Light higgsino-dominated NLSPs in semi-constrained NMSSM

In the semi-constrained next-to minimal supersymmetric standard model (scNMSSM, or NMSSM with non-universal Higgs mass) under current constraints, we consider a scenario where is the SM-like Higgs, is a singlino-dominated LSP; and are mass-degenerated, light, and higgsino-dominated next-to-lightest supersymmetric particles (NLSPs). We investigate the constraints of these NLSPs by searching for supersymmetry particles at the LHC Run-I and Run-II and discuss the possibility of discovering these NLSPs in the future. We arrive at the following conclusions: (i) With all data of Run I and up to data of Run II at the LHC, the search results by ATLAS and CMS still cannot exclude the higgsino-dominated NLSPs of . (ii) When the mass difference with is smaller than , and have opposite preferences with regard to decaying to or . (iii) When the mass difference between NLSP and LSP is larger than , most samples can be verified at the level with future data at the LHC. Meanwhile, with data at high-luminosity LHC (HL-LHC), almost all of the samples can be verified at the level, even if the mass difference is insufficient. (iv) The funnel and the funnel mechanisms for the singlino-dominated LSP annihilation cannot be distinguished by searching for NLSPs.

As an internal symmetry between fermions and bosons, supersymmetry (SUSY) is an attractive concept. In the framework of SUSY, the strong, weak, and hypercharge gauge couplings ( ) can be unified at the GUT scale ( ), and the large hierarchy problem between the electroweak and the Planck scales can be resolved. Further, with the R-parity conserved, the lightest SUSY particle (LSP) is stable and can be a good candidate for weakly-interacting-massive-particle (WIMP) dark matter (DM).
SUSY at the TeV scale is motivated by possible cancellation of quadratic divergences of the Higgs boson mass. The simplest implementation of SUSY is the minimal supersymmetric extension to the standard model (MSSM). Since the soft SUSY breaking parameters are totally free in the MSSM, a dynamic approach to obtain these parameters is more favored. In the minimal super-µ µĤ uĤd µ gravity (mSUGRA), the Käler potential is employed to yield minimal kinetic energy terms for MSSM fields, where all trilinear couplings, gaugino, and scalar mass parameters unify respectively at the GUT scale. The fully constrained MSSM (CMSSM) is the MSSM with the boundary conditions that are the same as in the mSU-GRA. However, to obtain a 125 GeV SM-like Higgs, the MSSM requires large one-loop radiative corrections to the Higgs mass, which renders the MSSM not natural. There is a so-called -problem [1] in the MSSM, where the superpotential contains a term , and is the only dimensionful parameter, which has to be chosen artificially.
In recent years, ATLAS and CMS collaborations have carried out numerous searches for SUSY particles, which pushed the gluino and squarks masses bounds in simple models up to several hundred GeV and even the TeV scale. Meanwhile, it remains possible for the electroweakino sector to be very light. The electroweakino sector of NMSSM was studied in [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], among which different search channels were provided, such as multileptons [35,36] and jets with missing transverse momentum ( ) [37]. These motivated us to verify the current status of higgsino in special SUSY models such as the scNMSSM, under direct-search constraints and their possibility of discovery by detailed simulation.
In this study, we discuss the light higgsino-dominated NLSPs (next-to-lightest supersymmetric particles) in the scNMSSM. We use the scenario of singlino-dominated and SM-like in the scan result in our former work on scNMSSM [26], where we considered the constraints including theoretical constraints of vacuum stability and Landau pole, experimental constraints of Higgs data, muon g-2, B physics, dark matter relic density, and direct searches, etc. Thus, in this scenario the and are higgsino-dominated, light and mass-degenerated NLSPs. We first investigate the constraints on these NLSPs, including searching for SUSY particles at the LHC Run-I and Run-II. We employ Monte Carlo algorithm to perform detailed simulations to impose these constraints. Then, we discuss the possibility of discovering the higgsino-dominated NLSPs in the future at the high-luminosity LHC (HL-LHC).
This paper is organized as follows. First, in Section 2, we briefly introduce the model of NMSSM and scNMSSM, especially in the Higgs and electroweakino sector. Later in Section 3, we discuss the constraints to the light higgsino-dominated NLSPs, and the possibility of discovering them at the HL-LHC. Finally, we draw our conclusions in Section 4.

Introduction to NMSSM and scNMSSM
The superpotential of NMSSM with symmetry : where superfields and are two complex doublet superfields, superfield is the singlet superfield, coupling constants and are dimensionless, and presents the Yukawa couplings of the and to the quark and lepton superfields. In electroweak symmetry breaking, the scalar component of superfields , , and obtain their vacuum expectation values (VEVs) , , and respectively. The relations between the VEVs are where the is the mass scale of a higgsino, like in the MSSM. In the following, for the sake of convenience, we refer to as . The soft SUSY breaking terms in the NMSSM are only different from the MSSM in several terms: where S, , and are the scalar components of the superfields, the is the soft SUSY breaking mass for singlet field S, and the trilinear coupling constants and have a mass dimension.
In the semi-constrained NMSSM (scNMSSM), the Higgs sector are considered non-universal, that is, the Higgs soft mass , , and are allowed to be different from , and the trilinear couplings , can be different from . Hence, in the scNMSSM, the complete parameter sector is usually chosen as:

H uĤdŜ
When the electroweak symmetry breaks, the scalar component of superfields , , and can be written as On the basis of , the CP-even scalar mass matrix is [45] L ∋ with where and . Indeed, here is a more common basis , where and the is the SM Higgs field. On the basis of , the scalar mass matrix is different from Eq. (7). However, since the rotation of the basis does not touch the third component , the will remain the same as in Eq. (10). The Higgs boson mass matrix on basis is given by [46] M 2 Subsequently, the Goldstone mode is dropped off, and the CP-odd scalar mass matrix on the basis of becomes [45] L ∋ 1 2 where The mass eigenstates of the CP-even Higgs ( ) and the CP-odd Higgs ( ) are obtained by where the matrix can diagonalize the mass matrix , and the matrix can diagonalize the mass matrix .

Electroweakino sector of NMSSM and scNMSSMχ
In the NMSSM, there are five neutralinos ( ), which are a mixture of (bino), (wino), , (higgsinos), and (singlino). On the gauge-eigenstate basis , the neutralino mass matrix takes the form [45] Chinese Physics C Vol. 44, No. 6 (2020) 061001 061001-3 where . To obtain the mass eigenstates, one may diagnolize the neutralino mass matrix where is the diagonal mass matrix, and the order of eigenvalues is . Meanwhile, the mass eigenstates are obtained In the scNMSSM, bino and wino were constrained to be very heavy because of the high mass bounds of gluino and the universal gaugino mass at the GUT scale. Thus, they can be decoupled from the light sector. Then, the following relations for the are found [47,48]: We assume that the lightest neutralino is the lightest supersymmetric particle (LSP) and makes up of the cosmic dark matter. If the LSP satisfies , we call it singlino-dominated. The coupling of such an LSP with the CP-even Higgs bosons is given by [47,48] In the singlino-dominated-LSP scenario, assuming , the mass of LSP can be written as . From Eq. (23), Eq. (10), and Eq. (21), one can find the sum rule [49,50]: In the case where singlet-like, sizable, , and are not too large, this equation can become The chargino sector in the NMSSM is similar to the neutralino sector. The charged higgsino , (with mass scale ) and the charged gaugino (with mass scale ) can combine, forming two couples of physical chargino . On the gauge-eigenstate basis , the chargino mass matrix is given by . (30) To obtain the chargino mass eigenstates, one can use two unitary matrices to diagonalize the chargino mass matrix by where indicates the diagonal mass matrix, and the order of eigenvalues is . Meanwhile, we obtain the mass eigenstates , .
In the scNMSSM, since , can be higgsinodominated, with a mass of approximately . With is singlino-dominated, can be higgsino-dominated, with masses that are nearly degenerate and approximately and . When is not large, i.e., smaller than the mass of other particles, the nearly-degenerate and are referred to as the next-to-lightest SUSY particles (NLSPs). In this study, we focus on the detection of the higgsino-dominated NLSPs ( and ) in the scNMSSM.

Light Higgsino-dominated NLSPs in scNMSSM
We employ the scan result from our previous study on scNMSSM [26]; however, we only consider the surviving samples with singlino-dominated ( ) as the LSP, and impose the SUSY search constraints with [51][52][53]. We perform the scan with the program [54][55][56][57], and considered the constraints there, including theoretical constraints of vacuum stability and the Landau pole, experimental constraints of Higgs data, muon g-2, B physics, dark matter relic density, and direct searches, etc 1) . We also use [58] to constrain the Higgs sector (with as the SM-like Higgs and ), and [59,60] to constrain SUSY particles. The detail of the constraints can be found in Ref. [26]. The scanned spaces of the parameters are: As shown in Ref. [26], in the surviving parameter space, In this study, we choose the surviving samples with LSP as singlino-dominated, and higgsino-dominated neutralino and chargino ( and ) as NLSPs. Thus, the samples with higgsino-dominated neutralinos as LSPs, or as NLSP, are discarded. In the following, we focus on the higgsino-dominated NLSPs in the scNMSSM, considering its constraints from direct search results at the LHC Run I and Run II, its production and decay, and its possibility of discovery at the HL-LHC in the future. Furthermore, we use [51][52][53] to impose these constraints of direct SUSY search results at the LHC, using all data at Run I and up to data at Run II [61][62][63][64][65][66]. For masses of GeV, the crosssections of the higgsino-dominated NLSPs can be sizeable, thus we pay special attention to the NLSPs. First, We use [67] to generate three types of tree level processes at 8 TeV and 13 TeV:

MadGraph
Since the cross-sections by the are at the tree After the simulation, we obtain a '.root' file. We use the to read this '.root' file. Then, we apply the same cuts in signal regions of the CMS and ATLAS experiments at 8 TeV and 13 TeV, using analysis cards that have been implemented in . At the last step, with the , we obtain an r-value for each sample, which is defined as where S is the total number of expected signal events, is the uncertainty of S, and is the experimentally measured at the 95% confidence limit of the signal events number. Hence, a model can be considered excluded at the 95% confidence level, if . If the in only one signal region, the model can also be excluded. We obtain , the maximal value of r in different signal regions. The model is excluded if . After using to verify our surviving samples, we note that most of the samples excluded are by the CMS analysis in multilepton final states at 13 TeV LHC with data [61-64, 72] 1) . We confirmed that the relevant mechanism is produced and each decaying to two bodies. Since the sleptons are heavier, the and mainly decay to LSP plus a W, or Z, or Higgs boson. The most effective processes excluding the samples are and .
̸ − → p T The search strategy for these two processes is three or more leptons and a large in the final state, since these channels are relatively cleaner than the jet channels at the LHC. The CMS searches related to our processes include the following signal regions (SR) SR-A, SR-C, and SR-F where is the angle between and .

fb −1 CheckMATE
139 fb −1 12.9 fb −1 139 fb −1 1) The CMS collaboration has not released update results with more data in the multilepton channel up to now. The cut scheme of ATLAS analysis in this channel is different, and we checked that the ATLAS result with data implemented in the is much weaker than the CMS result in constraining our samples, and thus even with data it has no significant impact on our final conclusion. We also have considered the CMS analysis for the compressed spectrum with data, and find that even with the current data it can not constrain our samples more.
Chinese Physics C Vol. 44, No. 6 (2020) 061001 • SR-C: events with two light leptons (e or ) forming an OSSF pair, and one candidate. The SR-C is divided into 18 bins, according to the invariant mass , the two-lepton 'stransverse mass' [73][74][75] instead of on the off-Z regions, and the . The is defined as where and are the transverse momenta for the two leptons, respectively, while and depict the random two components of the missing transverse momenta . This is used to suppress the SM background, since the large background is at low .
• SR-F: events with one light lepton (e or ) plus two candidates. SR-F is divided into 12 bins, according to , , and the . , and for compressed mass spectrum [89,90]. In their analyses, they considered simple models, where purely higgsino or wino NLSP produced in pair, each decaying to plus h, Z, or in 100%, such that the results do not apply to our samples directly. Imposing these new constraints is complex, and we plan to address them in our future studies.

Production and decay of Higgsino-dominated
For the surviving samples, we first investigated the production cross-sections of  ,  ,  ,  ,  , , , and at the 14 TeV LHC. Since the NLSPs are higgsino-dominated, the cross-sections are not significantly different from those of the pure higgsino production. Here, we only revise some of the old conclusions.ŝ • The cross-sections decrease quickly when masses increase, for the partonic Mandelstam variable increase, and the parton fluxes decrease.
• The cross-section of is approximately two times , for both and . The reason is that the LHC is proton-proton collider, and the parton distribution functions (PDF) for up quark are larger than for the down quark.χ where the matrix N is the neutralino mixing matrix. Since are higgsino-dominated, , from Eq. (30) we have , such that , and .χ The decay branching ratios of the NLSPs are shown in Fig. 1 and Fig. 2. From the left plots in Fig. 1, we see that the chargino decays to plus a W boson in 100%: when the mass difference between and is larger than , the W boson is a real one; whereas when the mass difference is insufficient, the W boson is a virtual one, or the decay is a three-body decay. The low mass difference is negative for us to search for the SUSY particles, since the leptons coming form a virtual W boson are very soft and hard to detect.χ The main decay modes of the neutralino ( ) are to a plus a Z boson or a Higgs boson. In the middle and right plots of Fig. 1 and in Fig. 2, we show the branching ratios of the neutralinos on the plane of vs , where . In these plots, we use the dashed line, , and the dotted line, , dividing the plane into three parts.  Fig. 2 shows that the mainly decays to the light Higgs boson .
• Case II: In the region , the neutralino can decay to plus a real Z boson or a light Higgs boson . As shown in the upper middle plot of Fig. 1, the mainly decay to a real Z boson plus . According to the upper right plot of Fig. 1 and lower left plot of Fig. 2, the mainly decays to a light Higgs boson plus .
• Case III: In the region , all these decay channels are opened. The mainly decays to and a 125 GeV SM-like Higgs boson , while mainly decays to and Z bosons.
In the channel ( ), like the , when the mass difference is insufficient, the Z boson also becomes a virtual one. In the channel ( ), where the Higgs boson can be or , and is the SM-like one. Both and mainly decay to , thus the background is sizable at the LHC. In the case that Higgs decay to WW, ZZ, or , and W or Z decays leptonically, this might contribute to the mul- Notably, when heavier neutralinos decay to the LSP, the and behave differently. Especially in the case II, prefers to decay to a Z boson and , ; while tends to decay to a light Higgs boson and , . The couplings and can be written as where the . When the two terms in Eq. (41) or Eq. (42) have different signs, and do not cancel each other, the couplings can be considerably larger than ; otherwise the cancel between the two terms can make smaller than . For some surviving samples, have the cancellation between the two terms, and that leads to small and large . Six benchmark points are listed in Table 1.

Possibility of future discovery at HL-LHC
We investigate the possibility of future detection of electroweakinos at the HL-LHC. We adopt the same analysis of the multilepton final state by CMS [61,62], only increasing the integrated luminosity from 35.9 to 300 , to estimate the possibility of discovery in the future. . Colors indicate branching ratios of chargino to plus W boson, and neutralino to plus Z boson, respectively. In the upper panel, the boson is a real one, and the decay is real two-body decay; while in the lower panel the boson is a virtual one and the decay is virtual three-body decay.
Chinese Physics C Vol. 44, No. 6 (2020) 061001 061001-7 We evaluate the signal significance by where S and B are the number of events from signal and background processes, respectively. ss mχ0 Fig. 3, we show on the planes of versus , , and respectively. We can see that most of the samples can be evaluated at the level when the mass difference between LSP and NLSPs is sufficient. However, there remain some samples that cannot be checked at level above 3 or 5 sigma. The main reason is that the mass spectra is compressed, such that the leptons from the decay of NLSPs are very soft. Because cut has to be very large at the LHC due to the large background, detecting soft particles is not easy. Combining with Figs. 1 and 2, we can learn the following facts: • If or ( ) decays to a virtual vector boson, or in the area upon the dashed line in all the plots, the signal significance is less than , and it is hard to check with data at the LHC.
• If or decays to a real vector boson, the area between the dashed and dotted line in left and middle plots, the signal significance can be larger than , which is easy to verify with data at the LHC. • The decay to a light Higgs , the area between the dashed and dotted line in the right plane, the signal significance is less than for some samples. This is because the light Higgs mainly decay to , which is hard to distinguish from the background.
• If ( ) decays to an SM-like Higgs, or in the area below the dotted line in middle and right plots, the signal significance is also larger than for most samples.
For the samples with insufficient mass difference between the NLSPs and LSP, the integrated luminosity 300 is not sufficient. Hence, we attempted to increase the luminosity to 3000 , the result is that nearly all samples can be checked with at 3000 . 1 ≃ m Fig. 4, we show the signal significance on the planes of versus . We can see that there are mainly two mechanisms for dark matter annihilation: the funnel, where , and the funnel where or . Unfortunately, searching for the NLSPs does not contribute to distinguishing these two   [26], the spinindependent cross-section of can be sizable, such that singlino-dominated dark matter may be accessible in the future direct detections, such as XENONnT and LUX-ZEPLIN (LZ-7 2T).

Conclusionsχ
We discussed the light higgsino-dominated NLSPs in the scNMSSM, which is also called the non-universal Higgs mass (NUHM) version of the NMSSM. We use the scenario with singlino-dominated and SM-like in the scan result of our previous study on scNMSSM, where we considered the constraints including theoretical constraints of vacuum stability and the Landau pole, experimental constraints of Higgs data, muon g-2, B physics, dark matter relic density and direct searches, etc. In our scenario, the bino and wino are heavy because of the high mass bound of gluino and the unification of gaugino masses at the GUT scale. Thus, the and are higgsino-dominated and mass-degenerated NLSPs.
We first investigate the direct constraints to these light higgsino-dominated NLSPs, including searching for the SUSY particle at the LHC Run-I and Run-II. We use Monte Carlo algorithm to perform detailed simulations to impose these constraints from the search of SUSY particles at the LHC. Then, we discuss the possibility of checking the light higgsino-dominated NLSPs at the HL-LHC in the future. We use the same analysis by increasing the integrated luminosity to 300 fb −1 and 3000 fb −1 .

∼ 200 GeV
Finally, we come to the following conclusions regarding the higgsino-dominated NLSPs in scNMSSM: 36  • The best channels to detect the NLSPs are though the real two-body decay and . When the mass difference is sufficient, most of the samples can be studied at the 5 level with future data at the LHC. For data at the LHC, nearly all of the samples can be studied at the level even if the mass difference is insufficient. • The funnel and the funnel are the two main mechanisms for the singlino-dominated LSP annihilating, which cannot be distinguished by searching for NLSPs.
We thank Yuanfang Yue, Yang Zhang, and Liangliang Shang for useful discussions.