Neutralinos betray their singlino nature at the ILC

It is one of the most challenging tasks at the Large Hadron Collider and at a future Linear Collider not only to observe physics beyond the Standard Model, but to clearly identify the underlying new physics model. In this paper we concentrate on the distinction between two different supersymmetric models, the MSSM and the NMSSM, as they can lead to similar low energy spectra. The NMSSM adds a singlet superfield to the MSSM particle spectrum and simplifies embedding a SM-like Higgs candidate with the measured mass of about 125.5 GeV. In parts of the parameter space the Higgs sector itself does not provide sufficient indications for the underlying model. We show that exploring the gaugino/higgsino sectors could provide a meaningful way to distinguish the two models. Assuming that only the lightest chargino and neutralino masses and polarized cross sections $e^+e^-\to \tilde{\chi}^0_i\tilde{\chi}^0_j$, $\tilde{\chi}^+_i\tilde{\chi}^-_j$ are accessible at the linear collider, we reconstruct the fundamental MSSM parameters $M_1$, $M_2$, $\mu$, $\tan\beta$ and study whether a unique model distinction is possible based on this restricted information. Depending on the singlino admixture in the lightest neutralino states, as well as their higgsino or gaugino nature, we define several classes of scenarios and study the prospects of experimental differentiation.


Introduction
The discovery of a neutral scalar particle [1,2] with mass ∼ 125.5 GeV at the Large Hadron Collider (LHC) [3,4] has opened a plethora of discussions about its identity. While the experimental uncertainty suggests the new particle to be the Standard Model Higgs boson, more data are still needed to precisely determine its branching ratios, the CP properties and the underlying model. The present results are in fact compatible with one of the most promising Beyond the Standard Model candidates: supersymmetry (SUSY) [5]. The latter solves -contrary to the Standard Model (SM) -the electroweak hierarchy puzzle, offers a dark matter candidate and is consistent with grand unification of the gauge couplings.
The most studied supersymmetric models are the Minimal Supersymmetric Standard Model (MSSM) [5] and its minimal extension, the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [6]. The NMSSM introduces a gauge singlet chiral supermultipletS that allows for a relaxation of the electroweak fine tuning conditions, compared to the MSSM. On the other hand, the -so far -negative result of LHC searches for physics beyond the Standard Model (BSM) [7,8] does not favor any of these models a priori.
In case of SUSY discovery at the LHC and/or at a linear collider (LC) it is therefore important to understand how to entail the underlying supersymmetric model, in particular how to distinguish between NMSSM and MSSM. These two models have indeed a very similar particle spectrum, with the exception for the superfieldŜ in the NMSSM that results in three additional states with respect to the MSSM: a CP-even Higgs, a CP-odd Higgs and a fifth neutralino.
It is therefore well-motivated to look at the Higgs sector, where the experiments are expected to give the most precise indications [9,10], and to complement the information by studying the (extended) neutralino sector of the NMSSM to look for deviations with respect to the MSSM.
In fact, the singlino admixtures of neutralino lightest states as well as the higgsino and gaugino components ofχ 0 1 allow to identify several classes of NMSSM scenarios. Scenarios where the singlino component in the light neutralinos is significant (light singlino scenarios), are often treated in the literature, featuring production cross sections and phenomenology different from the MSSM and are in principle easier to spot. If the singlino, however, is heavy and mainly present inχ 0 3 ,χ 0 4 orχ 0 5 , the phenomenology is more MSSM-like and we distinguish the cases where the main component of the lightest χ 0 1 is higgsino-like (light higgsino scenarios) or gaugino-like (light gaugino scenarios). Having a decoupled singlino may result in a scenario that is experimentally not distinguishable from the MSSM without further information about the heavier neutralino states and the Higgs sector. Our analysis confirms these hints, concluding that a light and accessible singlino is one of the most efficient ways for model distinction together with a light singlet scalar.
The paper is organized in the following way: first we introduce our proposed strategy to discriminate the different models in section 2 and describe the classes of scenarios in section 3. In that section we also try to clearly classify in which cases a unique distinction between both models is possible based only on the light electroweak states and to work out which further information is required in cases where the light sector alone does not provide sufficient information for a model discrimination. Therefore we perform scans in the (λ, κ)-plane, applying the most recent phenomenological and experimental constraints from colliders, including also dark matter experiments, and determine where the singlino admixtures are such that the NMSSM cannot be misinterpreted as MSSM. We summarize our results in 4 and list details and parameters on the models in the appendices A, B.

Strategy
As explained in the Introduction, the NMSSM adds to the MSSM an additional gauge singlet superfield S in the Higgs sector. The most studied version of the NMSSM has a Lagrangian with an accidental Z 3 symmetry, obtained from the scale invariant superpotential [6] S consists of a scalar Higgs singlet S and the singlinoS. The additional dimensional parameters A λ and A κ appear in the Higgs sector soft terms: The singlet S, see Eqs. (1) and (2), mixes due to the electroweak symmetry breaking with the MSSM Higgs doublets H u , H d , resulting in three CP-even neutral scalars h 1 , h 2 , h 3 and two CP-odd neutral scalars a 1 , a 2 . Correspondingly, the singlinoS mixes with the higgsinos and the gauginos, resulting in five neutralino mass eigenstates. Therefore, determining the nature of weakly coupling scalars or neutralinos is the first way to discriminate between NMSSM and MSSM.
In the light of the expected high accuracy in the Higgs sector measurements [10], it is a common practice to compare MSSM and NMSSM scenarios looking at the Higgs sector, in particular at the Higgs decays [18][19][20]. The case in which a very light CP-even and/or a light CP-odd scalars have high singlet component and allow new decay channels for the SM-like Higgs scalar affecting its decay width and branching ratios has been explored [21]. On the same footing, looking at the extended NMSSM neutralino sector is very well motivated, especially for linear collider phenomenology, due to the high precision in the electroweak sector. This can be crucial in case of relatively heavy singlet states in comparison with the SM-like Higgs, such that the observed Higgs sector can be interpreted within both the MSSM and the NMSSM. In such scenarios with heavy decoupled states, the corresponding signatures at the LHC would indeed be very similar in both models [10].
We are therefore interested to understand how much information can be obtained from the neutralino and chargino sector for the model distinction. In the MSSM, the parameters M 1 , M 2 , µ, tan β fully describe the chargino and neutralino sector. One should note that these are fundamental parameters without any assumption on the SUSY breaking scheme. Precise determination of these parameters is possible at a linear collider [11][12][13][14]22], provided thatχ 0 1 ,χ 0 2 andχ ± 1 can be produced at the LC and their masses as well as the polarized cross sections σ(e + e − →χ 0 1χ 0 2 ), σ(e + e − →χ + 1χ − 1 ) are measured. An accurate and rather model-independent determination of M 1 , M 2 , µ, tan β is performed by a χ 2 -minimisation that selects the parameters fitting the experimental results. Such analysis can be strengthened if the mass of the heavier neutralino states can be inferred from combined analyses of LHC and LC data [14].
The possibility of reconstructing the MSSM chargino-neutralino sector parameters can then be exploited as a tool for the distinction between the MSSM and the NMSSM [16]. Given experimental observation ofχ 0 1 ,χ 0 2 andχ ± 1 , a result of the χ 2 -fit that excludes the MSSM at 95% confidence level (C.L.), may suggest the NMSSM. It has indeed been shown [16] that relatively different mixing for MSSM and NMSSM scenarios can lead to very similar neutralino and chargino mass spectra in both models; this is of course also true in case of a scenarios with similar soft parameters and a decoupled singlet superfield.
Following this idea, we outline our strategy: • Scenario selection. We identify NMSSM scenarios that present a mass spectrum forχ ± 1 ,χ 0 1 , χ 0 2 and low Higgs spectrum that can be attributed also to a MSSM scenario. We calculate the corresponding NMSSM neutralino and chargino tree-level masses and polarized cross-sections for the processes e + e − →χ + 1χ − 1 and e + e − →χ 0 1χ 0 2 cf. (Figures 1 and 2). • Constraints. Each scenario has to fulfill a series of phenomenological and experimental constraints implemented in NMSSMTools-4.2.1, that includes NMHDECAY [23][24][25] and NMSDECAY [26,27]. These tools calculate the Higgs sector parameters, SUSY particle masses at the loop level and their decays, and confront them with limits from LEP, LHC and EW precision constraints. An interface to MicrOMEGAS [28] provides dark matter constraints, including the latest LUX [29] and Planck [30] results. The LSP relic density is required to be Ω LSP h 2 < 0.131, where h is the Hubble constant in units of 100 km/(s·Mpc). Higgs sector constraints are further controlled using HiggsBounds-4.0.0 [31] and HiggsSignals-1.0.0 [32], such that a scenario is accepted only if compatible with current data at the 95% (C.L.).
• Experimental assumption. We assume, for each NMSSM scenario, an observation ofχ ± 1 ,χ 0 1 andχ 0 2 at the ILC together with their total cross sections σ(e + e − →χ + 1χ − 1 ), σ(e + e − →χ 0 1χ 0 2 ) with electron-positron beam polarizations (P e − , P e + ) = (±0.9, ∓0.55) at √ s = 350 GeV (corresponding to the tt-threshold) and at √ s = 500 GeV. A precision of 0.5% on the masses and 1% on the cross sections is assumed [33,34]. If kinematically accessible, also mχ0 3 , and the processes e + e − →χ 0 1χ 0 3 , e + e − →χ 0 2χ 0 3 are considered. • χ 2 -fit to MSSM. The measured quantities and errors are used to perform a MSSM parameter determination through the χ 2 -fit following the recipe in [14], similarly to [16]. We apply the χ 2 -fit using Minuit [35], that minimizes the χ 2 function defined as where O i are the input observables, δO i are the associated experimental uncertainties andŌ i are the theoretical values of the observables calculated using the fitted MSSM parameters. The unknowns of the fit will be M 1 , M 2 , µ, tan β and mν e . 1 In the case of high tan β, its extraction 1 The mass mν e is related to the selectron masses by applying the SU(2) relation m 2 νe = m 2 e L + cos(2β) cos 2 θ W m 2 Z and mẽ L = mẽ R . could be difficult, and only a lower limit could be set. A fit that is not consistent with the MSSM 95% C.L., may give hints towards the NMSSM and model distinction. If this is not the case, more information is needed to be included to establish the nature of the observed model. The limiting (95% C.L.) value of χ 2 varies for different scenarios under consideration depending on the number of observables used in the fit.
• Information from the Higgs sector. If the singlet is relatively light and has a substantial mixing with the SM-like Higgs, one could observe deviations from the SM predictions that cannot be accommodated within the MSSM at the same time. In our case, we expect small departure from the SM values and we limit ourselves to comparing the NMSSM predictions to the SM model by doing a χ 2 -fit of the reduced couplings of the SM-like Higgs to g, γ, W, Z, b, c, τ . If the couplings do not differ too much from the SM, such a scenario could always be accommodated within the MSSM in the decoupling limit as well. Alternatively, one could consider a possibility of detection additional singlet-like states, but this analysis is beyond the scope of the current paper.

Classes of scenarios
The singlino (S) admixtures of the lightest neutralino statesχ 0 1 andχ 0 2 suggest the classification of NMSSM scenarios with the following limiting cases: A high singlino admixture inχ 0 1 and/orχ 0 2 as in case 1 may signal beyond-MSSM physics. A fit reconstructing the higgsino and gaugino components hypothesizing MSSM would give very different result with respect to the original NMSSM. In such a case, the outlined strategy for model distinction seems promising, see [16] and section 3.1, as different gaugino and neutralino admixtures lead to modified cross sections, production channels, as well as decays.   In cases 2 and 3, instead, both spectra and admixtures of the detected statesχ 0 1 andχ 0 2 could result in a MSSM-like phenomenology, therefore it is likely that the fit is still compatible with the MSSM, see subsections 3.2 and 3.3. In these cases one should ask how to efficiently integrate informations from heavier neutralino states, and/or from the Higgs sector.
Given a fixed µ eff = λs, the key parameters of the NMSSM neutralino sector are λ and κ as they regulate the singlino admixture in the mass eigenstates, see the NMSSM neutralino mass matrix, Eq. (18) in appendix A. In two heavy-singlino cases, see examples in subsections 3.2 and 3.3, we scan a grid of ten thousand points in the (λ, κ)-plane for values λ ∈ [0, 0.7] and κ ∈ [0, 0.7], to study how the model discrimination method works at the ILC along the (λ, κ)-plane, as the singlino admixtures vary. For each point passing the previous phenomenological and experimental constraints, we perform the χ 2 -fit described above. These scans allow to see how the singlino "mass" vary along the (λ, κ)-plane, and to observe areas in which the singlino is mostly very heavy and decoupled, areas in which the singlino is placed among the lightest neutralino states, as well as regions with mixed behaviour.

Light singlino scenario
As a first example, we choose an NMSSM scenario with winoχ 0 1 but with high singlino components iñ χ 0 2 (andχ 0 3 ). We refer to it as the light singlino scenario (LS). The lower neutralino/chargino spectrum can be reproduced by an MSSM scenario with different M 1 , M 2 , µ, tan β, see Table 1. Both for LS and the corresponding MSSM scenario we have M 1 > M 2 , as it is common in AMSB models. We set A λ = 4200 GeV and A κ = −200 GeV. For the remaining parameters of the NMSSM scenario, we refer to appendix B.1. A SM-like Higgs with m h = 125 GeV is reproduced.
The Higgs spectrum is given in Table 2. 2 The mass m h1 can be easily reproduced within the corresponding MSSM scenario with a proper choice of the stop soft parameters. The states h 2 and a 1 , being both ∼ 100% singlets, are not expected to be visible both at the LHC and ILC because they are not directly coupling to other particles and are relatively heavy. A detailed analysis could point a way to observe these states but this is beyond scope of this work.
The tree-level masses for the neutralino/chargino sector are listed in Table 3. The light part of the spectrum is nearly indistinguishable between the two models, withχ 0 1 ∼W . However, the other lighter statesχ 0 2 ,χ 0 3 feature different admixtures, see Table 4, leading to different production cross sections and relative importance of the production channels.
We take mẽ L = 303.5 GeV, assuming mẽ L = mẽ R and m 2 νe = m 2 e L + cos(2β) cos 2 θ W m 2 Z . The production cross sections are listed in Table 5. For the fit to the MSSM we only include NMSSM cross sections larger than 1 fb. The relatively light NMSSMχ 0 3 can be produced with a sizeable cross section at 500 GeV, therefore we also include σ(e + e − →χ 0 1χ 0 3 ) for P = (−0.9, 0.55) in the fit.   Table 3: Neutralino and chargino masses in the LS scenario and in the corresponding reference MSSM scenario. The mass difference mχ± 1 − mχ0 1 receives significant positive NLO corrections. Here, we only use tree-level masses, however for such a quasi-degenerate states the mass measurement usually has a larger uncertainty than the mass difference itself so in a more realistic setting one should use the mass difference as an input rather than the actual masses, see e.g. Ref. [36].
and tan β remains unconstrained. These parameters would be consistent with neutralino and chargino masses in the MSSM, listed in Table 6. The fit with 10−5 = 5 degrees of freedom (d.o.f.) gives χ 2 = 62.6, clearly stating that the hypothesized model (MSSM) is not compatible with the experimental data (with the 95% C.L. being χ 2 < 11.1). This could be additionally confirmed by the mass of the heavy neutralino, mχ0 4 , if it is eventually measured at the higher center-of-mass energy. Additionally, we note that the predicted mass of the heavy chargino, 3.2 Light higgsino scenario, µ eff < M 1 < M 2 We consider here an NMSSM scenario with a light higgsino (LH), whose chargino/neutralino parameters are: M 1 = 450 GeV, M 2 = 1600 GeV, µ eff = λ s = 120 GeV, tan β = 27 , while we have λ ∈ [0, 0.7] and κ ∈ [0, 0.7] as described above; µ eff is kept fixed by varying the singlet vacuum expectation value (vev) s. TheŜ soft parameters are A λ = 3000 GeV, A κ = −30 GeV. The first generation sfermion masses, needed for the production cross sections, are set to mẽ L = 303.5 GeV, mẽ L = mẽ R , m 2 νe = m 2 e L + cos(2β) cos 2 θ W m 2 Z , while squarks masses are > 1 TeV. For the full set of soft parameters, we refer to Appendix B.2. In Figure 3 we show the result of the scan in the NMSSM (λ, κ)-plane after our tests. Light-blueshaded area corresponds to points that pass DM constraints; 3 the points within purple-shaded boundary area pass the Higgs sector constraints from HiggsBounds and HiggsSignals. The solid red area is the region allowed by all the constraints, phenomenological and experimental ones, implemented within NMSSMTools, HiggsBounds and HiggsSignals.
As a reference MSSM scenario, we select the one with M 1 , M 2 , µ = µ eff , tan β and the slepton masses given in Eqs. (5) and (6), to show that the light neutralino spectrum and production cross sections, see Table 7, may be very similar to the analogue quantities in the LH-NMSSM scenario in the vast part of the (λ, κ)-plane, (cf. Figure 4(a) for mχ0 1 and Figure 5(a) for the corresponding cross sections). Regarding the Higgs sector, it is possible to get a MSSM counterpart with the same SM-Higgs mass and a similar spectrum for the other Higgs states (with the exception of the new singlet states) for each point in the (λ, κ)-plane of the LH scenario.
In Figure 4, the NMSSMχ 0 1 mass and its singlino component are shown. A negligible singlino component corresponds to a region in which the NMSSM mχ0 1 is very close to the MSSM value mχ0 1 = 114.8 GeV. Vice versa, with a higher singlino admixture the LSP mass, mχ0 1 , within NMSSM significantly decreases.   Likewise, the neutralino polarised production cross sections σ(e + e − →χ 0 1χ 0 2 ) decrease with respect to the MSSM value following larger singlino component inχ 0 1 , see Figure 5(a), as it is expected since the singlino does not couple directly to the gauge fields. The tree-level NMSSM chargino masses and production cross-sections, σ(e + e − →χ + 1χ − 1 ), depend only on M 2 , µ eff , tan β, therefore chargino production cross sections are identical to the MSSM values displayed in Table 7 along all the (λ, κ)-plane.
According to the recipe in Section 2, we assume for each point in the (λ, κ)-plane of the LH scenario the experimental measurement of: • mχ0 1 , mχ0 2 and mχ± 1 with an uncertainty of 0.5%.
In the regions in which the singlino component inχ 0 3 is higher, mχ0 3 is decreased andχ 0 2χ 0 3 may be kinematically accessible, see Figure 5(b). In these cases, ifχ 0 3 is detectable through its decays, we consider also mχ0 3 and σ(e + e − →χ 0 2χ 0 3 ). The productionχ 0 1χ 0 3 is negligible almost everywhere. With these assumptions, a χ 2 -fit to the MSSM gives the result displayed in Figure 6: the yellow areas correspond  to regions in the (λ, κ)-plane that are at 95% C.L. compatible with the MSSM, while in the black area MSSM is excluded. Therefore, a significant region of the parameter space, passing the implemented phenomenological and experimental constraints, can definitely be distinguished from the MSSM using collider observables. This is due to a higher singlino component in the neutralinoχ 0 3 (and partially iñ χ 0 1 as well, cf. Figure 4(b). We attempt here a reconstruction of the MSSM M 1 , M 2 , µ, tan β and mν e for two sample points in the (λ, κ)-plane of the LH scenario, relatively close to the boundary between the regions of compatibility from Figure 6.
Additional information from the heavier neutralino states, such asχ 0 3 (if its production is not already kinematically allowed at 500 GeV) orχ 0 4 , may help in reducing the region compatible with the MSSM. For example, given a (λ, κ) coordinate and the corresponding M 1 , M 2 , µ, tan β reconstructed from the fit, one can derive the masses of the heavier states and look for them at higher energies at the ILC or at the LHC, either confirming the fit to the MSSM or pinpointing the NMSSM.
As suggested in Section 2, our study may be extended by including information from the Higgs sector. A result of the naïve χ 2 -fit to the SM of the Higgs reduced couplings to g, γ, W, Z, b, c, τ , each defined as a ratio g h /g hSM between the SM-like Higgs coupling to the corresponding SM Higgs coupling, is shown in    4 In large part of the (λ, κ)-plane, the SM-like Higgs of the LH scenario is compatible with the SM (χ 2 14), corresponding to the MSSM-like area from the fit in Figure 6. A SM-like Higgs with a higher singlet component, see Figure 7(b), corresponds to a worse fit: there are two regions that are not compatible with the SM and have a different behaviour with respect of MSSM-like areas. The conclusion from this fit is therefore consistent with that of Figure 6 without clearly improving our analysis. Additional information about the NMSSM Higgs sector could obtained if new singlets are directly visible. This possibility opens up in a region with a higher singlino component inχ 0 3 , where the decays χ 0 3 →χ 0 1,2 a 1 become open. If the production cross section forχ 0 3 is non-negligible one could observe the pseudoscalar a 1 via its decays a 1 → bb. In Figure 8(a) we show an inclusive cross section for production of a 1 , where both production modes, e + e − →χ 0 1χ 0 3 and e + e − →χ 0 2χ 0 3 , for both polarisations has been added up together with decaysχ 0 3 →χ 0 1 a 1 andχ 0 3 →χ 0 2 a 1 . In certain regions of parameter space, with cross sections of order 10 fb, the new state should be clearly visible. This could serve as confirmation of the NMSSM, since the MSSM would be already excluded by the fit to other observables. As a reference, in Figure 8(b) we also show the mass of the pseudoscalar a 1 . Finally, we study an NMSSM scenario with light gauginos (LG), whose neutralino/chargino sector is given by: M 1 = 240 GeV, M 2 = 105 GeV, µ = µ eff = 505 GeV, tan β = 9.2 , with λ ∈ [0, 0.7] and κ ∈ [0, 0.7]. The singlet soft trilinear parameters are A λ = 3700 GeV, A κ = −40 GeV. The first generation sfermion masses are while squarks masses are > 1 TeV. For the full set of soft parameters, we refer to Appendix B.3. In Figure 9 we display the result of the scan in the NMSSM (λ, κ)-plane after our tests implemented within NMSSMTools, HiggsBounds and HiggsSignals. The colour conventions are the same as for the LH scenario, Section 3.2; for the LG scenario the regions allowed by the Higgs sector constraints from HiggsBounds and HiggsSignals overlap entirely those passing DM matter constraints.
A reference MSSM scenario with an almost indistinguishable lighter (tree-level) neutralino and chargino mass spectrum and production cross sections is found by choosing M 1 , M 2 , µ, tan β and the first generation slepton masses as in Eq. (9), see Table 12.   In the LG scenario mχ0 1 is very close to the reference MSSM value 99.5 GeV and it varies very mildly in the (λ, κ)-plane as the singlino component inχ 0 1 is approximately zero, see Figure 10. A similar reasoning applies to the production cross section σ(e + e − →χ 0 1χ 0 2 ), while the chargino production, σ(e + e − →χ + 1χ − 1 ), is exactly identical at the tree-level as explained in Section 3.2. We only use cross sections larger than 1 fb for χ 2 -fit to the MSSM. Figure 11(a) shows that our fit alone is not able to distinguish in this case between the two models, as basically every point in the allowed region is compatible with the MSSM.
As example we analyse the point LG1 with (λ, κ) = (0.2, 0.35) and the remaining parameters given by Eqs. (9) and (10) These values are remarkably close to the 'true' input parameters given by Eqs. (9) and (10). A naïve fit of the SM-like Higgs reduced couplings does not provide information useful for model distinction, see Figure 11(b), as they are always compatible with the SM, unlike in the LH scenario.  This behaviour can be understood by analysing the mixing within the neutralino sector. In the NMSSM, the singlino does not mix directly with gauginos but only indirectly via higgsino states, see Eq. (18) and Appendix A.1. If, like in the LG scenario, µ eff M 1 , M 2 , the mixing remains small even for a relatively light singlino. Therefore the properties of the light chargino and neutralino states, including masses and cross sections, remain very similar throughout the (λ, κ)-plane and cannot be distinguished from the MSSM case. In contrast to that, in the light singlino scenario from Section 3.

Conclusions and outlook
It will be very important to develop methods how to discriminate between the NMSSM and the MSSM at future experiments, as the two models may reproduce experimentally very similar light Higgs sectors as well as lower supersymmetric spectra. In this paper we have outlined a model distinction strategy that focuses on the neutralino and chargino sector and we have applied it to a series of NMSSM scenarios with different singlino, gaugino and higgsino properties. The idea is to assume that the lightest neutralino and chargino masses as well as their polarised pair production cross sections are measurable at a linear collider and to reconstruct the corresponding MSSM parameters, M 1 , M 2 , µ, tan β, via a χ 2 -fit. In case such a fit clearly excludes the MSSM hypothesis it would strongly point towards an extended model, preferably the NMSSM. Integrating the analysis with further information from the Higgs sector or from heavier neutralino resonances could confirm such a new model hypothesis. Throughout our study we have assumed to operate at the ILC with two different energy stages, namely at √ s = 350 and 500 GeV, using electron and positron beam polarisation with P = (±0.9, ∓0.55).
We have introduced three classes of scenarios with different phenomenological aspects concerning the model distinction: a light singlino, a light higgsino and a light gaugino scenario. We have first analysed an NMSSM scenario with singlino components in theχ 0 2 and a wino-like LSPχ 0 1 , i.e. with an inverted hierarchy of the gaugino mass parameters. Such a NMSSM scenario does not result in set of observables consistent with the MSSM. Accessing the mixing character of the heavier neutralinoχ 0 3 would confirm the situation and point to a model with an extended neutralino sector with respect to the MSSM.
In the class with light higgsinos, one usually has the hierarchy µ eff < M 1 < M 2 . In the corresponding NMSSM parameter space, a large part of the (λ, κ)-plane features the heavy and decoupled singlino while theχ 0 1 is higgsino-like. Such a model is indistinguishable from the MSSM. However, if a sufficient singlino admixture is present in the light neutralinos, the neutralino sector changes appreciably, allowing for a discrimination between the MSSM and the NMSSM. In some region of the parameter space additional pseudoscalar Higgs a 1 could also be observed. Precise measurement of the SM-like Higgs couplings would be beneficial for a confirmation of these conclusions.
As a third class we have chosen light gaugino scenarios again with an inverted hierarchy M 2 < M 1 < µ eff but with µ eff −M 1 ∼ O(250) GeV. In this way the singlino does not significantly mix with gauginos in the lightest neutralino states. In the light of our experimental assumptions, the low mass spectrum and production cross sections are not distinguishable from the MSSM ones all over the allowed (λ, κ)-plane. In this case analysing the SM-like Higgs couplings also does not provide further information.
Our studies show that the neutralino and chargino sector can provide the crucial information for the model distinction between the MSSM and the NMSSM. Such a discrimination depends on the gaugino mass hierarchies and the actual singlino admixture in the light neutralino states. Precise measurements and a model-independent analysis for the determination of the fundamental SUSY parameters are essential.
with the only difference that now µ is substituted by µ eff = λs, where s the vev of the singlet, and where v 2 u + v 2 d = v 2 = 2m 2 Z /(g 2 1 + g 2 2 ) ≈ (174 GeV) 2 . The NMSSM neutralino sector depends on two more singlet/singlino parameters with respect to the MSSM: λ, κ, while µ dependence is substituted by the dependence on the singlet vev s.

A.2 Z 3 -NMSSM Higgs sector
According to Ref. [6], for the Z 3 -invariant NMSSM, the part of the superpotential describing Higgs-Singlet (self) interactions is given by: while the Yukawa couplings are described by 2000 GeV 1800 GeV 1500 GeV 300 GeV 500 GeV 3700 GeV 2500 GeV 1500 GeV