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]


---
---

tempcntc citex\[\#1\]\#2[@fileswauxout tempcnta@tempcntb@neciteacite[forciteb:=\#2citeo]{}[\#1]{}]{} citeo[tempcnta&gt;tempcntbciteacitea[,]{} tempcnta=tempcntbtempcnta]{}

A search for single top production ( $e^{+}e^{-} \rightarrow t\bar{c}$ ) via Flavour Changing Neutral Currents (FCNC) was performed using the data taken by the DELPHI detector at LEP2. The data analysed have been accumulated at centre-of-mass energies ranging from 189 to 208 GeV. Limits at 95% confidence level were obtained on the anomalous coupling parameters $\kappa_{\gamma}$ and $\kappa_Z$.

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Introduction
============

Flavour Changing Neutral Currents (FCNC) are highly suppressed in the Standard Model (SM) due to the Glashow-Iliopoulos-Maiani (GIM) mechanism [@GIM]. However, small contributions appear at one-loop level (Br$(t\to(\gamma,g,Z) + c(u)) < 10^{-10}$) due to the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix [@SMFCNC]. Many extensions of the SM, such as supersymmetry [@supers] and multi-Higgs doublet models [@MHiggs], predict the presence of FCNC already at tree level. Some specific models [@Arbuzov] give rise to detectable FCNC amplitudes.

The most prominent signature for direct observation of FCNC processes at LEP is the production of a top quark together with a charm or an up quark in the process $e^+e^- \to t \bar{c}$ [^1]. The strength of the transitions $\gamma \to 
\mbox{f}\mbox{f}'$ and $Z \to \mbox{f}\mbox{f}'$ can be described in terms of the Lagrangian given in [@we]:

$$\begin{aligned}
\Gamma_{\mu}^{\gamma} & = & \kappa_{\gamma} \frac{ee_q}{\Lambda}\sigma_{\mu\nu}
\left( g_1 P_l + g_2 P_r \right) q^{\nu}, \\
\Gamma_{\mu}^Z & = & \kappa_Z \frac{e}{\sin 2\Theta_W}\gamma_{\mu} 
\left( z_1 P_l + z_2 P_r\right),\end{aligned}$$

where $e$ is the electron charge, $e_q$ the top quark charge, $\Theta_W$ is the weak mixing angle and $P_l$ ($P_r$) is the left (right) handed projector. The $\kappa_\gamma$ and $\kappa_Z$ are the anomalous couplings to the $\gamma$ and $Z$ bosons, respectively. $\Lambda$ is the new physics scale. A value of 175 GeV was used for numerical calculations throughout the paper. The relative contributions of the left and right handed currents are determined by the $g_i$ and $z_i$ constants which obey the constraints: $$\begin{aligned}
g_1^2 + g_2^2 = 1,\;\;\; z_1^2 + z_2^2 = 1.\end{aligned}$$ In the approach which gives the most conservative limits on the couplings, the interference term, which depends on $g_i$ and $z_i$, gives a negative contribution to the cross-section of the process $e^+e^- \to t\bar{c}$. This corresponds to the requirement [@we]: $$\begin{aligned}
  g_1 z_1 + g_2 z_2 = -1.\end{aligned}$$

The existence of anomalous top couplings to gauge bosons allows the top to decay through $t \to c \gamma$ and $t \to c Z$ in addition to the dominant decay mode $t \to b W$. This effect was taken into account in the evaluation of results. Numerical estimates of the expected number of events taking into account the limits on anomalous vertices set by the CDF collaboration [@CDF] can be found in [@we].

This paper is devoted to the search for FCNC processes associated to single top production at LEP ($e^+e^- \to t\bar{c}$). Limits are set on the anomalous couplings $\kappa_\gamma$ and $\kappa_Z$ in the most conservative approach. The $t$ quark is expected to decay predominantly into $Wb$, giving distinct signatures for the leptonic and hadronic $W$ decays. For each decay mode a dedicated analysis was developed. In the [*semileptonic channel*]{} two jets and one isolated lepton (from the $W$ leptonic decays, $W\to l\nu_l$) were searched for. In the [*hadronic channel*]{} four jets were required in the event (two of them from the $W$ hadronic decays, $W\to qq'$). A nearly background-free signature is obtained in the semileptonic channel, but the branching ratio is relatively low. In the hadronic channel, the $W$ decays give an event rate about two times higher, but the background conditions are less favourable.

The DELPHI data and simulated samples
=====================================

The data collected with the DELPHI detector [@DELPHI] at $\sqrt{s} = 189-208$ GeV, well above the $t\bar{c}$ production threshold, were used in this analysis. The integrated luminosity used for each centre-of-mass energy bin is given in Table \[tab:lum\]. The data collected in the year 2000 at energies up to 208 GeV are split into two energy bins 205 GeV and 207 GeV for centre-of-mass energies below and above 206 GeV, respectively. The 189, 192, 196, 200, 202, 205 and 207 GeV energy bins correspond to average centre-of-mass energies of 188.6, 191.6, 195.5, 199.5, 201.6, 204.8 and 206.6 GeV, respectively. While for the semileptonic channel the two last energy bins were considered separately, they were considered together in the hadronic channel.

The background process $e^+ e^- \rightarrow Z/\gamma \rightarrow q\bar q(\gamma)$ was generated with PYTHIA 6.125 [@pythia]. For $\mu^+\mu^-(\gamma)$ and $\tau^+\tau^-(\gamma)$, DYMU3 [@dymu3] and KORALZ 4.2 [@koralz] were used, respectively, while the generator [@bhwide] was used for Bhabha events. Simulation of four-fermion final states was performed using EXCALIBUR [@excalibur] and GRC4F [@grc4f]. Two-photon interactions giving hadronic final states were generated using TWOGAM [@twogam]. Signal events were generated by a standalone simulation program interfaced with PYTHIA 6.125 [@pythia] for quark hadronization. The generation of the signal events was performed with radiative corrections included. The SM contribution is known to be very small ($\mbox{Br}(t \to (\gamma,g,Z) + c(u)) < 10^{-10}$ [@SMFCNC]) and was not taken into account. Both the signal and background events were passed through the detailed simulation of the DELPHI detector and then processed with the same reconstruction and analysis programs as the real data.

0.45 cm

       $\sqrt{s}$ (GeV)         189    192    196    200    202    205    207
  --------------------------- ------- ------ ------ ------ ------ ------ ------
   Luminosity (pb ${}^{-1}$)   151.8   25.9   76.4   83.4   40.1   78.8   84.3

  : Luminosity collected by DELPHI and used in this analysis for each centre-of-mass energy (see text for details).[]{data-label="tab:lum"}

Hadronic channel
================

In the hadronic channel, the final state corresponding to the single top production is characterized by four jets: a $b$ jet from the top decay, a spectator $c$ jet and two other jets from the $W$ hadronic decay.

In this analysis the reconstructed charged particle tracks were required to fulfil the following criteria[^2]:

-   momentum $p > $ 0.4 GeV/c;

-   momentum error $\Delta p/p < 1$;

-   $R\phi$ impact parameter $< 4$ cm;

-   $z$ impact parameter $< 10$ cm.

Tracks seen by only the central tracking devices (Vertex Detector and Inner Detector) were rejected. Neutral clusters were required to have an energy of at least 400 MeV. Events with the visible energy $> 100$ GeV and at least 8 charged tracks were selected for further processing.

The information of the DELPHI calorimeters and tracking devices was used to classify charged particles as electrons or muons according to standard DELPHI algorithms [@DELPHI]. A well-identified lepton was designated as a “standard" lepton. Whenever some ambiguity persisted the lepton was called a “loose" lepton. To each lepton tag there corresponds a given detection efficiency and misidentification probability [@DELPHI]. Events with leptons with momenta above 20 GeV/$c$, identified as at least “standard" electrons or “loose" muons, were rejected.

The LUCLUS [@pythia] algorithm with $d_{join} = 6.5$ GeV/c was then applied to cluster the event into jets. Events with 4, 5, or 6 jets were selected and forced into a 4-jet topology. Each of the three most energetic jets must contain at least one charged particle. The preselection was completed by requiring the event visible energy and combined b-tag parameter [@vanina] to be greater than 130 GeV and $-1.5$, respectively. The energies and momenta of the jets were then rescaled by applying a constrained fit with $\mbox{NDF}=4$ imposing four-momentum conservation [@CFIT].

The assignment of jets to quarks is not straightforward as the kinematics of the event varies strongly with the energy. Near the $t\bar c$ production threshold both quarks are produced at rest and the subsequent top decay ($t\rightarrow Wb$) produces a high momentum $b$ quark. However, at higher LEP centre-of-mass energies the $c$ quark becomes more energetic with momentum values up to 30 GeV/c. Four different methods of jet assignment were considered:

-   the jet with highest b-tag parameter [@vanina] was the $b$ jet candidate and the least energetic jet (among the three remaining jets) was the $c$ jet candidate;

-   the most energetic jet was the $b$ jet candidate and the least energetic one was the $c$ jet candidate;

-   the jet with highest b-tag parameter was the $b$ jet candidate and two jets were assigned to the $W$ according to the probability of the 5-C constrained fit;

-   the most energetic jet was the $b$ jet candidate and two jets were assigned to the $W$ according to the probability of the 5-C constrained fit.

All the above studies were performed and the highest efficiency for the signal and strongest background suppression was obtained with the first method. This method was used in the hadronic analysis for all centre-of-mass energies. Method (2), well suited at the kinematic threshold of single-top production, was less efficient at the highest LEP energies because the energy of the $b$ jet becomes comparable to the energies of the other jets. After the preselection, signal and background-like probabilities were assigned to each event based on Probability Density Functions (PDF) constructed with the following variables:

-   the event thrust value [@tuning];

-   the event sphericity [@tuning];

-   the event b-tag calculated with the combined algorithm [@vanina];

-   the energy of the jet assigned as $b$ jet ($E_b$);

-   the energy of the most energetic jet in the event ($E_{max}$);

-   the ratio of the energies of the least and most energetic jets ($E_{min}/E_{max}$);

-   the invariant mass of the two jets assigned as originating from the $W$ decay ($M_W$);

-   the absolute value of the reconstructed $W$ momentum ($P_W$).

Examples of these distributions are shown in figures \[fig:oleg1\] and \[fig:oleg2\], after the preselection.

All eight PDF were estimated for the signal (${\cal P}_i^{signal}$) and background (${\cal P}_i^{back}$) distributions. They were used to construct the signal ${\cal{L_S}}$=$\prod_{i=1}^{8} {\cal P}_i^{signal}$ and background ${\cal{L_B}}$=$\prod_{i=1}^{8} {\cal P}_i^{back}$ likelihoods. A discriminant variable $$\begin{aligned}
W=\rm{ln}({\cal{L_S}} / {\cal{L_B}} )\end{aligned}$$ based on the ratio of the likelihoods was then constructed for each event.

Figure \[fig:oleg3\] shows the discriminant variable distribution and the number of accepted events, at $\sqrt{s}=205-207$ GeV, as function of signal efficiency for a top mass of 175 GeV/$c^2$. Events were selected by applying a cut on the discriminant variable $\rm{ln}(\cal{L}_S$/$\cal{L}_B)$, dependent on the centre-of-mass energy. Its value was chosen to maximize the efficiency for a low background contamination. The number of data events and expected background from the SM processes (mostly $WW$ background) passing the likelihood ratio selection are shown in Table \[tab:statistics2\] for all centre-of-mass energies, together with the signal efficiencies convoluted with the $W$ hadronic branching ratio. A general good agreement with the Standard Model expectations is observed.

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  ----------------- ------ ------------------- ------ --------------------------- --------------------------
   [$\sqrt{s}$ ]{}                                                                
        (GeV)        Data   Back. $\pm$ stat.   Data   Back.$\pm$stat.$\pm$syst.   $\epsilon\times Br.$ (%)
        $189$        568     530.6 $\pm$ 3.3     37    37.1 $\pm$ 1.4 $\pm$ 1.2    17.6 $\pm$ 0.5 $\pm$ 0.4
        $192$        106      91.4 $\pm$ 1.2     3      3.4 $\pm$ 0.4 $\pm$ 0.3    17.7 $\pm$ 0.5 $\pm$ 0.4
        $196$        266     253.1 $\pm$ 1.5     17    10.7 $\pm$ 0.4 $\pm$ 0.4    17.9 $\pm$ 0.6 $\pm$ 0.5
        $200$        251     265.0 $\pm$ 1.7     12    11.9 $\pm$ 0.5 $\pm$ 0.7    16.7 $\pm$ 0.5 $\pm$ 0.4
        $202$        134     133.3 $\pm$ 0.9     5      6.9 $\pm$ 0.3 $\pm$ 0.3    17.9 $\pm$ 0.6 $\pm$ 0.5
      $205-207$      486     544.1 $\pm$ 2.7     25    30.1 $\pm$ 0.9 $\pm$ 1.2    17.5 $\pm$ 0.5 $\pm$ 0.6
  ----------------- ------ ------------------- ------ --------------------------- --------------------------

  : Number of events in the hadronic analysis at the preselection and final selection levels, for different centre-of-mass energies. The efficiencies convoluted with the $W$ hadronic branching ratio ($Br$) are shown for a top-quark mass of 175 GeV/$c^2$. Statistical and systematic errors are also given (see section 5).[]{data-label="tab:statistics2"}

Semileptonic channel
====================

In the semileptonic channel, the final state corresponding to single top production is characterized by two jets (a $b$ jet from the top decay and a spectator $c$ jet) and at least one isolated lepton from the $W$ leptonic decay.

At the preselection level, events with an energy in the detector greater than 20% of the centre-of-mass energy and at least 7 charged particles were selected. The identification of muons relies on the association of charged particles to signals in the muon chambers and in the hadronic calorimeter and was provided by standard DELPHI algorithms [@DELPHI].

The identification of electrons and photons was performed by combining information from the electromagnetic calorimeter and the tracking system. Radiation and interaction effects were taken into account by an angular clustering procedure around the main shower  [@remclu].

Isolated leptons (photons) were defined by constructing double cones centered around the axis of the charged particle track (neutral cluster) with half-opening angles of 5$^\circ$ and 25$^\circ$ (5$^\circ$ and 15$^\circ$), and requiring that the average energy density in the region between the two cones was below 150 MeV/degree (100 MeV/degree), to assure isolation. In the case of neutral deposits, no charged particle with more than 250 MeV/c was allowed inside the inner cone. The energy of the isolated particle was then re-evaluated as the sum of the energies inside the inner cone. For well identified leptons or photons the above requirements were weakened. In this case only the external cone was used and the angle $\alpha$ was varied according to the energy of the lepton (photon) candidate, down to 2$^\circ$ for $P_{lep} \ge$ 70 GeV/c (3$^\circ$ for $E_\gamma \ge$ 90 GeV), with the allowed energy inside the cone reduced by sin$\alpha$/sin25$^\circ$ (sin$\alpha$/sin15$^\circ$).

Events with only one charged lepton and no isolated photons were selected. No other specific criteria were additionally applied to perform lepton flavour identification.

All other particles were then forced into jets using the Durham jet algorithm [@Durham], which is based on a scaled transverse momentum method. Two-jet events were selected by a cut on the value of the corresponding resolution variable $y$ at the transition between one and two jets: $-log_{10}(y_{2 
\rightarrow 1})\ge0.45$ . The most energetic particle in each jet had to be charged. It was required that the momenta of the lepton and jets were greater than 10 GeV/$c$ and 5 GeV/$c$, respectively. Polar angles of the lepton and of the two jets were required to be in the region 20$^{\circ} \le 
\theta_{lep} \le$ 160$^{\circ}$ and 10$^{\circ} \le \theta_{j1,j2} \le$ 170$^{\circ}$, respectively. The missing momentum polar angle had to be above 20$^{\circ}$ and below 160$^{\circ}$ and the combined b-tag parameter [@vanina] of the most energetic jet was required to be greater than $-1.1$.

The energies and momenta of the jets, the lepton and the momentum of the undetected neutrino (assumed to be the missing momentum) were calculated from four-momentum conservation with a constrained fit (NDF=1). Events with $\chi^2$ lower than 7 were accepted, provided the invariant mass of the neutrino and the isolated lepton was below 125 GeV/$c^2$. The most energetic jet was assigned to the $b$ quark and the second jet to the $c$ quark. The top mass was reconstructed as the invariant mass of the $b$ jet, the isolated lepton and the neutrino four-momenta.

Figures \[fig:fcnc\_m15\_exdis205\_lev1\_lslb-999.\_1\] and \[fig:fcnc\_m15\_exdis205\_lev1\_lslb-999.\_2\] show some relevant distributions for data and MC, after the preselection and for $\sqrt{s}=205-207$ GeV. The number of events at preselection and final selection levels are given in Table \[tab:stat\_semileptonic\] for each centre-of-mass energy. Most of the background comes from SM $e^+e^-\rightarrow WW$ events.

After the preselection, signal and background-like probabilities were assigned to each event (as for the hadronic channel) based on PDF constructed with the following variables:

-   momentum of the less energetic jet;

-   more energetic jet b-tag variable  [@vanina];

-   reconstructed mass of the two jets;

-   reconstructed top mass;

-   angle between the two jets;

-   lepton-neutrino invariant mass;

-   $q_l \cdot \cos \theta_l$, where $q_l$ is the charge and $\theta_l$ is the polar angle of the lepton;

-   $q_{j1} \cdot \cos \theta_{j1}$, where $q_{j1}=-q_{l}$ and $\theta_{j1}$ is the polar angle of the more energetic jet;

-   $p_{j1} \cdot [\sqrt{s} - p_{j1}(1-\cos \theta_{j1\, j2})]$, where $p_{j1}$ is the momentum of the more energetic jet and $\theta_{j1\, j2}$ is the angle between the two jets. This variable is proportional to $(m^2_t-m^2_W)/2.$ i.e., not dependent on the centre-of-mass energy.

The signal ($\cal{L}_S$) and background ($\cal{L}_B$) likelihoods were used on an event-by-event basis to compute a discriminant variable defined as  ln$(\cal{L}_S$/$\cal{L}_B)$. A loose cut on the signal likelihood was applied to the events. Figure \[fig:fcnc\_m15\_exdis205\_lev1\_lslb-999.\_3\] presents, after this cut, the discriminant variable distribution and the number of events accepted as a function of signal efficiency for $\sqrt{s}$=205-207 GeV (assuming a top mass of 175 GeV/$c^2$ for the signal). There is a general good agreement between the data and the SM predictions. The background distribution has a tail for higher values of the discriminant variable which goes below every data event. Correlations between the variables were studied. Their effect on the likelihood ratio is small.

Events were further selected by applying a cut on the discriminant variable  ln$(\cal{L}_S$/$\cal{L}_B)$, dependent on the centre-of-mass energy. Table \[tab:stat\_semileptonic\] shows the number of data and background events which passed the cut for the different centre-of-mass energies. The efficiencies convoluted with the $W$ leptonic branching ratio are also shown. The dominant backgrounds come from SM $e^+e^-\rightarrow WW$ and $e^+e^-\rightarrow ~q\bar q$ events.

0.45 cm

  ----------------- ------ ------------------- ------ --------------------------- --------------------------
   [$\sqrt{s}$ ]{}                                                                
        (GeV)        Data   Back. $\pm$ stat.   Data   Back.$\pm$stat.$\pm$syst.   $\epsilon\times Br.$ (%)
        $189$        102     120.7 $\pm$ 4.3     1      2.4 $\pm$ 0.7 $\pm$ 0.8    8.0 $\pm$ 0.3 $\pm$ 0.5
        $192$         24      21.5 $\pm$ 0.8     1      0.5 $\pm$ 0.1 $\pm$ 0.1    7.7 $\pm$ 0.9 $\pm$ 0.5
        $196$         72      76.2 $\pm$ 2.5     2      0.9 $\pm$ 0.3 $\pm$ 0.1    7.1 $\pm$ 0.9 $\pm$ 0.5
        $200$         95      87.6 $\pm$ 2.8     1      2.0 $\pm$ 0.5 $\pm$ 0.3    6.9 $\pm$ 0.3 $\pm$ 0.3
        $202$         40      42.2 $\pm$ 1.3     1      1.7 $\pm$ 0.3 $\pm$ 0.1    7.9 $\pm$ 0.4 $\pm$ 0.3
        $205$         90      90.0 $\pm$ 2.9     2      1.4 $\pm$ 0.4 $\pm$ 0.1    6.2 $\pm$ 0.3 $\pm$ 0.3
        $207$         71      90.2 $\pm$ 2.6     2      1.9 $\pm$ 0.5 $\pm$ 0.2    6.2 $\pm$ 0.3 $\pm$ 0.4
  ----------------- ------ ------------------- ------ --------------------------- --------------------------

  : Number of events in the semileptonic analysis at the preselection and final selection levels, for the different centre-of-mass energies. The efficiencies convoluted with the $W$ leptonic branching ratio are also shown for a top mass of 175 GeV/$c^2$. Statistical and systematic errors are given (see the systematic errors and limit derivation section).[]{data-label="tab:stat_semileptonic"}

Systematic errors and limit derivation
======================================

Studies of systematic errors were performed and their effect evaluated at the final selection level. The stability of the results with respect to variations on the selection criteria, the PDF definition, the different hadronization schemes and the uncertainty in top quark mass were studied.

Concerning the stability of the results, an independent (and large, compared to the resolution) variation on the selection criteria applied to analysis variables like the missing momentum polar angle, the combined b-tag of the most energetic jet, the $W$ mass, the Durham resolution variable, etc. was allowed. The most significant contributions gave a maximum error of 0.5 events and 0.3% for the expected background and efficiency, respectively. Different smoothing procedures were performed for the PDF definition and their effect is at most 0.5 events (0.4%) for the expected background (signal efficiency). Different hadronization schemes (string and independent) [@pythia] were studied for the signal and their effect contributes at most 0.1% for the signal efficiency error. The uncertainty on the top quark mass is the most important source of systematic errors. It affects not only the total production cross-section but also the kinematics of signal events. In terms of signal efficiency, its effect could be as high as 0.9% for the semileptonic channel (in the mass range between 170 GeV/$c^2$ and 180 GeV/$c^2$). The effects of such variations (added quadratically) on the final selection criteria are quoted as a systematic error in Tables \[tab:statistics2\] and  \[tab:stat\_semileptonic\].

The number of data and expected SM background events for the hadronic and semileptonic channels, the respective signal efficiencies and data luminosity collected at the various centre-of-mass energies were combined to derive limits in the ($\kappa_{\gamma}$, $\kappa_Z$) plane using a Bayesian approach [@Obraztsov_bayes]. In total, 13 independent channels (6 in the hadronic and 7 in the semileptonic modes) correspond to different $\sqrt{s}$ values. These channels are fitted simultaneously to extract the limits on the FCNC parameters. The total production cross-section and top FCNC decay widths dependence with $\kappa_{\gamma}$ and $\kappa_Z$ were properly considered  [@we] in the limit derivation.

The effect of systematic errors on the ($\kappa_\gamma,\kappa_Z$) plane limits was considered. Initial State Radiation (ISR) and QCD corrections [@QCDcorr] were also taken into account in the $t\bar c$ total production cross-section.

Figure \[fig:oleg4\] shows the 95% confidence level (C.L.) upper limits in the ($\kappa_{\gamma}$, $\kappa_Z$) plane obtained by this analysis. The different filled areas correspond to the allowed regions obtained for different top mass values and $\Lambda=175$ GeV. Due to the $s$-channel $Z$ dominance, the LEP2 data are less sensitive to the $\kappa_{\gamma}$ parameter than to $\kappa_Z$. The upper limits obtained by the CDF collaboration [@CDF] and ZEUS [@others_HERA] are also shown in the figure for comparison. The 95% C.L. upper limits on each coupling parameter, setting the other coupling to zero, are summarized in Table \[tab:resu\]. For comparison the values at $m_t=175$ GeV/$c^2$ are $\kappa_Z (\kappa_{\gamma} = 0)=0.434$ and $\kappa_{\gamma} (\kappa_Z = 0)=0.505$ if the Born level cross-section (without radiative corrections) is taken into account.

0.45 cm

          $m_t$ (GeV/$c^2$)            170     175     180
  ---------------------------------- ------- ------- -------
   $\kappa_Z (\kappa_{\gamma} = 0)$   0.340   0.411   0.527
   $\kappa_{\gamma} (\kappa_Z = 0)$   0.402   0.486   0.614

  : 95% C.L. upper limits derived from the combined hadronic and semileptonic channels at $\sqrt{s}= 189-208$ GeV for $\Lambda=175$ GeV.[]{data-label="tab:resu"}

Upper limits were also obtained by using only the hadronic and the semileptonic channels separately when radiative corrections to the total production cross-section were taken into account. The values at $m_t=175$ GeV/$c^2$ are $\kappa_Z (\kappa_{\gamma} = 0)=0.491$ ($0.547$) and $\kappa_{\gamma} (\kappa_Z = 0)=0.568$ ($0.625$) for the hadronic (semileptonic) channel alone.

Summary
=======

The data collected by the DELPHI detector at centre-of-mass energies ranging from 189 to 208 GeV were used to perform a search for FCNC $t\bar{c}$ production, in the hadronic and semileptonic topologies. No deviation with respect to the SM expectations was found. Upper limits on the anomalous couplings $\kappa_{\gamma}$ and $\kappa_Z$ were derived. A comparison with CDF  [@CDF] and ZEUS [@others_HERA] is also shown. Results on the search for single-top production were also obtained by the other experiments at LEP [@others_LEP].

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![Limits at 95% confidence level in the $\kappa_{\gamma}-\kappa_Z$ plane. The different curved and filled areas represent the regions allowed by DELPHI for different top quark masses. Radiative corrections were taken into account in the total production cross-section at LEP. The CDF and ZEUS allowed regions are also shown for a top quark mass of $175$ GeV/$c^2$. The ZEUS limits are scaled by a factor of $\sqrt{2}$ because of the difference in the Lagrangian definitions.[]{data-label="fig:oleg4"}](fig7_new_new.eps){width="14.5cm"}

[^1]: Throughout this paper the notation $t \bar{c}$ stands for $t \bar{c}$+$t \bar{u}$ and includes the charge conjugate contribution as well.

[^2]: The DELPHI coordinate system has the $z$-axis aligned along the electron beam direction, the $x$-axis pointing toward the centre of LEP and $y$-axis vertical. R is the radius in the $(x,y)$ plane. The polar angle $\Theta$ is measured with respect to the $z$-axis and the azimuthal angle $\phi$ is about $z$.