You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'The interplay between the LHC and the $e^+ e^-$ International Linear Collider (ILC) with $\sqrt{s}=500$ GeV might be crucial for the discrimination between the minimal and next-to-minimal supersymmetric standard model. We present an NMSSM scenario, where the light neutralinos have a significant singlino component, that cannot be distinguished from the MSSM by cross sections and mass measurements. Mass and mixing state predictions for the heavier neutralinos from the ILC analysis at different energy stages and comparison with observation at the LHC, lead to clear identification of the particle character and identify the underlying supersymmetric model.'
author:
- 'G. Moortgat-Pick'
- 'S. Hesselbach'
- 'F. Franke, H. Fraas'
title: |
    \
    Identifying the NMSSM by the interplay of LHC and ILC
---

CERN-PH-TH/2005-141\
TSL/ISV-2005-0293\
WUE-ITP-2005-008\
hep-ph/0508313

[**Identifying the NMSSM by the interplay of LHC and ILC[^1]**]{}

[*$^{1}$ TH Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland\
$^{2}$ High Energy Physics, Uppsala University, Box 535, S-75121 Uppsala, Sweden\
$^{3}$ Institut für Theoretische Physik und Astrophysik, Universität Würzburg,\
*]{}

The interplay between the LHC and the $e^+ e^-$ International Linear Collider (ILC) with $\sqrt{s}=500$ GeV might be crucial for the discrimination between the minimal and next-to-minimal supersymmetric standard model. We present an NMSSM scenario, where the light neutralinos have a significant singlino component, that cannot be distinguished from the MSSM by cross sections and mass measurements. Mass and mixing state predictions for the heavier neutralinos from the ILC analysis at different energy stages and comparison with observation at the LHC, lead to clear identification of the particle character and identify the underlying supersymmetric model.

INTRODUCTION
============

Supersymmetry (SUSY) is one of the most promising extensions to the Standard Model (SM). Since low-energy SUSY is broken there exist numerous free parameters that make it a highly challenging task to reveal the underlying model at the Large Hadron Collider (LHC) and at the International Linear Collider (ILC). It is planned that the ILC starts with an energy of $\sqrt{s}=500$ GeV, which will be upgraded to about 1 TeV [@ITRP]. However, already at the first energy stage, the ILC could reach higher energy up to about $\sqrt{s}=650$ GeV at cost of luminosity. In this study we sketch a possible motivation to apply this higher energy option. Particularly interesting are case studies which apply interplay of search strategies at the LHC and the ILC [@lhclc-report]. We extend in this paper the methods for combined LHC/ILC analyses developed in [@Moortgat-Pick:2005vs].

An interesting possibility for the determination of the supersymmetric model is to study the gaugino/higgsino particles, which are expected to be among the lightest supersymmetric particles. In this paper we consider two basic supersymmetric models: the minimal supersymmetric standard model (MSSM) and the next-to-minimal supersymmetric standard model (NMSSM). The MSSM contains four neutralinos $\tilde{\chi}^0_i$, the mass eigenstates of the photino, zino and neutral higgsinos, and two charginos $\tilde{\chi}^\pm_i$, being mixtures of wino and charged higgsino. The neutralino/chargino sector depends at tree level on four parameters: the U(1) and SU(2) gaugino masses $M_1$ and $M_2$, the higgsino mass parameter $\mu$, and the ratio $\tan\beta$ of the vacuum expectation values of the Higgs fields. For the determination of these parameters, straightforward strategies [@parameters; @choi] have been worked out even if only the light neutralinos and charginos $\tilde{\chi}^0_1$, $\tilde{\chi}^0_2$ and $\tilde{\chi}^\pm_1$ are kinematically accessible at the first stage of the ILC [@ckmz].

The NMSSM [@nmssm] is the simplest extension of the MSSM by an additional Higgs singlet field. New parameters in the neutralino sector are the vacuum expectation value $x$ of the singlet field and the trilinear couplings $\lambda$ and $\kappa$ in the superpotential, where the product $\lambda x = \mu_{\rm eff}$ replaces the $\mu$-parameter of the MSSM [@Franke; @Choi:2004zx]. The additional fifth neutralino may significantly change the phenomenology of the neutralino sector. In scenarios where the lightest supersymmetric particle is a nearly pure singlino, the existence of displaced vertices leads to a particularly interesting experimental signature [@singlinohugonie; @Singlinos]. In case only a part of the particle spectrum is kinematically accessible the distinction between the models may become challenging.

It has already been worked out that there exist MSSM and NMSSM scenarios with the same mass spectra of the light neutralinos but different neutralino mixing. In this case beam polarization is crucial for distinguishing the two models [@Sitges]. We present a scenario where all kinematically accessible neutralinos and charginos have similar masses and almost identical cross sections, within experimental errors, in MSSM and NMSSM. Although the second lightest neutralino in the NMSSM has a significant singlino component, the models cannot be distinguished by the experimental results at the LHC or at the ILC$_{500}$ with $\sqrt{s}=500$ GeV alone if only measurements of masses, cross sections and gaugino branching ratios are considered. Precision measurements of the neutralino branching ratio into the lightest Higgs particle and of the mass difference between the lightest and next-to-lightest SUSY particle [@gunion-mrenna] may give first evidence for the SUSY model but are difficult to realize in our case. Therefore the identification of the underlying model requires precision measurements of the heavier neutralinos by combined analyses of LHC and ILC as described in the following section.

CASE STUDY
==========

We study an NMSSM scenario with the parameters $$\begin{aligned}
&&M_1=360~\mbox{\rm GeV},\quad M_2=147~\mbox{\rm GeV},\quad \tan\beta=10,\quad
\lambda=0.5,\quad x=915~\mbox{\rm GeV},\quad \kappa=0.2.
\label{eq-para-nm}\end{aligned}$$ The hierarchy $M_1 > M_2$ of the U(1) and SU(2) mass parameters leads to very similar masses of the lightest neutralino $\tilde{\chi}_1^0$, which is assumed to be the lightest supersymmetric particle (LSP), and of the light chargino $\tilde{\chi}_1^\pm$. This mass degeneration is also typical for minimal anomaly mediated SUSY breaking (mAMSB) scenarios. Rather small mass differences may be resolved experimentally by applying the ISR method [@Hensel] at the linear collider [@gunion-mrenna] as well as at the LHC [@amsb-lhc].

The NMSSM parameters lead to the following gaugino/higgsino masses and eigenstates: $$\begin{aligned}
m_{\tilde{\chi}^0_1}=138~\mbox{\rm GeV}, \quad\quad 
\tilde{\chi}^0_1=(-0.02, +0.97, -0.20, +0.09, -0.07), \label{ez-chi01-nm} \\
m_{\tilde{\chi}^0_2}=337~\mbox{\rm GeV},\quad\quad
\tilde{\chi}^0_2=(+0.62, +0.14, +0.25, -0.31, +0.65), \label{ez-chi02-nm} \\
m_{\tilde{\chi}^0_3}=367~\mbox{\rm GeV}, \quad\quad
\tilde{\chi}^0_3=(-0.75, +0.04, +0.01, -0.12, +0.65), \label{ez-chi03-nm} \\
m_{\tilde{\chi}^0_4}=468~\mbox{\rm GeV}, \quad\quad
\tilde{\chi}^0_4=(-0.03, +0.08, +0.70, +0.70, +0.08), \label{ez-chi04-nm} \\
m_{\tilde{\chi}^0_5}=499~\mbox{\rm GeV}, \quad\quad
\tilde{\chi}^0_5=(+0.21, -0.16, -0.64, +0.62, +0.37), \label{ez-chi05-nm}\end{aligned}$$ where the neutralino eigenstates are given in the basis $(\tilde{B}^0, \tilde{W}^0, \tilde{H}^0_1,\tilde{H}^0_2, \tilde{S})$. As can be seen from eqs. (\[ez-chi02-nm\]) and (\[ez-chi03-nm\]), the particles $\tilde{\chi}^0_2$ and $\tilde{\chi}^0_3$ have a rather strong singlino admixture.

The Higgs sector does not allow the identification of the NMSSM [@NMSSMhiggs] if scalar and pseudoscalar Higgs bosons with dominant singlet character escape detection. A scan with NMHDECAY [@Ellwanger:2004xm] in our scenario over the remaining parameters in the Higgs sector, $A_\lambda$ and $A_\kappa$, results in parameter points which survive the theoretical and experimental constraints in the region $2740~\textrm{GeV} < A_\lambda < 5465$ GeV and $-553~\textrm{GeV} < A_\kappa < 0$. For $-443~\textrm{GeV} < A_\kappa
< -91$ GeV the second lightest scalar ($S_2$) and the lightest pseudoscalar ($P_1$) Higgs particle have very pure singlet character and are heavier than the mass difference $m_{\tilde{\chi}^0_3} - m_{\tilde{\chi}^0_1}$, hence the decays of the neutralinos $\tilde{\chi}^0_2$ and $\tilde{\chi}^0_3$, which will be discussed in the following, are not affected by $S_2$ and $P_1$, see figure \[higgs-sector\] (left panel). For our specific case study we choose $A_{\lambda}=4000$ GeV and $A_{\kappa}=-200$ GeV, which leads to $m_{S_2}=311$ GeV, $m_{P_1}=335$ GeV and $m_{S_3}$, $m_{P_2}$ and $m_{H^{\pm}}>4$ TeV. Furthermore the lightest scalar Higgs $S_1$ has MSSM-like character in this parameter range with a mass of about 124 GeV. Also the branching ratio of $\tilde{\chi}^0_2$ in the lightest Higgs particle differs only by a factor two in both scenarios. In case that a precise measurement of this BR is possible first hints for the inconsistency of the model could be derived at the ILC.

Strategy for the gaugino/higgsino sector
----------------------------------------

In our NMSSM scenario in the gaugino/higgsino sector only the light chargino $\tilde{\chi}^{\pm}_1$ and the light neutralinos $\tilde{\chi}_1^0$ and $\tilde{\chi}_2^0$ are accessible at the ILC$_{500}$. We calculate the masses of the charginos and neutralinos and the cross sections for the pair production of the light chargino $e^+e^- \rightarrow \tilde{\chi}^+_1
\tilde{\chi}^-_1$ and for the associated production of the light neutralinos $e^+e^- \rightarrow \tilde{\chi}_1^0 \tilde{\chi}_2^0$ with polarized and unpolarized beams.

The masses and cross sections in different beam polarization configurations provide the experimental input for deriving the supersymmetric parameters within the MSSM using standard methods [@choi; @ckmz]:

-   We assume an uncertainty of $\mathcal{O}(1-2\%)$ for the masses $m_{\tilde{\chi}^{\pm}_1}$, $m_{\tilde{\chi}^0_1}$, $m_{\tilde{\chi}^0_2}$, $m_{\tilde{\nu}_e}$, $m_{\tilde{e}_L}$ and $m_{\tilde{e}_R}$. The errors of the cross sections, $e^+e^- \rightarrow \tilde{\chi}^+_1 \tilde{\chi}^-_1$ and $e^+e^- \rightarrow \tilde{\chi}_1^0 \tilde{\chi}_2^0$, are composed of the error due to the mass uncertainties, polarization uncertainty and one standard deviation statistical error based on $\int {\cal L}=100$ fb$^{-1}$ for each polarization configuration. Deviations in the cross sections due to the polarization uncertainty of $\Delta P_{e^{\pm}}/P_{e^{\pm}}=0.5\%$ are generally small; it is expected that the error could even be reduced up to $\Delta P_{e^{\pm}}/P_{e^{\pm}}=0.2\%$–$0.1\%$, [@Power]. The assumed uncertainties in total are listed in table \[lc-input\].

-   From the chargino mass $m_{\tilde{\chi}^{\pm}_1}$ and the cross section $e^+e^- \rightarrow \tilde{\chi}^+_1 \tilde{\chi}^-_1$ measured at two energies, $\sqrt{s}=400$ GeV and $500$ GeV, we determine bounds for the elements $U_{11}$ and $V_{11}$ of the chargino mixing matrices: $$U^2_{11}=[0.84,1.0], \quad V^2_{11}=[0.83,1.0].$$ Polarized beams allow the resolution of ambiguities and the improvement of the accuracy.

-   Using the mixing matrix elements $U_{11}$ and $V_{11}$, the masses $m_{\tilde{\chi}^{\pm}_1}$, $m_{\tilde{\chi}^0_1}$ and $m_{\tilde{\chi}^0_2}$, and the cross sections for $e^+e^- \rightarrow \tilde{\chi}^0_1 \tilde{\chi}^0_2$, we derive constraints for the parameters $M_1$, $M_2$, $\mu$ and $\tan\beta$: $$\begin{aligned}
    M_1 &=& 377 \pm 42 \mbox{ GeV}, \label{MSSMresult1}\\
    M_2 &=& 150\pm 20 \mbox{ GeV}, \\
    \mu &=& 450 \pm 100 \mbox{ GeV}, \\
    \tan\beta &=& [1,30]. \label{MSSMresult4}\end{aligned}$$ Note that, in our scenario with $M_1 > M_2$, the crucial observable to determine the parameter $M_1$ is $m_{\tilde{\chi}^0_2}$ and not $m_{\tilde{\chi}^0_1}$ as often assumed. Such a hierarchy could be naturally embedded in mAMSB scenarios. For even larger $M_1\gg M_2$ the heavier neutralinos $\tilde{\chi}^0_{3,4}$ become crucial for $M_1$ determination see [@Moortgat-Pick:2005vs; @m1-paper]. Since the heavier neutralino and chargino states are not produced, some of the parameters —in our case $\mu$ and $\tan\beta$— can only be determined with a considerable uncertainty.

    Within these limits an explicit MSSM scenario, $$\begin{aligned}
    &&M_1=375~\mbox{\rm GeV},\quad M_2=152~\mbox{\rm GeV},\quad \tan\beta=8,\quad
    \mu=360~\mbox{\rm GeV},
    \label{eq-para-ms}\end{aligned}$$ leads to the same (lighter) neutralino/chargino masses and cross sections: $$\begin{aligned}
    m_{\tilde{\chi}^0_1}=138~\mbox{\rm GeV}, \quad\quad 
    \tilde{\chi}^0_1=(+0.03, -0.96, +0.26, -0.13), \label{ez-chi01-ms}\\
    m_{\tilde{\chi}^0_2}=344~\mbox{\rm GeV}, 
    \quad\quad
    \tilde{\chi}^0_2=(+0.72, +0.22, +0.48, -0.46), \label{ez-chi02-ms} \\
    m_{\tilde{\chi}^0_3}=366~\mbox{\rm GeV}, \quad\quad
    \tilde{\chi}^0_3= (-0.04, +0.10, -0.70, -0.71), \label{ez-chi03-ms} \\
    m_{\tilde{\chi}^0_4}=410~\mbox{\rm GeV}, \quad\quad
    \tilde{\chi}^0_4=(-0.70, +0.18, +0.47, -0.52), \label{ez-chi04-ms}\end{aligned}$$ where the neutralino mixing states are given in the basis $(\tilde{B}^0, \tilde{W}^0, \tilde{H}^0_1,\tilde{H}^0_2)$. Comparing eqs. (\[ez-chi01-ms\])–(\[ez-chi03-ms\]) with eqs. (\[ez-chi01-nm\])–(\[ez-chi03-nm\]) shows that the three lightest neutralino masses are the same within the experimental uncertainties. We checked that also the accessible cross sections at the ILC$_{500}$ and the BR’s of $\tilde{\chi}^0_2$ are consistent.

-   After the determination of the fundamental MSSM parameters we calculate the heavy chargino and neutralino masses and expected mixing characters. For the masses we obtain: $$m_{\tilde{\chi}^0_3} = 443\pm 107 \mbox{ GeV},\quad\quad 
    m_{\tilde{\chi}^0_4} = 490\pm 110 \mbox{ GeV}, \quad\quad
    m_{\tilde{\chi}^{\pm}_2} = 475 \pm 125 \mbox{ GeV}. \label{eq_mp3_nm}$$

The predicted gaugino admixture of $\tilde{\chi}^0_3$, $\tilde{\chi}^0_4$ within the allowed parameter ranges, eqs.(\[MSSMresult1\])–(\[MSSMresult4\]), are shown in figure \[higgs-sector\] (right panel). Obviously, the heavy neutralino $\tilde{\chi}^0_3$ should be almost a pure higgsino within the MSSM prediction. The predicted properties of the heavier particles can now be compared with mass measurements of such SUSY particles via the analysis of cascade decays at the LHC [@lhclc-report].

We emphasize that although we started with an NMSSM scenario where $\tilde{\chi}^0_2$ and $\tilde{\chi}^0_3$ have large singlino admixtures, the MSSM parameter strategy does not fail and the experimental results from the ILC$_{500}$ with $\sqrt{s}=400$ GeV and $500$ GeV lead to a consistent parameter determination in the MSSM. Hence in the considered scenario the analyses at the ILC$_{500}$ or LHC alone do not allow a clear discrimination between MSSM and NMSSM. All predictions for the heavier gaugino/higgsino masses are consistent with both models. However, the ILC$_{500}$ analysis predicts an almost pure higgsino-like state for $\tilde{\chi}^0_3$ and a mixed gaugino-higgsino-like $\tilde{\chi}^0_4$, see figure \[higgs-sector\] (right panel). This allows the identification of the underlying supersymmetric model in combined analyses at the LHC and the ILC$_{650}^{{\cal L}=1/3}$.

Interplay between LHC and ILC
-----------------------------

The expected large cross sections for squark and gluino production at the LHC give access to a large spectrum of coloured as well as non-coloured supersymmetric particles via the cascade decays. Heavy gaugino-states appear almost only in cascade decays and there exist some true simulations how to measure the heavier gauginos in such decays at the LHC [@Giacomo]. Particularly helpful for the identification of the particles involved in the cascades, e.g. for more model-independent analyses, are mass predictions from the ILC analysis which lead to an increase of statistical sensitivity for the LHC analysis and open the possibility of identifying even marginal signals in the squark cascades [@lhclc-report]. However, since higgsino-like charginos and neutralinos do not couple to squarks, their detection via cascade decays is not possible.

In our original NMSSM scenario the neutralinos $\tilde{\chi}^0_2$ and $\tilde{\chi}^0_3$ have a large bino-admixture and therefore appear in the squark decay cascades. The dominant decay mode of $\tilde{\chi}^0_2$ has a branching ratio $BR(\tilde{\chi}^0_2\to
\tilde{\chi}^{\pm}_1 W^{\mp})\sim 50\%$, while for the $\tilde{\chi}^0_3$ decays $BR(\tilde{\chi}^0_3\to
\tilde{\ell}^{\pm}_{L,R} \ell^{\mp})\sim 45\%$ is largest. Since the heavier neutralinos, $\tilde{\chi}^0_4$, $\tilde{\chi}^0_5$, are mainly higgsino-like, no visible edges from these particles occur in the cascades. It is expected to see the edges for $\tilde{\chi}^0_2 \to \tilde{\ell}^{\pm}_R \ell^{\mp}$, $\tilde{\chi}^0_2 \to \tilde{\ell}^{\pm}_L \ell^{\mp}$, $\tilde{\chi}^0_3 \to \tilde{\ell}^{\pm}_R \ell^{\mp}$ and for $\tilde{\chi}^0_3 \to \tilde{\ell}^{\pm}_L \ell^{\mp}$ [@Giacomo_private].

With a precise mass measurement of $\tilde{\chi}^0_1$,$\tilde{\chi}^0_2$, $\tilde{\ell}_{L,R}$ and $\tilde{\nu}$ from the ILC$_{500}$ analysis, a clear identification and separation of the edges of the two gauginos at the LHC is possible without imposing specific model assumptions. We therefore assume a precision of about 2% for the measurement of $m_{\tilde{\chi}^0_3}$, in analogy to [@Giacomo]: $$\begin{aligned}
\label{eq:mchi3LHC}
&& m_{\tilde{\chi}^0_3}=
367\pm 7 \mbox{ GeV}.\end{aligned}$$ The precise mass measurement of $\tilde{\chi}^0_3$ is compatible with the mass predictions of the ILC$_{500}$ but not with the prediction of the mixing character, see e.g. eq. (\[ez-chi03-ms\]). However, it is not clear that the measured particle at the LHC is indeed the $\tilde{\chi}^0_3$. Often in the constrained MSSM, as e.g. also in our MSSM comparison scenario, the second heaviest neutralino $\tilde{\chi}^0_3$ is nearly a pure higgsino and does not couple in the cascade decays. In those cases, the heaviest neutralino $\tilde{\chi}^0_4$ has frequently a sufficiently large gaugino component and could be measured in cascades, as shown in [@lhclc-report; @Giacomo].

Therefore is is inevitable to discuss the following cases of possible particle [*identification*]{} of the measured gaugino mass $m_{\tilde{\chi}^0_3}$ at the LHC:

-   interpretation of the measured particle as $\tilde{\chi}^0_3$ and feeding it back in the ILC analysis leads to improved parameter determination and mass prediction for $m_{\tilde{\chi}^0_4}$, $m_{\tilde{\chi}^{\pm}_2}$. Using eq. (\[eq:mchi3LHC\]) for the ILC$_{500}$ analysis leads in our case, after rechecking with the allowed cross sections of $\tilde{\chi}^0_1\tilde{\chi}^0_2$ and $\tilde{\chi}^{+}_1\tilde{\chi}^-_1$ production, to rather precise mass predictions: $$m_{\tilde{\chi}^0_4}=[384, 393]\mbox{ GeV \quad and \quad}
    m_{\tilde{\chi}^{\pm}_2}=[360, 380]\mbox{ GeV}.
    \label{eq:heavymass-pred}$$

-   interpretation of the measured particle as $\tilde{\chi}^0_4$ and feeding it back in the parameter determination of the ILC analysis leads to inconsistency with the measured cross sections of $\tilde{\chi}^0_1\tilde{\chi}^0_2$ and $\tilde{\chi}^{+}_1\tilde{\chi}^-_1$ production.

The combined LHC$\leftrightarrow$ILC$_{500}$ analysis leads therefore to a correct interpretation of the measured particles in the cascades. However, a neutralino $\tilde{\chi}^0_3$ with sufficiently large gaugino admixture to couple to squarks is incompatible with the allowed parameter ranges of eqs. (\[MSSMresult1\])–(\[MSSMresult4\]) in the MSSM, cf. figure \[higgs-sector\] (right panel).

We point out that a measurement of the neutralino masses $m_{\tilde{\chi}^0_1}$, $m_{\tilde{\chi}^0_2}$, $m_{\tilde{\chi}^0_3}$ which could take place at the LHC alone is not sufficient to distinguish the SUSY models since rather similar mass spectra could exist, cf. eqs. (\[ez-chi01-nm\])–(\[ez-chi03-nm\]) with eqs. (\[ez-chi01-ms\])–(\[ez-chi03-ms\]).

Therefore the cross sections in different beam polarization configurations at the ILC have to be included in the analysis. The combined results from the LHC and the ILC$_{500}$ analyses and the rather precise predictions for the missing chargino/neutralino masses, eq. (\[eq:heavymass-pred\]), constitute a serious motivation to apply immediately the low-luminosity but higher-energy option ILC$_{650}^{{\cal L}=1/3}$, which finally leads to the right identification of the underlying model. The expected polarized and unpolarized cross sections, including the statistical error on the basis of one third of the luminosity of the ILC$_{500}$, are given in Table \[tab\_heavy\]. The neutralino $\tilde{\chi}^0_3$ as well as the higgsino-like heavy neutralino $\tilde{\chi}^0_4$ and the chargino $\tilde{\chi}^{\pm}_2$ are now accessible at the ILC$^{{\cal L}=1/3}_{650}$. Already the high rates for $\tilde{\chi}^0_1\tilde{\chi}^0_3$ production give last true evidence for the obvious contraction with an corresponding MSSM scenario. Together with the mass measurements of $m_{\tilde{\chi}^0_4}=468$ GeV and $m_{\tilde{\chi}^{\pm}_2}=474$ GeV, which are also in strong disagreement with the mass prediction, eq. (\[eq:heavymass-pred\]), one has sufficient observables which point to the NMSSM. Extensions of existing fit programs for the NMSSM may lead to an exact resolution of the underlying parameters [@fit-porod].

CONCLUSIONS
===========

We have presented a scenario in the next-to-minimal supersymmetric standard model (NMSSM) that could not be distinguished from the MSSM at either the LHC or at the first stage of the International Linear Collider with $\sqrt{s}= 500$ GeV. It turns out that the most promising sector for distinction is the gaugino/higgsino sector. Although a light neutralino has a significant singlet component in the NMSSM, the masses of the accessible light neutralinos and charginos, as well as the production cross sections, lead to identical values in the two models within experimental errors. The comparison of the predicted masses and mixing character of the heavier neutralinos and charginos with the measured masses in combined analyses with the LHC followed by a precise measurement of the cross sections at the ILC at $\sqrt{s}= 650$ GeV leads to a clear identification of the supersymmetric model.

The exemplary scenario shows that the interplay between the two experiments could be crucial for the determination of the supersymmetric model. A possible feed-back of ILC$_{500}$/LHC results could motivate the immediate use of the low-luminosity option of the ILC at $\sqrt{s}=650$ GeV in order to resolve model ambiguities even at an early stage of the experiment and outline future search strategies at the upgraded ILC at 1 TeV.

  --------------------------------------------- --------------------- -------------------- ------------------------------------------------------------ --------------------
     $m_{\tilde{\chi}^0_1}$/GeV$=138\pm 2.8$                                                $\sigma(e^+e^-\to\tilde{\chi}^{0}_1\tilde{\chi}^{0}_2)$/fb  
     $m_{\tilde{\chi}^0_2}$/GeV$=337\pm 5.1$     $(P_{e^-},P_{e^+})$   $\sqrt{s}=400$ GeV                       $\sqrt{s}=500$ GeV                       $\sqrt{s}=500$ GeV
   $m_{\tilde{\chi}^{\pm}_1}$/GeV$=139\pm 2.8$       Unpolarized        $323.9 \pm 33.5$                         $287.5 \pm 16.5$                          $4.0 \pm 1.2$
       $m_{\tilde{e}_L}$/GeV$=240\pm 3.6$          $(-90\%,+60\%)$     $984.0 \pm 101.6$                         $873.9 \pm 50.1$                          $12.1 \pm 3.8$
       $m_{\tilde{e}_R}$/GeV$=220\pm 3.3$          $(+90\%,-60\%)$       $13.6 \pm 1.6$                           $11.7 \pm 1.2$                           $0.2 \pm 0.1$
      $m_{\tilde{\nu}_e}$/GeV$=226\pm 3.4$                                                                                                              
  --------------------------------------------- --------------------- -------------------- ------------------------------------------------------------ --------------------

  : Masses with 1.5% ($\tilde{\chi}^0_{2,3}$, $\tilde{e}_{L,R}$, $\tilde{\nu}_e$) and 2% ($\tilde{\chi}^0_1$, $\tilde{\chi}^{\pm}_1$) uncertainty and cross sections with an error composed of the error due to the mass uncertainties, polarization uncertainty and one standard deviation statistical error based on $\int {\cal L}=100$ fb$^{-1}$, for both unpolarized beams and polarized beams with $(P_{e^-},P_{e^+})=(\mp 90\%,\pm 60\%)$ and $\Delta P(e^\pm)/P(e^{\pm})=0.5\%$, in analogy to the study in [@lhclc-report]. \[lc-input\]

  -------------------------------- ---------------- ---------------------------------------------------------------------------------------- ------------ ---------------
                                                     $\sigma(e^+e^- \to \tilde{\chi}^{\pm}_1\tilde{\chi}^{\mp}_2)/$fb at $\sqrt{s}=650$ GeV               
                                                                                                                                                          
         Unpolarized beams          $12.2 \pm 0.6$                                       $5.5 \pm 0.4$                                        $\le 0.02$   $2.4 \pm 0.3$
   $P(e^-)=-90\%$, $P(e^+)=+60\%$   $36.9 \pm 1.1$                                       $14.8 \pm 0.7$                                       $\le 0.07$   $5.8 \pm 0.4$
   $P(e^-)=+90\%$, $P(e^+)=-60\%$   $0.6 \pm 0.1$                                        $2.2 \pm 0.3$                                        $\le 0.01$   $1.6 \pm 0.2$
  -------------------------------- ---------------- ---------------------------------------------------------------------------------------- ------------ ---------------

  : Expected cross sections for the associated production of the heavier neutralinos and charginos in the NMSSM scenario for the ILC$^{{\cal L}=1/3}_{650}$ option with one sigma statistical error based on $\int {\cal L} = 33$ fb$^{-1}$ for both unpolarized and polarized beams. \[tab\_heavy\]

(10,5) (-3.4,4.55)[$m_{H}/$GeV]{} (-.5,4.1)[$P_1$]{} (-.5,1.8)[$S_2$]{} (-.5,.8)[$S_1$]{} (0,-.5)[$A_{\kappa}$/GeV]{} (-3,-1)[![image](0206-higgsmass.epsy){width="70mm"}]{} (8.7,3.5)[ and [$\tilde{\chi}^0_4$]{}]{} (4.9,4.6)[$m_{\tilde{\chi}^0_i}$/GeV]{} (4.5,1.2)[LHC:]{} (4.5,.9)[measurement]{} (4.95,.6)[of $m_{\tilde{\chi}^0_i}\rightarrow$]{} (7.5,5)[Contradiction within MSSM]{} (8.8,4.5)[ILC: prediction of]{} (8.8,4)[mixing character of]{} (8.3,-.5)[gaugino character]{} (6,-1)[![image](0206-chi3chi4){width="70mm"}]{}

We are very grateful to G. Polesello and P. Richardson for constructive discussions. S.H. is supported by the Göran Gustafsson Foundation. This work is supported by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00149 and by the Deutsche Forschungsgemeinschaft (DFG) under contract No. .

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[^1]: Contribution to the proceedings of the ‘2005 International Linear Collider Workshop - Stanford, U.S.A.’