Electron and Muon Anomalous Magnetic Moments in the Inverse Seesaw Extended NMSSM

The recently improved observation of the fine structure constant has led to a negative $2.4\sigma$ anomaly of electron $g-2$. Combined with the long-existing positive $4.2\sigma$ discrepancy of the muon anomalous magnetic moment, it is interesting and difficult to explain these two anomalies with a consistent model without introducing flavor violations. We show that they can be simultaneously explained in the inverse seesaw extended next-to-minimal supersymmetric standard model (ISS-NMSSM) by the Higgsino--sneutrino contributions to $(g-2)_e$ and $(g-2)_\mu$. The spectrum features prefer light $\mu$, which can predict $m_Z$ naturally, and it is not difficult to obtain a $\tau$-type sneutrino dark matter candidate that is compatible with the observed dark matter relic density and the bounds from dark matter direct detection experiments. Due to the compressed spectra and the undetectable decay mode of selectrons, they can evade the current Large Hadron Collider (LHC) constraints.

There is insufficient evidence to show that these two anomalies are indeed signs of new physics (NP). The discovery level confirmation of ∆a µ , for example, requires efforts from the currently running E989 experiment at the Fermilab and the future J-PARC experiment and also progress in reducing the theoretical uncertainty. Providing a common explanation to these two anomalies in an NP model is very challenging. In general, in a complete renormalizable model, a can only be a quantum loop effect, because it comes from a dimension-5 operator. In a generic NP model without flavor violation, the new contribution to the anomalous magnetic moment a NP is proportional to the mass square of the lepton times an NP factor R NP . Taking  which is difficult to achieve from a common physical origin. At present, there have already been several discussions offering combined explanations of the experimental results for electron and muon anomalous magnetic moments . Among these discussions, the supersymmetry (SUSY) framework includes a chiral enhancement factor tan β, which has shown promising results [48]. Reference [30] argued that the combined explanation in the SUSY framework needs relatively large non-universal trilinear A terms and also requires a flavor violation (for a more detailed discussion, see Refs. [49,65]). Due to the constraint from the lepton flavor violating process, Ref. [47] examined the minimal flavor violation within the minimal supersymmetric standard model (MSSM) and found its compatibility with the Higgs mediation scenario. However, since the value of parameter µ needs to be at O(100 TeV), the parameter space of the explanation in Ref. [47] is unattractive.
More recently, Ref. [44] argued that in the MSSM without any flavor violation, a combined explanation can be achieved by setting the conditions that sgn(M 1 µ) < 0 and sgn(M 2 µ) > 0. The corresponding result features very light selectrons and wino-like charginos, which avoid the Large Hadron Collider (LHC) constraint due to their degenerate spectra. The solution of Ref. [44] is impressive, but it also has two unsatisfactory characteristics. One is that the solution prefers heavy Higgsinos with masses µ ∼ O (1 TeV). This leads to a relatively fine-tuned electroweak sector. In general, µ should be close to the Z boson mass m Z to avoid large cancellation when predicting the observed value of m Z = 91.2 GeV [66][67][68][69][70]. µ ∼ 1 TeV often induces tuning on the order of 1/10000 to predict m Z . The other is that wino-like particles are too light due to the current restrictions of the LHC direct SUSY searches. The wino exclusion planes reported by ATLAS and CMS within the simplified model framework are appropriate for the scenario of Ref. [44]. According to Fig. 8 in the CMS report [71], for example, the benchmark points in Ref. [44] with M 2 ∼ 200 GeV are on the verge of being excluded by the multi-lepton plus E miss T signal via the electroweakino channel pp →χ ± 1χ 0 2 . In our previous work [72], we investigated the observation that in the inverseseesaw mechanism extended next-to-minimal supersymmetric standard model (ISS-NMSSM), due to the O(0.1) level Yukawa coupling Y ν of the Higgs field to the right-handed neutrino, the Higgsino-sneutrino (HS) loop can be a new source of a µ to explain ∆a µ . Unlike the MSSM, the newly introduced HS contribution a HS in the ISS-NMSSM prefers a light µ. The sign of a HS is determined by the mass mixing effect of sneutrino fieldsν L ,ν R , andν x for a given flavor , not by the mixing of charginos or neutralinos. In the ISS-NMSSM explanation, the masses of wino-like particles can be much heavier than the current LHC bounds. One can also assume one generation of sneutrinos to be the lightest supersymmetric particles (LSPs), which act as a dark matter (DM) candidate coannihilating with Higgsinos to achieve the observed relic density. Due to the singlet nature, the DM-nucleus scattering cross section is naturally suppressed below the current experimental detection limits [72][73][74]. In this case, the neutral Higgsinos are the next-to-lightest supersymmetric particles (NLSPs) that decay into the invisible final states of the collider (H 0 → νν). The charged Higgsino decays into a soft charged lepton and DM (H ± → ±ν ). Due to the lower production rate than that of winos and the degenerate mass spectrum, the current LHC data still allow a low Higgsino mass of around 100 GeV. Thus, compared with the MSSM framework, the ISS-NMSSM is more natural for providing common explanations for ∆a e and ∆a µ .
In this work, we investigate this issue by applying the ISS-NMSSM to explain ∆a e and ∆a µ . The remainder of this paper is organized as follows. First, we briefly introduce the ISS-NMSSM and the properties of leptonic g −2 in Sec 2. We then scan the parameter space that explains both the electron and muon g−2 discrepancies and analyze the characteristics of the input parameters and particle mass spectrum in Sec. 3. In Sec. 4, we find that our scenario can be embedded into a τ -type sneutrino, which co-annihilates with a Higgsino to achieve the observed DM relic density and does not conflict with the current DM direct search observations. In Sec. 5, we show the impact of the current LHC SUSY particle direct searches. Finally, we draw conclusions in Sec. 6.
2 Inverse seesaw mechanism extended next-to-minimal supersymmetric standard model and the lepton g − 2 2.1 Brief introduction to inverse seesaw next-to-minimal supersymmetric standard model The complete definition of the ISS-NMSSM Lagrangian, such as quantum number setting, can be found in Ref. [75]. Here, we only briefly introduce the basic idea of the "inverse-seesaw" extension and the neutrino sector. The "inverse-seesaw" mechanism is added to the NMSSM framework by introducing two gauge singlet superfieldsν andX with opposite lepton numbers L = −1 and L = 1, respectively [76]. With the assumptions of R-parity conservation, not introducing the ∆L = 1 lepton number violation, the superpotential W is given as follows: (2.1) The first line of Eq. (2.1) is the standard NMSSM superpotential. The soft breaking terms of the ISS-NMSSM are given as follows: where V NMSSM is the NMSSM soft breaking term, andν R andx are the scalar parts of superfieldsν andX, respectively. The dimensional parameter µ X is a small Xtype neutrino mass term, which often is treated as an effective mass parameter to obtain the tiny masses of active neutrinos. The introduction of µ X violates the lepton number due to the ∆L = 2 term µ XXX and the Z 3 symmetry of the superpotential. After the electroweak symmetry breaking, the 9 × 9 complex and symmetric neutrino mass matrix M ISS in the basis (v L , v * R , x) reads neutrino Dirac mass matrices. The M ISS can be diagonalized by a 9 × 9 unitary matrix U ν according to This M ν is diagonalized by the unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U PMNS : In this work, for our phenomenological purpose, we use the µ X -parametrization scheme that was introduced in [77] 1,2 to reproduce low-energy neutrino data Y ν and λ N in this work are assumed to be flavor diagonal, so the neutrino oscillation data is attributed only to the flavor nondiagonal parameter µ X . In this case, the unitary constraint of neutrino sector can be translated into a constraint on the input parameters [74] λ Ne Y νe These inequalities indicate that, for given λ N and the Higgs sector parameters λ, tan β, and µ, this unitary constraint allows Y νe to be greater than Y νµ , which is good for explaining ∆a e . The ISS mechanism preserve an approximate lepton number conservation. Equation (2.7) indicates that the light neutrino masses require the magnitude of lepton number breaking parameter µ X to be highly suppressed. Numerically, both µ X and the soft breaking mass term B µ X are extremely small (|µ X | O(KeV) and |B µ X | O(100 GeV 2 )), and they just slightly split the complex sneutrino field into the CP -even part and CP -odd part. When studying a , the influence of these two nonvanishing parameters can be ignored [72]. Because the dimension of the scan parameters is too high, for the sake of simplicity, we simply set the values of µ X and B µ X to zero in the following discussion. 1 A detailed discussion of µ X -parametrization can also be found in [78]. In principal, Y ν in the µ X -parametrization can be arbitrarily large except for that the perturbativity of the theory requires |Y ν | 2 /4π ≤ 1.5. 2 The neutrino oscillation data can also be reconstruct by the Casas-Ibarra parametrization, in which the neutrino oscillations are generated by the off-diagonal term in m D [79]. Because of the special role of parameter Y ν in this work, the µ X parametrization scheme is more intuitive than the Cassas-Ibarra parametrization scheme.
The SUSY particles of particular importance to a are sleptons˜ , -type sneutrinos, neutralinosχ 0 i , and charginosχ ± i . The neutralino and chargino sector in the ISS-NMSSM were same as that of the NMSSM. On the basis of φ 0 = (B,W 3 ,H 0 d ,H 0 u ,S) T , the symmetric neutralino mixing matrix M 0 is as follows: where the Higgsino mass µ = λv s is an effective µ term after the electroweak symmetry breaking, and v u , v d , and v s represent the vacuum expectation values (VEVs) of the Higgs field H u , H d , and S, respectively. The mass eigenstatesχ 0 i = N ij φ 0 j are arranged in ascending order of mass. With the basis φ + = (W + ,H + u ) and , the chargino mass term is given by φ − M ± φ + + H.c. with the mass matrix (2.10) The corresponding mass eigenstates are defined bỹ The symmetric mass matrix M 2 for slepton˜ for each flavor in the (˜ L ,˜ R ) basis is , (2.12) and the corresponding rotation matrix is represented by X . In terms of the particle composition, the ISS-NMSSM differs from the NMSSM [80] only in the neutrino sector. For each generation = e, µ, τ , the sneutrino fields are the mixtures of left-handed sneutrinoν L , right-handed sneutrinoν R , and x-type sneutrinoν x . On the basis of φ ν = (ν L ,ν R ,ν x ), the symmetric mass matrix M 2 ν is given by (2.13) The mass eigenstate of one generation sneutrino isν i = j Z ij φ ν,j , with Z denoting the unitary matrix to diagonalize M 2 ν . From Eq. (2.13), one can easily find that the diagonal elements can be adjusted by the soft breaking parameters M 2 L , M 2 ν , and M 2 X . For the off-diagonal elements, Higgs VEV terms, such as Y ν v u and λ N v s , provide a scale for the mixing of the three fields of left-handed, right-handed, and x-type sneutrinos. The relative signs and the magnitude of different sneutrino components can be adjusted by two A-term soft breaking parameters A ν and A N .

Lepton anomalous magnetic moment in ISS-NMSSM
The lepton anomalous magnetic moment a always corresponds to lepton chiralityflipping interactions. In the ISS-NMSSM, the chirality of the -lepton number can be flipped by Y e or Y ν . All the SM-like diagrams (the -lepton number is carried only by lepton and/or neutrino ν ) involve only SM particles, so their contribution to a is identical to the SM prediction a SM . Therefore, the SUSY contribution a SUSY , in which the -lepton number is carried also by a scalar lepton˜ and/or -type sneutrinõ ν , provides the source of the observed anomaly ∆a .
The general one-loop SUSY contribution to a in the ISS-NMSSM is given as follows [72]: (2.14) Here, i = 1, · · · , 5 and j = 1, 2, respectively, denote the neutralino and chargino indices, l = 1, 2 denotes the slepton index, n = 1, 2, 3 denotes the sneutrino index, and The kinematic loop function F (x) is normalized with condition F (1) = 1, which is given as follows: Figure 1. One-loop diagram of Higgsino-sneutrino contribution, the additional contribution to a SUSY in the ISS-NMSSM compared to the MSSM.
Comparing with the MSSM contribution to a given in [81], the ISS-NMSSM contribution contains an extra HS term. As depicted in Fig. 1, it is easy to understand the leading behaviour with the help of diagram that are written in terms of interaction eigenstates, where the insertions of mass and mixing terms and lepton number chirality flips. Accordingly, this HS contribution a HS can be approximately expressed as GeV, and the function xF C 2 (x) increases monotonically with x, and its value ranging from 0 to 1.5. Equation (2.17) shows that a HS is enhanced by the factor cos β ≈ 1/ tan β for large tan β. A relatively large Y ν is expected to achieve a larger a HS . The lepton chirality flipping in the HS contribution is reflected in the left-right mixing term Z n1 Z n2 for each sneutrinoν n . If the xfield-dominated sneutrino is too heavy, then the contribution from the left-handed dominated sneutrino and the right-handed dominated state to a cancel each other because Z 11 Z 12 ≈ −Z 21 Z 22 . This cancellation can be alleviated by rendering a sizable mixing between x-field and the right-handed sneutrino field, which, as shown in Eq. (2.13), can be obtained by choosing a large mixing term λ N v s . Furthermore, A light µ is favored for large a HS , so the small λ at O(0.01) often has a larger a HS . This small λ is also preferred by the leptonic unitary condition in Eq. (2.8).
Looking back to the MSSM explanation to both anomalies, the first difficulty is that the most of the relevant parameters (except the lepton soft trilinear breaking term A e ) are lepton flavor independent. Specifically, the full expression of a SUSY in MSSM is very similar to that in ISS-NMSSM, and the difference is that c L in Eq. (2.15) of MSSM does not contain the Y ν V j2 term. The µ parameter governs the HiggsinoH u −H d transition and the dominant part of scalar lepton mixing term, so that the MSSM contribution is proportional to µ tan β. Compared with µ tan β , A e is usually ignored in the left-handed and right-handed scalar lepton mass mixing. The most attractive property of the HS term in explaining both anomalies in ISS-NMSSM is that the left-right mixing Z n1 Z n2 is positively related to A ν , which is lepton flavor relevant. The magnitude and sign of the HS contribution can be adjusted by A ν . Numerically, an |A ν | at O(100 GeV) to O(1 TeV) is sufficient for a HS to explain both anomalies. Furthermore, the weakening of mass relevance between left-handed dominated slepton˜ L and left-handed dominated sneutrinoν L gives ISS-NMSSM parameter space more room for interpreting both anomalies.
3 Features of combined explanation to ∆a e and ∆a µ in ISS-NMSSM The relevant SUSY particle masses have different contributions to a SUSY . To reveal the detailed features of a SUSY in the ISS-NMSSM, which is covered up by the complexity of the loop functions F (x) in Eq. (2.16), we use the MultiNest technique [82,83] to scan the ISS-NMSSM parameter space with the following parameter values and ranges: 0 < λ < 0.7, |κ| < 0.7, 100 GeV < µ < 500 GeV, tan β = 60, where all parameters are defined at the scale of 1 TeV. All the other soft breaking parameters, like those first two generation squark, are fixed at a common value of 3 TeV. The prior probability distribution function (PDF) of these inputs are set as uniform distributions, and the n live parameter, which indicates the number of the active points to determine the iso-likelihood contour in the MultiNest algorithm iteration, is set at 10000. The likelihood function χ 2 in the scan is taken as is a standard Gaussian form of two anomalies. χ 2 Higgs requires the sample in the parameter space given by Eq. (3.1) to predict an SM-like Higgs boson compatible with current experimental observations using the HiggsSignals-2.2.3 code [84][85][86] and to satisfy the constraints of a direct search of the Higgs boson using the HiggsBounds-5.3.2 code [87]. χ 2 veto is introduced to ensure that the LSP is a Higgsino, the electroweak vacuum is stable, sneutrino fields have not developed nonzero VEVs, and the unitary constraint of Eq. (2.8) is satisfied. If the parameter point satisfies the above assumption, χ 2 veto is equal to 0; otherwise, χ 2 veto = 10000. Technically, the introduction of χ 2 veto highlights the characteristics of the parameter space of ISS-NMSSM explaining the two anomalies, and minimizes the influence of other phenomenological constraints on the statistics of the scanning result. After several repeated scans, we checked that the statistical distribution of the parameters in the results is reproducible.
In the numerical calculation, an ISS-NMSSM model file is generated by the SARAH-4.14.3 package [88,89], the particle spectra of the parameter samples are calculated by the SPheno-4.0.3 [90,91] and FlavorKit [92] packages, and electroweak vacuum stability and sneutrino stability are tested by the Vevacious [93,94] package, in which the tunneling time from the input electroweak potential minimum to the true vacuum is obtained via the CosmoTransitions package [95].  • The distributions of the input parameters M 2 , λ, µ, Y νe , Y νµ , and A νe show some trends that verify the discussion in the previous section. The narrow PL picks of 100 GeV µ 150 GeV and Y νe ≥ 0.7 reflect the properties of Eq. (2.17) and the difficulty of explaining ∆a e . The wider credible region of the µ-type input parameter than that of e-type parameter also confirms this feature.
• For most input parameters, the credible regions are smaller than the confidence interval. The PL values are very close to 1 almost in the entire space. This implies that the value of these parameters are not affected by the two anomalies.
• µ ∼ 110 GeV and 0.015 λ 0.05 cause the masses of all the singlet Higgs particles to be on the order of several TeV. There were 7989 samples obtained that explained two anomalies within the 2σ range in total. In Fig. 5, we plot the mass distributions of the SUSY particles via violin plots 3 , where sleptons and sneutrinos are labeled by their dominating components. The masses of the Higgsino-dominated triplet states were around 110 GeV, and the wino particles were often heavier than 700 GeV. The masses of left-handed selectronẽ L andμ L were distributed around 400 GeV, and the right-handed sleptons were relatively heavy. In contrast to the MSSM spectrum, the mass ofν e L can be much lower than that ofẽ L in the ISS-NMSSM. The Higgs and sparticle spectra for two typical parameter points are shown in Fig. 6. M 1 may play a crucial role in the mass splitting of Higgsino-dominated neutralinos and chargino. Taking the mixing terms as a perturbation and calculating the neutralino and chargino masses to the first order in perturbation theory, the mass splitting between Higgsino-dominated electroweakinos are approximately given as follows: where mS = 2κv s is the singlino mass. These formulae indicate that the effect of M 1 is negligibly small when |M 1 | is extremely large, while as it approaches zero from above (below), it can enhance (decrease) the splitting significantly. This characteristic is shown in Fig. 7, where we projected the scanned samples on the ∆m(χ ± 1 ,χ 0 1 ) − ∆m(χ 0 2 ,χ 0 1 ) plane with the color bar denoting the M 1 value. This figure reveals that ∆m(χ ± 1 ,χ 0 1 ) 4 GeV and ∆m(χ 0 2 ,χ 0 1 ) 10 GeV when |M 1 | = 1 TeV. When  M 1 200 GeV, the mass splittings increase to several tens of GeV, and when M 1 −200 GeV, they decrease to less than 1 GeV.

Dark matter phenomenology
In the ISS-NMSSM explanation of the two leptonic anomalous magnetic moments ∆a e and ∆a µ , the Higgsino mass is less than 200 GeV and acts as the LSP. Such a light Higgsino is not a good DM candidate due to its spin-dependent and spinindependent scattering rates with nuclei larger than the current experimental limits [67,78,[97][98][99][100]. In the ISS-NMSSM, a right-handed-field-or x-field-dominated sneutrino can also serve as the DM candidate. The sizable neutrino Yukawa coupling Y ν contributes significantly to the DM-nucleon scattering rate, so that e-type and µ-type sneutrinos cannot act as DM candidates. Fortunately, Y ν and λ N are chosen to be flavor diagonal, and the flavor mixing effect introduced by µ X can be ignored. This leaves us room for taking the lightest τ -type sneutrinos,ν τ 1 , and its charge conjugate sate,ν τ * 1 , as feasible DM candidates [73][74][75]. 4 The candidates should be lighter thanχ 0 1 and dominated in components by the right-handed fieldν R , thexfield, or their mixture. In addition, given the preferred parameters in the previous section, one can further restrict their properties.

Sneutrino dark matter
First, we consider the annihilation of τ -type sneutrinos, whose thermally averaged cross section at the freeze-out temperature should satisfy σv F ∼ 3 × 10 −26 cm 3 /s to obtain the measured abundance. As was pointed out in Ref. [74], the DM has three types of popular annihilation channels: where h s and A s denote the singlet-dominated CP -even and CP -odd Higgs bosons, respectively. Since the singlet Higgs dominated particles are much heavier than the DM, this annihilation processes are kinematically forbidden.
τ * 1 →ν hνh , where ν h represents any of the heavy neutrinos. This process is mainly proceeded via h s in the s-channel and/or via the singlino-dominated neutralino in the t-channel. Evidently, the annihilation cross section is suppressed by the heavy mediator mass. 4 In this study, we takeν τ 1 as a complex field by setting B ν X = 0. Consequently,ν τ * 1 also acts as a DM candidate. The ISS-NMSSM is then a two-component DM theory [74]. We add that DM physics requires non-trivial configurations in the theory's parameter space.
Next, we consider the DM-nucleon scattering proceeded by the t-channel exchange of the CP -even Higgs bosons and Z boson. The spin-dependent cross section is vanishing, and the spin-independent (SI) cross section is [74] where σ h N and σ Z N with N = p, n denote the Higgs-mediated and the Z-mediated contribution, respectively. For the preferred parameters in the last section, σ h N and σ Z N are approximated by where Cντ * 1ν τ 1 [H 0 u ] represents the coupling of the DM pair to the CP -even H 0 u field and takes the following form: Noting that Z τ 11 is proportional to Y ντ , one can conclude that the scattering is suppressed in the case of a small Y ντ and λ Nτ . This case is favored by current and future DM direct detection experiments [102]. In the ISS-NMSSM, the coupling of the DM to electroweakinos are

Effect of dark matter embedding on sparticle signal at the LHC
where P is a unitary matrix to diagonalize the neutrino mass matrix in one-generation bases (ν τ L , ν τ R , x τ ). These expressions show that the coupling strengths are determined by Y ντ and λ Nτ , and they vanish when Y ντ = 0. This characteristic has crucial applications in the phenomenology at the LHC. Concretely speaking, in the case where Y ντ and λ Nτ are tiny and the DM achieves the measured abundance by the co-annihilation mechanism, the collider signal of the sparticle is roughly identical to that of the NMSSM withχ 0 1 acting as the LSP [73]. This conclusion can be understood from the following aspects: • The coannihilation mechanism is effective only when mχ0 1 mντ 1 . • Sinceχ 0 1 of the NLSP decays asχ 0 1 →ν τ 1 ν τ , it appears as a missing track in the LHC detectors.
•χ 0 2 may decay byχ 0 2 →ν τ 1 ν τ andχ 0 2 →χ 0 1 Z * →χ 0 1 ff . The two-body decay is always suppressed by the tiny coupling strength, while the phase space may suppress the three-body decay. We checked that in the case of Y ντ = λ Nτ = 10 −3 and mχ0 2 − mχ0 1 2 GeV, these two decay branching ratios are comparable in size, but the current LHC has no detection capability for such a small mass splitting of Higgsino-dominated triplet states [103]. In the case of mχ0 2 − mχ0 1 > 2 GeV,χ 0 2 mainly decays by the three-body channel, so its signal is identical to that of the NMSSM withχ 0 1 as the LSP.
• The situation ofχ ± 1 is quite similar to that ofχ 0 2 except that it decays bỹ • For the other sparticles, their interactions withν τ 1 are weak, and thus, their decay chains do not change significantly.
• Due to the constraints of DM direct detection experiments, the values of Y ντ and λ Nτ tend to be relatively small. However, in order to satisfy the DM freezeout mechanism, it is generally believe that the value of Y ντ and λ Nτ should be greater than 0.001. We checked that the lifetime of NLSPχ 0 1 is always less than 10 15 GeV −1 . 5 Table 1 shows the details of a sample obtained in the previous section, including the mass spectra and decay modes of some moderately light sparticles. In Fig. 8, we show the decay path of these sparticles to illustrate their properties further. This result indicates that the collider signature will not change after being embedded into the DM. Hence a natural way of a τ -type sneutrino DM to achieve the measured relic density is coannihilation with Higgsino. In practice, the Yukawa parameters Y ντ and λ Nτ should be small to suppress the DM-nucleon scattering rate, the DM mass mντ 1 should be slightly lighter than the Higgsino mass, which is mντ 1 ≈ 100 GeV. Since the embedded DM will increase the dimension of the scanning parameters, so we will ignore the DM constraints when studying the collider phenomenology. Figure 5 shows that to simultaneously explain ∆a e and ∆a µ , very light electroweakinos and sleptons are necessary in many samples. Generally, such light sparticles are strongly constrained by the current LHC SUSY searches. In a specific SUSY model, the mass hierarchy and decay modes of sparticles can be significantly different from the simplified model on which the experimental interpretations are based. In this 5 In some models of right-handed sneutrino as LSP [104,105], the stauτ 1 decays into the righthanded sneutrino viaτ 1 → W ( * )ν R driven by the tiny neutrino Yukawa coupling Y ν sin β ∼ 10 −13 . Depending on the decay modes and mixing inτ andν sectors, the lifetime ofτ 1 ranges from a few seconds to over ten years. However, the stau NLSP will not be a long-lived particle in our theory. One reason is that Y ντ should not so small to keep the LSPν 1 in thermal equilibrium in the early universe. Another reason is that the neutralino NLSP is usually predicted by the DM physics, and there is no requirement on the stau mass.  Table 1. Input parameters, mass spectrum, and decay modes of the sample in Fig. 8 before and after inserting τ -type sneutrino DM.

Constraints from LHC sparticle searches
section, we elaborated on the experimental searches we considered, discussed the sparticle decay modes on the ISS-NMSSM parameter space in interpreting ∆a e and ∆a µ , and summarized the impact of current LHC sparticle searches on the interpretation.
The cross section of √ s = 8, 13 TeV were normalized at the NLO using the Prospino2 package [111]. The Monte Carlo events were generated by MadGraph aMC@NLO [112,113] with the PYTHIA8 package [114] for parton showering and hadronization. The event files were then input into CheckMATE for analysis with Delphes [115] for detector simulation.
In addition to the analysis that has been implemented before, we added the following newly released LHC analyses to CheckMATE.
• ATLAS search for chargino and neutralino production using recursive jigsaw reconstruction in three-lepton final states [116]: this analysis is optimized for signals fromχ 0 2χ ± 1 production with on shell W Z decay modes. The signal regions (SRs) are split into a low-mass region (jet veto) and the initial state radiation (ISR) region (contains at least one energetic jet) using a variety of kinematic variables, including the dilepton invariant mass m , the transverse mass m T , and variables arising from the application of the emulated recursive jigsaw reconstruction technique. The smallness of the mass splittings lead to events with lower-p T leptons or smaller E miss T in the final state. Cuts in the low-mass SR are designed to reduce the W Z background and the number of fake or nonprompt leptons, and cuts in the ISR region requiring large E miss T to identify events have a real E miss T source. This search is sensitive to samples with relative light winos.
• ATLAS search for chargino and slepton pair production in two lepton final states [103]: this analysis targets pair production of charginos and/or sleptons decaying into final states with two electrons or muons. Signal events are required to have an exactly opposite-sign (OS) signal lepton pair with a large invariant mass m > 100 GeV to reduce diboson and Z + jets backgrounds. SRs are separated into same-flavor and different-flavor categories with variables m , the stransverse mass m T2 [117], E miss T and E miss T significance, and the number of non-b-tagged jets. The sensitivity of this analysis to the slepton mass can reach 700 GeV, and that to the chargino mass can reach about 1 TeV (420 GeV) of the decay modeχ ± • ATLAS search for electroweak production of supersymmetric particles with compressed mass spectra [118]: this was optimized on a simplified model of mass-degenerated Higgsino triplets that assumedχ 0 2χ ± 1 production followed by the decaysχ ± 1 → W * χ0 1 andχ 0 2 → Z * χ0 1 . It is also sensitive to the degenerate slepton-LSP mass spectrum. The selected events have exactly two OS same-flavor leptons or one lepton plus at least one OS track, and at least one jet is required. The preselection requirements include the requirements that the invariant mass m is derived from the J/ψ meson mass window, that E miss T is greater than 120 GeV, and that the p T of the leading jet is larger than 100 GeV. After applying the preselection requirements, SRs are further optimized for the specific SUSY scenario into three categories: SR-E (for electroweakino recoiling against ISR), SR-VBF [electroweakino produced through vector boson fusion (VBF)], and SR-S (sleptons recoiling against ISR). A variety of kinematic variables and the recursive jigsaw reconstruction technique are used to identify the SUSY signals. Assuming Higgsino production, this search occurs at the minimum mass ofχ 0 2 at 193 GeV at a mass splitting of 9.3 GeV.
• ATLAS search for chargino and neutralino pair production in final state with three-leptons and missing transverse momentum [119]: this search targets chargino-neutralino pair production decaying via W Z, W * Z * , or W h into three-lepton final states. This analysis uses the full LHC run II dataset. The simplified model has anχ 0 2 mass of up to 640 GeV for on shell W Z decay mode with masslessχ 0 1 , up to 300 GeV for the off shell W Z decay mode, and up to 185 GeV for the W h decay mode with anχ 0 1 mass below 20 GeV.
• ATLAS search for supersymmetric states with a compressed mass spectrum [120]: this analysis uses the OS lepton pair and large E miss T , searching for the electroweakino and slepton pair production with a compressed mass spectrum. Two sets of SRs are constructed separately for the production of electroweakinos and sleptons. The electroweakino SRs require the invariant mass of the lepton pair m to be less than 60 GeV, and the slepton SRs require the stransverse mass m mχ T2 to be greater than 100 GeV, where the hypothesized mass of the LSP m χ is equal to 100 GeV. The most sensitive location of the mass splitting is at about 5-10 GeV. The 95% confidence level exclusion limits of the Higgsino, wino, and slepton are up to 145, 175, and 190 GeV, respectively.
• CMS combined search for charginos and neutralinos [71]: various simplified models of the SUSY are used in this combined search to interpret the results. Related to our work, the simplified model scenario interpretation of χ 0 2χ ± 1 with decaysχ 0 2 → Z ( * )χ0 1 /hχ 0 1 andχ ± 1 → W ( * )χ0 1 represents the most stringent constraints from the CMS to date for electroweakino pair production. Compared with the results of individual analyses, this interpretation improves the observed limit inχ ± 1 to about 650 GeV for W Z topology.
• ATLAS search for electroweakino production in W h final states [121]: this was optimized on a simplified model that assumedχ 0 2χ ± 1 production with decay modesχ ± 1 → W ±χ0 1 andχ 0 2 → hχ 0 1 . Signal events were selected with exactly one lepton, two b-jets requiring 100 GeV < m bb < 140 GeV and E miss T > 240 GeV, a transverse mass of the lepton-E miss T system m T greater than 100 GeV, and a "contransverse mass" m CT [122,123] greater than 180 GeV. Masses of the winos up to 740 GeV are excluded at 95% confidence level for the massless LSP.
• CMS search in final states with two OS same-flavor leptons, jets, and missing transverse momentum [124]: This search is sensitive to the on shell and off shell Z boson from BSM processes and to direct slepton production. Search regions are split into on-Z SRs, off-Z SRs, and slepton SRs via various kinematic variables, including the invariant mass of the lepton pair m , M T2 , the scalar sum of jet p T , the missing transverse momentum E miss T , the number of jets n j , and the number of b-tagging jets n b . The result interpretation using the simplified model assuming direct slepton pair production with 100% decay into dilepton final states shows that the probing limit of the slepton mass m˜ is up to 700 GeV. Certainly, this search is sensitive to the sparticles in the ISS-NMSSM interpretation of ∆a .
Appendix A shows a part of the validation table of the analyses above. We used the R values obtained from CheckMATE to apply the LHC constraints. Here, R ≡ max{S i /S obs i,95 } for individual analysis, in which S i represents the simulated event number of the i th SR or bin of the analysis, and S obs i,95 is the 95% confidence level upper limit of the event number in the corresponding SR or bin. The combination procedure of the CMS electroweakino search [71] was also performed though the CL s method [125] with RooStats [126] using the likelihood function described previously [71].
The impacts of LHC constraints on the samples were relatively strong, as 7029 samples were excluded, and only 960 samples survived, corresponding to a total posterior probability of 11.2%. In Fig. 9, we plot the SUSY particle mass of the surviving samples and excluded samples via a split violin plot. 6 The LHC constraints have not significantly changed the distributions of the SUSY particle mass. In Fig. 10, we plot the samples on the mχ0 1 − ∆m(χ 0 1 ,χ ± 1 ) plane and the mẽ L − mμ L plane with M 1 valued color. The detection ability of the LHC was mainly affected by the decay modes of SUSY particles.
1. For the light left-handed slepton pair production processes, • whenν e 1 is very light, as shown by the samples in Fig. 8, the dominant decay mode wasẽ L → Wν e 1 (the sample in Fig. 8 for example), wherẽ ν e 1 contains large left-handed ingredients 7 . Such samples are difficult to detect by the current LHC experiments. In the right plane of Fig. 10 and Fig. 9, we find that the mẽ L values of the surviving samples can reach 200 GeV, and such a lightẽ L is good for explaining ∆a e .
• when |M 1 | is lighter than mẽ L or mν L , ifẽ L → Wν e 1 is kinematically forbidden, as shown in the right panel of Fig. 6, the dominated decay mode of˜ L is into plus a bino-like neutralinoχ 0 . In this case, the LHC slepton pair production searches provide the most sensitivity to the slepton mass. Figure 10 shows that samples withμ L < 400 GeV and small |M 1 | have difficulty escaping the LHC constraints.
• when |M 1 | is very large andν e 1 is very heavy, the left-handed slepton will decay into χ 0 1 , χ 0 2 , and νχ ± 1 , as depicted in the left panel of Fig. 6. Because the decay products ofχ ± 1 are too soft, the LHC constraints for these samples are weaker than those in the light bino case.
2. For light right-handed slepton pair production processes, the production cross section is about 2.7 times smaller than that of left-handed slepton. The righthanded sleptons mainly decay into χ 0 1 and χ 0 2 . Only a few samples contain lightμ R orẽ R , so the right-handed slepton had little effect on the result.
3. For the Higgsino-dominated electroweakino pairχ 0 2χ ± 1 production process, the analysis in Ref. [118] provides a strong constraint on the samples featured by the small positive M 1 , as shown in the left plane of Fig. 10. 4. For the wino-dominated neutralino/chargino pair production process, the explanation of the two lepton anomalies require M 2 > 400 GeV, and in about more than 97% samples, the wino-like particle mass is greater than 700 GeV. Therefore, in this study, because the decay modes of the winos are more complicated, the winos did contribute to the results, but they were not the main effect.
In conclusion, except for some samples whereẽ L decayed to Wν e 1 , the samples with positive M 1 were strongly restricted by the current LHC experiments. The searches   Fig. 11. After considering the constraints from the LHC, there were still a large number of samples that could simultaneously fit ∆a e and ∆a µ at the 1σ level.
Before we conclude this section, we briefly comment that the signal of the spar-ticles may differ significantly from that of the NMSSM in the case of sizable Y ντ and λ Nτ [73][74][75]. In general, because the decay chain becomes lengthened and the signal contains at least two τ leptons, the sparticles of the ISS-NMSSM are more challenging to detect at the LHC than those of the NMSSM. We drew this conclusion by globally fitting the ISS-NMSSM with various experimental data and studying the scanned samples, similar to our previous work [74]. Therefore, we may overestimate the LHC constraints in this work, but this does not affect the main conclusion that the ISS-NMSSM can easily explain both anomalies.
When this work was about to be finished, a new measurement of fine structure constant, via the rubidium atom, was reported [127]: What needs to be emphasized is that there is a 5.4σ discrepancy between two measurements α −1 (Cs) and α −1 (Rb) [127]. It is currently suspected that this difference may be caused by speckle or by a phase shift during the measurement, which requires further study. Currently, the difference of α(Rb) and α(Cs) are in dispute. We are more concerned about the impacts on the theory. If the future experimental study confirm the correctness of α(Rb), that is there is no significance excess of a e within 2σ confidence level, then our previous work [72] shows that in ISS-NMSSM has a large parameter space to explain ∆a µ . Especially, the HS contribution alone can explain the central value of ∆a µ , and µ can be larger than 500 GeV (see the benchmark point in Table 2 in [72]) correspondingly. On the other hand, if α(Cs) is confirmed in the future, then this work provides a supersymmetric solution without introducing leptonic flavor violation to explain ∆a e and ∆a µ simultaneously.

Summary
We proposed scenarios in the ISS-NMSSM that explain both electron and muon g −2 anomalies without introducing leptonic flavor violation. The results are summarized as follows: • The HS loop induced by the neutrino Yukawa coupling Y ν plays an important role in the explanation. The advantage of this explanation is that the sign of the HS contribution to a can be determined by the sign of A ν , which greatly reduces the correlation between a e and a µ . A larger HS contribution corresponds to a light µ, which is natural for predicting m Z .
• The features of the sparticle spectrum preferentially had large tan β, light Higgsino mass µ 110 GeV, and heavy wino M 2 600 GeV. Moreover, the masses of the left-handed selectron and smuon were around 400 and 500 GeV, respectively, and the singlet Higgs-dominated particles are often heavier than 1 TeV. M 1 affects the mass splitting of Higgsino triplets and the decay mode of the sleptons.
• The mass spectrum can be introduced into a right-handed or x-field-dominated τ -type sneutrino as a proper DM candidate, which coannihilated with Higgsinos to obtain the observed DM relic density. As indicated by the previous discussion, choosing λ Nτ and Y ντ to be less than about 0.001 can avoid the change of the LHC signal caused by the introduction of the DM, and the corresponding DM direct detection cross section is much smaller than the detection ability of the current experiment.
• The signals of the electroweakino/slepton pairs produced at the LHC are sensitive to the parameter space explaining the two anomalies, especially for a positive M 1 . However, due to the compressed mass spectrum, the insensitive decay mode ofẽ L → Wν e 1 , and the very heavy wino, the surviving samples can satisfy the current LHC constraints. Table 2. Cut-flow validation of ATLAS analysis [103] for mass point m(χ ± 1 ,χ 0 1 ) = (300, 50) GeV in search channel ofχ ± 1χ ∓ 1 production with W -boson-mediated decays. The yields in "Baseline" of "CheckMATE" are normalized to "Baseline" of "ATLAS". "Efficiency" is defined as the ratio of the event number passing though the cut-flow to the event number of the previous event.