Light Singlino Dark Matter at the LHC

Light singlino-like neutralino is found to be a very promising candidate for DM in the allowed parameter space of the NMSSM. The DM annihilation process takes place via light Higgs bosons which are natural in this model. Identifying the allowed region of parameter spaces including various constraints, the detection prospect of such light DM candidate and Higgs bosons are explored at the LHC with its high luminosity options. Light Higgs bosons and the DM candidate, the lightest singlino-like neutralino are indirectly produced at the LHC via the SM Higgs production and its subsequent decays. Jet substructure techniques are used to tag boosted Higgs. We observed that a reasonable signal significance, more than 5$\sigma$ can be achieved corresponding to integrated luminosity options ${\cal L}=$300 $\rm fb^{-1}$ and 3000 $\rm fb^{-1}$, for the range of Higgs bosons and neutralino masses compatible with low mass DM solution.


Introduction
Understanding the nature of dark matter(DM) candidate is of great interest in present day of particle physics, particularly, in the context of beyond standard model(BSM) physics. Huge efforts are in place by various experiments to search for DM candidate via direct and indirect manner [1,2]. Unfortunately, still the candidate of DM remains elusive. Very recent observations from PLANCK [3] predict the limits of relic density at 2σ as, Ωh 2 = 0.12 ± 0.001. (1.1) It is observed that the DM annihilation cross section at the weak scale naturally predicts relic density consistent with this PLANCK data. Currently, searches for DM candidates are one of the most exciting and challenging programs. Numerous dedicated experiments including Large Hadron Collider (LHC) are involved in this endeavour, and have made considerable progress. However, all negative results in direct searches of DM experiments, lead to stringent limits on DM-nucleon scattering cross sections in terms of DM particle masses [4][5][6][7][8][9]. As we know, because of the non-relativistic nature of DM candidate, the DM-nucleon scattering cross section can be separated into two parts, spin independent(SI) and spin dependent(SD). The spin independent(SI) part is mediated by scalars and increases with the mass of nucleon, whereas the spin dependent(SD) process involving axial-vector coupling with nuclear spin is mediated by gauge bosons. Obviously, SD cross section is larger than SI because of the suppressed coupling due to light quark masses [10][11][12]. Recent measurements by XENON1T experiment reported an upper limit of the DM-nucleon SI elastic scattering cross section at 4.1×10 −47 cm 2 and 2×10 −44 cm 2 corresponding to DM particle masses of 30 GeV and 6 GeV respectively [4]. These are the most stringent limits to date, whereas limits from LUX [5] and PANDA [6] are not competitive. With the detector upgrade in XENONnT experiment, the sensitivity is expected to improve by an order of magnitude [4]. It is to be noted that the XENON1T experiment is not sensitive to very lower range(<6 GeV) DM particle masses. However, there are few other experiments which are sensitive to this lower mass range of DM [7,[13][14][15]. For instance, DarkSide-50 experiment searches for DM candidate covering the mass range < ∼ 20 GeV, and lack of observation of any signal event leads to an exclusion limit on DM-nucleon SI cross section at 10 −41 cm 2 at 90% C.L corresponding to DM particle of mass 1.8 GeV [7]. Similarly, the SD DMproton and DM-neutron scattering cross sections are also constrained for a reasonably wide range of DM particle masses. The exclusion limit on SD DM-neutron scattering cross section at 95% C.L. also come from XENON1T, which predicts at 90% CL, an upper limit 6.3×10 −42 cm 2 for 30 GeV DM particle mass, and it increases further to 3×10 −39 cm 2 for 6 GeV mass [8]. The most stringent SD cross section limits to date on DM-proton scattering cross section at 90% C.L ∼ 3 × 10 −41 cm 2 for 20 GeV DM particle mass come from PICO-60 [9]. Apart from these direct searches, DM candidates are also explored indirectly at the LHC experiment. The DM particle produced in protonproton collision at the LHC leaves an imbalance of missing energy signature in the detector because of its extreme weak interaction with matters. Hence, the final state consisting a hard missing energy along with a recoil of visible energy is assumed to be a classical signature of DM. Currently, in both CMS and ATLAS experiments, searching for signature of DM candidates are treated as a very high priority analysis [16]. However, from the non-observation of any signal events in data, model dependent limits of DM particle masses are set by both CMS and ATLAS experiments [17,18]. Evidently, even in the presence of stringent constraints on DM particle masses from all direct and indirect searches, as discussed above, still a considerable range of lower (∼ few GeV) and higher (O(100) GeV) masses are not ruled out by data. Naturally, this phenomena attracts special attention to look for models, which can offer viable DM candidates compatible with data. Motivated by this observation, in this paper we try to find models of DM particle masses corresponding to this lower range of masses, which can provide solution consistent with all constraints due to direct and indirect searches, as discussed above [4-9, 17, 18].
Variety of well motivated BSM predicts plethora of cold and warm DM particle candidates [19]. Among them, the most popular and widely studied DM models are offered by minimal supersymmetric standard model (MSSM). In MSSM, the most popular candidate for DM with conserved R-parity is the lightest neutralino ( χ 0 1 ), a majorana spin 1/2 particle. In order to obtain right relic density (Eq. 1.1), the DM candidate is favoured to be the Higgsino like, and of the mass range ∼1 TeV [20][21][22]. Interestingly, like MSSM, in the theory of next-to-minimal supersymmetric standard model(NMSSM) [23][24][25][26], the lightest neutralino also appears to be a potential DM candidate. In the NMSSM, the Higgsino mass term(µ) is generated dynamically, in order to cure the µ-problem [27] by adding a singlet field with two Higgs doublet, and extending the Higgs sector resulting in seven Higgs bosons states. Because of interplay between model parameters, one or two of the Higgs boson states can be very light, even less than the mass of SM like Higgs boson, without violating any collider constraints [28][29][30][31][32][33][34]. Furthermore, the singlino, fermionic superpartner of singlet field extends the neutralino sector with five physical states, where the lightest neutralino state which may be a dominantly singlino like, plays role as a DM candidate. In particular, even with a very low mass (∼ few GeV), the singlino-like neutralino appears to be a viable dark matter candidate without violating any existing constraints predicted by DM experiments [35][36][37][38][39][40][41]. In such a scenario, the possible DM annihilation process occurs via light Higgs bosons yielding cross section consistent with the relic density given by Eq. 1.1. This phenomena resembles the scenario of Higgs portal model, where light Higgs boson acts as a portal between the SM and non-SM sector [42]. Moreover, the DM(singlino)-nucleon scattering cross sections, both SI and SD, satisfy experimental limits predicted by direct searches, thanks to the presence of appropriate singlino composition in the lightest neutralino state. In this regard, the immediate and pertinent question to ask is about the prospect of detecting the signal of this very low mass singlino like DM candidate at the LHC. In literature, quite a few studies exist in this context [40,41,[43][44][45][46]. The objective of this present study is to revisit this DM solution in the framework of NMSSM, and then explore the detection prospect of such scenario at the LHC for the current and future luminosity options. More precisely, our goal is to find discovery potential of very light neutralino and Higgs boson states at the LHC, which in combination provide a DM solution.
At the LHC, the direct production of light singlino state and singlet like Higgs bosons, having negligible coupling with fermions and gauge bosons are very much suppressed. In such a scenario, these particles can be produced indirectly via the production of some other particles which are having non negligible coupling with these states [34,41,47,48]. For example, in this paper, we consider the production of non SM like light Higgs bosons via the decays of SM Higgs boson which is produced through standard mechanism. Subsequently, light singlino states are produced via the decay of light non SM like Higgs bosons. It is to be noted that the corresponding branching ratios(BR) of all these decay modes are very much sensitive to model parameters, which will be discussed in detail in later sections.
The SM Higgs boson is considered to be produced via gluon-gluon fusion, which is the dominant production mechanism [49,50]. In order to give a boost to the final state, the SM Higgs boson is produced exclusively along with a jet. Consequently, the lighter Higgs boson states originating from SM Higgs boson of mass 125 GeV are moderately boosted (p T ∼ 30 − 40 GeV), and so the decay products from those states emerge as a single fat jet. Using jet substructure technique this "Higgs jet" (HJ) is tagged where the two subjets are likely to be b-like [51,52]. In summary, we focus on the signal final state consisting a HJ and missing energy, along with at least one untagged QCD jet. Considering this signal final state, we perform detail signal and background simulation, and predict signal sensitivity for 300 fb −1 and 3000 fb −1 luminosity options.
The paper is organised as follows. In section 2, review the NMSSM model very briefly, discussed the region of parameter space compatible with relic density and DMnucleon scattering constraints as well. The relevant range of parameters are identified through numerical scan and presented. Signal and background simulation is presented in section 3, followed by a discussion on results. Finally, summary is presented in section 5.
2 The NMSSM Model and Dark Matter relic density In this section, we briefly outline NMSSM model set up relevant to our scenario, which provides very light singlet-like Higgs bosons and a very light singlino-like neutralino as a DM candidate with right relic density (Eq.1.1). The NMSSM contains an additional gauge singlet superfield (S) along with two Higgs doublet superfields (H u and H d ). The corresponding Z 3 -invariant superpotential is given by [23][24][25][26]53], where λ and κ are the dimensionless couplings, and W MSSM represents the part of the superpotential in MSSM counting Higgs doublets but without µ-term. In addition, two soft terms, λA λ SH u H d and 1 3 κA κ S 3 are also included. The Yukawa like term with coupling λ generates the Higgsino mass term, µ eff = λv s , where v s is the vacuum expectation value (VEV) acquired by the singlet superfield. The dynamic generation of µ-term, the key aspect for the motivation of NMSSM, prevents it to acquire large value [27]. The Higgsino mass term is expected to be at the level of electroweak scale to obtain appropriate electroweak symmetry breaking [25]. On the other hand, phenomenologically, µ eff is restricted to be µ eff > ∼ 100 GeV, due to the chargino mass limit predicted by LEP experiment [54].
As mentioned before, the enlarged Higgs sector of NMSSM consists of seven physical Higgs bosons, 3 CP even states (H 1 , H 2 , H 3 , assuming m H 1 < m H 2 < m H 3 ) and 2 CP odd states (A 1 , A 2 , assuming m A 1 < m A 2 ) and 2 charged Higgs(H ± ) boson states. Masses and couplings of these Higgs bosons are determined by model parameters. Higgs sector is briefly revisited here with the aim to identify respective ranges of corresponding parameters to our interest. The 3 CP even Higgs states are described by 3×3 mass matrices in the basis ψ R ≡ (H uR , H dR , S R ), the real parts of Higgs fields. The elements of mass matrix are given by [26], Here tanβ is the ratio of VEVs of neutral components of two Higgs doublet. The masses of 3 CP-even Higgs boson states can be obtained by diagonalising the mass matrix by an orthogonal matrix (S ij ; i,j=1-3), and hence physical states (H i ) become the admixture of weak Higgs boson states as, Notably the lightest CP-even Higgs boson mass is found to be bounded by [55,56], at the tree level. Notice that the extra contribution lifts the tree level Higgs boson mass substantially, and hence may not require huge contribution from higher order correction [57]. As a consequence, a wide region of parameter space which is less constrained can easily accommodate one of the CP even Higgs boson (primarily either H 1 or H 2 ) state as the SM like Higgs boson with a mass ∼ 125 GeV. This feature makes the NMSSM very attractive after the discovery of the SM Higgs boson at the LHC [31,[57][58][59][60].
In CP-odd sector, eliminating the goldstone modes, the elements of 2×2 mass matrix for CP odd Higgs boson states in ψ I ≡ (A, S I ) basis is given as, Similarly, diagonalising this mass matrix by an orthogonal (P ij , i,j = 1,2) matrix, the masses of the two physical CP-odd states (A 1 , A 2 ) can be obtained, and hence the corresponding composition of physical states are given as, Interestingly, unlike the MSSM, in NMSSM, the physical Higgs boson states contain a fraction of singlet component (S I ) which does not couple with fermions and gauge bosons. Of course, the content of singlet component in physical states is very much parameter space sensitive.
The Higgs sector and the corresponding masses and composition of physical states are described by six parameters: Dependence on squark masses and other trilinear terms ( A-terms) occurs via radiative corrections [61].
The fermionic superpartner,S of singlet field, mixes with Higgsinos extending the neutralino mass matrix to 5 × 5, in the basis (−iB,−iW 3 ,H 0 u ,H 0 d ,S) and it is presented as, with s β ≡ sinβ, c β ≡ cosβ, M 1 and M 2 are the masses ofB andW 3 gauginos respectively, v u and v d are the VEVs for neutral components of H u and H d fields, and constrained to be v 2 u + v 2 d = v 2 , and tan β = vu v d , and g 1 and g 2 are weak couplings. The masses of 5 neutralino states, mχ0 i (i = 1, .., 5) can be obtained by diagonalising the mass matrix M N by an orthogonal matrix N 5×5 as, The analytical expressions of mχ0 i and the corresponding physical states exist in the literature for the MSSM [62,63], and as well as for the NMSSM [64,65]. The masses and couplings of neutralinos are very sensitive to NMSSM specific parameters, in particular λ, κ and v s or µ eff , along with M 1 and M 2 . Moreover, these parameters (except M 1 and M 2 ) are also strongly connected with the Higgs sector (Eq. 2.2 -2.4), and play important roles, along with A λ and A κ in determining the masses and mixings of Higgs bosons.
As stated earlier, the goal of this study is to provide a very low mass DM solution within the framework of the NMSSM. With this motivation, we try to identify the corresponding ranges of relevant model parameters compatible with all existing experimental data.
In our proposed solution, DM annihilation takes place via s-channel mediated by light Higgs scalars giving a pair of fermions in the final state [66][67][68], The DM annihilation rate is primarily sensitive to the interaction between neutralino and Higgs boson, and their relative mass difference. The Higgs-neutralino-neutralino couplings are given as [26,36], Here N 15 presents the singlino composition of LSP, whereas S i3 and P i2 stand for the singlet components of H i and A i respectively. Parameters λ and κ, which are connected with the singlino mass and its composition, are found to be very sensitive to the annihilation cross-section due to the above couplings(Eq. 2.10 and 2.11). The analytical expression for cross-section of annihilation processes are presented in Appendix-A. As indicated, the right relic density corresponding to lower range ( < ∼ 20 GeV) of DM masses can be achieved requiring neutralino and Higgs boson states singlino and singlet dominated respectively (N 15 ,P 12 , S 13 ∼ 1), i.e. g χ 0 The DM-nucleon scattering cross-sections, both σ SI and σ SD mediated by Higgs scalars and gauge bosons respectively, are given in Appendix-B. From direct searches, allowed spin-independent cross-section corresponding to DM masses of our interest, varies from ∼ 10 −44 cm 2 − 10 −46 cm 2 , which is achievable through the adjustments of coupling g χ 0 Again, we observed that a singlino like lightest neutralino and singlet dominant light Higgs bosons are most favoured. It suggests that the light singlino like DM candidate requires singlet dominated light Higgs boson states in order to have right relic density and DM-nucleon scattering cross-section [3,4,[7][8][9]. Therefore, the preferred parameter space favouring our scenario should provide, (a) a light singlino LSP, (b) light singlet-like Higgs bosons states.
A closer look at the neutralino mass matrix reveals few features of neutralino masses and mixings [53]. For instance, absence of mixing terms between singlino and gaugino fields implies no interaction between singlino like neutralino and gaugino like or gauge boson states. Notice that the mixing between singlet and doublet Higgs fields is decided by λv cos β or λv sin β (Eq. 2.7). Among the five neutralino states, two of them remain to be gaugino like if, |M 1,2 −µ eff | ≥ M Z , the mass of Z-boson. For a decoupling scenario, 2|κ|v s << µ eff , M 1,2 , the mass of singlino like neutralino turns out to be ∼ 2|κ|v s , and dominantly a singlino like. On the other hand, since µ eff or λv s ∼ O(100) GeV, hence for smaller values of λ < ∼0.1, the typical value of v s is expected to be large ∼ O(1) TeV. Therefore, for a very light singlino like LSP, |κ| should lie within the range of ∼ 10 −3 .
For higher values of λ ∼ 0.1, it is possible to accommodate comparatively lower values of v s , with little larger values of |κ|. In fact, the mass of singlino like LSP, m χ 0 1 ∼ 2 κ λ µ eff becomes small for κ λ ∼ 10 −2 . On contrary, for 2|κ|v s >> M 1,2 , µ eff , singlino like state becomes very heavy, and decouples from other neutralino states which consist only Higgsino and gaugino components like MSSM scenario. The other NMSSM parameters A κ and A λ which are not related with neutralino masses and mixings at the tree level, are expected to be restricted due to the requirements of very light singlet-like Higgs bosons. Following Eq. 2.4, the lighter CP odd state(A 1 ) is found to be singlet-like for decoupling type of scenario such as [ −3A κ κv s [53]. Moreover, as required above, |κ|v s cannot be large, so a moderate range (O(10) GeV) of A κ is required to obtain a light A 1 state. For the CP even Higgs sector, the spectrum of relevant parameters corresponding to our interest can be understood following a sum rule obtained using the tree level masses of H 1 and H 2 . This sum rule reads as [53], , which we also require for our proposed collider searches. The third CP even physical Higgs state H 3 , seems to be very massive and decoupled for large values of A λ . Finally, with all these above arguments corresponding to our proposed scenario, we conclude as: • very light singlino like LSP requires very small |κ|v s , with κ/λ ∼ 10 −2 , • requirement of very light Higgs boson states to be singlet-like, leads A λ to be very large(few TeV), but A κ not necessarily to be very large, but with a relative sign opposite to κ.

Parameter scan
Regions of parameters interesting to us are identified performing a numerical scan using NMSSMTools [70,71], interfaced with micrOMEGAs [35,[72][73][74] for calculation of DM observables. For the random scan, the numerical ranges of six sensitive parameters (Eq.2.6) are set as: We first performed a scan for a very wide range of these set of parameters, and then focus only on the above narrow range which is relevant to the signal phenomenology to be studied in this paper. The A-term for third generation(A t ) plays an important role in predicting the mass of the SM like Higgs boson [26,61] and is varied for a wider range, while setting other 3rd generation trilinear parameters as, In order to reduce the number of parameters to vary, all soft masses for left and right handed squarks for the first two generations are assumed as, The gaugino masses M 1 , M 2 and M 3 , which are important for chargino and neutralino sectors are set to be within the range, Slepton masses of first two generations are fixed to, While performing numerical scan, various constraints, theoretical and as well as experimental, included in NMSSMTools5.5.0 [70,71] are examined, and accordingly mass points are rejected or accepted. Precision measurements of the SM-like Higgs boson are used to constrain the model along with the mass requirement of 125 ± 3 GeV. In addition, limits on supersymmetric particles obtained at LEP, and Tevatron experiments, and as well as at the LHC are also imposed. Various measurements in flavour physics are also used to check the consistency of mass points. Of course, since lightest neutralino is assumed to be a DM candidate, it is also ensured that the selected mass points are consistent with PLANCK [3] constraint and Direct searches [4][5][6][7][8][9]. In the following, we present the allowed range of sensitive parameters, which are mentioned in the previous section. We focus the region of parameters which provide the mass of the lightest singlino-like neutralino up to 25 GeV and lightest Higgs bosons almost twice the singlino mass. In Fig. 1, the spin independent(SI) DM-nucleon crosssections are presented(dotted) for a range of neutralino masses up to 25 GeV and it is also subject to XENON1T and PICO constriants [4,9]. It clearly demonstrates that lightest neutralino, even with very low mass, can emerge as a viable DM candidate in NMSSM scenario. In Fig. 2, we show the dependence of lightest neutralino mass corresponding to interesting range shown in Fig. 1, on κ λ and µ eff . As anticipated, preferred values are |κ| ∼ 10 −3 and λ ∼ 10 −1 , whereas µ eff < ∼ 1 TeV, which is not expected to be very large. The tri-linear parameters A κ and A λ , play a very crucial role along with κ and λ, in determining the masses of Higgs bosons [53], in particular, m H 1 and m A 1 . In Fig. 3, the available region in the A κ − κ and A λ − κ plane, relevant to our scenario, are presented along with µ eff . As argued above, the large values of A λ ∼ O(1000) GeV and A κ ∼ O(10) GeV, for a very small value of κ, are required to achieve light singlet like Higgses and as well light singlino LSP interesting to us. Value of |κ| ∼ 0 is not permissible and symmetric nature of distribution arises because of dependence of magnitude of κ. We have checked that corresponding to this parameter space (Fig. 1, 2, 3), the singlet composition in lighter Higgs boson states, and singlino content in lightest neutralino, both are at the level of 95% or more.
Branching fractions for H SM → H 1 H 1 /A 1 A 1 , and subsequent decays H 1 /A 1 → χ 0 1 χ 0 1 or ff decide the signal rate. We observe that for favoured range of parameters such as λ, κ, A λ and A κ , as discussed above, the BR(H SM → H 1 H 1 /A 1 A 1 ) ∼ 10% or less, which is much below the upper limit of BR(H SM → BSM), constrained by Higgs data, and given by [75], BR BSM < 0.26 at 95% C.L. (3.8) Branching ratio of light Higgs bosons decay to LSP is also very sensitive to λ and κ, as evident from Eq.2.12 and 2.13. Substantial amount of singlet composition in light Higgs boson state and singlino content in LSP favour this decay channel. However, even a little presence of doublet components in light Higgs bosons enhance the decay rate in fermionic channel (ff ). Corresponding to our interesting region of parameters, the BR(H 1 /A 1 → χ 0 1 χ 0 1 ) appears to be quite reasonable, and sometimes it turns out to be around ∼ 70-80%.

Signal and Background
In this section, we present the discovery potential of singlino-like DM signal at the LHC with the CM energy √ s = 14 TeV with few luminosity options. We consider the production of light singlet-like Higgs bosons via the non standard decay channel of the SM Higgs, H SM → H 1 H 1 /A 1 A 1 , where the mass of H 1 or A 1 is less than the half of the mass of the SM Higgs boson. Subsequently, the lighter Higgs boson states assumed to decay to lightest neutralino pair (H 1 /A 1 → χ 0 1 χ 0 1 ) with a reasonable BR depending on the model parameter space, whereas the other competitive decay modes are to heavy fermions, like bb when kinematically accessible, otherwise τ τ . In order to have harder final state particles, we focus exclusive H SM + 1 jet process. As we know, the most dominant process of Higgs production proceeds via heavy top quark loop leading, gg → H SM [49,50]. An additional jet originates in next-to-leading order(NLO) perturbative QCD with significant increase of cross section, either from initial gluons or the heavy quarks inside the loop, leading to gg → H SM + g. Hence, the signal process to our interest appears to be, Hence we focus on signal final state comprising missing energy, which is a characteristic of DM signature, along with a reconstructed Higgs boson mass accompanied with at least one untagged jet.
The separation between decay products from lighter Higgs boson is given by [51], implying they are collimated for larger p T and/or lower mass of parent particle, where z is the fraction of momentum of Higgs boson carried by one of the decay product. In Fig. 4, we demonstrate the transverse momentum of lighter Higgs boson originating from SM Higgs decay (left) and the separation (Eq. 4.2) between their decay products (right) for three sets of Higgs boson masses. Clearly, the lighter states are more boosted and as expected, and their decay products are more collimated than those from higher states. These characteristic kinematic features are exploited in simulation to isolate signal. Armed with this observation, simulation is performed for signal events for three range of masses of H 1 or A 1 , as: i) low mass region : m H 1 /A 1 ≤ 10 GeV, ii) moderate mass region : 10 GeV ≤ m H 1 /A 1 ≤ 30 GeV, and iii) high mass region : 30 GeV ≤ m H 1 /A 1 ≤ 60 GeV. Notably, as stated above, for very 'low' and 'moderate' mass regions, the decay products, either τ τ or bb pair appear to be very collimated, and emerge as a single 'Higgs jet'(HJ) with constituents either two b-like or τ -like subjets depending on the decay modes. Hence, instead of tagging individual τ -jet or b-jet, which is challenging in this present scenario, 'Higgs jet' is tagged to classify signal from the background. On contrary, tagging HJ is not very effective for "high mass region", since decay products emerge with a wider separation. In this case, we observed that even losing signal events due to tagging of HJ, still it is found to be very useful to reduce the SM backgrounds substantially. Hence, in summary, simulation is performed for three categories: We discuss signal selection strategy for lower mass range of Higgs bosons, Eq. 4.3, in a later subsection separately.
The dominant sources of SM backgrounds corresponding to the signal processes(Eq. 4.4 and 4.5) are due to the processes: pp → tt, Wbb + jets, Zbb + jets, (4.6) Neutrinos originating from W or Z decay contribute to missing transverse energy (E / T ). We also checked the background contribution from WZj, ZZj, H SM Wj and H SM Zj and found to be very small due to comparatively very low cross sections and respective branching ratios.
For the sake of illustration, six benchmark points (BP), as shown in Table 1, compatible with various experimental data are chosen to simulate signal process. These BP are selected such that 2m χ 0 1 ∼ m H 1 /A 1 and covering mass ranges as required in Eq. 4.3 -4.5. Notice that for all cases, H 2 turns out to be the SM-like Higgs boson and decays to a pair of non SM-like Higgs bosons states H 2 → H 1 H 1 /A 1 A 1 , with a BR ranging from ∼ 0.01% to 10%, which is within the constrain given by (Eq. 3.8). As mentioned before, light Higgs bosons, mainly decay to either in bb or χ 0 1 χ 0 1 channel, which we require for our signal process. All background processes, except tt, are generated using Madgraph5-aMC@NLO-2.6.4 [76] and PYTHIA8 [77,78] for subsequent showering and hadronization, while tt events are simulated using PYTHIA8. The signal events are generated using PYTHIA8 inputting masses and branching ratios of SUSY particles and Higgs bosons through SLHA file [79] which is generated using NMSSMTools. In order to take detector effects, generated events for both signal and backgrounds are passed through Delphes-3.4.2 [80] using CMS detector card. The Delphes objects, namely eflows are used for analysis.
In simulation, events are selected adopting the following strategy.
• Lepton veto: Events consisting leptons are vetoed out. Leptons are selected with p T > 10 GeV and |η| <2.5. It reduces the background events significantly without losing any signal.
• HJ selection: The e-flow objects (e-flow tracks, e-flow photons and e-flow neutral hadrons) of Delphes are given as input to Fastjet3.3.2 [81] to construct fat jets. The Cambridge-Aachen [82] algorithm is used setting the jet size parameter R=1 and 1.6 for moderate and high mass regions (Eq.4.2) of lighter Higgs bosons respectively. The Fatjets are selected with p J T >40 GeV and |η| <4.0. In order to tag Fat jets with two subjets, those are then passed through mass-drop Tagger (MDT) [51,83] with µ =0.667 and y cut >0.01. The subjets of 'tagged fat jet' are further matched with the b-quarks of the events which are selected with a minimum p T cut of 0.5 GeV and |η| < 2.5 with a matching cone ∆R <0.3, where ∆R = (η q − η j ) 2 + (φ q − φ j ) 2 ; η q , η j are pseudorapidities and φ q , φ j are azimuthal angles of parton level b-quark and jet respectively. If both of the subjets are found to be b-like satisfying matching criteria, then it is claimed to be tagged as the HJ (J bb ). We found, the tagging efficiency of HJ is around 30% for lower range of light Higgs boson mass and goes down to around 15% for higher mass range. The mass of HJ is depicted in Fig. 5 for three sample of Higgs bosons masses. Clearly, the mass peaks are observed at the given input masses. However, peaks are observed to be more broader for higher Higgs boson masses. In the same figure, the corresponding distributions from backgrounds are also shown, which are not showing clearly any peaks, as expected. Note that the presence of J bb with a peak in its mass distribution is the characteristics of our signal events.
• Non tagged jets: After tagging J bb , non tagged jets are constructed out of remaining hadrons in the events using Anti-k T [84] algorithm with a jet size parameter R=0.5. The reconstructed jets are selected with p j T >20 GeV and |η| <4.0.
• Missing transverse momentum(E / T ): The missing transverse momentum is constructed by vector addition of momenta of all visible particles, i.e. p T = − p i T , where i runs over all constructed collection from the Detector. Delphes stores E / T of each events taking into account detector effects.

Signal for low mass of H 1 /A 1
In this sub-section, we discuss the search strategy of signal process, Eq. 4.3, which is very challenging owing to the fact that the masses of intermediate Higgs bosons are too low to have energetic decay products. We consider the decay mode of Higgs bosons to a pair of τ leptons. In order to avoid huge irreducible QCD background, H 1 /A 1 → bb decay channel was not considered even it is allowed to decay, and notably, for the same reason, the hadronic mode of tau leptons leading τ jets are not also simulated. Hence, in this scenario, we concentrated on the final state following Eq. 4.3 as, Note that the combined BR for both the τ leptons decaying leptonically is very small (∼ 12%). Moreover, leptons are too soft with a very low p T ∼ . The dominating sources of SM backgrounds are due to the processes, inclusive Drell-Yan, tt, and electroweak processes W+jets, WW+jets, WZ+jets. Carrying out a very naive simulation for both signal and background, we try to find the signal sensitivity. For all backgrounds processes except tt, matrix elements are generated in MadGraph5aMC@NLO-2.6.4(MG5NLO), then showering and hadronization are performed using PYTHIA8 as before. The tt events are fully generated using PYTHIA8. In simulation, leptons(both e and µ) are selected with p T ≥ 10 GeV and |η| <2.5. 2 Requirement of isolated leptons reduces the signal event significantly. The two leptons originating from τ pairs are not expected to be widely separated. In our simulation, we ensure isolated leptons by checking e-flow objects of Delphes using following criteria as, where p R<0.2 T is the sum of transverse momentum of all particles which are within ∆R < 0.2 with respect to lepton momentum direction. It also ensures that both the signal leptons are separated by ∆R >0.2. Construction E / T and jets (including b-jets) are same as before, and performed by Delphes.

Results and Discussion
Identifying various distinguishing features of signal process, we impose few selection cuts to eliminate background events. For example, the characteristics of HJ mass (m J bb ) distribution, as shown in Fig. 5, are very different for backgrounds and signal processes. Therefore, a background rejection cut setting as, m J bb < 30 GeV for lower mass range, 30 < m J bb < 60 GeV for high mass range, (5.1) is very effective, in particular, for eliminating tt background by 70-80%.
2 Experimentally lepton trigger of low p T are to be used  Figure 6: Transverse mass between J bb and E / T (Eq. 5.2) (left) and R(Eq.5.3)(right) for signal, bbZ + jets and tt.
Evidently, the transverse mass between HJ and E / T is restricted by the SM Higgs boson mass in signal, as shown in Fig. 6 (left), which is not the case for backgrounds. Hence an upper cut on it as, is found to be very useful in background suppression. Another interesting observable is found to be very helpful in suppressing the top background which is defined as [85], where n min j is the minimum number of jets required in event selection and H T = Obviously, by construction 0 < R ≤ 1 where n min j is set equal to 1 for signal event selection.
Distribution of R is expected to be on higher side for signal, since it is not very jetty, whereas for tt it is expected to be on lower side, as shown in Fig. 6. Therefore, a selection on R >0.5 suppresses a good fraction of top events and to some extent Zbb + jets events for moderate mass regions.
Cross section yields for signal corresponding to benchmark points, and background processes after each set of cuts are presented in Tables 2 and 3, which are subject to two different sets of cuts(Eq. 5.1) on m J bb , according to the range of light Higgs boson mass(Eq. 4.4 and 4.5).
The first row presents the leading order(LO) cross section with the center of mass energy √ s = 14 TeV setting NNPDF23LO [86] for parton distribution and choosing dynamic scale ( m 2 + p 2 T ) computed by Madgraph5-aMC@NLO-2.6.4 [76]. Cross sections for the background processes(Zbb+jets, Wbb+jets) are computed in Madgraph5-aMC@NLO-2.6.4 in five flavour scheme and subject to cuts, p b T >20 GeV, p j T >20 GeV, ∆R(b, b) >0.1 and ∆R(j, j) >0.4. Higher order effects to all these cross sections are taken into account through K-factors, as defined, K = σ NLO σ LO . These K-factors are obtained by computing respective cross sections using MCFM [87][88][89][90]. The K-factors of the processes, Zbb + jets and Wbb + jets, are considered to be the same as for the processes Zbb and Wbb, which are found to be ∼1.7 and ∼2.6 respectively, and are in close agreement with Ref [91]. For tt, K-factor=1.4 is used [92,93]. For signal, K-factors are found to be ∼ 1.8 using MCFM, very close to quoted values in Ref. [94]. These K-factors are taken into account in Tables 2 and 3 while computing final yields at the end.
Events are required to contain at least one jet with cuts p j T >20 GeV and |η| < 3 and vetoed out if there be any lepton. The E / T cut is useful in reducing the backgrounds, in particular due to the process with a Z and W boson in the final state, however, it costs signal also by almost a factor of 2, even it is more severe for signal corresponding to lower mass ∼15 GeV. Notice that the selection of HJ, and respective mass window suppresses backgrounds substantially, by almost two orders of magnitude, while signal events remains less affected. A cut on transverse mass, Eq. 5.2, is very effective in suppressing the backgrounds without costing signal events too much, as seen in both tables. Finally, as expected, the cut on R is very effective in suppressing top background further by about ∼ 50%.    Finally, in order to obtain final cross section yields, we take into account p Tdependent b-tagging efficiency ( b ) [95]. For tt event we use b = 0.66, whereas for other cases b = 0.55. The total background cross section is found to be 750 fb and 730 fb corresponding to two sets of selections as described in Tables 2 and 3 respectively. We summarize signal significances, as defined S/ √ B corresponding to five benchmark points in Table 4, for two choices of integrated luminosities L =300 fb −1 and 3000 fb −1 . It is to be noted that in background estimation, the contribution due to QCD is not taken into account, where jets and mis-measurement of jets can fake as b-jets and E / T respectively, which is beyond the scope to simulate in this current analysis. Remarkably, the significances are more than 5σ even for lower luminosity option.  Table 5). Cross-sections (LO) shown in the 1st row are computed using MadGraph5 aMC@NLO-2.6.4(MG5NLO) subject to cut p j T > 20 GeV, whereas in the subsequent rows, those are presented after each set of selection cuts, as shown. Notice the severe effect of selection cut of invariant mass of lepton pair. Finally, at the last row, we present cross sections, multiplying respective K-factors in order to take care of higher order effects. Similar K-factors are used for signal process and tt process, whereas for DY process it is taken to be 1.3 [96]. For electroweak processes, W+jets, WW+jets, WZ+jets K-factors are considered to be 1.42 [97], 1.8 [98] and 2.07 [99] respectively. We find the dominant background contribution are mainly due to the tt, DY and W+jets process. We have also checked the background contribution due to Υ and J/ψ production process, and found to be negligible attributing to the comparatively harder E / T cut. The total background cross section are found to be ∼ fb and signal significance turns out to be, S √ B ∼ 6(19) for integrated luminosity options 300 fb −1 (3000 fb −1 ).

Summary
Various experiments for DM searches have excluded a substantial range of masses DM particle candidates. However, the DM candidate with very low mass is still a viable option to explain the right relic density of our universe. In this study, we explore the scenario with very light DM candidate in the framework of the NMSSM which attempts to address the µ-problem of the MSSM by adding one additional singlet Higgs scalar with the two Higgs doublets. In this model the lightest neutralino, assumed to be the LSP of very lower mass, is offered as a DM candidate. The dominant presence of singlino composition in the lightest neutralino helps to evade constraints on DM-nucleon scattering cross section imposed by several experiments. In this proposed scenario, the DM annihilation takes place primarily via resonant process mediated by singlet-like light Higgs bosons, which decays to a pair fermions in the final state. Thus suppressed interaction between singlino-like neutralino and singlet-like Higgs scalars is responsible to overcome the constraint due to observed relic abundance. Remarkably, the light non-SM like Higgs bosons play a role as a portal between the non-SM and the SM sectors present in the initial and final states of the annihilation process respectively.
Detailed numerical scan of model parameters is performed taking into account various existing experimental constraints to identify compatible region corresponding to our proposed DM solution. This numerical study indicates that the range of NMSSM parameters are of the range, κ ∼ 10 −3 − 10 −2 , λ ∼ 10 −1 , |A κ | ∼ 10 − 100 GeV and A λ > ∼ 800GeV, which are very close to our understanding, as discussed in sections 2 and 3. Those allowed regions of parameters are demonstrated for the sake of illustrations.
There are various interesting phenomenological implications of singlino-like DM candidate at the LHC experiments, which are complimentary to direct searches of it in recoil experiments. In this current study, we have explored the discovery potential of such low mass DM candidate at the LHC corresponding to its high luminosity options. We consider the production of DM particle through SM Higgs production. The SM Higgs boson produced via the standard dominant gluon-gluon fusion process, decays to a pair of light non-SM like Higgs bosons. Subsequently, one of the light Higgs bosons decays to a pair of DM particle resulting in missing energy, whereas the other one decays primarily to a pair of, either b quarks or τ leptons, depending on its mass. In order to make the final state more boosted, we require one extra jet accompanied with SM Higgs boson production. Since the parent SM Higgs boson is moderately boosted, the pair of final state fermions are not expected to be well separated, and appear as a single jet, namely HJ. Therefore, the signal final state is characterized by a HJ, and missing transverse energy accompanied with at least one untagged jet. The HJ is tagged by employing sophisticated MD technique. For lower range of lighter Higgs boson mass (<10 GeV), we consider its decay to a pair of τ leptons, which eventually decay to leptonic channel leading final state with two leptons of opposite charge along with missing transverse energy and at least one untagged jet. For the sake of presentation of signal sensitivity, six benchmark points are selected covering all possible mass ranges. Detailed simulation for both the signal and backgrounds are carried out taking into account the detector effects using Delphes. Studying signal and background event characteristics, we have developed search strategy to suppress background contribution corresponding to a given range of light Higgs boson masses. We found that for medium and higher combination of LSP and light boson masses as presented by benchmark points, the sensitivity is more than 5σ with an integrated luminosity L=300 fb −1 , and for very high integrated luminosity option, L=3000 fb −1 , the sensitivity goes up further. This study clearly indicates that the discovery potential for most of the mass range which are consistent with DM solution is very promising with the reasonably high luminosity options of the LHC. We have also carried out a simulation for lower mass range, less than 10 GeV, of Higgs boson in leptonic final states. Our naive study shows a very promising results of achieving signal sensitivity with a reasonable significance. It is to be noted that in this study we do not estimate any systematics, which is out of the scope of the present study. We conclude that the singlino-like LSP may be a very good viable candidate for DM candidate corresponding to lower mass mass range, and its signature at the LHC is very robust with a very promising discovery potential for future LHC options.
A.2 DM annihilation through s-channel pseudo-scalar light Higgs: Using similar notations, only replacing H 1 by A 1 (light pseudoscalar Higgs), we have the squared amplitude given by [100,101]: , ω A 1 ff = g 2 ffA 1 g 2 3) The coupling gχ0 1χ 0 1 A 1 is given in Eq. 2.11, and g ffA 1 has similar structure as Eq.7.2 except components of pseudoscalar mass matrix P ij (Eq. 2.4), replacing S ij . Then the "thermally averaged pair-annihilation cross-section times velocity", σv , can be obtained as [100] σv = 1 m 2 Where ω(s) is ω H 1 ff (s) or ω A 1 ff (s) and T is temperature.

B.1 SI cross-section
So the effective spin independent interaction can be written as : L SI = λ Nχ χψψ (7.6) In our case, this spin-independent scattering cross-section (σ SI ) ofχ 0 1 with nuclei dominantly happens through exchange of scalar Higgs bosons. Whenχ 0 1 is singlino-like, we can write the scattering cross-section approximately as [36] σ SI

B.2 SD cross-section
The effective lagrangian in this case can be written as: ζ Nχ γ µ γ 5 χψγ µ γ 5 ψ (7.8) Here DM-nucleon scattering can be mediated in t-channel by Z-boson or squark mediator (I denote it as V, with mass m V ) . The cross-section in this case is becomes : N |qq| N 2 ζ p S A p + ζ n S A n 2 (7.9) Where J A is the angular momentum of the nucleus with A nucleons and S A N are the expectation value of the spin content of nucleon type N (n or p).