Impact of leptonic unitarity and dark matter direct detection experiments on the NMSSM with inverse seesaw mechanism
Junjie Cao, Yangle He, Yusi Pan, Yuanfang Yue, Haijing Zhou, Pengxuan Zhu
IImpact of leptonic unitarity and dark matter direct detection experiments on the NMSSM with inverse seesaw mechanism
Junjie Cao a,b , Yangle He a , Yusi Pan a , Yuanfang Yue a , Haijing Zhou a ,Pengxuan Zhu aa School of Physics, Henan Normal University, Xinxiang 453007, China b Center for High Energy Physics, Peking University, Beijing 100871, China
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
In the Next-to-Minimal Supersymmetric Standard Model with the in-verse seesaw mechanism to generate neutrino masses, the lightest sneutrino may actas a feasible dark matter candidate in vast parameter space. In this case, the small-ness of the leptonic unitarity violation and the recent XENON-1T experiment canlimit the dark matter physics. In particular, they set upper bounds of the neutrinoYukawa couplings λ ν and Y ν . We study such effects by encoding the constraints ina likelihood function and carrying out elaborated scans over the parameter space ofthe theory with the Nested Sampling algorithm. We show that these constraints arecomplementary to each other in limiting the theory, and in some cases, they are verystrict. We also study the impact of the future LZ experiment on the theory. a r X i v : . [ h e p - ph ] D ec ontents h s scenario 183.3 Results for the massive h s scenario 25 As the most popular ultraviolet-complete Beyond Standard Model, the MinimalSupersymmetric Standard Model (MSSM) with R-parity conservation predicts twokinds 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 super-symmetric particle (LSP) predicted a much smaller relic abundance than the mea-sured value as well as an unacceptably tremendous DM-nucleon scattering rate dueto 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 DMcandidate, and consequently, it was studied intensively since then. However, withthe rapid progress in DM direct detection (DD) experiments in recent years, thecandidate became more and more tightly limited by the experiments [5–8] assumingthat 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) [10–12], where the sneutrinos are purely left-handed, and the neutralinoDM candidate may be either bino- or singlino-dominated [13]. In this context, we– 1 –evived the idea of the sneutrino DM in a series of works [8, 14–16]. In particular, mo-tivated by the phenomenology of the neutrino oscillations, we extended the NMSSMwith the inverse seesaw mechanism by introducing two types of gauge singlet chiralsuperfields ˆ ν R and ˆ X for each generation matter, which have lepton numbers -1 and1, respectively, and their fermion components corresponded to the massive neutrinosin literatures [14]. Subsequently, we studied in detail whether the ˜ ν R (the scalarcomponent of ˆ ν R ) or ˜ x (the scalar component of ˆ X ) dominated sneutrino could actas a feasible DM candidate [14]. We were interested in the inverse seesaw mechanismbecause it was a TeV scale physics to account for the neutrino oscillations and maybeexperimentally testable soon. We showed by both analytic formulas and numericalcalculations that the resulting theory (abbreviated as ISS-NMSSM hereafter) wasone 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, alot of authors have showed interest in the sneutrino DM in recent years [17–36], butnone of them considered the same theoretical framework as ours.In the NMSSM, the introduction of the singlet field ˆ S can solve the µ problemof the MSSM [13], enhance the theoretical prediction of the SM-like Higgs bosonmass [37–39], as well as enrich the phenomenology of the NMSSM significantly (see,for example, Ref. [40–45]). In the ISS-NMSSM, the ˆ S field also plays extraordinaryroles in generating the massive neutrino mass by the Yukawa interaction λ ν ˆ S ˆ ν R ˆ X and making the sneutrino DM compatible with various measurements, especially theDM DD experiments [14]. There are at least two aspects in manifesting the latterrole. One is that the newly introduced heavy neutrino superfields are singlet underthe gauge group of the SM model. Thus, they can interact directly with ˆ S by theYukawa couplings [14]. In this case, the sneutrino DM candidate ˜ ν , the singlet dom-inated scalars h s and A s , and the massive neutrinos ν h compose a roughly secludedDM sector where the annihilations ˜ ν ˜ ν ∗ → A s A s , h s h s , ν h ¯ ν h can produce the mea-sured relic density (In the ISS-NMSSM, these annihilations proceed by quartic scalarinteractions, s -channel exchange of h s and t/u -channel exchange of the sneutrinosor the singlino-dominated neutralino). Since this sector communicates with the SMsector by the small singlet-doublet Higgs mixing (dubbed by Higgs-portal in litera-tures [46]) and/or by the massive neutrinos (neutrino-portal [47–50]), the scatteringof the DM with nucleons is naturally suppressed, which coincides with current DMDD results. The other aspect is that the singlet-dominated Higgs scalars can me-diate the transition between ˜ ν pair and the higgsino pair, and consequently, theseparticles were in thermal equilibrium in early Universe before their freeze-out fromthe thermal bath. If their mass splitting is less than about 10%, the number densityof the higgsinos can track that of ˜ ν during the freeze-out [51] (in literatures such aphenomenon was called co-annihilation [52]). Since, in this case, the couplings of ˜ ν 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– 2 –iggsino mass that appears in the coupling of ˜ ν ∗ ˜ ν state with Higgs bosons, andhence, there is no tension between the DM DD experiments and the naturalness forthe mass of the Z boson [14].In the ISS-NMSSM, the rates of the DM annihilation and the DM-nucleon scat-tering depend on the coupling strength of ˜ ν interacting with Higgs fields, i.e., theYukawa couplings λ ν and Y ν (the coefficient for ˆ ν L · ˆ H u ˆ ν R interaction) and their cor-responding soft-breaking trilinear parameters A λ ν and A Y ν . They also depend on theHiggs mass spectrum and the mixing between the Higgs fields that are ultimatelydetermined by the parameters in the Higgs sector [14]. As such, the DM physics isquite complicated and is difficult to understand intuitively. This fact inspired us tostudy the theory from different aspects, e.g., from the features of the DM-nucleonscattering [8, 14], and its capability to explain the muon anomalous magnetic mo-mentum [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 recentXENON-1T experiment should limit them [54]. We also noted that the upper boundon the unitarity violation in neutrino sector sets a specific correlation between thecouplings 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 theleptonic unitarity and current and future DM DD experiments on the sneutrino DMsector. We will show that they are complementary to each other in limiting thetheory, and in some cases, the constraints are rather tight. It is evident that sucha study helps improve the understanding of the theory, and may be treated as apreliminary work before more comprehensive studies in the future.We organize this work as follows. In section 2, we briefly introduce the theoryof 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 ourconclusions in section 4. Since the ISS-NMSSM has been introduced in detail in [8, 14], we only recapitulateits key features in this section.
The renormalizable superpotential and the soft breaking terms of the ISS-NMSSMtake following form [14] W = (cid:20) W MSSM + λ ˆ s ˆ H u · ˆ H d + 13 κ ˆ s (cid:21) + (cid:20) µ X (cid:98) X (cid:98) X + λ ν ˆ s ˆ ν R (cid:98) X + Y ν ˆ l · ˆ H u ˆ ν R (cid:21) ,L soft = (cid:104) L soft MSSM − m S | S | − λA λ SH u · H d − κ A κ S (cid:105) − (cid:20) m ν ˜ ν R ˜ ν ∗ R + m x ˜ x ˜ x ∗ + 12 B µ X ˜ x ˜ x + ( λ ν A λ ν S ˜ ν ∗ R ˜ x + Y ν A Y ν ˜ ν ∗ R ˜ lH u + h.c.) (cid:21) , – 3 –here W MSSM and L soft MSSM represent the corresponding terms of the MSSM withoutthe µ -term. The terms in the first brackets on the right side of each equation makeup 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 bracketsare needed to implement the supersymmetric inverse seesaw mechanism. Coefficientssuch as the neutrino mass term µ X , the Yukawa couplings λ ν and Y ν , and the soft-breaking parameters A λ ν , A Y ν , B µ X , m ˜ ν , and m ˜ x are all 3 × µ X is dimensional, and it parameterizes the effect of lepton number violation (LNV).Since this matric arises from the integration of massive particles in the high-energyultraviolet theory with LNV interactions (see, for example, [56–58]), its magnitudeshould be small. Based on a similar perspective, the matrix B µ X is also theoreticallyfavored to be suppressed.It is the same as the NMSSM that the ISS-NMSSM predicts three CP-evenHiggs bosons, two CP-odd Higgs bosons, a pair of charged Higgs bosons, and fiveneutralinos. Throughout this work, we take λ , κ , tan β ≡ v u /v d , A λ , A κ , and µ ≡ λv s / √ v u ≡ √ (cid:104) H u (cid:105) , v d ≡ √ (cid:104) H d (cid:105) , and v s ≡ √ (cid:104) S (cid:105) denote the vacuum expectation values (vev) of the fields H u , H d , and S ,respectively. The elements of the CP-even Higgs fields’ squared mass matrix in thebases ( S ≡ cos β Re[ H u ] − sin β Re[ H d ], S ≡ sin β Re[ H u ] + cos β Re[ H d ], S ≡ Re[ S ])are given by [13, 39] M = 2 µ ( λA λ + κµ ) λ sin 2 β + 12 (2 m Z − λ v ) sin β, M = −
14 (2 m Z − λ v ) sin 4 β, M = −√ λA λ + 2 κµ ) v cos 2 β, M = m Z cos β + 12 λ v sin β, M = v √ λµ − ( λA λ + 2 κµ ) sin 2 β ] , M = λA λ sin 2 β µ λv + µλ ( κA κ + 4 κ µλ ) , (2.1)where S denotes the heavy doublet Higgs field with a vanishing vev, S represents theSM Higgs field with its vev v ≡
246 GeV, M is the mass of S at tree level withoutconsidering its mixing with the other bases, and M characterizes the mixing of S with the singlet field S .The squared mass matrix in Eq. (2.1) can be diagonalized by a unitary matrix U , and its eigenstates h i with i = 1 , , h i = (cid:88) j =1 U ij S j , (2.2)– 4 –here h i are labelled in an ascending mass order, i.e. m h < m h < m h . Then thecouplings of h i to vector bosons W and Z and fermions u and d quarks, which arenormalized to their SM predictions, take the following form [59]¯ C h i V ∗ V = U i , ¯ C h i ¯ uu = U i cot β + U i , ¯ C h i ¯ dd = − U i tan β + U i . (2.3)Obviously, U i (cid:39) h i are far dominated by S , andconsequently, ¯ C h i V ∗ V (cid:39) ¯ C h i ¯ uu (cid:39) ¯ C h i ¯ dd (cid:39)
1. We call h i as the SM-like Higgs boson.Similarly, the elements of the CP-odd Higgs fields’ squared mass matrix are [13] M P, = 2 µ ( λA λ + κµ ) λ sin 2 β , M P, = ( λA λ + 3 κµ ) sin 2 β µ λv − µλ κA κ , M P, = v √ λA λ − κµ ) , (2.4)in the bases ( A ≡ cos β Im[ H u ] + sin β Im[ H d ], Im[ S ]). As a result, the two CP-oddmass eigenstates A and A are the mixtures of A and Im[ S ]. The charged Higgsare given by H ± = cos βH ± u + sin βH ± d , and their masses are m H ± = 2 µ ( λA λ + κµ ) / ( λ sin 2 β ) + v ( g / − λ ).Concerning the neutralinos, they are the mixtures of the bino field ˜ B , the winofield ˜ W , the Higgsinos fields ˜ H d and ˜ H u , and the singlino field ˜ S . In the bases ψ = ( − i ˜ B , − i ˜ W , ˜ H d , ˜ H u , ˜ S ), their mass matrix is given by [13] M = M − g (cid:48) v d √ g (cid:48) v u √ M gv d √ − gv u √ − µ − λv u − λv d κλ µ , (2.5)where M and M are soft breaking masses of the gauginos. It can be diagonalizedby a rotation matrix N so that the mass eigenstates are˜ χ i = N i ψ + N i ψ + N i ψ + N i ψ + N i ψ . (2.6)It is evident that N i and N i characterize the ˜ H d and ˜ H u components in ˜ χ i , respec-tively, and N i 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 u ]-dominated by the LHC data when tan β (cid:29) λ ˆ s ˆ H u · ˆ H d , thedoublet-singlet Higgs mixing as well as the radiative correction from top/stoploops [37–39]. In the following, we denote this state as h .– 5 – ight h s scenario with A λ = 2000 GeV Massive h s scenario with A λ = 2000 GeVtan β λ κ β λ κ A t A κ -680.4 µ A t A κ -120.4 µ m h s m h m H m h m h s m H m A s m A H m H ± m A s m A H m H ± m ˜ χ m ˜ χ -206.1 m ˜ χ ± m ˜ χ m ˜ χ -341.3 m ˜ χ ± U -0.018 U -0.261 U U U -0.999 U U -0.005 U U U U -0.003 U -0.999 U U U U -0.999 U -0.000 U -0.006¯ C h s V ∗ V -0.261 ¯ C h s ¯ uu -0.263 ¯ C h s ¯ dd -0.038 ¯ C h s V ∗ V -0.003 ¯ C h s ¯ uu -0.003 ¯ C h s ¯ dd -0.162¯ C hV ∗ V C h ¯ uu C h ¯ dd C hV ∗ V -0.999 ¯ C h ¯ uu -0.999 ¯ C h ¯ dd -1.00¯ C HV ∗ V C H ¯ uu C H ¯ dd -12.4 ¯ C HV ∗ V -0.000 ¯ C H ¯ uu -0.035 ¯ C H ¯ dd C h s h s h s C h s h s h C h s hh -0.460 ¯ C h s h s h s -163.4 ¯ C h s h s h C h s hh -1.591 N -0.696 N N -0.147 N N -0.641 N N -0.699 N -0.709 N -0.055 N -0.702 N -0.709 N -0.010 Table 1 . Specific configuration of the Higgs and neutralino sectors for the scenarios dis-cussed in the text, and their prediction on the properties of the Higgs bosons and neu-tralinos such as the mass spectrum and the couplings of the Higgs bosons with differentparticles, ¯ C ijk , which are normalized to their corresponding SM predictions. Parameterswith mass dimensions are in the unit of GeV. Other fixed parameters that are not listedin the table include m ˜ q = 2000 GeV for flavor universal soft-breaking masses of squarks, M = M = 2000 GeV and M = 5000 GeV for gaugino masses, A i = 0 for all soft-breaking trilinear coefficients except for A λ and A t , and [ Y ν ] , = 0 .
01, [ λ ν ] , = 0 . m ˜ ν ] , = [ m ˜ x ] , = 2000 GeV for the parameters of the first two generations ofthe sneutrinos. All these parameters are defined at the scale Q = 1000 GeV. Besides, theHiggs masses and U ij are obtained by setting [ Y ν ] = [ λ ν ] = 0, and sneutrino loop effectsmay slightly change them when varying the parameters in the sneutrino DM sector. • In most cases, the heavy doublet-dominated CP-even state is mainly composedof the field Re[ H d ]. It roughly degenerates in mass with the doublet-dominatedCP-odd state and also with the charged states. The LHC search for extra Higgsbosons and the B -physics measurements requires these states to be heavier thanabout 500 GeV [60]. We represent them by H , A H , and H ± . • Concerning the singlet-dominated states, they may be very light without con-flicting with any collider constraint. As we introduced before, these states mayappear as the final state of the sneutrino pair annihilation or mediate the an-nihilation, and thus, they can play a vital role in the sneutrino DM physics. Inthis work, we label these states by h s and A s . • The lightest neutralino ˜ χ is Higgsino dominated if | µ | < | M | , | M | and | κ/λ | <
1. In this case, | N | (cid:39) | N | (cid:39) √ / h s and the SM-like Higgs boson h correspond to the lightest and the next-to-lightestCP-even Higgs bosons h and h . The S component of h s is measured by the ro-tation element U , which is determined by the elements M and M in Eq.(2.1).We dub this scenario light h s scenario. By contrast, we call the second scenario asthe massive h s scenario. It predicts h = h , h s = h , and U to characterize the S component in h s . Besides, we note that triple Higgs interactions may play an essen-tial role in the sneutrino DM annihilation. So in addition to the couplings ¯ C h i V ∗ V ,¯ C h i ¯ uu and ¯ C h i ¯ dd , 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 anddenoted by ¯ C h s h s h s , ¯ C h s h s h and ¯ C h s hh , respectively. These strengths are obtained bythe formulas in [13]. They are characterized by | ¯ C h s h s h s | (cid:29) | ¯ C h s h s h | , | ¯ C h s hh | , which isevident by the superpotential and the soft breaking terms of the ISS-NMSSM. In the interaction bases ( ν L , ν ∗ R , x ), the neutrino mass matrix is given by [14] M ISS = M TD M D M R M TR µ X , (2.7)where both the Dirac mass M D = Y ν v u / √ M R = λ ν v s / √ × × U ν to obtain three light neutrinos ν i ( i = 1 , ,
3) and six massiveneutrinos 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: (cid:0) U † ν (cid:1) × = (cid:32) ˆ U × X × Y × Z × (cid:33) . (2.8)The sub-matrix ˆ U × encodes the neutrino oscillation information and it is deter-mined by the neutrino experimental results.Alternatively, one can get the analytic expression of the light active neutrinos’mass matrix from Eq. (2.7) in the limit (cid:107) µ X (cid:107) (cid:28) (cid:107) M D (cid:107) (cid:28) (cid:107) M R (cid:107) , where (cid:107) M (cid:107) isdefined by (cid:107) M (cid:107) ≡ (cid:112) Tr( M † M ) for an arbitrary matrix M . The result is M ν = (cid:104) M TD M T − R (cid:105) µ X (cid:2) ( M − R ) M D (cid:3) + O ( µ X ) ≡ F µ X F T + O ( µ X ) . (2.9)where F = M TD M T − R , and its elements’ magnitude is of the order (cid:107) M D (cid:107) / (cid:107) M R (cid:107) . This3 × U T PMNS M ν U PMNS = diag( m ν , m ν , m ν ) . (2.10)– 7 –ue to the mixings among the states ( ν L , ν ∗ R , x ), the matrix ˆ U in Eq.(2.8) does notcoincide with U PMNS . Instead, they are related by [61]ˆ U (cid:39) (cid:18) − F F † (cid:19) U PMNS ≡ ( − η ) U PMNS , (2.11)where η = F F † is a measure of the non-unitarity for the matrix ˆ U . A recent globalfit of the theory to low energy experimental data reveals that [61] (cid:112) | η | ee < . , (cid:113) | η | µµ < . , (cid:112) | η | ττ < . , (cid:113) | η | eµ < . , (cid:112) | η | eτ < . , (cid:113) | η | µτ < . . (2.12)We call these inequalities as the leptonic unitarity constraint.Eq. (2.9) indicates that the tininess of the active neutrino masses in the in-verse seesaw mechanism is due to the smallness of the lepton-number violatingmatrix µ X and the suppression factor (cid:107) M D (cid:107) / (cid:107) M R (cid:107) . For (cid:107) µ X (cid:107) (cid:46) O (keV) and (cid:107) M R (cid:107) ∼ O (TeV), the magnitude of the Dirac Yukawa coupling Y ν may reach orderone in predicting m ν i ∼ . Y ν may conflict with the unitarityconstraint once the Majorana mass M R is specified. So the constraint must be takeninto account in phenomenological study.In the following, we will discuss the application of the unitarity constraint in DMphysics. After noticing that the neutrino oscillation phenomenon can be explained bychoosing an appropriate µ X [55, 62], we assume flavor diagonal Y ν and λ ν to simplifythe DM physics (see discussion below). We then determine µ X by the formula [55, 62] µ X = M TR m T − D ˆ U ∗ × Diag( m ν , m ν , m ν ) ˆ U † × m D − M R , where m ν i and ˆ U (cid:39) U PMNS take the values extracted from relevant neutrino exper-iments. With the assumption, the neutrino oscillation is solely attributed to thenon-diagonality of µ X , and the unitarity constraint in Eq. (2.12) becomes (cid:12)(cid:12)(cid:12)(cid:12) [ λ ν ] µ [ Y ν ] λv u (cid:12)(cid:12)(cid:12)(cid:12) > . , (cid:12)(cid:12)(cid:12)(cid:12) [ λ ν ] µ [ Y ν ] λv u (cid:12)(cid:12)(cid:12)(cid:12) > . , (cid:12)(cid:12)(cid:12)(cid:12) [ λ ν ] µ [ Y ν ] λv u (cid:12)(cid:12)(cid:12)(cid:12) > . . (2.13)These inequalities reveal that the ratio [ λ ν ] / [ Y ν ] may be significantly smaller than[ λ ν ] / [ Y ν ] and [ λ ν ] / [ Y ν ] for fixed λ , µ , and v u , or equally speaking, [ Y ν ] maybe much larger than [ Y ν ] and [ Y ν ] when λ ν is proportional to identity matrix.Concerning the LNV coefficients µ X and B µ X , one should note two points. One isthat B µ X can induce an effective µ X through sneutrino-singlino loops to significantlyaffect the active neutrino masses by Eq. (2.9). We estimate the correction by themass insertion method, which was widely used in B physics study. We find δµ X ∼ π λ ν M B µ X M λ ν M , (2.14)– 8 –here M parameterizes the mixing of the field ˜ ν ∗ R with the field ˜ x , and M SUSY represents the sparticles’ mass scale. Under the premise that Y ν , λ ν , and B µ X areflavor diagonal, M can be roughly flavor diagonal, too (see the discussion in thenext section). So one can study the correction in one generation case. The result is δµ X (keV) ∼ . × (cid:18) λ ν . (cid:19) (cid:18) M M (cid:19) (cid:18) B µ X / GeV M SUSY / GeV (cid:19) , (2.15)which indicates that B µ X / GeV may be comparable with M SUSY / GeV in getting δµ X ∼ λ ν = 0 . M (cid:46) . × M . Alternatively, if λ ν = 0 . M = 0 . × M , the approximation requires B µ X / GeV < . × M SUSY / GeV toget δµ X ∼ B µ X ’s magnitude.In our study, we limit B µ X ≤
100 GeV for simplicity. The other point is that theLNV coefficients may induce sizable neutrinoless double beta decay since the inverseseesaw scale may be around several hundred GeV and the Yukawa couplings Y µ and λ ν may be moderately large. As indicated in [63], because µ X is related to theactive neutrino mass, the decay rate is below current experiment sensitivity whenthe massive neutrinos are heavier than 1 GeV. So there is no need to consider theconstraint in our study. If the sneutrino fields are decomposed into CP-even and CP-odd parts˜ ν L = 1 √ φ + iσ ) , ˜ ν ∗ R = 1 √ φ + iσ ) , ˜ x = 1 √ φ + iσ ) , (2.16)the squared mass of the CP-even fields is given by m ν = m m m m ∗ m m m ∗ m ∗ m , (2.17)in the bases ( φ , φ , φ ), where m = 14 (cid:104) v u Re (cid:16) Y ν Y ∗ ν (cid:17) + 4Re (cid:16) m l (cid:17)(cid:105) + 18 (cid:16) g + g (cid:17)(cid:16) − v u + v d (cid:17) ,m = − v d v s Re (cid:16) λY ∗ ν (cid:17) + 1 √ v u Re (cid:16) Y ν A Y ν (cid:17) ,m = 12 v s v u Re (cid:16) Y ν λ ∗ ν (cid:17) ,m = 14 (cid:104) v s Re (cid:16) λ ν λ ∗ ν (cid:17) + 2 v u Re (cid:16) Y ν Y ∗ ν (cid:17) + 4Re (cid:16) m ν (cid:17)(cid:105) ,m = 18 (cid:110) − v d v u λλ ν + 2 (cid:104)(cid:16) − v d v u λ + v s κ (cid:17) λ ∗ ν + v s κλ ν (cid:105) + √ v s (cid:104) − (cid:16) µ X λ ∗ ν (cid:17) + 4Re (cid:16) A ∗ λ ν λ ν (cid:17)(cid:105)(cid:111) ,m = 18 (cid:104) v s Re (cid:16) λ ν λ ∗ ν (cid:17) − (cid:16) B µ X (cid:17) + 8Re (cid:16) µ X µ ∗ X (cid:17) + 8Re (cid:16) m x (cid:17)(cid:105) . (2.18)– 9 –hese formulas indicate the following facts: • The squared mass is a 9 × φ , φ , φ ) bases. Itinvolves a series of 3 × Y ν , λ ν , A Y ν , A λ ν , µ X , B µ X , m ˜ l , m ˜ ν , and m ˜ x . Among these matrices, only µ X must be flavornon-diagonal to account for the neutrino oscillations, but since its magnitudeis less than 10 keV [62], it can be neglected. Thus, if there is no flavor mixingsfor the other matrices, the squared mass is flavor diagonal, and one can adoptone-generation ( φ , φ , φ ) bases in studying the mass. In this work, we onlyconsider the third generation sneutrinos as DM candidates. This is motivatedby that both the unitarity bound and the LHC constraint in sparticle searchare weakest for the third generation [14]. When we mention the sneutrinoparameters hereafter, we are actually referring to their 33 elements. Underthe assumption, the squared mass is diagonalized by a 3 × V , which parameterizes the chiral mixings between the fields φ , φ and φ .Consequently, the sneutrino mass eigenstates are given by ˜ ν 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 afterproperly rotating the phase of the fields ˆ ν R and ˆ X . • The mixing of φ with the other fields is determined by Y ν and A ν . As Y ν approaches zero, | m | and | m | diminish monotonically, and so is | V | whichrepresents the ˜ ν L component in the lightest sneutrino state ˜ ν R, . In the extremecase Y ν = 0, all these quantities vanish and ˜ ν R, is merely the mixture of φ and φ . Furthermore, if λ ν /λ is moderately large, the first term in m and m maybe far dominant over the other contributions so that m (cid:39) m . This resultsin maximal mixing between φ and φ and ˜ ν R, (cid:39) / √ φ − sgn( m ) φ ] [14].This is a case encountered frequently in our study.Similarly, one may adopt the one-generation ( σ , σ , σ ) 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 by ˜ ν I,i = V (cid:48) ij σ j ( i, j = 1 , , V (cid:48) denotes the rotation of the CP-odd fields. Given that B µ X represents thedegree of the LNV and is theoretically small, we are particularly interested in thefollowing two cases: • The extreme case of B µ X = 0 where any CP-even sneutrino state is accom-panied by a mass-degenerate CP-odd state. In this case, any sneutrino masseigenstate corresponds to a complex field and it has an anti-particle [64]. Con-cerning the sneutrino DM ˜ ν , we have ˜ ν R, ≡ Re[˜ ν ], ˜ ν I, ≡ Im[˜ ν ], V ij = V (cid:48) ij ,and ˜ ν and its anti-particle ˜ ν ∗ contribute equally to the relic density. This caseis actually a two-component DM theory. It is notable that the ˜ ν ∗ ˜ ν Z couplingis proportional to | V | and it contributes to the scattering of ˜ ν with nucleons.– 10 –his effect is important when | V | ∼ .
01 (discussed below). It is also notablethat the ˜ ν ∗ ˜ ν A i coupling vanishes since it is induced only by the LNV effect. • A more general case satisfying | B µ X | (cid:46)
100 GeV . It has four distinctivefeatures. First, since m ˜ ν R,i > m ˜ ν I,i when B µ X >
0, the DM candidate ˜ ν is identified as the ˜ ν I, state with a definite CP number -1. The oppositeconclusion applies to B µ X < m ˜ ν R, − m ˜ ν I, (cid:39) . B µ X = 100 GeV and m ˜ ν R, = 100 GeV, and so are the rotations V and V (cid:48) .These sneutrino states compose a pseudo-complex particle [58, 67, 68]. Third,given the approximate mass degeneracy, ˜ ν R, and ˜ ν I, always co-annihilated inearly 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 couplesonly to a pair of sneutrino states with opposite CP numbers. It also contributeslittle to the DM annihilation because the ˜ ν R, ˜ ν I, Z coupling is suppressed by afactor V ∗ V (cid:48) (cid:39) | V | .We fix B µ X = 0 or B µ X = −
100 GeV in this work. In either case, the ˜ ν ∗ ˜ ν h i ( h i = h s , h, H ) coupling coefficient is given by C ˜ ν ∗ ˜ ν h i = C ˜ ν ∗ ˜ ν Re[ H d ] ˜ U i + C ˜ ν ∗ ˜ ν Re[ H u ] ˜ U i + C ˜ ν ∗ ˜ ν Re[ S ] ˜ U i , where ˜ U diagonalizes the CP-even Higgs fields’ squared mass in (Re[ H d ] , Re[ H u ] , Re[ S ])bases [38, 39], and C ˜ ν ∗ ˜ ν s on the right side denotes the sneutrino coupling to the scalarfield s . For the one-generation sneutrino case, C ˜ ν ∗ ˜ ν s is given by C ˜ ν ∗ ˜ ν Re[ H d ] = λY ν v s V V + λλ ν v u V V −
14 ( g + g ) v d V V ,C ˜ ν ∗ ˜ ν Re[ H u ] = λλ ν v d V V − √ Y ν A Y ν V V − Y ν v u V V − λ ν Y ν v s V V − Y ν v u V V + 14 ( g + g ) v u V V ,C ˜ ν ∗ ˜ ν Re[ S ] = λY ν v d V V − κλ ν v s V V − √ λ ν A λ ν V V + √ λ ν µ X V V − λ ν Y ν v u V V − λ ν v s ( V V + V V ) . (2.19)These formulas indicate that the parameters Y ν , λ ν , A Y ν , and A λ ν affect not only thesneutrino 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 ν and m x affect only the latter property. For typical values of the parameters inEq. (2.19), e.g., tan β (cid:29) | V | < . Y ν , κ, λ, λ ν ∼ O (0 .
1) and λ ν v s , λv s , A Y ν , A λ ν ∼O (100 GeV), C ˜ ν ˜ ν s is approximated by C ˜ ν ∗ ˜ ν Re[ H d ] (cid:39) λY ν v s V V + λλ ν v u V V ,C ˜ ν ∗ ˜ ν Re[ H u ] (cid:39) −√ λ ν A Y ν V V − λ ν Y ν v s V V − Y ν v u V V ,C ˜ ν ∗ ˜ ν Re[ S ] (cid:39) − κλ ν v s V V − √ λ ν A λ ν V V − λ ν v s . (2.20)– 11 –t is estimated that | C ˜ ν ∗ ˜ ν Re[ H d ] | , | C ˜ ν ∗ ˜ ν Re[ H u ] | (cid:46)
10 GeV and C ˜ ν ∗ ˜ ν Re[ S ] (cid:46)
100 GeVin most cases, which reflects that | C ˜ ν ∗ ˜ ν Re[ S ] | may be much larger than the othertwo couplings. The basic reason is that ˜ ν is a singlet-dominated scalar, so it cancouple directly to the field S and the mass dimension of C ˜ ν ∗ ˜ ν Re[ S ] is induced by v s or A λ ν . By contrast, the other couplings emerge only after the electroweak symmetrybreaking when V = 0, and their mass dimension originates from v u . In the B µ X = 0 case, both ˜ ν and ˜ ν ∗ act as the DM candidate. Their annihilationincludes those initiated by ˜ ν ˜ ν ∗ , ˜ ν ˜ ν , and ˜ ν ∗ ˜ ν ∗ state, and the co-annihilation of ˜ ν and ˜ ν ∗ with the other sparticles. Considering the numerousness of the annihilationchannels and the complexity of this issue, we will only discuss the channels frequentlymet in our study (see footnote 2 of this work for more details), which are [14, 65]:(1) ˜ ν ˜ H, ˜ ν ∗ ˜ H → XY and ˜ H ˜ H (cid:48) → X (cid:48) Y (cid:48) , where ˜ H and ˜ H (cid:48) denote Higgsino-dominated neutralinos or charginos, and X ( (cid:48) ) and Y ( (cid:48) ) represent any possibleSM particles, the massive neutrinos or the Higgs bosons if the kinematics are ac-cessible. More specifically, the channels ˜ ν ˜ H → W l, Zν, hν ( l and ν denote anypossible lepton and neutrino, respectively) proceed by the s -channel exchangeof neutrinos, and the t/u channel exchange of sleptons or sneutrinos. The pro-cesses ˜ H ˜ H (cid:48) → f ¯ f (cid:48) , V V (cid:48) , hV ( f and f (cid:48) denote quarks or leptons, and V and V (cid:48) represent SM vector bosons) proceed by the s -channel exchange of vectorbosons or Higgs bosons, and the t/u channel exchange of sfermions, neutralinosor charginos. This annihilation mechanism is called co-annihilation [51, 52].(2) ˜ ν ˜ ν ∗ → ss ∗ ( s denotes a light Higgs boson), which proceeds through any rele-vant quartic scalar coupling, the s -channel exchange of CP-even Higgs bosons,and the t/u -channel exchange of sneutrinos.(3) ˜ ν ˜ ν ∗ → ν h ¯ ν h 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.(4) ˜ ν ˜ ν → ν h ν h and ˜ ν ∗ ˜ ν ∗ → ¯ ν h ¯ ν h , which mainly proceed through the t/u -channelexchange of a singlino-dominated neutralino due to its majorana nature.(5) ˜ ν ˜ ν ∗ → V V ∗ , V s , f ¯ f , which proceeds mainly by the s -channel exchange of CP-even Higgs bosons. They are important if one of the bosons is at resonance.Under specific parameter configurations, these channels can be responsible forthe 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 getthe measured density. This mechanism works only when the mass splitting– 12 –etween ˜ H and ˜ ν is less than about 10%, and a specific channel’s contributionto the density depends not only on its cross-section but also on the mass split-ting. To illustrate this point, we assume that the DM annihilations comprisethose initiated by ˜ ν ˜ ν , ˜ ν ˜ ν ∗ , ˜ ν ∗ ˜ ν ∗ , ˜ ν ˜ χ , ˜ ν ∗ ˜ χ , and ˜ χ ˜ χ states, and denote thecross-sections of these channels by σ AB with A, B = ˜ ν , ˜ ν ∗ , ˜ χ . The effectiveannihilation rate at temperature T is then given by [52] σ eff = 14 (cid:18)
11 + (1 + ∆) / e − x ∆ (cid:19) × (cid:8) σ ˜ ν ˜ ν + 2 σ ˜ ν ˜ ν ∗ + σ ˜ ν ∗ ˜ ν ∗ +4 (cid:16) σ ˜ ν ˜ χ + σ ˜ ν ∗ ˜ χ (cid:17) (1 + ∆) / e − x ∆ + 4 σ ˜ χ ˜ χ (1 + ∆) e − x ∆ (cid:111) , (2.21)where ∆ ≡ ( m ˜ χ − m ˜ ν ) /m ˜ ν parameterizes the mass splitting and x ≡ m ˜ ν /T .This formula indicates that the ˜ ν ˜ χ and ˜ χ ˜ χ channel’s contributions are sup-pressed by factors e − x ∆ and e − x ∆ , respectively. So they become less and lesscritical as m ˜ ν deviates from m ˜ χ . Besides, the formulae of the density in [52]indicate that the density depends on the sneutrino parameters only through m ˜ ν and σ eff . In the extreme case of σ ˜ ν ˜ ν (cid:39) σ ˜ ν ˜ ν ∗ (cid:39) σ ˜ ν ˜ χ (cid:39) λ ν and Y ν are sufficiently small, the ˜ χ ˜ χ annihilation is solely responsible forthe measured density through tuning the value of m ˜ ν . This situation wasintensively studied in [8]. We will present such examples in Section III. • Barring the co-annihilation, ˜ ν ˜ ν → ss ∗ is usually the most crucial channel inaffecting the density if the kinematics are accessible. In particular, the process˜ ν ˜ ν → h s h s can be solely responsible for the measured density if the Yukawacoupling λ ν is moderately large. We exemplify this point by considering thelight h s scenario in Table 1. From the Higgs boson and sneutrino mass spectrumand the ˜ ν ’s couplings to h s , one can learn that the annihilation proceeds mainlyby the s -channel exchange of h s , t/u -channel exchange of ˜ ν , and ˜ ν ˜ ν ∗ h s h s quartic scalar coupling. As a result, the cross-section of the annihilation nearthe freeze-out temperature is approximated by [14, 65] σv (cid:39) a + bv , (2.22)where a = (cid:113) − m h s /m ν πm ν (cid:12)(cid:12)(cid:12)(cid:12) C ˜ ν ˜ ν h s h s − C ˜ ν ˜ ν h s C h i h s h s m ν − m h s + 2 C ν ˜ ν h s m ν − m h s (cid:12)(cid:12)(cid:12)(cid:12) ,b = (cid:18) −
14 + m h s m ν − m h s ) (cid:19) × a − (cid:113) − m h s /m ν π × (cid:26) C ν ˜ ν h s C h s h s h s (4 m ν − m h s ) − C ˜ ν ˜ ν h s C h s h s h s C ν ˜ ν h s (10 m ν − m h s )(4 m ν − m h s ) (2 m ν − m h s ) + 2 C ν ˜ ν h s (2 m ν − m h s ) − C ˜ ν ˜ ν h s h s (cid:18) C ˜ ν ˜ ν h s C h s h s h s (4 m ν − m h s ) − C ν ˜ ν h s (2 m ν − m h s ) (cid:19)(cid:27) . – 13 –he measured density then requires a + 3 b/ (cid:39) . × − cm because weare considering a two-component DM theory [69, 70]. This requirement limitsthe ˜ ν ’s couplings to h s or for the fixed parameters in Table 1, ultimately theYukawa coupling λ ν since the cross-section is very sensitive to λ ν . We estimatethat λ ν ∼ . m ˜ ν = 130 GeV can account for the measured density. • The process ˜ ν ˜ ν ∗ → ν h ¯ ν h could be responsible for the density when m ˜ ν > ν h , m ˜ ν < h s , and the co-annihilation mechanism did not work. This processproceeded 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: (cid:104) σv (cid:105) T f ∼ (cid:18) C ˜ ν ˜ ν ∗ h s C ¯ ν h ν h h s m ν − m h s (cid:19) , (2.23)which implies that the density limits non-trivially λ ν , m ˜ ν and m h s . • About the other channels, they usually played a minor role in determining thedensity. So we leave the discussion of them in our future works.Concerning the B µ X (cid:54) = 0 case, either ˜ ν R, or ˜ ν I, acts as the DM candidate. Sincethe mass splitting between the DM ˜ ν and ˜ ν (cid:48) (the partner of ˜ ν with a different CPnumber) is small, ˜ ν always co-annihilated with ˜ ν (cid:48) to get the measured density. Therelevant annihilation included ˜ ν ˜ ν (cid:48) and ˜ ν (cid:48) ˜ ν (cid:48) initiated processes, and they proceededin a way similar to the previous discussion. We confirmed that the density is insen-sitive to B µ X for | B µ X | ≤
100 GeV , which can be inferred from Eq. (2.21). We alsoverified that the cross-section of the DM annihilation today is insensitive to B µ X . In the B µ X (cid:54) = 0 case, the scattering of ˜ ν with nucleon N ( N = p, n ) proceedsby the t/u -channel exchange of the CP-even Higgs bosons. Consequently, the spinindependent (SI) cross-section is given by [14] σ SI˜ ν − N = F ( N )2 u g µ m N πm W × (cid:40)(cid:88) i (cid:20) C ˜ ν ∗ ˜ ν h i m h i m ˜ ν ( U i sin β + U i cos β F Nd F Nu ) (cid:21)(cid:41) , where µ red = m N / (1 + m N /m ν ) ( N = p, n ) represents the reduced mass of thenucleon with m ˜ ν , F Nu = f Nu + f NG and F Nd = f Nd + f Ns + f NG are nucleon formfactors with f Nq = m − N (cid:104) N | m q q ¯ q | N (cid:105) and f NG = 1 − (cid:80) q f Nq for q = u, d, s [2]. With thedefault setting of the package micrOMEGAs [71–73] for nucleon sigma terms, i.e., σ πN = 34 MeV and σ = 42 MeV [74] , one can conclude F pu (cid:39) .
15 and F pd (cid:39) . It is notable that σ was replaced by the strangeness-nucleon sigma term, σ s ≡ m s / ( m u + m d ) × ( σ πN − σ ) (cid:39) . × ( σ πN − σ ), in recent calculation of the nucleon form factor [72]. Compared withthe previous calculation, this treatment changes significantly the strange quark content in nucleon N , f Ns , but it change little F Nu and F Nd . – 14 –or protons. Instead, if σ πN = 59 MeV [75–77] and σ = 57 MeV [78] are adopted,the form factors become F pu (cid:39) .
16 and F pd (cid:39) .
13. These results reflect that differentchoices of σ πN and σ can induce uncertainties of O (10%) in F pu and F pd , and it doesnot drastically change the cross-section. Besides, the default setting also predicts F nu (cid:39) .
15 and F nd (cid:39) .
14 for neutrons, which implies the relation σ SI˜ ν − p (cid:39) σ SI˜ ν − n forthe Higgs-mediated contribution.To clarify the features of the cross-section, we assume m H ± (cid:38) S = sin β Re[ H u ] + cos β Re[ H d ]and the singlet field Re[ S ]. We then calculate the scattering amplitude by the massinsertion method to get the following result: σ SI˜ ν − N (cid:39) F ( N )2 u g µ m N πm W (125 GeV) × (cid:18)
125 GeV m h (cid:19) × (cid:18) C ˜ ν ∗ ˜ ν Re[ S ] m ˜ ν × δ sin θ cos θ − cos βC ˜ ν ∗ ˜ ν Re[ H d ] + sin βC ˜ ν ∗ ˜ ν Re[ H u ] m ˜ ν × (1 + δ sin θ ) (cid:19) (cid:39) . × − cm × (cid:18)
125 GeV m h (cid:19) × (cid:18) C ˜ ν ∗ ˜ ν Re[ S ] m ˜ ν × δ sin θ cos θ − cos βC ˜ ν ∗ ˜ ν Re[ H d ] + sin βC ˜ ν ∗ ˜ ν Re[ H u ] m ˜ ν × (1 + δ sin θ ) (cid:19) , (2.24)where δ = m h /m h s −
1, and θ is the mixing angle of the S field with Re[ S ] toform mass eigenstates. This formula reveals that if the terms in the second bracketsare on the order of 0.1, which can be achieved if λ ν and/or Y ν in Eq. (2.20) aresufficiently large, the cross-section may reach the sensitivity of the recent XENON-1T experiment [54]. We will discuss this issue later.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 totalSI cross-section in this case is obtained by averaging over ˜ ν N and ˜ ν ∗ N scatteringsand the interferences between the Z - and the Higgs-exchange diagrams for the twoscatterings have opposite signs [79], the SI cross-section is given by [4] σ SI N ≡ σ SI˜ ν − N + σ SI˜ ν ∗ − N σ hN + σ ZN , (2.25)where σ hN is the same as before and the Z -mediated contributions are σ Zn ≡ G F V π m n (1 + m n /m ˜ ν ) , σ Zp ≡ G F V (4 sin θ W − π m p (1 + m p /m ˜ ν ) , (2.26)with G F and θ W denoting the Fermi constant and the weak angle, respectively. Since σ Zn is larger than σ Zp 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– 15 –f the DMs with xenon nucleus as σ SIeff = ( σ SI˜ ν − Xe + σ SI˜ ν ∗ − Xe ) / (2 A ), where A denotesthe mass number of the xenon nucleus, and calculate it by σ SIeff = 0 . σ SI p + 0 . σ SI n + 0 . (cid:113) σ SI p σ SI n , (2.27)where the three coefficients on the right side are obtained by averaging the abundanceof different xenon isotopes in nature. It is evident that the effective cross-section isidentical 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 spindependent cross-section is always zero, and its SI cross-section is usually much smallerthan that of the neutralino DM in the MSSM and NMSSM, which was discussed indetail in Ref.[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. In this section, we clarify the impact of the leptonic unitarity and current and futureDM DD experiments on the sneutrino DM sector under the premise that the theorypredicts the right density and the photon spectrum from the DM annihilation in dwarfgalaxies is compatible with the Fermi-LAT observation. Since the singlet-dominatedHiggs 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 cansimplify greatly the analysis of the impact and make the underlying physics clear.We also emphasize that the two scenarios were obtained by scanning intensively theparameters in the Higgs and DM sectors . They agree well with the latest Higgs dataof the LHC if the exotic decays h → ν h ¯ ν h , ˜ ν ˜ ν ∗ are kinematically forbidden. Thiswas confirmed by the packages HiggsSignal-2.4.0 [80] and
HiggsBounds-5.7.0 [81].
The procedure of our study is as follows. We constructed a likelihood function ofthe DM physics to guide sophisticated scans over the sneutrino parameters for eitherscenario. With the samples obtained in the scans, we plotted the profile likelihoodmap in different two-dimensional planes to illustrate its features and underlyingphysics. We express the likelihood function as L DM = L Ω ˜ ν × L DD × L ID × L Unitary , (3.1) With the parameter scan strategy reported in [15] for the Type-I NMSSM, we explored theparameter space of the ISS-NMSSM which takes tan β , λ , κ , A t , A κ , µ , λ ν , Y ν , A λ ν , A Y ν , m ˜ ν , and m ˜ x as inputs. For either the light or the massive h s scenario, we have studied more than fifty millionsamples. The settings in Table 1 were chosen from the samples that best fit the experimental data.We will present the analysis of these samples elsewhere. – 16 –here L Ω ˜ ν , 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 ofdwarf galaxies, and the unitarity constraint, respectively. They are given by • L Ω ˜ ν is Gaussian distributed, i.e., L Ω ˜ ν = e − [Ωth − Ωobs]22 σ , (3.2)where Ω th denotes the theoretical prediction of the density Ω ˜ ν h , Ω obs = 0 . σ = 0 . × 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]: L DD = e − (cid:18) σ SIeff δσ (cid:19) . (3.3)In this formula, σ SIeff is defined in Eq. (2.27) and its error bar δ σ is evaluatedby δ σ = (cid:112) U L σ / . + (0 . σ SIeff ) , where U L σ denotes experimental upper lim-its on the scattering cross-section at 90% C.L. and 0 . σ SIeff parameterizes thetheoretical uncertainty of σ SIeff . • L ID is calculated by the likelihood function proposed in [84, 85] with the dataof the Fermi-LAT collaboration taken from [86, 87]. • The likelihood function of the unitarity constraint in Eq. (2.13) is as follows: L Unitary = exp [ − (cid:18) r − . . r (cid:19) ] if r ≤ .
41 if r > . r ≡ λ ν µ/ ( Y ν λv u ) and 0 . r parameterizes total uncertainties.In addition, we abandoned samples that open up the decays h → ν h ¯ ν h , ˜ ν ˜ ν ∗ . Inpractice, this was completed by setting the likelihood value to be e − if any of thedecays were kinematically accessible. In fitting the ISS-NMSSM to experimental data, the total likelihood function is calculated by L tot = L Higgs × L DM , where L Higgs represents the Higgs physics function. Given χ ≡ − L , onecan infer that χ = χ + χ , and the 2 σ confidence interval defined below Eq. (3.5) satisfies χ − χ , min ≡ δχ + δχ ≤ . , where δχ ≡ χ − χ , min and δχ ≡ χ − χ , min . For the Higgs parameter settingsin Table 1, δχ vanishes because the settings correspond to the scenarios’ best points whenthe decays are forbidden. It increases if the decays are open and their branching ratios graduallyincrease. Since we hope to determine the interval only by the DM physics in the following study,we kinematically shut down the decays. – 17 –o make the conclusions in this study complete, we adopt the MultiNest algo-rithm [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 thealgorithm to be 10000. This parameter represents the number of active or live pointsused to determine the iso-likelihood contour in each scan’s iteration [88, 89]. Thelarger it is, the more elaborated the scan becomes. The output of the scans includesthe Bayesian evidence defined by Z ( D | M ) ≡ (cid:90) P ( D | O ( M, Θ)) P (Θ | M ) (cid:89) d Θ i , where P (Θ | M ) represents the prior PDF of the inputs Θ = (Θ , Θ , · · · ) in a model M , and P ( D | O ( M, Θ)) ≡ L (Θ) denotes the likelihood function involving theoreticalpredictions of observables O and their experimental measurements D . Computa-tionally, the evidence is an averaged likelihood that depends on the priors of thetheory’s input. In comparing different scenarios of the theory, the larger Z is, themore readily the corresponding scenario is consistent with the data.The output of the scan also includes the profile likelihood (PL) defined in fre-quentist statistics as the most significant likelihood value [15, 90]. For example,two-dimensional (2D) PL is defined by L (Θ A , Θ B ) = max Θ , ··· , Θ A − , Θ A +1 , ··· , Θ B − , Θ B +1 , ··· L (Θ) , (3.5)where the maximization is obtained by varying the parameters other than Θ A andΘ B . The PL reflects the preference of the theory on the parameter (Θ A , Θ B ), orin other words, the capability of the parameter to account for experimental data.Sequentially, one can introduce the concept of confidence interval (CI) to classify theparameter region by how well the points in it fit the data. For example, the 1 σ and2 σ CIs for the 2D PL are defined by satisfying χ − χ ≤ . χ − χ ≤ . χ ≡ − L (Θ A , Θ B ) and χ is the minimal value of χ for thesamples obtained in the scan.In this work, we utilized the package SARAH-4.11.0 [91–93] to build the modelfile 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. h s scenario Given the information of the light h s scenario in Table 1, the Higgs-mediated SIcross-section in Eq. (2.24) is approximated by σ SI˜ ν − N (cid:39) . × − cm × (cid:18) . C ˜ ν ∗ ˜ ν Re[ S ] m ˜ ν + C ˜ ν ∗ ˜ ν Re[ H u ] + 0 . C ˜ ν ∗ ˜ ν Re[ H d ] m ˜ ν (cid:19) . (3.6)– 18 – igure 1 . The profile likelihoods of the function L DM in Eq. (3.1) for the light h s scenario,projected onto λ ν − m ˜ ν plane. The upper panels are the results for the B µ X (cid:54) = 0 case,where the bounds of the DM-nucleon scattering cross-section were taken from the XENON-1T (2018) experiment (left panel) and the future LZ experiment (right panel), respectively.The lower panels are same as the upper panels except that they are for the B µ X = 0case. Since χ (cid:39) σ and 2 σ confidence interval satisfy χ (cid:39) . χ (cid:39) .
18 and are marked with white and redsolid line, respectively. This figure reflects the preference of the DM measurements on theparameters λ ν and m ˜ ν . Since C ˜ ν ∗ ˜ ν Re[ S ] may be larger than C ˜ ν ∗ ˜ ν Re[ H u ] and C ˜ ν ∗ ˜ ν Re[ H d ] by two orders, the firstterm in the brackets can be comparable with the other contributions. To clarify theimpact of the unitarity and DM DD experiments on the theory, we performed fourindependent scans over the following parameter space:0 ≤ Y ν , λ ν ≤ . , ≤ m ˜ ν , m ˜ x ≤
500 GeV , | A Y ν | , | A λ ν | ≤ ,
400 GeV ≤ m ˜ l ≤ , (3.7)where m ˜ l denoted the common soft breaking mass of three-generation sleptons andits lower bound was motivated by the non-observation of slepton signals at the LHCRun-II. For the first scan, we fixed B µ X = −
100 GeV and used the XENON-1T’s– 19 – igure 2 . Same as Fig.1, but for the profile likelihood projected onto σ SI˜ ν − p − m ˜ ν plane. bound on the SI cross-section to calculate the L DD . The second scan was same asthe first one except that we adopted the sensitivity of the LZ experiment. The lasttwo scans differed from the previous ones only in that we set B µ X = 0. As explainedbefore, the setting induces an additional Z -mediated contribution to the DM-nucleonscattering so that the constraints of the DD experiments are strengthened.With the samples obtained in the scans, we show different 2D PL maps in Figures1 to 5. Fig. 1 and 2 plot the CIs on λ ν − m ˜ ν and σ SI˜ ν − p − m ˜ ν planes. They showthe following features: • m ˜ ν is concentrated on the range from 120 to 181 GeV. Specifically, m ˜ ν isclose to m ˜ χ for 172 GeV (cid:46) m ˜ ν (cid:46)
181 GeV, and the DM achieves the correctdensity mainly through the ˜ χ ˜ χ annihilation (see discussions about the co-annihilation in Section 2.4 and details of the points in subsequent Table 2). Inthis case, the density is insensitive to the parameter λ ν . Thus, λ ν varies withina broad range from 0.15 to 0.6 in Fig. 1, where the lower limit forbids thedecay h → ν h ¯ ν h kinematically, and the upper bound comes from the DM DDexperiments (discussed below). For the other mass range, the DM obtains thecorrect density mainly through the annihilations ˜ ν ˜ ν ∗ → h s h s , h s h, hh . Thisrequires λ ν (cid:38) .
26, which can be understood from the discussion of Eq. (2.22).– 20 – igure 3 . Same as Fig.1, but for the profile likelihood projected onto Y ν − λ ν plane, wherethe red line denotes the leptonic unitarity bound. • Fig. 2 indicates that the SI cross-section of the DM-nucleon scattering may beas low as 10 − cm over the entire mass range. It reflects that the theory hasmultiple mechanisms to suppress the scattering, which becomes evident by theapproximation in Eq. (3.6) and was recently emphasized in [8]. • Although λ ν > . σ CIs of the four cases, respectively.Careful comparisons of the left and right panels revealed that it was due tothe DD experiments’ constraint on the co-annihilation region. Besides, westudied the Bayesian evidences Z i ( i = 1 , , ,
4) of the four cases and foundln Z = − . δ ≡ ln Z − ln Z = 1 . δ ≡ ln Z − ln Z = 0 .
54, and δ ≡ ln Z − ln Z = 1 .
43. These results reveal at least two facts. On the oneside, the Jeffreys’ scale δ [96, 97] reflects that current XENON-1T experimenthas no significant preference of the B µ X (cid:54) = 0 case to the B µ X = 0 case [98]. Onthe other side, δ and δ show that the Bayesian evidence (or equivalentlythe averaged L DM ) is reduced by a factor of more than 40%. It implies that asizable portion of the parameter space will become disfavored once the future– 21 – igure 4 . Same as Fig.1, but for the profile likelihood projected onto Y ν − m ˜ l plane. LZ experiment improves the XENON-1T’s sensitivity by 50 times. This featureis also reflected in Fig. 1 and Fig. 2 by the sizable shrink of the 1 σ CIs.In order to better understand Fig. 1, we describe how we obtained it. FromEq. (3.5), the 2D PL L ( λ ν , m ˜ ν ) is given by L ( λ ν , m ˜ ν ) = max Y ν ,A λν , ··· L DM ( λ ν , Y ν , A λ ν , A Y ν , m ˜ ν , m ˜ x , m ˜ l ) . (3.8)In plotting the figure, we implemented the maximization over the parameters Y ν , A λ ν , A Y ν , m ˜ ν , m ˜ x , and m ˜ l by three steps. First, we split the λ ν − m ˜ ν plane into 80 × λ ν and m ˜ ν , even though the other parameters maydiffer significantly. Finally, we select the maximum likelihood value of the samples ineach box as the PL value. These procedures imply that the CIs are not necessarilycontiguous, instead they usually distributed in isolated islands [15, 90]. Besides, weemphasize that χ (cid:39) igure 5 . Same as Fig.1, but for the profile likelihood projected onto V − m ˜ l plane. Next, we study 2D PL on Y ν − λ ν plane. The results are shown in Fig. 3 where thered dashed line denotes the correlation λ ν µ/ ( Y ν λv u ) = 9 . λ ν = 2 . Y ν from the unitarity constraint. This figure shows that Y ν is maximized at 0.17 when λ ν (cid:39) .
52 and it is upper bounded only by the unitarity. The reason is that theunitarity requires λ ν (cid:38) . Y 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 bycontrast, the unitarity limits Y ν .We also plot 2D PLs on Y ν − m ˜ l and V − m ˜ l planes in Fig. 4 and Fig. 5, respec-tively. Fig. 4 indicates that the 2 σ CI in each panel occupies a roughly rectangulararea on the Y ν − m ˜ l plane. This result reflects that L DM is insensitive to parameter m ˜ l . It can be understood from the following two aspects. One is that L DM relies on m ˜ l mainly through V by the ˜ ν ˜ ν ∗ h i coupling in Eq. (2.19). The other is that m ˜ l and V are weakly correlated, which can be inferred by the expression of m and m in Eq. (2.18) and is shown numerically in Fig. 5 and Fig. 10. Specifically, for the B µ X (cid:54) = 0 case, both the annihilation and the scattering are insensitive to V since itsmagnitude is small, and so is L DM . This property determines that the allowed rangeof Y ν is roughly independent of m ˜ l , 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 by– 23 – ight h s scenario Massive h s scenario P P P P Y ν Y ν λ ν λ ν A Y ν A Y ν A λ ν A λ ν -288.8 -295.0 m ˜ ν m ˜ ν m ˜ x m ˜ x m ˜ l m ˜ l m ˜ ν m ˜ ν V -0.002 -0.007 V -0.014 -0.031 V V -0.717 -0.726 V -0.851 0.717 V h h C ˜ ν ˜ ν ∗ h s C ˜ ν ˜ ν ∗ h s -53.54 -134.9 C ˜ ν ˜ ν ∗ h C ˜ ν ˜ ν ∗ h σ SI ˜ ν − p . × − . × − σ SI ˜ ν − p . × − . × − σ SI ˜ ν − n . × − . × − σ SI ˜ ν − n . × − . × −
16% ˜ χ ˜ χ → W + W −
11% ˜ χ ˜ χ − → d ¯ u
42% ˜ ν ˜ χ → W + τ −
13% ˜ χ ˜ χ − → d ¯ u
11% ˜ χ ˜ χ − → s ¯ c
21% ˜ ν ˜ χ → Zν τ annihilation 89% ˜ ν ˜ ν ∗ → h s h s
13% ˜ χ ˜ χ − → s ¯ c annihilation 11% ˜ χ ˜ χ → W + W −
19% ˜ ν ˜ χ → hν τ processes 10% ˜ ν ˜ ν ∗ → h s h
11% ˜ χ ˜ χ → ZZ processes 8 .
9% ˜ χ ˜ χ − → b ¯ t .
6% ˜ χ ˜ χ − → d ¯ u .
9% ˜ χ ˜ χ − → b ¯ t .
4% ˜ χ ˜ χ → ZZ .
6% ˜ χ ˜ χ − → s ¯ c ... . . . ... . . . ... . . . Table 2 . Detailed information of the points in the light h s scenario (left side of the table)and the massive h s scenario (right side of the table) with the setting B µ X = 0. The numberbefore each annihilation process represents the fraction of its contribution to the total DMannihilation cross section at the freeze-out temperature. Parameters in mass dimensionare in unit of GeV, and the DM-nucleon scattering cross section are in unit of cm . the formula in Eq. (2.26), the XENON-1T experiment has required | V | (cid:46) .
02 andthis upper bound is very insensitive to m ˜ l . In this case, one may replace m ˜ l by V asa theoretical input so that L DM does not depend on m ˜ l any more. This feature againleads to the conclusion that the allowed range of Y ν is roughly independent of m ˜ l .We add that the tight experimental constraint on the mixing V for the B µ X = 0case was also discussed in [99]. We also add that one may fix m ˜ l in performing globalfit of the ISS-NMSSM to experimental data due to the insensitivity of L DM to m ˜ l .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 featuresfurther. For the point P1, the DMs annihilated mainly by ˜ ν ˜ ν ∗ → h s h s , h s h to getthe density. The process ˜ ν ˜ ν ∗ → hh is unimportant because | C h s hh | is significantlysmaller than | C h s h s h s | and | C h s h s h | , and also because the phrase space of the finalstate is relatively small. By contrast, the DMs got their right relic density mainlyby the Higgsino pair annihilation for the point P2, and the mass splitting is ∆ ≡ m ˜ χ − m ˜ ν (cid:39) ν ˜ ν ∗ → h s h s ,and consequently, its effect is negligibly small. Besides, both the points predict– 24 – igure 6 . Same as Fig.1, but for the massive h s scenario defined in Table 1. Y ν ∼ .
01. As a result, V ’s magnitude is only a few thousandths, and the DM-neutron scattering rate is not much larger than the DM-proton scattering rate. Weverified that, once we set B µ X = −
100 GeV, the two rates became roughly equal. h s scenario In the massive h s scenario, the Higgs-mediated SI cross-section is given by σ SI˜ ν − N (cid:39) . × − cm × (cid:18) . C ˜ ν ∗ ˜ ν Re[ S ] m ˜ ν + C ˜ ν ∗ ˜ ν Re[ H u ] + 0 . C ˜ ν ∗ ˜ ν Re[ H d ] m ˜ ν (cid:19) , when one takes the parameters in Table 1. In large λ ν and Y ν case, e.g., λ ν (cid:38) . Y ν (cid:38) .
4, the typical sizes of C ˜ ν ∗ ˜ ν Re[ S ] and C ˜ ν ∗ ˜ ν Re[ H u ] are 100 GeV and 10 GeV,respectively. Thus, the first term in the brackets is no longer more critical than theother terms, and the σ SI˜ ν − N for m ˜ ν (cid:39)
300 GeV may reach 10 − cm only in optimalcases. Consequently, the XENON-1T experiment scarcely limit the B µ X (cid:54) = 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 independentscans over the parameter region in Eq. (3.7), and projected the PL onto differentplanes. The results are presented from Fig. 6 to Fig. 10 in a way similar to those forthe light h s scenario. These figures indicate the following facts:– 25 – igure 7 . Same as Fig.2, but for the massive h s scenario defined in Table 1. • Since the unitarity for the parameters in Table 1 requires only λ ν (cid:38) . Y ν , Y ν may be comparable with λ ν in size. As a result, the SI cross-section is sensitiveto both λ ν and Y ν , which is different from the light h s scenario. • Since the B µ X (cid:54) = 0 case is hardly limited by the XENON-1T experiment, both λ ν and Y ν may be larger than 0.4, which is shown in the upper left panel ofFig. 8. However, with the experimental sensitivity improved or the Z -mediatedcontribution considered in the B µ X = 0 case, the DM DD experiments becomepowerful enough to limit λ ν and Y ν . In this case, Y ν (cid:38) . • ˜ ν obtained the correct density through the co-annihilation with ˜ χ , which isreflected by the range of m ˜ ν in Fig. 6. We will take the points P3 and P4 inTable 2 as examples to show more details of the annihilation later.We confirmed that δ ≡ ln Z − ln Z = 0 . δ ≡ ln Z − ln Z = 0 .
62 and δ ≡ ln Z − ln Z = 0 .
56 in the massive h s scenario. Similar to the analysis ofthe light h s scenario, the smallness of δ and δ reflects that the LZ experimentcan not improve the constraint of the XENON-1T experiment on the scenario– 26 – igure 8 . Same as Fig.3, but for the massive h s scenario defined in Table 1. significantly, and the smallness of δ reflects that the XENON-1T experimentdoes not show significant preference of the B µ X (cid:54) = 0 case to the B µ X = 0 case. • Concerning the other features of the massive h s scenario, such as the suppres-sion of the SI cross-section and the correlation of m ˜ l with Y ν and V , they aresimilar 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 pairthat is responsible for the measured density, and the corresponding mass splitting isabout 5 GeV. By contrast, the ˜ ν ˜ H annihilation mainly accounts for the latter pointdensity, and the mass splitting reaches about 19 GeV. The difference is caused by thefact that P4 takes a relatively large Y ν and a lighter m ˜ l , making the ˜ ν ˜ H annihilationmore critical. Besides, it is notable that both the points predict Y ν (cid:38) .
18 to inducea sizable ˜ ν L component in ˜ ν , e.g., | V | > .
01. Consequently, Z boson can mediatea large DM-neutron scattering so that σ SI˜ χ − n (cid:29) σ SI˜ χ − p . Such a significant differencedisappears if one sets B µ X (cid:54) = 0.In summary, both λ ν and Y ν are more constrained in the light h s scenario thanin the massive h s scenario. The unitarity always plays a vital role in limiting Y ν – 27 – igure 9 . Same as Fig.4, but for the massive h s scenario defined in Table 1. except for the case shown in the last panel of Fig. 8, where the LZ experiment maybe more critical in limiting Y ν . We emphasize that the tight DD constraint on the B µ X (cid:54) = 0 case of the light h s scenario arises from that h s is light and it contains sizabledoublet components. In this case, the coupling C ˜ ν ∗ ˜ ν Re[ S ] contributes significantly tothe scattering rate.Before we end this section, we emphasize that the parameter points discussed inthis 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 andsquarks because these particles are too massive. Concerning the Higgsino-dominatedparticles, they may be detectable at the 8 TeV and 13 TeV LHC since their productionrates reach 100 fb. We scrutinized the property of the points in B µ X = 0 case andfound that they all predictBr( ˜ χ , → ˜ ν ¯ ν τ ) = Br( ˜ χ , → ˜ ν ∗ ν τ ) (cid:39) , Br( ˜ χ ± → ˜ ν ( ∗ )1 τ ± ) (cid:39) , (3.9)due to the Yukawa interaction Y ν ˆ l · ˆ H u ˆ ν R in the superpotential. In this case, the mostpromising way to explore the two scenarios at the LHC is to search the Di- τ plus miss-ing momentum signal through the process pp → ˜ χ ± ˜ χ ∓ → ( τ ± E MissT )( τ ∓ E MissT ) [14,15]. So far, the ATLAS collaboration has finished three independent analyses of the– 28 – igure 10 . Same as Fig.5, but for the massive h s scenario defined in Table 1. signal based on 20 . . m ˜ ν and m ˜ χ ± becomes narrow. As far as the light and massive h s scenarios are concerned, the analysis can not exclude at 95% confidence level thepoints satisfying m ˜ ν (cid:38)
100 GeV and m ˜ ν (cid:38)
200 GeV, respectively. So we concludethat the LHC analyses do not affect the results presented in this work.
Motivated by the increasingly tight limitation of the DM DD experiments on the tra-ditional neutralino DM in the natural MSSM and NMSSM, we extended the NMSSMby the inverse seesaw mechanism to generate the neutrino mass in our previous stud-ies [8, 14, 16], and studied the feasibility that the lightest sneutrino acts as a DMcandidate. A remarkable conclusion for the theory is that experimental constraintsfrom both the collider and DM search experiments are relaxed significantly. Conse-– 29 –uently, large parameter space of the NMSSM that has been experimentally excludedresurrects as physical points in the extended theory. In particular, the higgsino massmay be around 100 GeV to predict Z-boson mass naturally. This feature makes theextension attractive and worthy of a careful study.We realized that sizable neutrino Yukawa couplings λ ν and Y ν contributed signif-icantly to the DM-nucleon scattering rate. Thus, the recent XENON-1T experimentcould limit them. We also realized that the unitarity in the neutrino sector set aspecific correlation between the couplings λ ν and Y ν , which in return limited theparameter space of the ISS-NMSSM. Since these issues were not studied before, weinvestigated the impact of the leptonic unitarity and current and future DM DDexperiments on the sneutrino DM sector in this work. Specially, we considered thelight and massive h s scenarios after noticing that the singlet dominated Higgs playsa vital role in both the DM annihilation and the DM-nucleon scattering. For eachscenario, we studied the B µ X (cid:54) = 0 and B µ X = 0 case separately. Their differencecomes from that Z boson can mediate the DM-nucleon scattering for the B µ X = 0case, and thus, the experimental constraints on it are much tighter.In this work, we encoded the experimental constraints in a likelihood functionand performed sophisticated scans over the vast parameter space of the model bythe 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 limi-tation is powerful when h s is light and contains sizable doublet components. • As an useful complement to the DM DD experiments, the unitarity alwaysplays 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 dependenceon the left-handed slepton soft mass m ˜ l . This property implies that one mayfix m ˜ l in surveying the phenomenology of the ISS-NMSSM by scanning inten-sively its parameters and considering various experimental constraints. Thistreatment does not affect the comprehensiveness of the results. • The DM experiments tightly limit the left-handed sneutrino component inthe sneutrino DM, e.g., if one considers the XENON-1T experiment’s results, | V | (cid:46) .
15 for the B µ X (cid:54) = 0 case and | V | (cid:46) .
02 for the B µ X = 0 case;these upper bounds become 0.10 and 0.01, respectively, if one adopts the LZexperiment’s sensitivity.Finally, we briefly discuss the phenomenology of the ISS-NMSSM. The sparti-cles’s signal in this theory may be distinct from those in traditional supersymmetrictheories, and so is the strategy to look for them at the LHC. This feature can be– 30 –nderstood as follows: since the sneutrino DM carries a lepton number, and in mostcases has feeble interactions with particles other than the singlet-dominated Higgsboson and the massive neutrinos, the sparticle’s decay chain is usually long, andits final state contains at least one τ or ν τ . In addition, the decay branching ratiodepends not only on particle mass spectrum but also on new Higgs couplings, suchas 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, oneusually encounters the following two situations: • The DM co-annihilated with the Higgsino-dominated particles. This situationrequires the mass splitting ∆ ≡ m ˜ χ − m ˜ ν to be less than about 10 GeV.Consequently, the Higgsino-dominated particles usually appear as missing mo-mentum at the LHC due to the roughly degenerate mass spectrum. As pointedout in [8], this situation’s phenomenology may mimic that of the NMSSM withthe Higgsino-dominated ˜ χ as a DM candidate. • The singlet-dominated particles ˜ ν , h s , A s , and ν h compose a secluded DMsector where the DM was mainly annihilated by any of the channels ˜ ν ˜ ν ∗ → A s A s , h s h s , ν h ¯ ν h . It communicates with the SM sector by the Higgs-portal orthe neutrino-portal. As we introduced before, this situation constrains theYukawa coupling λ ν tightly in getting the measured density, but it has nolimitation on the splitting between m ˜ ν and the Higgsino mass. As mentionedbefore, 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 2 τ plus missing momentum signalof the process pp → ˜ χ ± ˜ χ ∓ → ( τ ± E MissT )( τ ∓ E MissT ) in testing the theory. Unlike thecolored sparticles that may be very massive, light Higgsinos are favored by naturalelectroweak symmetry breaking. As a result, they are expected to be richly pro-duced at the LHC. For the secluded DM case, ATLAS analyses have excluded someparameter space discussed at the end of the last section. With the advent of theLHC’s high luminosity phase, more parameter space will be explored. For exam-ple, we once compared the ATLAS analyses of the signal at the 13 TeV LHC with36 . − and 139 fb − data [101, 102]. We found the excluded region on m ˜ ν − m ˜ χ ± plane expanded from m ˜ χ (cid:46)
45 GeV to m ˜ χ (cid:46)
110 GeV for m ˜ χ ± = 200 GeV, andfrom m ˜ χ (cid:46)
120 GeV to m ˜ χ (cid:46)
200 GeV for m ˜ χ ± = 300 GeV. Concerning the co-annihilation case, it is hard for the LHC to detect the signal due to the compressedspectrum, but the future International Linear Collider may be capable of doing sucha job (see, for example, the study in [103] for the compressed spectrum case). Weemphasize that, different from the prediction of the MSSM, m ˜ χ ± may be significantlylarger than m ˜ χ in the ISS-NMSSM due to the mixing of ˜ H u,d with ˜ S in Eq. (2.5).As a result, the splitting between m ˜ χ ± and m ˜ ν can reach 20 GeV (see the points in– 31 –able 2), and it becomes even larger as the parameter λ increases. This feature isbeneficial for the signal’s detection. Acknowledgement
This work is supported by the National Natural Science Foundation of China (NNSFC)under grant No. 11575053 and 12075076.
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