Impact of leptonic unitarity and dark matter direct detection experiments on the NMSSM with inverse seesaw mechanism

In the Next-to-Minimal Supersymmetric Standard Model with the inverse seesaw mechanism to generate neutrino masses, the lightest sneutrino may act as a feasible dark matter candidate in vast parameter space. In this case, the smallness of the leptonic unitarity violation and the recent XENON-1T experiment can limit the dark matter physics. In particular, they set upper bounds of the neutrino Yukawa couplings λν and Yν. We study such effects by encoding the constraints in a likelihood function and carrying out elaborated scans over the parameter space of the theory with the Nested Sampling algorithm. We show that these constraints are complementary to each other in limiting the theory, and in some cases, they are very strict. We also study the impact of the future LZ experiment on the theory.


Introduction
As the most popular ultraviolet-complete Beyond Standard Model, the Minimal Supersymmetric Standard Model (MSSM) with R-parity conservation predicts two kinds of electric neutral, possibly stable and weakly interactive massive particles, namely, sneutrino and neutralino, which may act as dark matter (DM) candidates [1,2]. In the 1990s, it was proven that the left-handed sneutrino as the lightest supersymmetric particle (LSP) predicted a much smaller relic abundance than the measured value as well as an unacceptably tremendous DM-nucleon scattering rate due to its interaction with the Z boson [3,4]. This fact made the lightest neutralino (usually with the bino field as its dominant component) the only reasonable DM candidate, and consequently, it was studied intensively since then. However, with the rapid progress in DM direct detection (DD) experiments in recent years, the candidate became more and more tightly limited by the experiments [5][6][7][8] assuming that it was fully responsible for the measured relic density and the higgsino mass µ was less than 300 GeV, which was favored to predict the Z boson mass naturally [9]. These conclusions apply to the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [10][11][12], where the sneutrinos are purely left-handed, and the neutralino DM candidate may be either bino-or singlino-dominated [13]. In this context, we revived the idea of the sneutrino DM in a series of works [8,[14][15][16]. In particular, motivated by the phenomenology of the neutrino oscillations, we extended the NMSSM with the inverse seesaw mechanism by introducing two types of gauge singlet chiral superfieldsν R andX for each generation matter, which have lepton numbers −1 and 1, respectively, and their fermion components JHEP12(2020)023 corresponded to the massive neutrinos in literatures [14]. Subsequently, we studied in detail whether theν R (the scalar component ofν R ) orx (the scalar component ofX) dominated sneutrino could act as a feasible DM candidate [14]. We were interested in the inverse seesaw mechanism because it was a TeV scale physics to account for the neutrino oscillations and maybe experimentally testable soon. We showed by both analytic formulas and numerical calculations that the resulting theory (abbreviated as ISS-NMSSM hereafter) was one of the most economic framework to generate the neutrino mass and, meanwhile, to reconcile the DM DD experiments naturally [8,14]. We add that, besides us, a lot of authors have showed interest in the sneutrino DM in recent years [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], but none of them considered the same theoretical framework as ours.
In the NMSSM, the introduction of the singlet fieldŜ can solve the µ problem of the MSSM [13], enhance the theoretical prediction of the SM-like Higgs boson mass [37][38][39], as well as enrich the phenomenology of the NMSSM significantly (see, for example, refs. [40][41][42][43][44][45]). In the ISS-NMSSM, theŜ field also plays extraordinary roles in generating the massive neutrino mass by the Yukawa interaction λ νŜνRX and making the sneutrino DM compatible with various measurements, especially the DM DD experiments [14]. There are at least two aspects in manifesting the latter role. One is that the newly introduced heavy neutrino superfields are singlet under the gauge group of the SM model. Thus, they can interact directly withŜ by the Yukawa couplings [14]. In this case, the sneutrino DM candidateν 1 , the singlet dominated scalars h s and A s , and the massive neutrinos ν h compose a roughly secluded DM sector where the annihilationsν 1ν * 1 → A s A s , h s h s , ν hνh can produce the measured relic density (In the ISS-NMSSM, these annihilations proceed by quartic scalar interactions, s-channel exchange of h s and t/u-channel exchange of the sneutrinos or the singlino-dominated neutralino). Since this sector communicates with the SM sector by the small singlet-doublet Higgs mixing (dubbed by Higgs-portal in literatures [46]) and/or by the massive neutrinos (neutrino-portal [47][48][49][50]), the scattering of the DM with nucleons is naturally suppressed, which coincides with current DM DD results. The other aspect is that the singlet-dominated Higgs scalars can mediate the transition betweenν 1 pair and the higgsino pair, and consequently, these particles were in thermal equilibrium in early Universe before their freeze-out from the thermal bath. If their mass splitting is less than about 10%, the number density of the higgsinos can track that ofν 1 during the freeze-out [51] (in literatures such a phenomenon was called co-annihilation [52]). Since, in this case, the couplings ofν 1 with SM particles is usually very weak, the scattering is again naturally suppressed. We emphasize that, in either case, the suppression of the scattering prefers a small higgsino mass that appears in the coupling ofν * 1ν 1 state with Higgs bosons, and hence, there is no tension between the DM DD experiments and the naturalness for the mass of the Z boson [14].
In the ISS-NMSSM, the rates of the DM annihilation and the DM-nucleon scattering depend on the coupling strength ofν 1 interacting with Higgs fields, i.e., the Yukawa couplings λ ν and Y ν (the coefficient forν L ·Ĥ uνR interaction) and their corresponding softbreaking trilinear parameters A λν and A Yν . They also depend on the Higgs mass spectrum and the mixing between the Higgs fields that are ultimately determined by the parameters in the Higgs sector [14]. As such, the DM physics is quite complicated and is difficult to JHEP12(2020)023 understand intuitively. This fact inspired us to study the theory from different aspects, e.g., from the features of the DM-nucleon scattering [8,14], and its capability to explain the muon anomalous magnetic momentum [53] or other anomalies at the LHC [16]. In this work, we noted that a large λ ν or Y ν can enhance the DM-nucleon scattering rate significantly, so the recent XENON-1T experiment should limit them [54]. We also noted that the upper bound on the unitarity violation in neutrino sector sets a specific correlation between the couplings λ ν and Y ν [55], which can limit the parameter space of the ISS-NMSSM. Since these issues were not studied before, we decided to survey the impact of the leptonic unitarity and current and future DM DD experiments on the sneutrino DM sector. We will show that they are complementary to each other in limiting the theory, and in some cases, the constraints are rather tight. It is evident that such a study helps improve the understanding of the theory, and may be treated as a preliminary work before more comprehensive studies in the future.
We organize this work as follows. In section 2, we briefly introduce the theory of the ISS-NMSSM. In section 3, we describe the strategy to study the constraints, present numerical results and reveal the underlying physics. Finally, we draw our conclusions in section 4.

NMSSM with inverse seesaw mechanism
Since the ISS-NMSSM has been introduced in detail in [8,14], we only recapitulate its key features in this section.

Basics of the ISS-NMSSM
The renormalizable superpotential and the soft breaking terms of the ISS-NMSSM take following form [14] where W MSSM and L soft MSSM represent the corresponding terms of the MSSM without the µ-term. The terms in the first brackets on the right side of each equation make up the Lagrangian of the NMSSM that involves the Higgs coupling coefficients λ and κ and their soft-breaking parameters A λ and A κ . The terms in the second brackets are needed to implement the supersymmetric inverse seesaw mechanism. Coefficients such as the neutrino mass term µ X , the Yukawa couplings λ ν and Y ν , and the soft-breaking parameters A λν , A Yν , B µ X , mν, and mx are all 3×3 matrices in flavor space. Besides, among the parameters in the superpotential, only the matrix µ X is dimensional, and it parameterizes the effect of lepton number violation (LNV). Since this matric arises from the integration of massive JHEP12(2020)023 particles in the high-energy ultraviolet theory with LNV interactions (see, for example, [56][57][58]), its magnitude should be small. Based on a similar perspective, the matrix B µ X is also theoretically favored to be suppressed.
It is the same as the NMSSM that the ISS-NMSSM predicts three CP-even Higgs bosons, two CP-odd Higgs bosons, a pair of charged Higgs bosons, and five neutralinos. Throughout this work, we take λ, κ, where S 1 denotes the heavy doublet Higgs field with a vanishing vev, S 2 represents the SM Higgs field with its vev v ≡ 246 GeV, M 22 is the mass of S 2 at tree level without considering its mixing with the other bases, and M 23 characterizes the mixing of S 2 with the singlet field S 3 .
The squared mass matrix in eq. (2.1) can be diagonalized by a unitary matrix U , and its eigenstates h i with i = 1, 2, 3 are obtained by where h i are labelled in an ascending mass order, i.e. m h 1 < m h 2 < m h 3 . Then the couplings of h i to vector bosons W and Z and fermions u and d quarks, which are normalized to their SM predictions, take the following form [59] Obviously, U i2 1 if the components of the particle h i are far dominated by S 2 , and consequently, Similarly, the elements of the CP-odd Higgs fields' squared mass matrix are [13] M 2 P,11 = 2µ(λA λ + κµ) λ sin 2β , Concerning the neutralinos, they are the mixtures of the bino fieldB 0 , the wino fieldW 0 , the Higgsinos fieldsH 0 d andH 0 u , and the singlino fieldS 0 . In the bases , their mass matrix is given by [13] where M 1 and M 2 are soft breaking masses of the gauginos. It can be diagonalized by a rotation matrix N so that the mass eigenstates arẽ It is evident that N i3 and N i4 characterize theH 0 d andH 0 u components inχ 0 i , respectively, and N i5 denotes the singlino component.
In this work, we use the following features in the Higgs and neutralino sectors: • A CP-even state corresponds to the SM-like Higgs boson discovered at the LHC. This state is favored to be Re[H 0 u ]-dominated by the LHC data when tan β 1, and its mass may be significantly affected by the interaction λŝĤ u ·Ĥ d , the doublet-singlet Higgs mixing as well as the radiative correction from top/stop loops [37][38][39]. In the following, we denote this state as h.
• In most cases, the heavy doublet-dominated CP-even state is mainly composed of the field Re[H 0 d ]. It roughly degenerates in mass with the doublet-dominated CP-odd state and also with the charged states. The LHC search for extra Higgs bosons and the B-physics measurements requires these states to be heavier than about 500 GeV [60]. We represent them by H, A H , and H ± .
• Concerning the singlet-dominated states, they may be very light without conflicting with any collider constraint. As we introduced before, these states may appear as the JHEP12(2020)023 final state of the sneutrino pair annihilation or mediate the annihilation, and thus, they can play a vital role in the sneutrino DM physics. In this work, we label these states by h s and A s .
We add that, to study the property of the sneutrino DM, we consider two benchmark scenarios where all the input parameters for the Higgs and neutralino sectors are fixed. The details of the scenarios are presented in table 1. For the first scenario, h s and the SM-like Higgs boson h correspond to the lightest and the next-to-lightest CP-even Higgs bosons h 1 and h 2 . The S 2 component of h s is measured by the rotation element U 12 , which is determined by the elements M 2 23 and M 2 33 in eq. (2.1). We dub this scenario light h s scenario. By contrast, we call the second scenario as the massive h s scenario. It predicts h = h 1 , h s = h 2 , and U 22 to characterize the S 2 component in h s . Besides, we note that triple Higgs interactions may play an essential role in the sneutrino DM annihilation. So in addition to the couplingsC h i V * V ,C h iū u andC h id d , we also list in table 1 the coupling strengths for h s h s h s , h s h s h and h s hh interactions, which are normalized to the triple Higgs coupling in the SM and denoted byC hshshs ,C hshsh andC hshh , respectively. These strengths are obtained by the formulas in [13]. They are characterized by |C hshshs | |C hshsh |, |C hshh |, which is evident by the superpotential and the soft breaking terms of the ISS-NMSSM.

Leptonic unitarity
In the interaction bases (ν L , ν * R , x), the neutrino mass matrix is given by [14] where both the Dirac mass M D = Y ν v u / √ 2 and the Majorana mass M R = λ ν v s / √ 2 are 3 × 3 matrix in the flavor space. One can diagonalize this mass matrix by a 9 × 9 unitary matrix U ν to obtain three light neutrinos ν i (i = 1, 2, 3) and six massive neutrinos ν h as mass eigenstates, i.e., U * ν M ISS U † ν = diag(m ν i , m ν h ), and decompose U ν into the following blocks: The sub-matrixÛ 3×3 encodes the neutrino oscillation information and it is determined by the neutrino experimental results. Alternatively, one can get the analytic expression of the light active neutrinos' mass matrix from eq. (2.   where F = M T D M T −1 R , and its elements' magnitude is of the order M D / M R . This 3 × 3 matrix can be diagonalized by the unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, i.e., Due to the mixings among the states (ν L , ν * R , x), the matrixÛ in eq. (2.8) does not coincide with U PMNS . Instead, they are related by [61] where η = 1 2 F F † is a measure of the non-unitarity for the matrixÛ . A recent global fit of the theory to low energy experimental data reveals that [61] 2|η| ee < 0.050, 2|η| µµ < 0.021, 2|η| τ τ < 0.075, 2|η| eµ < 0.026, 2|η| eτ < 0.052, 2|η| µτ < 0.035. (2.12) We call these inequalities as the leptonic unitarity constraint.

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Eq. (2.9) indicates that the tininess of the active neutrino masses in the inverse seesaw mechanism is due to the smallness of the lepton-number violating matrix µ X and the sup- , the magnitude of the Dirac Yukawa coupling Y ν may reach order one in predicting m ν i ∼ 0.1 eV. However, a large Y ν may conflict with the unitarity constraint once the Majorana mass M R is specified. So the constraint must be taken into account in phenomenological study.
In the following, we will discuss the application of the unitarity constraint in DM physics. After noticing that the neutrino oscillation phenomenon can be explained by choosing an appropriate µ X [55,62], we assume flavor diagonal Y ν and λ ν to simplify the DM physics (see discussion below). We then determine µ X by the formula [55,62] where m ν i andÛ U PMNS take the values extracted from relevant neutrino experiments.
With the assumption, the neutrino oscillation is solely attributed to the non-diagonality of µ X , and the unitarity constraint in eq. (2.12) becomes These inequalities reveal that the ratio 33 may be significantly smaller than Concerning the LNV coefficients µ X and B µ X , one should note two points. One is that B µ X can induce an effective µ X through sneutrino-singlino loops to significantly affect the active neutrino masses by eq. (2.9). We estimate the correction by the mass insertion method, which was widely used in B physics study. We find where M 2 parameterizes the mixing of the fieldν * R with the fieldx, and M SUSY represents the sparticles' mass scale. Under the premise that Y ν , λ ν , and B µ X are flavor diagonal, M 2 can be roughly flavor diagonal, too (see the discussion in the next section). So one can study the correction in one generation case. The result is SUSY , the approximation requires B µ X /GeV 2 < 0.1 × M SUSY /GeV to get δµ X ∼ 1 keV. These estimations provide an upper bound on B µ X 's magnitude. In our study, we limit B µ X ≤ 100 GeV 2 for simplicity. The other point is that the LNV coefficients may induce sizable neutrinoless double beta decay since the inverse seesaw scale may be around several hundred GeV and the Yukawa couplings Y µ and λ ν may be moderately large. As indicated JHEP12(2020)023 in [63], because µ X is related to the active neutrino mass, the decay rate is below current experiment sensitivity when the massive neutrinos are heavier than 1 GeV. So there is no need to consider the constraint in our study.

Properties of sneutrino dark matter
If the sneutrino fields are decomposed into CP-even and CP-odd parts the squared mass of the CP-even fields is given by These formulas indicate the following facts: • The squared mass is a 9 × 9 matrix in three-generation (φ 1 , φ 2 , φ 3 ) bases. It involves a series of 3 × 3 matrices in the flavor space, such as Y ν , λ ν , A Yν , A λν , µ X , B µ X , ml, mν, and mx. Among these matrices, only µ X must be flavor non-diagonal to account for the neutrino oscillations, but since its magnitude is less than 10 keV [62], it can be neglected. Thus, if there is no flavor mixings for the other matrices, the squared mass is flavor diagonal, and one can adopt one-generation (φ 1 , φ 2 , φ 3 ) bases in studying the mass. In this work, we only consider the third generation sneutrinos as DM candidates. This is motivated by that both the unitarity bound and the LHC constraint in sparticle search are weakest for the third generation [14]. When we mention the sneutrino parameters hereafter, we are actually referring to their 33 elements. Under the assumption, the squared mass is diagonalized by a 3 × 3 unitary matrix V , which parameterizes the chiral mixings between the fields φ 1 , φ 2 and φ 3 .

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Consequently, the sneutrino mass eigenstates are given byν R,i = V ij φ j with i, j = 1, 2, and 3. We add that Y ν and λ ν are real and positive numbers after properly rotating the phase of the fieldsν R andX. • The mixing of φ 1 with the other fields is determined by Y ν and A ν . As Y ν approaches zero, |m 12 | and |m 13 | diminish monotonically, and so is |V 11 | which represents theν L component in the lightest sneutrino stateν R,1 . In the extreme case Y ν = 0, all these quantities vanish andν R,1 is merely the mixture of φ 2 and φ 3 . Furthermore, if λ ν /λ is moderately large, the first term in m 22 and m 33 may be far dominant over the other contributions so that m 22 m 33 . This results in maximal mixing between φ 2 and φ 3 andν R, 1 . This is a case encountered frequently in our study.
Similarly, one may adopt the one-generation (σ 1 , σ 2 , σ 3 ) bases to study the CP-odd sneutrino's mass, which is the same as eq. (2.17) except for the substitution B µ X → −B µ X . The mass eigenstates are then given byν where V denotes the rotation of the CP-odd fields. Given that B µ X represents the degree of the LNV and is theoretically small, we are particularly interested in the following two cases: • The extreme case of B µ X = 0 where any CP-even sneutrino state is accompanied by a mass-degenerate CP-odd state. In this case, any sneutrino mass eigenstate corresponds to a complex field and it has an anti-particle [64]. Concerning the sneutrino , andν 1 and its anti-particlẽ ν * 1 contribute equally to the relic density. This case is actually a two-component DM theory. It is notable that theν * 1ν 1 Z coupling is proportional to |V 11 | 2 and it contributes to the scattering ofν 1 with nucleons. This effect is important when |V 11 | ∼ 0.01 (discussed below). It is also notable that theν * 1ν 1 A i coupling vanishes since it is induced only by the LNV effect.
• A more general case satisfying |B µ X | 100 GeV 2 . It has four distinctive features. First, since mν R,i > mν I,i when B µ X > 0, the DM candidateν 1 is identified as theν I,1 state with a definite CP number −1. The opposite conclusion applies to B µ X < 0 case. Second, any CP-even state is slightly different from its CP-odd partner in mass, e.g., mν R,1 − mν I, 1 0.2 GeV when B µ X = 100 GeV 2 and mν R,1 = 100 GeV, and so are the rotations V and V . These sneutrino states compose a pseudo-complex particle [58,67,68]. Third, given the approximate mass degeneracy,ν R,1 andν I,1 always co-annihilated in early universe to affect the DM density. We will discuss this issue later. Finally, Z boson does not mediate the DM-nucleon scattering any more since it couples only to a pair of sneutrino states with opposite CP numbers. It also contributes little to the DM annihilation because theν R,1νI,1 Z coupling is suppressed by a factor V * 11 V 11 |V 11 | 2 .
We fix B µ X = 0 or B µ X = −100 GeV 2 in this work. In either case, theν *  [38,39], and Cν * 1ν 1 s on the right side denotes the sneutrino coupling to the scalar field s. For the one-generation sneutrino case, Cν * 1ν 1 s is given by These formulas indicate that the parameters Y ν , λ ν , A Yν , and A λν affect not only the sneutrino interactions but also their mass spectrum and mixing. In particular, a large λ ν or Y ν can enhance the coupling significantly. Instead, the soft-breaking masses m 2 ν and m 2 x affect only the latter property. For typical values of the parameters in eq. (2.19), e.g., It is estimated that |Cν * 100 GeV in most cases, which reflects that |Cν * 1ν 1 Re[S] | may be much larger than the other two couplings. The basic reason is thatν 1 is a singlet-dominated scalar, so it can couple directly to the field S and the mass dimension of Cν * 1ν 1 Re[S] is induced by v s or A λν . By contrast, the other couplings emerge only after the electroweak symmetry breaking when V 11 = 0, and their mass dimension originates from v u .

Relic density of sneutrino dark matter
In the B µ X = 0 case, bothν 1 andν * 1 act as the DM candidate. Their annihilation includes those initiated byν 1ν * 1 ,ν 1ν1 , andν * 1ν * 1 state, and the co-annihilation ofν 1 andν * 1 with the other sparticles. Considering the numerousness of the annihilation channels and the complexity of this issue, we will only discuss the channels frequently met in our study (see footnote 2 of this work for more details), which are [14,65]: (1)ν 1H ,ν * 1H → XY andHH → X Y , whereH andH denote Higgsino-dominated neutralinos or charginos, and X ( ) and Y ( ) represent any possible SM particles, the massive neutrinos or the Higgs bosons if the kinematics are accessible. More specifically, the channelsν 1H → W l, Zν, hν (l and ν denote any possible lepton and neutrino, respectively) proceed by the s-channel exchange of neutrinos, and the t/u channel exchange of sleptons or sneutrinos. The processesHH → ff , V V , hV (f JHEP12(2020)023 and f denote quarks or leptons, and V and V represent SM vector bosons) proceed by the s-channel exchange of vector bosons or Higgs bosons, and the t/u channel exchange of sfermions, neutralinos or charginos. This annihilation mechanism is called co-annihilation [51,52].
(2)ν 1ν * 1 → ss * (s denotes a light Higgs boson), which proceeds through any relevant quartic scalar coupling, the s-channel exchange of CP-even Higgs bosons, and the t/u-channel exchange of sneutrinos.
(3)ν 1ν * 1 → ν hνh via the s-channel exchange of CP-even Higgs bosons or the t/u-channel exchange of neutralinos, where ν h denotes a massive neutrino.
1 →ν hνh , which mainly proceed through the t/u-channel exchange of a singlino-dominated neutralino due to its majorana nature. Under specific parameter configurations, these channels can be responsible for the DM density precisely measured by the Planck experiment [66]. In this aspect, we have the following observations (see footnote 2 for more explanations): • In most cases, the DMs annihilated mainly through the co-annihilation to get the measured density. This mechanism works only when the mass splitting betweenH andν 1 is less than about 10%, and a specific channel's contribution to the density depends not only on its cross-section but also on the mass splitting. To illustrate this point, we assume that the DM annihilations comprise those initiated byν 1ν1 ,ν 1ν * 1χ 0 1 states, and denote the cross-sections of these channels by σ AB with A, B =ν 1 ,ν * 1 ,χ 0 1 . The effective annihilation rate at temperature T is then given by [52] σ eff = 1 4 (1 + ∆) 3/2 e −x∆ + 4σχ0 Besides, the formulae of the density in [52] indicate that the density depends on the sneutrino parameters only through mν 1 and σ eff . In the extreme case of σν 1ν1 σν 1ν * 1 σν 1χ 0 1 0 realized when λ ν and Y ν are sufficiently small, theχ 0 1χ 0 1 annihilation is solely responsible for the measured density through tuning the value of mν 1 . This situation was intensively studied in [8]. We will present such examples in section 3.
• Barring the co-annihilation,ν 1ν1 → ss * is usually the most crucial channel in affecting the density if the kinematics are accessible. In particular, the processν 1ν1 → h s h s can be solely responsible for the measured density if the Yukawa coupling λ ν is moderately large. We exemplify this point by considering the light h s scenario in table 1. From the Higgs boson and sneutrino mass spectrum and theν 1 's couplings to h s , one can learn that the annihilation proceeds mainly by the s-channel exchange of h s , t/u-channel exchange ofν 1 , andν 1ν * 1 h s h s quartic scalar coupling. As a result, the cross-section of the annihilation near the freeze-out temperature is approximated by [14,65] σv a + bv 2 , The measured density then requires a + 3b/25 4.6 × 10 −26 cm 2 because we are considering a two-component DM theory [69,70]. This requirement limits theν 1 's couplings to h s or for the fixed parameters in table 1, ultimately the Yukawa coupling λ ν since the cross-section is very sensitive to λ ν . We estimate that λ ν ∼ 0.4 for mν 1 = 130 GeV can account for the measured density.
• The processν 1ν * 1 → ν hνh could be responsible for the density when mν 1 > ν h , mν 1 < h s , and the co-annihilation mechanism did not work. This process proceeded mainly by the s-channel exchange of h s , and consequently, the cross-section at the freeze-out temperature T f takes the following form: which implies that the density limits non-trivially λ ν , mν 1 and m hs .
• About the other channels, they usually played a minor role in determining the density. So we leave the discussion of them in our future works.
Concerning the B µ X = 0 case, eitherν R,1 orν I,1 acts as the DM candidate. Since the mass splitting between the DMν 1 andν 1 (the partner ofν 1 with a different CP number) is small,ν 1 always co-annihilated withν 1 to get the measured density. The relevant annihilation includedν 1ν 1 andν 1ν 1 initiated processes, and they proceeded in a way similar to the previous discussion. We confirmed that the density is insensitive to B µ X for |B µ X | ≤ 100 GeV 2 , which can be inferred from eq. (2.21). We also verified that the cross-section of the DM annihilation today is insensitive to B µ X .

DM-nucleon scattering
In the B µ X = 0 case, the scattering ofν 1 with nucleon N (N = p, n) proceeds by the t/u-channel exchange of the CP-even Higgs bosons. Consequently, the spin independent (SI) cross-section is given by [14] σ SĨ , n) represents the reduced mass of the nucleon with mν 1 , where δ = m 2 h /m 2 hs − 1, and θ is the mixing angle of the S 2 field with Re[S] to form mass eigenstates. This formula reveals that if the terms in the second brackets are on the order of 0.1, which can be achieved if λ ν and/or Y ν in eq. (2.20) are sufficiently large, the cross-section may reach the sensitivity of the recent XENON-1T experiment [54]. We will discuss this issue later. 1 It is notable that σ0 was replaced by the strangeness-nucleon sigma term, σs ≡ ms/(mu + m d ) × (σπN − σ0) 12.4 × (σπN − σ0), in recent calculation of the nucleon form factor [72]. Compared with the previous calculation, this treatment changes significantly the strange quark content in nucleon N , f N s , but it change little F N u and F N d .

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Concerning the B µ X = 0 case, where the DM corresponds to a complex field, the Z-boson also mediates the elastic scattering of the DM with nucleons. Since the total SI cross-section in this case is obtained by averaging overν 1 N andν * 1 N scatterings and the interferences between the Z-and the Higgs-exchange diagrams for the two scatterings have opposite signs [79], the SI cross-section is given by [4] where σ h N is the same as before and the Z-mediated contributions are with G F and θ W denoting the Fermi constant and the weak angle, respectively. Since σ Z n is larger than σ Z p by a factor around 100, σ SI n may differ significantly from σ SI p . Correspondingly, one may define the effective cross-section for the coherent scattering of the DMs with xenon nucleus as σ SI eff = (σ SĨ where A denotes the mass number of the xenon nucleus, and calculate it by where the three coefficients on the right side are obtained by averaging the abundance of different xenon isotopes in nature. It is evident that the effective cross-section is identical to σ SI p if σ SI p = σ SI n , and it is related directly with the bound of the XENON-1T experiment [54]. Before concluding the introduction of the sneutrino DM, we add that its spin dependent cross-section is always zero, and its SI cross-section is usually much smaller than that of the neutralino DM in the MSSM and NMSSM, which was discussed in detail in refs. [8,14]. As a result, the extension is readily consistent with the XENON-1T experiment except for large λ ν and/or Y ν case studied in this work.

Constraints on sneutrino DM sector
In this section, we clarify the impact of the leptonic unitarity and current and future DM DD experiments on the sneutrino DM sector under the premise that the theory predicts the right density and the photon spectrum from the DM annihilation in dwarf galaxies is compatible with the Fermi-LAT observation. Since the singlet-dominated Higgs boson, h s , plays a vital role in the density and the DM-nucleon scattering, we study the DM physics in both the light and the massive h s scenarios in table 1. We emphasize that fixing the parameters in the Higgs and neutralino sectors can simplify greatly the analysis of the impact and make the underlying physics clear. We also emphasize that the two scenarios were obtained by scanning intensively the parameters in the Higgs and DM sectors.

Research strategy
The procedure of our study is as follows. We constructed a likelihood function of the DM physics to guide sophisticated scans over the sneutrino parameters for either scenario. With the samples obtained in the scans, we plotted the profile likelihood map in different two-dimensional planes to illustrate its features and underlying physics. We express the likelihood function as where L Ων 1 , L DD , L ID , and L Unitary describe the relic density, the current XENON-1T experiment [54] or the future LZ experiment [82], the Fermi-LAT observation of dwarf galaxies, and the unitarity constraint, respectively. They are given by where Ω th denotes the theoretical prediction of the density Ων 1 h 2 , Ω obs = 0.120 represents its experimental central value [66], and σ = 0.1×Ω obs is the total (including both theoretical and experimental) uncertainty of the density.
• L DD takes a Gaussian distributed form with a mean value of zero [83]: In this formula, σ SI eff is defined in eq. (2.27) and its error bar δ σ is evaluated by δ σ = U L 2 σ /1.64 2 + (0.2σ SI eff ) 2 , where U L σ denotes experimental upper limits on the scattering cross-section at 90% C.L. and 0.2σ SI eff parameterizes the theoretical uncertainty of σ SI eff . • L ID is calculated by the likelihood function proposed in [84,85] with the data of the Fermi-LAT collaboration taken from [86,87].

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In addition, we abandoned samples that open up the decays h → ν hνh ,ν 1ν * 1 . 3 In practice, this was completed by setting the likelihood value to be e −100 if any of the decays were kinematically accessible.
To make the conclusions in this study complete, we adopt the MultiNest algorithm [88,89] to implement the scans. We take the prior probability density function (PDF) of the input parameters uniformly distributed and set nlive parameter of the algorithm to be 10000. This parameter represents the number of active or live points used to determine the iso-likelihood contour in each scan's iteration [88,89]. The larger it is, the more elaborated the scan becomes. The output of the scans includes the Bayesian evidence defined by where P (Θ|M ) represents the prior PDF of the inputs Θ = (Θ 1 , Θ 2 , · · · ) in a model M , and P (D|O(M, Θ)) ≡ L(Θ) denotes the likelihood function involving theoretical predictions of observables O and their experimental measurements D. Computationally, the evidence is an averaged likelihood that depends on the priors of the theory's input. In comparing different scenarios of the theory, the larger Z is, the more readily the corresponding scenario is consistent with the data. The output of the scan also includes the profile likelihood (PL) defined in frequentist statistics as the most significant likelihood value [15,90]. For example, two-dimensional (2D) PL is defined by where the maximization is obtained by varying the parameters other than Θ A and Θ B . The PL reflects the preference of the theory on the parameter (Θ A , Θ B ), or in other words, the capability of the parameter to account for experimental data. Sequentially, one can introduce the concept of confidence interval (CI) to classify the parameter region by how well the points in it fit the data. For example, the 1σ and 2σ CIs for the 2D PL are defined by satisfying χ 2 − χ 2 min ≤ 2.3 and χ 2 − χ 2 min ≤ 6.18, respectively, where χ 2 ≡ −2 ln L(Θ A , Θ B ) and χ 2 min is the minimal value of χ 2 for the samples obtained in the scan. In this work, we utilized the package SARAH-4.11.0 [91][92][93] to build the model file of the ISS-NMSSM, the SPheno-4.0.3 [94] code to generate its particle spectrum, and the package MicrOMEGAs 4.3.4 [71,73,95] to calculate the DM observables. 3 In fitting the ISS-NMSSM to experimental data, the total likelihood function is calculated by Ltot = L Higgs × L DM , where L Higgs represents the Higgs physics function. Given χ 2 ≡ −2 ln L, one can infer that χ 2 tot = χ 2 Higgs + χ 2 DM , and the 2σ confidence interval defined below eq. (3.5) satisfies d ] by two orders, the first term in the brackets can be comparable with the other contributions. To clarify the impact of the unitarity and DM DD experiments on the theory, we performed four independent scans over the following parameter space: where ml denoted the common soft breaking mass of three-generation sleptons and its lower bound was motivated by the non-observation of slepton signals at the LHC Run-II.
For the first scan, we fixed B µ X = −100 GeV 2 and used the XENON-1T's bound on the SI cross-section to calculate the L DD . The second scan was same as the first one except that we adopted the sensitivity of the LZ experiment. The last two scans differed from the previous ones only in that we set B µ X = 0. As explained before, the setting induces an additional Z-mediated contribution to the DM-nucleon scattering so that the constraints of the DD experiments are strengthened.
With the samples obtained in the scans, we show different 2D PL maps in figures 1 to 5. Figure 1 and 2 plot the CIs on λ ν − mν 1 and σ SĨ ν 1 −p − mν 1 planes. They show the following features: • mν 1 is concentrated on the range from 120 to 181 GeV. Specifically, mν 1 is close to mχ0 1 for 172 GeV mν 1 181 GeV, and the DM achieves the correct density mainly through theχ 0 1χ 0 1 annihilation (see discussions about the co-annihilation in section 2.4 and details of the points in subsequent table 2). In this case, the density is insensitive to the parameter λ ν . Thus, λ ν varies within a broad range from 0.15 to 0.6 in figure 1, where the lower limit forbids the decay h → ν hνh kinematically, and the upper bound comes from the DM DD experiments (discussed below). For the other mass range, the DM obtains the correct density mainly through the annihilations ν 1ν * 1 → h s h s , h s h, hh. This requires λ ν 0.26, which can be understood from the discussion of eq. (2.22).
• Figure 2 indicates that the SI cross-section of the DM-nucleon scattering may be as low as 10 −49 cm 2 over the entire mass range. It reflects that the theory has multiple mechanisms to suppress the scattering, which becomes evident by the approximation in eq. (3.6) and was recently emphasized in [8].
• Although λ ν > 0.6 is allowed by the setting in eq. of the left and right panels revealed that it was due to the DD experiments' constraint on the co-annihilation region. Besides, we studied the Bayesian evidences Z i (i = 1, 2, 3, 4) of the four cases and found ln Z 1 = −55.6, δ 12 ≡ ln Z 1 − ln Z 2 = 1.0, δ 13 ≡ ln Z 1 − ln Z 3 = 0.54, and δ 34 ≡ ln Z 3 − ln Z 4 = 1.43. These results reveal at least two facts. On the one side, the Jeffreys' scale δ 13 [96,97] reflects that current XENON-1T experiment has no significant preference of the B µ X = 0 case to the B µ X = 0 case [98]. On the other side, δ 12 and δ 34 show that the Bayesian evidence (or equivalently the averaged L DM ) is reduced by a factor of more than 40%. It implies that a sizable portion of the parameter space will become disfavored once the future LZ experiment improves the XENON-1T's sensitivity by 50 times. This feature is also reflected in figure 1 and figure 2 by the sizable shrink of the 1σ CIs. In order to better understand figure 1, we describe how we obtained it. From eq. (3.5), the 2D PL L(λ ν , mν 1 ) is given by A Yν , mν, mx, ml). (3.8) In plotting the figure, we implemented the maximization over the parameters Y ν , A λν , A Yν , mν, mx, and ml by three steps. First, we split the λ ν − mν 1 plane into 80 × 80 equal boxes, i.e., we divided each dimension of the plane by 80 regular bins. Second, we fit the samples obtained in the scan into each box. Consequently, samples in each box correspond to roughly equal λ ν and mν 1 , even though the other parameters may differ significantly. Finally, we select the maximum likelihood value of the samples in each box as the PL value. These procedures imply that the CIs are not necessarily contiguous, instead they usually distributed in isolated islands [15,90]. Besides, we emphasize that χ 2 min 0 for the best point in the scans. This is because the DM experiments are independent and consistent with each other, and the ISS-NMSSM can explain them well.

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Next, we study 2D PL on Y ν − λ ν plane. The results are shown in figure 3 where the red dashed line denotes the correlation λ ν µ/(Y ν λv u ) = 9.4 or equivalently λ ν = 2.9Y ν from the unitarity constraint. This figure shows that Y ν is maximized at 0.17 when λ ν 0.52 and it is upper bounded only by the unitarity. The reason is that the unitarity requires JHEP12(2020)023 λ ν 2.9Y ν , so the SI cross-section is much more sensitive to λ ν than to Y ν . Consequently, the DD experiments set the upper bound of λ ν and by contrast, the unitarity limits Y ν .
We also plot 2D PLs on Y ν −ml and V 11 −ml planes in figure 4 and figure 5, respectively. Figure 4 indicates that the 2σ CI in each panel occupies a roughly rectangular area on the Y ν − ml plane. This result reflects that L DM is insensitive to parameter ml. It can be understood from the following two aspects. One is that L DM relies on ml mainly through V 11 by theν 1ν * 1 h i coupling in eq. (2.19). The other is that ml and V 11 are weakly correlated, which can be inferred by the expression of m 12 and m 13 in eq. (2.18) and is shown numerically in figure 5 and figure 10. Specifically, for the B µ X = 0 case, both the annihilation and the scattering are insensitive to V 11 since its magnitude is small, and so is L DM . This property determines that the allowed range of Y ν is roughly independent of ml, and thus explains the rectangular shape. For the B µ X = 0 case, although the effective cross-section in eq. (2.27) is sensitive to V 11 by the formula in eq. (2.26), the XENON-1T experiment has required |V 11 | 0.02 and this upper bound is very insensitive to ml. In this case, one may replace ml by V 11 as a theoretical input so that L DM does not depend on ml any more. This feature again leads to the conclusion that the allowed range of Y ν is roughly independent of ml. We add that the tight experimental constraint on the mixing V 11 for the B µ X = 0 case was also discussed in [99]. We also add that one may fix ml in JHEP12(2020)023 performing global fit of the ISS-NMSSM to experimental data due to the insensitivity of L DM to ml. Such a treatment affects little the generality of the fit results.
In table 2, we present the details of two points to illustrate the scenario's features further. For the point P1, the DMs annihilated mainly byν 1ν * 1 → h s h s , h s h to get the density. The processν 1ν * 1 → hh is unimportant because |C hshh | is significantly smaller than |C hshshs | and |C hshsh |, and also because the phrase space of the final state is relatively small. By contrast, the DMs got their right relic density mainly by the Higgsino pair annihilation for the point P2, and the mass splitting is ∆ ≡ mχ0 1 −mν 1 7 GeV. We confirmed that, due to the specific parameter setting of P2, there is cancellation between different contributions to the processν 1ν * 1 → h s h s , and consequently, its effect is negligibly small. Besides, both the points predict Y ν ∼ 0.01. As a result, V 11 's magnitude is only a few thousandths, and the DM-neutron scattering rate is not much larger than the DM-proton scattering rate. We verified that, once we set B µ X = −100 GeV, the two rates became roughly equal.

Results for the massive h s scenario
In the massive h s scenario, the Higgs-mediated SI cross-section is given by · · · · · · · · · · · · · · · u ] are 100 GeV and 10 GeV, respectively. Thus, the first term in the brackets is no longer more critical than the other terms, and the σ SĨ ν 1 −N for mν 1 300 GeV may reach 10 −46 cm 2 only in optimal cases. Consequently, the XENON-1T experiment scarcely limit the B µ X = 0 case. This situation is significantly different from the light h s scenario.
Similar to the analysis of the light h s scenario, we performed four independent scans over the parameter region in eq. (3.7), and projected the PL onto different planes. The results are presented from figure 6 to figure 10 in a way similar to those for the light h s scenario. These figures indicate the following facts: • Since the unitarity for the parameters in table 1 requires only λ ν 1.3 Y ν , Y ν may be comparable with λ ν in size. As a result, the SI cross-section is sensitive to both λ ν and Y ν , which is different from the light h s scenario. However, with the experimental sensitivity improved or the Z-mediated contribution considered in the B µ X = 0 case, the DM DD experiments become powerful enough to limit λ ν and Y ν . In this case, Y ν 0.4 may contradict the experiments, which is indicated by the other panels of figure 8.
•ν 1 obtained the correct density through the co-annihilation withχ 0 1 , which is reflected by the range of mν 1 in figure 6. We will take the points P3 and P4 in table 2 as examples to show more details of the annihilation later.
We confirmed that δ 12 ≡ ln Z 1 − ln Z 2 = 0.2, δ 13 ≡ ln Z 1 − ln Z 3 = 0.62 and δ 34 ≡ ln Z 3 − ln Z 4 = 0.56 in the massive h s scenario. Similar to the analysis of the light h s scenario, the smallness of δ 12 and δ 34 reflects that the LZ experiment can not improve the constraint of the XENON-1T experiment on the scenario significantly, and the smallness of δ 13 reflects that the XENON-1T experiment does not show significant preference of the B µ X = 0 case to the B µ X = 0 case.
• Concerning the other features of the massive h s scenario, such as the suppression of the SI cross-section and the correlation of ml with Y ν and V 11 , they are similar to those of the light h s scenario. We do not discuss them anymore.
Next, let us study two representative points, P3 and P4, of the massive h s scenario in table 2. For the former point, it is the annihilation of the Higgsino pair that is responsible JHEP12(2020)023 for the measured density, and the corresponding mass splitting is about 5 GeV. By contrast, theν 0 1H annihilation mainly accounts for the latter point density, and the mass splitting reaches about 19 GeV. The difference is caused by the fact that P4 takes a relatively large Y ν and a lighter ml, making theν 0 1H annihilation more critical. Besides, it is notable that both the points predict Y ν 0.18 to induce a sizableν L component inν 1 , e.g., |V 11 | > 0.01. Consequently, Z boson can mediate a large DM-neutron scattering so that σ SĨ In summary, both λ ν and Y ν are more constrained in the light h s scenario than in the massive h s scenario. The unitarity always plays a vital role in limiting Y ν except for the case shown in the last panel of figure 8, where the LZ experiment may be more critical in limiting Y ν . We emphasize that the tight DD constraint on the B µ X = 0 case of the light h s scenario arises from that h s is light and it contains sizable doublet components. In this case, the coupling Cν * 1ν 1 Re[S] contributes significantly to the scattering rate. Before we end this section, we emphasize that the parameter points discussed in this work are consistent with the LHC results in searching for sparticles. Specifically, for the parameters in table 1, it is evident that the LHC fails to detect gluinos and squarks because these particles are too massive. Concerning the Higgsino-dominated particles, they may be detectable at the 8 TeV and 13 TeV LHC since their production rates reach 100 fb. We JHEP12(2020)023 scrutinized the property of the points in B µ X = 0 case and found that they all predict Br(χ 0 1,2 →ν 1ντ ) = Br(χ 0 1,2 →ν * 1 ν τ ) 50%, Br(χ ± 1 →ν ( * ) 1 τ ± ) 100%, (3.9) due to the Yukawa interaction Y νl ·Ĥ uνR in the superpotential. In this case, the most promising way to explore the two scenarios at the LHC is to search the Di-τ plus missing momentum signal through the process pp →χ ± 1χ ∓ 1 → (τ ± E Miss T )(τ ∓ E Miss T ) [14,15]. So far, the ATLAS collaboration has finished three independent analyses of the signal based on 20.3 fb data at the 8 TeV LHC [100], 36.1 fb data at the 13 TeV LHC [101], and 139 fb data at the 13 TeV LHC [102], respectively. We repeated these analyses by elaborated Monte Carlo simulations, like what we did for the first two analyses in [14,15]. We found that the tightest constraint on the two scenarios comes from the last analysis, and its efficiency in detecting the signal decreases gradually as the gap between mν 1 and mχ± 1 becomes narrow. As far as the light and massive h s scenarios are concerned, the analysis can not exclude at 95% confidence level the points satisfying mν 1 100 GeV and mν 1 200 GeV, respectively. So we conclude that the LHC analyses do not affect the results presented in this work.

Conclusion
Motivated by the increasingly tight limitation of the DM DD experiments on the traditional neutralino DM in the natural MSSM and NMSSM, we extended the NMSSM by the JHEP12(2020)023 inverse seesaw mechanism to generate the neutrino mass in our previous studies [8,14,16], and studied the feasibility that the lightest sneutrino acts as a DM candidate. A remarkable conclusion for the theory is that experimental constraints from both the collider and DM search experiments are relaxed significantly. Consequently, large parameter space of the NMSSM that has been experimentally excluded resurrects as physical points in the extended theory. In particular, the higgsino mass may be around 100 GeV to predict Z-boson mass naturally. This feature makes the extension attractive and worthy of a careful study.
We realized that sizable neutrino Yukawa couplings λ ν and Y ν contributed significantly to the DM-nucleon scattering rate. Thus, the recent XENON-1T experiment could limit them. We also realized that the unitarity in the neutrino sector set a specific correlation between the couplings λ ν and Y ν , which in return limited the parameter space of the ISS-NMSSM. Since these issues were not studied before, we investigated the impact of the leptonic unitarity and current and future DM DD experiments on the sneutrino DM sector in this work. Specially, we considered the light and massive h s scenarios after noticing that the singlet dominated Higgs plays a vital role in both the DM annihilation and the DMnucleon scattering. For each scenario, we studied the B µ X = 0 and B µ X = 0 case separately. Their difference comes from that Z boson can mediate the DM-nucleon scattering for the B µ X = 0 case, and thus, the experimental constraints on it are much tighter. JHEP12(2020)023 In this work, we encoded the experimental constraints in a likelihood function and performed sophisticated scans over the vast parameter space of the model by the Nested Sampling method. The results of our study are summarized as follows: • The XENON-1T experiment set an upper bound on the couplings λ ν and Y ν , and the future LZ experiment will improve the bound significantly. The limitation is powerful when h s is light and contains sizable doublet components.
• As an useful complement to the DM DD experiments, the unitarity always plays a vital role in limiting Y ν . It becomes more and more powerful when v s approaches v from top to bottom.
• The parameter space favored by the DM experiments shows a weak dependence on the left-handed slepton soft mass ml. This property implies that one may fix ml in surveying the phenomenology of the ISS-NMSSM by scanning intensively its parameters and considering various experimental constraints. This treatment does not affect the comprehensiveness of the results. the B µ X = 0 case and |V 11 | 0.02 for the B µ X = 0 case; these upper bounds become 0.10 and 0.01, respectively, if one adopts the LZ experiment's sensitivity.
Finally, we briefly discuss the phenomenology of the ISS-NMSSM. The sparticles's signal in this theory may be distinct from those in traditional supersymmetric theories, and so is the strategy to look for them at the LHC. This feature can be understood as follows: since the sneutrino DM carries a lepton number, and in most cases has feeble interactions with particles other than the singlet-dominated Higgs boson and the massive neutrinos, the sparticle's decay chain is usually long, and its final state contains at least one τ or ν τ . In addition, the decay branching ratio depends not only on particle mass spectrum but also on new Higgs couplings, such as Y ν and λ ν . As a result, sparticle's phenomenology is quite complicated [8,16]. Depending on the mechanism by which the DM obtained the correct density, one usually encounters the following two situations: • The DM co-annihilated with the Higgsino-dominated particles. This situation requires the mass splitting ∆ ≡ mχ0 1 −mν 1 to be less than about 10 GeV. Consequently, the Higgsino-dominated particles usually appear as missing momentum at the LHC due to the roughly degenerate mass spectrum. As pointed out in [8], this situation's phenomenology may mimic that of the NMSSM with the Higgsino-dominatedχ 0 1 as a DM candidate.

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• The singlet-dominated particlesν 1 , h s , A s , and ν h compose a secluded DM sector where the DM was mainly annihilated by any of the channelsν 1ν * 1 → A s A s , h s h s , ν hνh . It communicates with the SM sector by the Higgs-portal or the neutrino-portal. As we introduced before, this situation constrains the Yukawa coupling λ ν tightly in getting the measured density, but it has no limitation on the splitting between mν 1 and the Higgsino mass. As mentioned before, the signals of the sparticles in this situation are complicated. However, systematic researches on this subject are still absent.
We suggest experimentalists to look for the 2τ plus missing momentum signal of the process pp →χ ± 1χ ∓ 1 → (τ ± E Miss T )(τ ∓ E Miss T ) in testing the theory. Unlike the colored sparticles that may be very massive, light Higgsinos are favored by natural electroweak symmetry breaking. As a result, they are expected to be richly produced at the LHC. For the secluded DM case, ATLAS analyses have excluded some parameter space discussed at the end of the last section. With the advent of the LHC's high luminosity phase, more parameter space will be explored. For example, we once compared the ATLAS analyses of the signal at the 13 TeV LHC with 36.1 fb −1 and 139 fb −1 data [101,102]. We found the excluded region on mν 1 − mχ± 1 plane expanded from mχ0 Concerning the co-annihilation case, it is hard for the LHC to detect the signal due to the compressed spectrum, but the future International Linear Collider may be capable of doing such a job (see, for example, the study in [103] for the compressed spectrum case). We emphasize that, different from the prediction of the MSSM, mχ± 1 may be significantly larger than mχ0 1 in the ISS-NMSSM due to the mixing ofH u,d withS in eq. (2.5). As a result, the splitting between mχ± 1 and mν 1 can reach 20 GeV (see the points in table 2), and it becomes even larger as the parameter λ increases. This feature is beneficial for the signal's detection.