M A 1 =H 1 ≳ 15 GeV region (as shown in Fig. 8), however, it could be observed that a major region of the currently allowed parameter space lies outside its reach. It is to be noted that, along with light scalars and pseudoscalars, the particle spectrum of the parameter space under study also features light electroweakinos. Therefore, the question that arises next is, whether it would be possible to probe the parameter space through direct electroweakino searches at the future colliders. The next section intends to provide a conservative answer to this question.

VII. FUTURE REACH OF DIRECT ELECTROWEAKINO SEARCHES AT HL-AND HE-LHC
The ATLAS Collaboration has made MSSM projections for the wino reach of the HL-LHC via trilepton searches pp →χ 0 2χ1 AE , assuming thatχ 1 [120,121]. Hereafter, we will refer to these as the WZ-and WH 125 -mediated channels, respectively. In Sec. VII A, we perform our own analysis of the wino reach in the WZ-mediated channel, and compare our results with those in [121]. Our purpose in doing so is to assess how our "theorist's computation" does, vis a vis the ATLAS analysis. We then project the reach for doublet Higgsino production via the WZ-as well as the WH 125 -channels, assuming that these Higgsinos directly decay to a lighter LSP. 6 This is, of course, not possible in the MSSM with our assumption of a heavy bino. We perform this analysis as a prelude to our goal of mapping out how well experiments at the HL-LHC will be able to probe the NMSSM parameter space with a light LSP even though this analysis covers a much larger range of masses for the LSP. In Sec. VII B, we translate the discovery projections (and exclusion reach) derived in Sec. VII A to the currently allowed NMSSM parameter space of interest. In Sec. VII C, we perform another collider study to obtain the discovery reach (and exclusion regions) of Higgsinos at the HE-LHC. These are then translated in Sec. VII D to the corresponding reach in the parameter space of the NMSSM.
A simplified model with degenerate Higgsino-likẽ χ 0 2 ,χ 0 3 ,χ 1 AE , and binolikeχ 0 1 is considered while evaluating the HL-LHC and HE-LHC projections. Within the simplified framework, the relevant branching fractions: Brðχ 0 3 =χ 0 2 →χ 0 1 ZÞ or Brðχ 0 3 =χ 0 2 →χ 0 1 H 125 Þ, and, Brðχ 1 AE → χ 0 1 W AE Þ, are assumed to be 100%. Note that in the NMSSM, this assumption remains valid for the lighter chargino but not for the heavier chargino or for neutralinos. Indeed thẽ χ 0 2 ,χ 0 3 andχ 0 4 are an admixture of the winos and Higgsinos while the LSP is a singlino resulting in variedly different values of the relevant branching fractions. Moreover, additional decay channels into light Higgses are possible. In order to translate the projected discovery/exclusion regions on to the NMSSM parameter space, we first map out the efficiency grid for the corresponding search strategy in the doublet-Higgsino LSP mass plane. Since this grid is determined largely by the kinematics, the details of the composition of the parent Higgsinos or the daughter LSP are completely irrelevant: the efficiency grid is determined by the kinematics and the branching fractions of the parent Higgsinos to decay to the LSP and the associated boson (W, Z or H 125 ).

A. Direct electroweakino searches at HL-LHC
As stated previously, the aim of the present subsection is twofold: to derive the projected discovery reach and exclusion range for direct Higgsino searches (pp → χ 0 2χ1 AE þχ 0 3χ1 AE ) in the WZ mediated and WH 125 mediated 3l þ = E T final state at the HL-LHC, and, map out the efficiency grid for the WZ and WH 125 -mediated channels that we will use to delineate the region of NMSSM parameter space that can be probed at the luminosity and energy upgrades of the LHC.

WZ-mediated 3l + = E T channel at HL-LHC
We begin by considering the trilepton signal produced via pp →χ 0 2χ1 We assume that the W and Z bosons are close to their mass shell and focus on the trilepton += E T final state from their leptonic decays to electrons, muons and taus. We use PYTHIA-6 [122,123]  AE ) has been computed at next to leading order (NLO) using Prospino [124][125][126], while the branching fraction ofχ 0 2 =χ 0 3 → Zχ 0 1 andχ 1 AE → W AEχ0 1 has been considered to be 100%.
A signal event is required to have three isolated leptons (l ¼ e, μ) in the final state. The lepton isolation criteria requires the ratio: p charged Standard model backgrounds to the WZ-mediated 3l þ = E T final state mainly come from: on-shell and off-shell WZ, ttV (V ¼ W AE ; Z), triple vector boson (VVV) production and ZZ processes. The ttV and the ZZ samples have been produced by decaying Vð¼ W AE ; ZÞ and the Z bosons, respectively, via leptonic decay modes. In the remaining sections of this work, the ttV and the ZZ backgrounds have always been generated in a similar way with the V and the Z bosons, respectively, decaying leptonically. The leading order (LO) MADGRAPH5_aMC@NLO cross section values for ttV and the ZZ process considered in this work accounts for the aforementioned decay channels. Events containing a btagged jet with p T > 30 GeV and jηj < 2.5 are also rejected in order to suppress the ttV background. 7 These background events have been generated using the MADGRAPH5_aMC@NLO [123,128] framework and for both, the signal as well as the background, detector effects are simulated using DELPHES-3.4.2 [129].
A cut based analysis is performed by considering 8 signal regions with different set of cuts on the following kinematic variables:  andχ 0 1 in BPB1, the Z boson produced from the cascade decay ofχ 0 3 =χ 0 2 carries a relatively larger boost causing the ΔR SFOS distribution of BPB1 to peak at smaller values. The SFOS lepton pairs for the background events are however, produced with relatively larger angular separation. The same is reflected in the ΔR SFOS distribution of BPB1 and the background events shown in the lower left panel of Fig. 9.
We optimize the signal significance of the 8 benchmark points by applying various combination of selection cuts. The signal significance is computed as: where, S and B represents the signal and background yields, respectively, while sys un corresponds to the systematic uncertainty taken to be 5%. The selection cuts corresponding to the 8 signal regions, obtained from the cut-based analysis, have been listed in Table I. The figures on the right panel of Fig. 9 have been obtained on further imposing the selection cuts labeled SRB1. The efficiency grids for SRB1, SRC1, SRE1 and SRG1 are illustrated in the four panels of Fig. 10 where the efficiency values are shown as a color palette in the M˜χ0 2 ;χ 0 3 ;χ 1 AE -M˜χ0 1 plane. The efficiency (Eff.) of a signal region for a particular signal point is computed by taking the ratio of the number of signal events which pass through the signal region selection cuts to the total number of generated signal events. We note that SRB1, SRD1 and SRG1 cuts were particularly optimized to maximize the signal significance of signal processes with larger mass difference between the directly produced doublet Higgsinos and the LSP (BPB1, BPD1 and BPG1, respectively), while SRA1, SRC1 and SRF1 were optimized to maximize the signal significance of such signal events where the mass difference between the directly produced Higgsinos and the LSP was just above the Z mass threshold (BPA1, BPC1 and BPF1, respectively). On the other hand, SRE1 and SRH1 were optimized for signal process with intermediate mass difference between the NLSP Higgsinos and the LSP (BPE1 and BPH1, respectively).
The signal yield is computed as S ¼ σ˜χ0 where L corresponds to the integrated luminosity (3000 fb −1 for HL-LHC) and l 0 ¼ e, μ, τ. The values of Brðχ 0 2 ;χ 0 3 → Zχ 0 1 Þ and Brðχ 1 AE → W AEχ0 1 Þ are assumed to be 100%, while SM values for BrðZ → l 0 l 0 Þ and BrðW → l 0 νÞ are considered. Similarly, in the calculation of background yields, the leading order (LO) cross section computed by MADGRAPH5_aMC@NLO are considered. The background yields for WZ, ttV, VVV and ZZ processes, corresponding to the 8 different signal regions, are listed in Table II. The leading order cross section labeled WZ refers to the tree-level cross section for lνll production from MADGRAPH5_aMC@NLO with generator-level cuts p T > 5 GeV and jηj < 3.5 on the leptons. These include contributions from off-shell W, Z and γ amplitudes. 8 The offshell W contributions are crucial for obtaining a reliable estimate of the background remaining after a hard cut on the transverse mass M l W T , which is very efficient in reducing the background where the W boson is close to its mass shell.
As we have already mentioned, the ATLAS collaboration has also derived the projected exclusion and discovery contours at 95% C.L. from direct wino searches in the WZ mediated 3l þ = E T searches at the HL-LHC (see Fig. 9 from [121]). In order to check the consistency of our analysis setup, we rederive the projection contours for direct wino searches in the WZ mediated 3l þ = E T final state at the HL-LHC by using the signal regions: SRA1-SRH1, which were obtained through optimization of the signal significances for BPA1-BPH1, respectively. The corresponding projection contours derived by using our analysis setup are illustrated in Fig. 11, in the M˜χ0 2 ;χ 1 AE -M˜χ0 1 mass plane. The projection contours in Fig. 11 display a potential to discover (exclude) a winolikeχ 1 AE ;χ 0 2 up to a mass of ∼960ð1150Þ GeV for a massless LSP. This is comparable to the corresponding projections of ATLAS which are respectively ∼950ð1110Þ GeV. Our discovery (exclusion) projections degrade by less than ∼50ð100Þ GeV for LSP masses up to 400 (600) GeV. Next, we move on to derive the projected reach of direct doublet Higgsino searches in the WZ mediated 3l þ = E T final state at the HL-LHC.
The projected exclusion and discovery contours from direct Higgsino searches in the WZ mediated 3l þ = E T final state at HL-LHC are illustrated in Fig. 12 with the same color code as Fig. 11. While deriving the HL-LHC projections, we compute the value of signal significance at a grid point for all 8 signal regions and the maximum among them is ascribed to S σ . We conclude that HL-LHC will be capable of discovering (excluding) pure doublet Higgsinos up to 820 (1000) GeV for a massless binolikeχ 0 1 through direct Higgsino searches in the WZ mediated 3l þ = E T final state. AE -M˜χ0 1 plane. Note that the z-axis scale for SRC1 is different from the other three frames. 8 In the remainder of this article, WZ background will refer to lνll production process with the aforesaid generator level cuts and will include contribution from off-shell W, Z and γ amplitudes. An event is required to have exactly three isolated leptons (l ¼ e, μ) with p T > 10 GeV and must lie within a pseudorapidity range of jηj < 2.47 (for electrons) and jηj < 2.4 (for muons). The lepton isolation criteria requires the ratio p charged  regions considered for the cut-based analysis, are tabulated. The leading order (LO) cross sections generated by MADGRA-PH5_aMC@NLO are also shown. The ttV background has been generated with V decaying leptonically while the generic sample of triple vector boson production has been used in the analysis. Here, we have followed the lepton isolation criteria considered in [120]. and M inv OS;min . Here, M l i T (i ¼ 1, 2, 3) corresponds to the transverse mass of the i th lepton and = E T system, while M inv OS;min is the invariant mass of the opposite sign (OS) lepton pair with minimum ΔR separation. The choice of signal regions is motivated by a similar analysis in [120] which probes the future prospect of direct wino searches at the HL-LHC. The list of selection cuts for the four optimized signal regions have been listed in Table III.
The background yields corresponding to the four signal regions are shown in Table IV along with the LO MADGRAPH5_aMC@NLO cross section values. The background estimates for WZ, WH 125 , ttV, VVV and ZZ have been obtained by passing the simulated background samples through the signal region cuts. The tree-level cross section for WH 125 has been computed using MADGRAPH5_aMC@NLO with generator level cuts: p T > 5 GeV and jηj < 3.5 on the leptons and includes contribution from off-shell W amplitudes. 10 In order to correctly estimate the background yield, it is essential to consider the contribution from off-shell W due to the hard cuts on M l i T (i ¼ 1, 2, 3). The two final states of H 125 from the WH 125 background process which dominantly contributes to the 3l þ = E T signal are H 125 → WW Ã → l 0 ν l 0 l 0 ν l 0 and H 125 → τ þ τ − . We generate these two final states separately.
We should mention that the ATLAS study [120] reports a substantial background from inclusive tt production tripling (doubling) the background in the signal region SRA2 (SRB2) but with not very significant contributions in the SRC2 and SRD2 signal regions. We anticipate that ttW; W → lν and both tops decaying leptonically is the dominant physics source of isolated trileptons from tt production. The trilepton topology can also result if a lepton from the semileptonic decay of one of the bottom quarks is accidentally isolated, or if one of jets is misidentified as an lepton. Evaluation of these detectordependent backgrounds is beyond the scope of our study, but we mention it for completeness. We have been unable to ascertain the origin of the large background reported in Ref. [120] and do not include it in the rest of our analysis (except where we mention it in our discussion of Fig. 14  below). The efficiency maps of SRA2, SRB2, SRC2 and SRD2 for the WðH 125 → WW Ã → l 0 νl 0 νÞ mediated 3l þ = E T final state are shown in Fig. 13.
The projected exclusion contour for direct Higgsino searches in the 3l þ = E T final state is illustrated in light blue color in Fig. 14. The contribution from WðH 125 → WW Ã → lνlνÞ mediated and WðH 125 → τ þ τ − Þ mediated 3l þ = E T signal processes have been added together in order to compute the signal yield. The signal significance for all four signal regions is computed, and the maximum among them is considered in the derivation of the projection contours shown in Fig. 14. Furthermore, we do not consider any systematic uncertainty in the current analysis since we do not perform the signal-background optimization of our own and use the signal region cuts from [120]. Our results indicate that the searches in the WH 125 mediated 3l þ = E T final state has the potential to discover (exclude) Higgsinos up to 400(590) GeV for a masslessχ 0 1 . We mentioned previously that the signal regions SRA2, SRB2, SRC2 and SRD2 were motivated from Ref. [120] which probes the future reach of direct wino searches at the HL-LHC. We remark that if we include the tt background yields corresponding to the afore-mentioned signal regions from Ref. [120] (this double counts the ttW background), the projected discovery (exclusion) reach shown by the solid   10 Henceforth, the WH 125 background process will refer to the off-shell and on-shell WH 125 production process generated with the aforesaid generator level cuts. (derived in Sec. VII A) on to the parameter space under study.

B. Impact of projected electroweakino search limits from HL-LHC
Because we have set the bino mass parameter to be 2 TeV,χ 0 2 ;χ 0 3 ;χ 0 4 as well asχ 1 AE andχ 2 AE are either doublet Higgsino-like, winolike or a wino-Higgsino admixture. In the present scenario, the direct chargino-neutralino pair production modes which could potentially lead to WZ or AE . The direct production cross section for each of these chargino-neutralino pairs is computed by scaling the pure Higgsino production cross section (computed at NLO using Prospino) with the respective reduced squared Wχ 0 iχj AE couplings. The reduced squared couplings have the following form: where V, U are the chargino mixing matrices. V j1 =U j1 and V j2 =U j2 represents the wino and Higgsino component, respectively, in the jth chargino. N i2 represents the wino admixture while N i3=i4 corresponds to the Higgsino admixture in the ith neutralino. In the case of direct production of aχ 0 iχj AE pair whereχ 0 i and the two chiral components ofχ j AE have dominant wino composition, N 2 i3 þ N 2 i4 ; V 2 j2 and U 2 j2 will be ≪ 1, while N 2 i2 ; V 2 j1 and U 2 j1 will be ∼1. Similarly, in the case of direct production of Higgsino-like charginoneutralino pair, N 2 i3 þ N 2 i4 ; V 2 j2 and U 2 j2 will be ∼1, while the other three components will be negligible. From Eq. (18), it can also be observed that the direct wino production cross section is roughly twice the direct Higgsino production cross section at the LO. However, the arguments of the simplified scenario will not hold for our allowed parameter space points since the neutralinos and charginos are an admixture of both winos and Higgsinos. Thereby, all terms in Eq. (18) could be nonnegligible (depending upon the electroweakino mixing structure) and contribute toward the computation of the scaled production cross section. The production cross sections obtained are multiplied by the relevant branching fractions needed to obtain the 3l þ = E T final state. The next step involves the computation of signal yield for all the optimized signal regions. For the WZ mediated 3l þ = E T final state we compute the signal yields for all 8 signal regions, SRA1, SRB1, SRC1, SRD1, SRE1, SRF1, SRG1 and SRH1 prescribed in Sec. VII A 1 while for the WH 125 mediated search we use the four regions SRA2, SRB2, SRC2 and SRD2. The only ingredient left in the computation of the signal yield is the efficiency of the particular signal region, where efficiency is defined as the ratio of the number of events which are allowed by the selection cuts of a particular signal region to the total number of generated events. The efficiency is a function of the masses of χ 0 4 =χ 0 3 =χ 0 2 =χ 2 AE =χ 1 AE andχ 0 1 , and, is extracted from the signal region efficiency maps, some of which are shown in Fig. 10 (for WZ mediated 3l þ = E T searches) and Fig. 13 (for WH 125 mediated 3l þ = E T searches). The signal significance is then computed using Eq. (17) by adopting the signal region that yields the highest value of S σ . Parameter space points which generate a signal significance >2 (>5) are considered to be within the projected exclusion (discovery) reach of HL-LHC.
The points shown in Fig. 15 correspond to the allowed parameter space points obtained after imposing all constraints specified in Sec. IV, see also Fig. 1. The parameter space points which fall within (outside) the projected exclusion reach of direct Higgsino searches in the WZ mediated 3l þ = E T final state at HL-LHC are shown in pale blue (dark blue) color in Fig. 15, while the green colored points fall within the projected discovery reach.
Whether or not the signal is observable is the result of an intricate interplay between the production cross sections, determined by the doublet Higgsino and wino masses, and the signal efficiencies shown in Fig. 10. In the green region with μ ∼ M 2 ∼ 500 GeV, the efficiencies in Fig. 10 are relatively large (the red/orange region) for small values of reduces except for SRA1/SRB1 type cuts, for which efficiencies are small. 11 We observe a dark blue colored protrusion region in the M 2 ∼ 150 GeV region from μ ∼ 300 GeV to μ ∼ 400 GeV. The signal significance of the majority of the points in this region is marginally less than 2σ on account of the relatively smaller efficiency from SRB1.
In order to understand the underlying features, we chose two benchmark points, BP A (μ ∼ 717 GeV, M 2 ∼ 1244 GeV) and BP B (μ ∼ 717 GeV, M 2 ∼ 400 GeV), from the allowed parameter space. The masses and the composition ofχ 0 1 ;χ 0 2 ;χ 0 3 ;χ 0 4 ;χ 1 AE andχ 2 AE 12 for BP A and BP B are shown in Table V and Table VI  Singlino fraction inχ 0 1 : 0.99    11 Note that the scale is different in the efficiency maps of SRB1, SRC1, SRE1 and SRG1 in Fig. 10. 12 Note that the neutralinos in our parameter space have a negligible bino admixture. 13 The projected discovery range and exclusion regions for direct Higgsino searches in the WZ and WH 125 mediated 3l þ = E T final state at the HE-LHC ( ffiffi ffi s p ¼ 27 TeV, L ¼ 15 ab −1 ) and its translation on to the allowed parameter space has been studied in the next section (Sec. VII C).
with the relevant branching fractions of the charginos and neutralinos.
In the case of BP A ,χ 1 AE ;χ 0 2 ;χ 0 3 have a dominant Higgsino composition whileχ 0 4 ;χ 2 AE are wino dominated and much heavier. Theχ 0 1 , on the other hand, is singlino dominated.Theχ 0 2 andχ 0 3 decay into Z þχ 0 1 and H 125 þχ 0 1 with a branching fraction of ∼50% and ∼43%, respectively, in each mode (shown in Table V). BP A yields a signal significance of 3.8 in direct Higgsino searches in the WZ mediated 3l þ = E T searches at the HL-LHC, and falls short of the discovery reach of the HL-LHC. In case of BP A , although Brðχ 0 3 =χ 0 2 →χ 0 1 H 125 Þ is comparable with Brðχ 0 3 =χ 0 2 →χ 0 1 ZÞ, the signal significance of BP A in the WH 125 mediated 3l þ = E T final state is ≲2 due to the larger cascade decay chain in the latter case which reduces the signal significance to 0.4.
In the case of BP B ,χ 0 2 is winolike and decays dominantly intoχ 0 1 H 125 (Brðχ 0 2 →χ 0 1 H 125 Þ ∼ 82%) while the branching ratio intoχ 0 1 Z is only 4%. Theχ 1 AE which is also winolike decays into W AEχ0 1 with 100% branching fraction. Among the six possible combinations of chargino neutralino pairs, theχ 0 2χ1 AE pair has by far the highest cross section ∼104 fb and thereby, plays the most significant role in signal yield computation. The signal significance of BP B in the WZ mediated 3l þ = E T search channel is ∼1.5, and therefore, falls marginally outside the projected exclusion reach of HL-LHC. However, considering the large branching fraction ofχ 0 2 → H 2χ 0 1 , it is expected that direct searches in the WH 125 channel will be more effective in probing BP B . The reach of the HL-LHC for NMSSM doublet Higgsinos via the WH 125 mediated 3l þ = E T search channel is illustrated in Fig. 16, where we plot the allowed points from Fig. 1, with the same color convention as in Fig. 15. Indeed BP B falls within the projected discovery reach of this search with a signal significance of 5.3.
Another important aspect to be noted for the BP B benchmark point is the presence of other cascade decay modes with considerable branching ratios. For instance,χ 0 3 decays to Zχ 0 2 whileχ 0 2 in addition to its decay toχ 0 1 H 125 , also decays toχ 0 1 H 1 with a branching ratio ∼14%. We see from Table VI   Branching ratioχ 0 2 →χ 0 1 Z (0.04),χ strategies devised to optimize the 3l þ = E T searches will not be particularly efficient in the presence of these cascade decay channels and Figs. 15 and 16 may not provide the complete picture. Dedicated searches beyond the scope of this paper will be needed to explore these novel signals.
It can also be observed from Fig. 16 that future direct searches in the WH 125 mediated 3l þ = E T channel are more effective in probing the parameter space region with M 2 ≲ μ. However, below M 2 ≲ 125 GeV, an on-shell H 125 cannot be produced resulting in zero signal efficiency, and therefore this channel cannot be used. It is very striking to see the complementarity in the search power via the WZ and the WH 125 -mediated trilepton search channels exhibited in Fig. 15 and Fig. 16, respectively. For instance, in the dark blue coloured band near M 2 ∼ 180 GeV and μ ∼ 300-400 GeV in Fig. 15, the signal significance is less than 2, while in the same region in Fig. 16, the signal is above the discovery limit! Likewise, in much of the green region at large values of M 2 where the signal can be discovered via the WZ-mediated channel (Fig. 15), the signal is unobservable via the WH 125 channel in Fig. 16. To obtain a better understanding of what is happening in the low M 2 region, we examine a representative benchmark point BP C (see Table VII). BP C features a winolikeχ 0 2 ;χ 1 AE and thus a large production cross section of pp →χ 0 2χ1 AE (1593 fb), a large branching fraction ofχ 0 2 →χ 0 1 H 2 (92%) andχ 1 AE → χ 0 1 W AE (100%). However, due to the small doublet Higgsino component inχ 0 2 , the branching ratio ofχ 0 2 →χ 0 1 Z is only 4%. Additionally, the signal regions in Table I generate relatively lower efficiencies in the M NLSP ∼ 180 GeV region compared to the smaller and larger values of M˜χ0 2 , from SR1A and SR1B. Consequently, the signal significance of BP C in the WZ channel is only ∼1.3 and thereby falls outside the projected reach of HL-LHC. BP C is, however, within the projected discovery reach of HL-LHC in the WH 125 channel where the signal significance is 8.8.
An examination of Figs. 15 and 16 shows that it should be possible to discover SUSY with >5σ confidence at the HL-LHC over almost the entire allowed parameter space (with the exception of the island at large values of M 2 and μ near the benchmark point BP A ) if it is realized as in the NMSSM scenario with m˜χ0 1 < 62.5 GeV.

C. Electroweakino searches at the HE-LHC
We have just seen that the HL-LHC will be able to probe most of the NMSSM parameter space via 3l þ = E T searches via at least one or the other of the WZ or WH 125 mediated channels. Here, we turn to the exploration of the capabilities of the proposed HE-LHC for the corresponding search. Our motivation for this is two-fold. First, it is clear that the signal might be just above the discovery limit at the HL-LHC so that a larger signal may be obtained at the proposed energy upgrade, allowing for a detailed study of the new physics. Second, we want to map out the NMSSM region where the signal would be discoverable via both the WZ and the WH 125 mediated channels, since observations in multiple channels would clearly help to elucidate its origin. To this end, we devise new strategies to isolate the SUSY signal from the background, and delineate the corresponding 5σ discovery reach and 2σ exclusion regions for the cases of both the WZ and the WH 125 mediated 3l þ = E T channels. We recognize that other superpartners would also be accessible, but for definiteness focus only on the golden trilepton signal from the lightest ino states.

WZ mediated 3l + = E T searches at the HE-LHC
We select events with exactly three isolated leptons, where lepton isolation as described in Sec. VII A 1 is used. Two isolated leptons are required to form a SFOS lepton pair with invariant mass in the range M Z AE 10 GeV. In the presence of two such SFOS lepton pairs, the one which minimizes the transverse mass of the non-SFOS lepton and = E T is chosen as the correct SFOS pair. In addition, a veto on b jet with p T > 30 GeV and jηj < 2.5 is also applied. Onshell and off-shell WZ, VVV, ttV and ZZ are the background sources. Signal events have been generated by varying M˜χ0 1 over the range of 0-1000 GeV with a step size of 30 GeV while M˜χ0 Based on this analysis, we choose the ten signal regions that optimize the significance of the signal for a variety of ino and LSP masses (though, of course, for the NMSSM analysis of interest, the LSP is light). The selection cuts are listed in Table VIII.
As an illustration, we show the = E T and ΔR SFOS distributions for the benchmark case BPH3 together with those for the main SM backgrounds (for HE-LHC) in Fig. 17. The signal event distributions are represented in a black colored shade while the distributions for WZ, VVV, ttV and ZZ backgrounds are shown in blue, orange, green and red colors, respectively. Note that BPH3 is the signal benchmark point with intermediate mass gap between the NLSP and the LSP, and was used to derive the optimized set of selection cuts corresponding to SRH3. The plots in the left panel of Fig. 17 have been obtained by imposing the requirement of three isolated leptons in the final state, presence of at least one SFOS pair of leptons with invariant mass in the range M Z AE 10 GeV, and the absence of any b jet with p T ≥ 30 GeV and jηj < 2.5. The distribution in the right hand panels of Fig. 17 are obtained after all the SRH3 cuts in Table VIII. As before, the dominant physics SM backgrounds to the hadronically quiet 3l þ = E T signal is expected to come from WZ (which includes W ðÃÞ Z and W ðÃÞ γ Ã ) production, ttV production, VVV production and ZZ production. The corresponding background levels obtained using MADGRAPH5_aMC@NLO are shown in Table IX We show the projected exclusion region (S σ > 2σ, blue color) and the projected discovery region (S σ > 5σ, grey color) obtained from doublet Higgsino searches in the WZ mediated 3l þ = E T final state at the HE-LHC in Fig. 19. It can be seen from Fig. 19 that HE-LHC will be able to discover (exclude) Higgsinolikeχ 0 3 ;χ 0 2 andχ 1 AE with masses up to ∼1330ð1660Þ GeV if the LSP is light.     Although not directly germane to the NMSSM signal with a light LSP, we find it interesting to show the event distribution of p l 1 T and = E T for BPD4 at the HE-LHC. The signal distribution is shown by the black-hatched region in Fig. 20 together with the distributions of the relevant background processes (ttV: green color, WH 125 : red color, VVV: orange color, and, WZ: blue color). The distributions on the left panels of Fig. 20 have been obtained by requiring the presence of exactly three isolated leptons in the final state, the absence of any SFOS lepton pair with invariant mass in the range M Z AE 12 GeV and b − jet veto. In the case of BPD4, the large mass difference betweeñ χ 0 3 ;χ 0 2 ;χ 1 AE andχ 0 1 results in the leading p T lepton getting produced with a larger boost and therefore peaks at a higher value (around 200 GeV) and has a flatter distribution as compared to the background processes. The larger mass difference between the NLSPs and the LSP also results in a = E T distribution which extends to much larger values (>900 GeV). The = E T distribution for the background processes, on the other hand, falls down more steeply. The figures in the right panel of Fig. 21 have been obtained after passing through the SRD4 cuts.
The event distribution of p l 1 T and ΔR min OS at the HE-LHC for BPE4 and the relevant background processes have been shown in Fig. 21 following the color code of Fig. 20. The left panels of Fig. 21 have been obtained by imposing the following criteria: presence of three isolated leptons, veto on SFOS pair with invariant mass M Z AE 12 GeV and b-jet veto. Due to a smaller mass difference between thẽ χ 0 3 ;χ 0 2 ;χ 1 AE andχ 0 1 in BPE4, the leading p T lepton is produced with a smaller boost and peaks at a slightly lower p T (around 50 GeV) which is roughly similar to the background processes (all of which peak around 50-100 GeV). Similarly, the ΔR OS min distribution for BPE4 falls off before ≲3.0 unlike the WZ, WH 125 , ttV and VVV backgrounds which extend up to ≳4.0. The plots on the right panels of Fig. 20 show the event distribution obtained upon the further imposition of SRE4 cuts.
The LO background cross sections and the background yields corresponding to the 7 signal regions are shown in Table XI. We also show the efficiency grids for SRA4, SRB4, SRC4 and SRD4 for the H 125 → WW Ã scenario in   Fig. 24. Comparing with Fig. 15, we see that the HE-LHC provides a larger discovery opportunity than the HL-LHC for the detection of NMSSMinos with a light LSP. Nearly all the NMSSM allowed points are within the discovery reach of the HE-LHC. The parameter space points shown in Fig. 24 correspond to the allowed points obtained after imposing all the current constraints discussed in Sec. IV. The green colored points in Fig. 24 are within the projected discovery reach (signal significance >5), while the pale blue colored points are within the projected exclusion reach (signal significance >2) of direct Higgsino searches in the WZ mediated 3l þ = E T final state at HE-LHC. The dark blue colored points, however, will be undetectable to HE-LHC, these correspond to a small band of points at M 2 ∼ 200 GeV. We note that the signal cases, and hence the cuts, that we have chosen in Table VIII are not designed to probe the very light wino region where the lightest NLSP mass is around M NLSP ∼ 180-200 GeV resulting in reduced signal efficiency in the dark blue region. Furthermore, the lightest NLSP in the dark blue region is dominantly wino in nature resulting in smallχ 0 2 →χ 0 1 Z branching rates. We have checked that even so, these points only narrowly escape the projected exclusion. We fully expect that choosing additional signal regions to optimize the signal significance around M NLSP ∼ 190 GeV will prove effective in extending the exclusion/discovery region of the HE-LHC to this region of parameter space.
Before concluding this section, we also study the future reach of direct Higgsino searches in the WH 125 mediated 3l þ = E T final state at the HE-LHC. The results are illustrated in Fig. 25 using the color code from Fig. 24. It can be seen from Fig. 25 that the projected reach of HE-LHC extends much beyond that of the HL-LHC (shown in Fig. 16) for the detection of NMSSM-inos. Here, the dark blue colored points are concentrated in the M 2 ≲ 125 GeV (where the decay to H 125 is kinematically forbidden). The projected discovery reach covers the rest of the NMSSM parameter space. We see that a combination of doublet Higgsino searches at the HE-LHC in the WH 125 and WZ mediated 3l þ = E T channels will probe the entire region of the currently allowed NMSSM parameter space with light neutralinos at the >5σ level.
Even though discovery may be possible also at the HL-LHC, the reader may find it interesting to compare the signal size after the final cuts at these facilities. We show such a comparison in Table XII for the benchmark points BP A , BP B and BP C in Figs. 15 and 16, and 6 other benchmark points (BP D , BP E , BP F , BP G , BP H and BP I ), for both the WZ and the WH 125 mediated decays. 14 The corresponding signal significance is shown in parenthesis. We see that the accumulated signal is appreciably larger at the HE-LHC. This larger data sample should enable detailed studies of event shapes and distributions, and help zero in on the origin of the new physics.

VIII. CONCLUSION
In this paper, our goal was to analyze the current status and future prospects of a light neutralino DM (M˜χ0 1 < M H 125 =2) within the NMSSM framework. Previous studies of light neutralinos in the MSSM have shown that such scenarios could be entirely probed at future direct detection experiments such as Xenon-nT, with potential complementary signatures in invisible Higgs decays. We therefore concentrated only on regions of the NMSSM with features clearly distinct from those of the MSSM regions explored before. In particular we focused our study on the case where a new light singlet Higgs was present (with M A 1 ; M H 1 < 122 GeV). Such scenarios can provide a new mechanism for efficient neutralino annihilation in the early Universe via the exchange of a light scalar and/or pseudoscalar Higgs at roughly twice the mass of the neutralino DM. Thus in the NMSSM framework neutralino DM at the GeV scale can satisfy the relic density constraint.
After imposing relevant current constraints from collider and astrophysical experiments, the allowed parameter FIG. 24. The currently allowed parameter space points are shown in the M 2 − μ plane. The pale blue and green colored points fall within the projected exclusion and discovery reach, respectively, of direct Higgsino searches in the WZ mediated 3l þ = E T channel at the HE-LHC.
FIG. 25. The allowed parameter space points have been shown in the M 2 − μ plane. The pale blue and green colored points fall within the projected exclusion and discovery reach, respectively, of direct electroweakino searches in the WH 125 mediated 3l þ = E T channel at the HE-LHC. The dark blue colored points fall outside the HE-LHC's projected reach. 14 The model parameters for these benchmark points are shown in Table XIII as an Appendix. space of the model was identified. This parameter space features both a singlino-dominated LSP and a singletlike light Higgs. We showed that direct detection experiments and Higgs invisible width measurements at future colliders such at the ILC, CEPC and FCC have the potential to probe some of the model's parameter space, however two regions in particular remain out of reach. The first correspond to a neutralino with a mass below about 10 GeV where not only direct detection experiments lose their sensitivity but also where the Higgs invisible width can often be very small due to the singlino nature of the LSP. The second had a LSP mass near 62 GeV. For such a neutralino with a mass close to M H 125 =2, the relic density constraint can be satisfied even when the coupling of the neutralino LSP to the Higgs is strongly suppressed, thus reducing the direct detection signal as well as the Higgs invisible width. A significant fraction of the first region is also out of reach of searches for light scalars/pseudoscalars in SM-like Higgs decays at the future HL-LHC since those searches which typically involve a decay channel of a light Higgs into b-quarks lose their sensitivity when M A 1 ;H 1 ≲ 11 GeV, which in our framework is linked to M DM < 5.5 GeV. Thus a large fraction of the NMSSM parameter space with light neutralinos will remain out of reach of these future searches.
We found that these difficult-to-probe regions came with light electroweakinos, in particular with NLSP winos barely above 100 GeV or a NLSP Higgsino as light as 400 GeV, hence have the potential to be discovered at future runs of the LHC. After recasting and validating the current searches for light electroweakinos in the 3l þ = E T final state from neutralino-chargino pairs in the WZ and WH 125 channel, we investigated how these searches could probe the allowed parameter space of the model at the future highluminosity and high-energy extension of the LHC. We showed that over a wide range of parameters, the doublet Higgsino could be probed at the HL-LHC via the WZ mediated channel, at least at the 2σ level, but for the most part even at the 5σ level. Moreover, we found that the WH 125 mediated channel was truly complementary particularly in the wino NLSP region (M 2 < μ) where the WZ channel had the least power. We found that over almost all the allowed NMSSM parameter space with a light LSP, experiments at the HL-LHC would be able to discover (5σ) SUSY via at least one of the two channels. We have limited our study to the golden 3l þ = E T signal from electroweakino production. It may be worth examining other signal channels which occur at larger rates, but also have larger backgrounds.
The higher energy and luminosity available at the HE-LHC guarantees discovery via both the WZ and WH 125 mediated trilepton channels over essentially the entire allowed parameter space. We also highlighted some regions where additional search channels, involving the decays of neutralinos into light scalar/pseudoscalar Higgses could offer distinctive probes of the NMSSM electroweakino sector. The tools developed for this analysis are not restricted to NMSSM scenarios under consideration and the recasting presented here can be applied to other prospective studies of electroweakino searches at the HL-LHC and HE-LHC where the LSP neutralinos can be heavier than 60 GeV.