The Light Higgsino-dominated NLSPs in the Semi-constrained NMSSM

In the semi-constrained NMSSM (scNMSSM, or NMSSM with non-universal Higgs mass) under current constraints, we consider a scenario where $h_2$ is the SM-like Higgs, $\tilde{\chi}^0_1$ is singlino-dominated LSP, $\tilde{\chi}^{\pm}_1$ and $\tilde{\chi}^0_{2,3}$ are mass-degenerated, light and higgsino-dominated NLSPs (next-to-lightest supersymmetric particles). We investigate the constraints to these NLSPs from searching for SUSY particles at the LHC Run-I and Run-II, discuss the possibility of discovering these NLSPs in the future, and come to the following conclusions regarding the higgsino-dominated $100\sim200{\rm GeV}$ NLSPs in scNMSSM: (i) Among the search results for electrowekinos, the multilepton final state constrain our scenario most, and can exclude some of our samples. While up to now, the search results by Atlas and CMS with Run I and Run II data at the LHC can still not exclude the higgsino-dominated NLSPs of $100\sim200{\rm GeV}$. (ii) When the mass difference with $\tilde{\chi}^0_{1}$ is smaller than $m_{h_2}$, $\tilde{\chi}^0_{2}$ and $\tilde{\chi}^0_{3}$ have opposite preference on decaying to $Z/Z^*$ or $h_1$. (iii) The best channels to detect the NLSPs are though the real two-body decay $\tilde{\chi}^{\pm}_{1} \to \tilde{\chi}^0_1 W^{\pm}$ and $\tilde{\chi}^0_{2,3} \to \tilde{\chi}^0_1 Z/h_2$. When the mass difference is sufficient, most of the samples can be checked at $5\sigma$ level with future $300{~\rm fb}^{-1}$ data at the LHC. While with $3000 {~\rm fb}^{-1}$ data at the High Luminosity LHC (HL-LHC), nearly all of the samples can be checked at $5\sigma$ level even if the mass difference is insufficient. (iv) The $a_1$ funnel and the $h_2/Z$ funnel are the two main mechanisms for the singlino-dominated LSP annihilation, which can not be distinguished by searching for NLSPs.


Introduction to NMSSM and scNMSSM
The superpotential of the NMSSM with Z 3 symmetry : where the superfieldsĤ u andĤ d are two complex doublet superfields, the superfieldŜ is the singlet superfield, the coupling constrats λ and κ are dimensionless, and the W M SSM | µ=0 is actually the Yukawa couplings of theĤ u andĤ d to the quark and lepton superfields. When electroweak symmetry breaking, the scalar component of superfieldsĤ u ,Ĥ d and S get their vacuum expectation values(VEVs) v u , v d and v s respectively. The relations between the VEVs are GeV (2.3) µ ef f = λv s , (2.4) where the µ ef f is the mass of Higgsino, like in the MSSM. In the following, for the sake of convenience, we refer to µ as µ ef f . The soft SUSY breaking terms in the NMSSM is only different from the MSSM in several terms: (2.5) where the S, H u and H d is the scalar component of the superfields, the m 2 S is the soft SUSY breaking mass for single field S, and the trilinear coupling constants A λ and A κ have mass dimension.
In the semi-constrained NMSSM (scNMSSM), the Higgs sector are considered nonuniversal, that is, the Higgs soft mass m 2 Hu ,m 2 H d and m 2 S are allowed to be different from M 2 0 , and the trilinear couplings A λ , A κ can be different from A 0 . Hence, in the scNMSSM, the complete parameter sector is usually chosen as: λ, κ, tanβ, µ, A λ , A κ , A 0 , M 1/2 , M 0 . (2.6)

The Higgs sector of NMSSM and scNMSSM
When the electroweak symmetry breaking, the scalar component of superfieldsĤ u ,Ĥ d and S can be written as  13) and s β = sinβ, c β = cosβ. Actually, there is a more common basis H 1 , H 2 , S R , where 2.15) and the H 2 is the SM-like Higgs field. In the basis (H 1 , H 2 , S R ), the scalar mass matrix is different from eq.(2.9). But, since the rotation of the basis do not touch the third component S R , the M 2 S,S R S R will keep the same as in eq. (2.13). The Higgs boson mass matrix M 2 S in basis (H 1 , H 2 , S R ) is given by [16] And comparing eq.(2.21) with eq.(2.13), it's not hard to get M 2 S ,S R S R = M 2 S,S R S R . In order to get the physical CP-odd scalar Higgs bosons, one can rotate the Higgs fields, Then the Goldstone mode can be dropped off, and the CP-odd scalar mass matrix in the basis (A, S I ) become [15] The mass eigenstates of the CP-even Higgs h i (i = 1, 2, 3) and the CP-odd Higgs 26) where the matrix S ij can diagonalize the mass matrix M 2 S , and the matrix P ij can diagonalize the mass matrix M 2 P .

The electroweakino sector of NMSSM and scNMSSM
In the NMSSM, there are five neutralinosχ 0 i (i = 1, 2, 3, 4, 5), which are the mixture of B (bino),W 3 (wino),H 0 d ,H 0 u (higgsino) andS (singlino). In the gauge-eigenstate basis ψ 0 = (B,W 3 ,H 0 d ,H 0 u ,S), the neutralino mass matrix takes the form [15] To get the mass eigenstates, one can diagonalize the neutralino mass matrix Mχ0 where M D χ 0 means the diagonal mass matrix, and the order of eigenvalues is mχ0 1 < mχ0 2 < mχ0 3 < mχ0 4 < mχ0 5 . Meanwhile, one can get the mass eigenstates 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 GUT scale, thus they can be decoupled from the light sector. Then the following relations for the N ij can be can be found [17]: We assume the lightest neutralinoχ 0 1 is the lightest supersymmetric particle (LSP) and makes up of the cosmic dark matter. If the LSPχ 0 1 satisfies N 2 15 > 0.5, we call it singletdominated. And the coupling of such an LSP with the CP-even Higgs bosons is given by [17] In the singlet-dominated-LSP scenario, taking N 11 = N 12 = 0, the mass of LSP can be written as mχ0 1 ≈ Mχ 0 ,SS = 2κv s . And from eq.(2.27), eq.(2.13) and eq.(2.25), one can find the sum rule [18]: In the case that h 1 singlet-like, tanβ sizable, λ, κ and A λ not too large, this equation can become m 2 The Chargino sector in the NMSSM is very similar to neutralino sector. The charged HiggsinoH + u ,H − d (with mass scale around µ) and the charged gauginoW ± (with mass scale M 2 ) can also mixed respectively, forming two couples of physical chargino χ ± 1 , χ ± 2 . In the gauge-eigenstate basis (W ± ,H ± u,d ), the chargino mass matrix is given by [15] To obtain the chargino mass eigenstates, one can use two unitary matrix to diagonalize the chargino mass matrix where M D χ ± means the diagonal mass matrix, and the order of eigenvalues is mχ± 1 < mχ± 2 . Meanwhile, we can get the mass eigenstates (2.36) In the scNMSSM, since M 2 µ, χ ± 1 can be higgsino-dominated, with mass around µ. With χ 0 1 singlino-dominated, χ 0 2,3 can be higgsino-dominated, with masses nearly degenerate also around µ, and with N 2 23 +N 2 24 > 0.5. Then with µ not large, smaller than other sparticle mass, the nearly-degenerate χ ± 1 and χ 0 2,3 can be called the next-to-lightest SUSY particles (NLSPs). In this work, we will focus on the detection of the higgsino-dominated NLSPs (χ ± 1 and χ 0 2,3 ) in the scNMSSM.

The Light Higgs-dominated NLSPs in scNMSSM
In this work, we use the scan result in our former work about scNMSSM [7], but only consider the surviving samples with singlino-dominated χ 0 1 (|N 15 | 2 > 0.5) as the LSP, and impose the SUSY search constraints with CheckMATE [19]. We did the scan with the program NMSSMTools-5.4.1 [20], and considered the constraints there, 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. We also use HiggsBounds-5.1.1beta [21] to constrain the Higgs sector (with h 2 as the SM-like Higgs and 123 < m h 2 < 127 GeV), and SModelS-v1.1.1 [22] to to constrain the SUSY particles. The scanned spaces of the parameters are: Since in scNMSSM the bino and wino are heavy because of the high bounds of gluino mass, and the χ 0 1 LSP is singlino-dominated in our samples, the neutralino and chargino NLSPs (χ ± 1 and χ 0 2,3 ) are Higgsino dominated in this work. In the following, we focus on the Higgsino-dominated NLSPs in the scNMSSM, considering the constraints from the recent search results at the LHC Run I and Run II, and possibility of discovery at the HL-LHC in the future.

Constraints from the recent search results at the LHC
Unlike colored particles, the production rates of electroweakinos are very low at the LHC. But they can mainly decay to leptons plus E / T , and the SM backgrounds in these leptons channels are relatively cleaner than jet channels at the LHC. In recent years, Atlas and CMS collaborations have released several search results with the LHC Run-I & Run-II data in such channels as 2 + E / T [23], 3 + E / T [24,25] , 2γ + E / T [26] and Higgs+E / T [27]. In their analyses, they considered simple models, where purely higgsino or wino NLSP produced in pair, each decaying to χ 0 1 plus h, Z, or W ± in 100%. In this work, we use these result to constrain our specific surviving samples in scN-MSSM. We consider the production and decay ofχ + 2,3 at the LHC, using CheckMATE 2.0.26 [19] to impose these constraints.
Firstly, We use MadGraph5_aMC@NLO 2.6.6 [28] to generate three types tree level process at 8 TeV and 13 TeV: Since the cross sections by the MadGraph are at tree level, we multiply them by a NLO Kfactor calculated with the Prospino2 [29]. Then, we use the PYTHIA 8.2 [30] to deal with particle decay, parton showering, and hardronization, use Delphes 3.4.1 [31] to simulate the detector response, and use the anti-k T algorithm [32] for jet clustering.
After the simulation, we can get a '.root' file. We use the CheckMATE2 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, by using analysis cards which have been implement in CheckMATE2. At the last step, with the CheckMATE2 we get a r-value for each samples, which is defined as where S is the total number of expected signal events, ∆S is the uncertainty of S, and S 95 Exp. is the experimentally measured 95% confidence limit of signal events number. So, a model can be considered excluded at 95% confidence level, if r ≥ 1. If the r ≥ 1 in only one signal region, the model can also be excluded. We can get r max , the maximal value of r in different signal regions. The model is excluded if r max ≥ 1.
In Fig.1, we show the production cross sections ofχ + 1χ 0 • The relation between the cross sections and the masses of final states is clear shown in all of these plots, that is, the masses of final states particles are heavier, its production i need the initial state to be ud or dū, respectively. So, the reason is that the LHC is proton-proton collider, and the proton is a bound state of uud, so the parton distribution functions (PDF) for up quark is more than down quark, that leads to a linear relation of 2 times.
• From the third and fourth plots in lower panel, we can see that the production cross section ofχ 0 2χ 0 2 (i = 2 or 3) is very small, only a few fb. The reason is that the squarks are very heavy, so σ(pp →χ 0 iχ 0 i ) mainly contribute from s channel through Z boson resonance. The coupling of Z −χ 0 i −χ 0 j is given by where the matrix N is neutralino mixing matrix. And we can see that if After using CheckMATE to checking our surviving samples, we notice that most of the samples excluded are by the CMS analysis in multilepton final states at 13 TeV LHC with 35.9 fb −1 data [24]. We checked that the relevant mechanism isχ ± 1χ 0 2 produced and each decaying to 2 body. Since the sleptons are heavier, theχ ± 1 andχ 0 2 mainly decay to theχ 0 1 LSP plus a W , or Z, or Higgs boson. The most effective process excluding the samples are pp →χ ± 1 (W ±χ0 1 )χ 0 2 (Zχ 0 1 ) and pp →χ ± 1 (W ±χ0 1 )χ 0 2 (h 2χ 0 1 ). The searching strategy for these two process is three or more leptons plus large E / T in the final state. The CMS searches related to our process included the following signal regions (SR) SR-A, SR-C and SR-F • SR-A: events with three light leptons (e or µ), two of which forming an opposite sign same-flavor (OSSF) pair. The SR-A is divided into 44 bins, according to the invariant mass of OSSF pair M , the third lepton's transverse mass M T and the missing energy E / T .
• SR-C: events with two light leptons (e or µ) forming an OSSF pair, and one τ h candidate. The SR-C is divided into 18 bins, according to the invariant mass M , the two-lepton 'stransverse mass' M T 2 ( 1 , 2 ) [33] instead of M T on the off-Z regions, and the E / T . The M T 2 is defined as: where the E / T 1 and E / T 2 stand for the missing transverse energy for the two leptons respectively. And it's used to suppressed the SM background, since the large tt background is at low M T 2 .
• SR-F: events with one electron or muon plus two τ h candidates. SR-F is divided into 12 bins, according to M , M T 2 ( , τ 1 ) and the E / T .
In Fig.2, we show the branching ratios ofχ + 1 on the plane of mχ0 1 vs mχ+ 1 . We can see that, the charginoχ + 1 decay toχ 0 1 plus a W boson in 100%: when the mass difference betweenχ + 1 andχ 0 1 is greater than m W , the W boson is a real one; while when the mass difference is insufficient, the W boson is a virtual one, that is, the decay is a three-body decayχ ± 1 → νχ 0 1 . 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.
The main decay modes of neutralinoχ 0 i (i = 2, 3) are to aχ 0 1 plus a Z boson or a Higgs boson. In Fig.3  • Case II: In the region m Z ≤ mχ+ 1 − mχ0 1 < m h 2 , the neutralinoχ 0 i can decay toχ 0 1 plus a real Z boson or a light Higgs boson h 1 . As showed in the first and third plots on the upper panel, theχ 0 2 mainly decay to a real Z boson withχ 0 1 . While according to the first and third plots on the lower panel, theχ 0 3 mainly decay to a light Higgs boson h 1 plus a LSP.
• Case III: In the region mχ+ 1 − mχ0 1 ≥ m h 2 , the neutralinoχ 0 i can decay toχ 0 1 plus a 125 GeV SM-like Higgs boson h 2 , which is showed in the fourth plot (upper and lower panels).
In the channelχ 0 i →χ 0 1 Z (i = 2, 3), like theχ ± 1 → W ±χ0 1 , when the mass difference is insufficient the Z boson becomes a virtual one, which make it hard to detect. In the channel where the Higgs boson can be h 1 or h 2 and h 2 is the SM-like one. Both h 1 and h 2 mainly decay to bb, thus the tt background is sizable at the LHC. In the case that Higgs decay to WW, ZZ, or ττ, and W or Z decays leptonically, it might contribution to the multilepton final state. Since the light Higgs h 1 is highly singlet-dominated, thẽ χ 0 i →χ 0 1 h 1 is very hard to contribute to the multilepton signal regions. Thus only thẽ χ 0 i →χ 0 1 h 2 can contribute to the multilepton signal regions visibly. It is worth to mention that, when the heavier neutralinos decay to theχ 0 1 LSP, theχ 0 2 andχ 0 3 behave differently. Especially in the case II,χ 0 2 prefers to decay to a Z boson plus χ 0 1 , Br(χ 0 2 →χ 0 1 Z) > Br(χ 0 2 →χ 0 1 h 1 ); whileχ 0 3 tends to decay to a light Higgs boson h 1 plusχ 0 1 , Br(χ 0 3 →χ 0 1 Z) < Br(χ 0 3 →χ 0 1 h 1 ). The couplings C h 1χ 0 2χ 0 1 and C h 1χ 0 3χ 0 1 can be written down as where the N 11 , N 12 , N 21 , N 22 , N 31 and N 32 was set to 0 since the wino and bino are very heavy and decoupled in the scNMSSM, and the S 11 and S 12 was set to 0 since |S 13 | |S 11 |, |S 12 |. λ/ √ 2 1 and √ 2κ 1, so the couplings C h 1χ 0 2χ 0 1 and C h 1χ 0 2χ 0 1 are both very small and roughly the same. While the couplings C Zχ 0 2χ 0 1 and C Zχ 0 3χ 0 1 can be different from each other according to eq.(3.4), which can be approximated to where the g 2 /c W ∼ 1. When the two terms in eq. (3.8) or eq.(3.9) have different sign, and do not cancel with each other, the couplings C Zχ 0 iχ 0 1 can be much larger than C h 1χ 0 iχ 0 1 ; otherwise the cancel between the two terms can make C Zχ 0 iχ 0 1 smaller than C h 1χ 0 3χ 0 1 . For some surviving samples, C Zχ 0 3χ 0 1 have the cancellation between the two terms, and that leads to small Br(χ 0 3 →χ 0 1 Z) and large Br(χ 0 3 →χ 0 1 h 1 ). Six benchmark points are listed in the Table 1. Table 1. Masses and branching ratios for 6 benchmark points in the scNMSSM. The signal significances in the last line are calculated with the luminosity of 300 fb −1 , and with similar analysis of multi-lepton final state as in Ref. [24].

Possibility of discovery at the HL-LHC in the future
In this part, we investigate the possibility of detect electrowekinos in the future High Luminosity LHC (HL-LHC). We adopt the same analysis of multilepton final state by CMS [24], only increasing the integrate luminosity from 35.9 fb −1 to 300 fb −1 , to see the possibility of discovery in the future. And we evaluate the signal significance by 3.10) where S and B are the number of events from signal and background process respectively. In the Fig.4, we show ss on the planes of mχ0 1 versus mχ± 1 , mχ0 2 and mχ0 3 respectively. We can see that most of the samples can be checked at 5σ level when the mass difference between LSPχ 0 1 and NLSPsχ ± 1 ,χ 0 2,3 is sufficient. However, there are still some samples can not be checked at level above 3 or 5 sigma. The main reasons is that the mass spectra is compacted, so that the leptons from the decay of NLSPs are very soft. Because P T cut has to be very large at the LHC due to the large background, detecting soft particles is not easy. Combining with Fig.2 and 3, we can learn the following facts: • Ifχ ± 1 orχ 0 i (i = 2, 3) decays to a virtual vector boson, the area over the dash line in all the planes, the signal significance is less than 5σ, and it is hard to check at the LHC with 300 fb −1 data.
• Ifχ ± 1 orχ 0 2 decays to a real vector boson, the area between the dash and dotted line in left and middle the planes, the signal significance can be larger than 5σ, and it is easy to check at the LHC with 300 fb −1 data.  (right) planes. The colors indicates the signal significance, where red represents ss < 3σ, green represents 3σ < ss < 5σ, and gray represents 5σ < ss. In the left plane, the dash line indicates the mass difference equal to m W , mχ± In the middle and right planes, the dash line and dotted line indicate the mass difference equal to m Z and m h2 respectively, that is, mχ0 i − mχ0 1 = m Z and mχ0 i − mχ0 1 = m h2 , where i = 2, 3 for the middle and right planes respectively.  Figure 5. The samples in the m a1 versus mχ0 1 plane. The color convention is the same as in Fig.4. The dash, dotted and dash dotted lines indicate 2mχ0 1 , equal to m a1 , m h2 , and m Z respectively.
• Theχ 0 3 decay to a light Higgs h 1 , the area between the dash and dotted line in the right plane, the signal significance is less than 5σ for some samples. The reason is that the decay width of light Higgs h 1 has been constrained by many experiments, so it's hard to detect it.
• Ifχ 0 i (i = 2, 3) decays to a SM-Like Higgs, the area under the dotted line in middle and right planes, it is also have ss > 5σ for most samples.
For the samples with insufficient mass difference between the NLSPs and the LSP, the integrate luminosity at 300 fb −1 is not enough. So we also tried to increase the luminosity to 3000 fb −1 , the result is that almost all samples can be checked with ss > 5 at 3000 fb −1 .
In Fig.5, we show ss on the planes of mχ0 1 versus m a 1 . We can see that there are mainly two mechanisms for dark matter annihilation: the a 1 funnel where 2mχ0 1 m a 1 , and the h 2 /Z funnel where 2mχ0 1 m h 2 or 2mχ0 1 m Z . Unfortunately, searching for the NLSPs is helpless to distinguish these two mechanisms.

Conclusions
In this work, we have 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 of singlino-dominatedχ 0 1 and SM-like h 2 in the scan result in our former work on scNMSSM, 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. In our scenario, the bino and wino are heavy because the high mass bound of gluino and the unification of gaugino masses at GUT scale. Thus theχ ± 1 andχ 0 2,3 are higgsino-dominated and mass-degenerated NLSPs.
We first investigate the constraints to these light higgsino-dominated NLSPs, including searching for SUSY particle at the LHC Run-I and Run-II. We use Monte Carlo to do the detailed simulations to add these constraints from search SUSY particles at LHC. Then we discuss the possibility of discovering the higgsino-dominated NLSPs at the HL-LHC in the future. We use the same analysis by increasing the integrate luminosity to 300 fb −1 and 3000 fb −1 .
Finally, we come to the following conclusions regarding the higgsino-dominated 100 ∼ 200 GeV NLSPs in scNMSSM or NUHM-NMSSM: • Among the search results for electrowekinos, the 'multi-lepton final state' constrain our scenario most, and can exclude some of our samples. While up to now, the search results by Atlas and CMS with Run I and Run II data at the LHC can still not exclude the light higgsino-dominated NLSPs of 100 ∼ 200 GeV.
• When the mass difference withχ 0 1 is smaller than m h 2 ,χ 0 2 andχ 0 3 have different preference on decaying to Z/Z * or h 1 .
• The best channels to detect the NLSPs are though the real two-body decayχ ± 1 →χ 0 1 W andχ 0 2,3 →χ 0 1 Z/h 2 . When the mass difference is sufficient, most of the samples can be checked at 5 σ level with future 300 fb −1 data at the LHC. While with 3000 fb −1 data at the LHC, nearly all of the samples can be checked at 5σ level even if the mass difference is insufficient.
• The a 1 funnel and the h 2 /Z funnel are the mainly two mechanisms for the singlinodominated LSP annihilation, which can not be distinguished by searching for NLSPs.