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]


[Determination of the strangeness contents of light-flavour isoscalars from its production rates in hadronic Z decays measured at LEP]{}

Institute for High Energy Physics, RU-142284, Protvino, Russia

A new phenomenological approach is suggested to determine the strangeness contents of light-flavour isoscalars. This approach is based on phenomenological laws of hadron production related to the spin, isospin, strangeness content and mass of the particles. The “effective” numbers of $s$ and $\bar{s}$ quarks in the isoscalar partners $i_1$ and $i_2$ are given by the nonstrange-strange mixing angle $\varphi$: $k(i_1)=2\sin^2\varphi$ and $k(i_2)=2\cos^2\varphi$. From the total production rates per hadronic Z decay of all light-flavour hadrons measured so far at LEP the values for $k$ are found to be: $k(\eta)$ $\equiv$ 2$-$$k(\eta^{\,\prime})$ = 0.91$\pm$0.12, $k(\phi)$ $\equiv$ 2$-$$k(\omega)$ = 1.94$\pm$0.09, $k(f_2^{\prime})$ $\equiv$ 2$-$$k(f_2^{~})$ = 1.84$\pm$0.21 and $k(f_0)$ = 0.09$\pm$0.13. Our results on the $\eta$–$\eta^{\,\prime}$, $\omega$–$\phi$ and $f_2^{~}$–$f_2^{\prime}$ isoscalar mixing are consistent with the present experimental evidence. Quite remarkably, our value for $k(\eta)$ corresponds to the singlet-octet mixing angle $\theta_P$ = $-$12.4$^{\circ}$$\pm$3.5$^{\circ}$. The obtained strangeness content of the $f_0$(980) scalar/isoscalar is not consistent with the values supported by different model studies. However, taking the $f_0$(980) in our analysis with the mass of the bare state ($K$-matrix pole) $f_0^{bare}$(720$\pm$100), the mixing angle is found to be: $\vert\varphi_S^{\,bare}\vert$ = 73$^{\circ}$$\pm$7$^{\circ}$$\pm$24$^{\circ}$, in good agreement with the prediction of the $K$-matrix analysis.

The quark contents of pseudoscalar ($P$), vector ($V$), tensor ($T$) and scalar ($S$) mesons have been discussed many times from the beginning of the creation of unitary SU(3)-flavour symmetry. This is a quite interesting question because the quark contents of the lightest isoscalars are different from the prediction of the SU(3) quark model. The phenomenological studies of hadronic processes involving isoscalars usually make assumptions about their quark compositions. Therefore, to the understanding of the quark model and QCD, it is very important to determine the SU(3)-breaking hadronic parameters which define these quark compositions.

In terms of the SU(3) singlet and octet basis states $$\label{eq1}
\eta_{1} = (u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}, ~~~~~
\eta_{8} = (u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$$ the quark contents of the physical $\eta$ and $\eta^{\,\prime}$ states are given by the singlet-octet mixing angle $\theta_P$. Assuming the orthogonality of the physical states and no mixing with other isoscalars and glueballs the flavour wave functions of the $\eta$ and $\eta^{\,\prime}$ pseudoscalars are defined to be: $$\label{eq2}
\eta~ = \eta_8\cdot\cos\theta_P - \eta_1\cdot\sin\theta_P,$$ $$\label{eq3}
\eta^{\,\prime} = \eta_8\cdot\sin\theta_P + \eta_1\cdot\cos\theta_P.$$ To determine the [*strangeness*]{} contents of the same isoscalars it is more convenient to use the so-called nonstrange-strange quark basis ($n\bar{n} = (u\bar{u}+d\bar{d})/\sqrt{2}$ and $s\bar{s}$): $$\label{eq4}
\eta~ = n\bar{n}\cdot\cos\varphi_P - s\bar{s}\cdot\sin\varphi_P,$$ $$\label{eq5}
\eta^{\,\prime} = n\bar{n}\cdot\sin\varphi_P + s\bar{s}\cdot\cos\varphi_P,$$ where $\varphi_P$ is the nonstrange-strange mixing angle (with $\theta_P = \varphi_P - \arctan\sqrt{2}$). The quark contents of the $\omega$–$\phi$ and $f_2^{~}$–$f_2^{\prime}$ isoscalars are defined in a way analogous to the $\eta$–$\eta^{\,\prime}$ case, replacing $\eta \rightarrow \phi (f_2^{\prime})$, $\eta^{\,\prime} \rightarrow \omega (f_2^{~})$ in Eqs. (\[eq2\])–(\[eq3\]) and $\eta \rightarrow \omega (f_2^{~})$, $\eta^{\,\prime} \rightarrow \phi (f_2^{\prime})$ in Eqs. (\[eq4\])–(\[eq5\]). The flavour wave function of the $f_0$(980) scalar meson is written as $$\label{eq6}
f_0 = n\bar{n}\cdot\cos\varphi_S + s\bar{s}\cdot\sin\varphi_S.$$

The values of the mixing angles have been estimated from different phenomenological and theoretical analyses (see Refs. [@I1]-[@I9] and references therein). The phenomenological estimations of the pseudoscalar mixing angle $\theta_P$ (or $\varphi_P$) use the available world data on the following decay processes: strong decays of tensors and higher-spin mesons ($M_{J>2}$) into pseudoscalar pairs, radiative transitions between vectors and pseudoscalars, two-photon annihilation decays, leptonic decays of vectors, $J/\psi$ decays into a vector plus a pseudoscalar, radiative $J/\psi$ decays, semileptonic $D_s$ decays plus transition form factors $\eta/\eta^{\,\prime} \rightarrow \gamma\gamma^*$. There are also phenomenological analyses of $\theta_P$ which use the data on the quasi-two-body $\pi^-p$ and $\bar{p}p$ reactions. The values for $\theta_P$ (and $\varphi_P$) obtained from the above highlighted processes are given in Table \[tab1\] together with the corresponding references.

  process                                                                                    $\varphi_P$($\,^{\circ}$)   $\theta_P$($\,^{\circ}$)       Ref.
  ----------------------------------------------------------------------------------------- --------------------------- -------------------------- ---------------
  $T(2^{++}) \rightarrow PP$                                                                         $42\pm 2$                  $-13\pm 2$              [@R1]
                                                                                                   $43.1\pm 3.0$              $-11.6\pm 3.0$           [@R10]
  $M_{J>2} \rightarrow PP$                                                                           $41\pm 4$                  $-14\pm 4$              [@R1]
  $V \rightarrow P\gamma$;  $P \rightarrow V\gamma$                                                $37.7\pm 2.4$              $-17.0\pm 2.4$            [@R2]
                                                                                                   $35.3\pm 5.5$              $-19.4\pm 5.5$           [@R10]
  $P \rightarrow \gamma\gamma$                                                                     $41.3\pm 1.3$              $-13.4\pm 1.3$            [@R1]
                                                                                                   $36.3\pm 2.0$              $-18.4\pm 2.0$            [@R3]
  $P \rightarrow V\gamma / \gamma\gamma$;  $V \rightarrow P\gamma / e^+e^-$                        $43.1\pm 0.8$              $-11.6\pm 0.8$           [@R12]
  $J/\psi \rightarrow VP$                                                                          $35.5\pm 1.4$              $-19.2\pm 1.4$            [@R4]
                                                                                                   $35.6\pm 1.4$              $-19.1\pm 1.4$            [@R5]
                                                                                                   $37.8\pm 1.7$              $-16.9\pm 1.7$            [@R1]
                                                                                                   $39.9\pm 2.9$              $-14.8\pm 2.9$           [@R10]
  $J/\psi \rightarrow P\gamma$                                                                     $39.0\pm 1.6$              $-15.7\pm 1.6$           [@R10]
  $D_s \rightarrow P\,e\,\nu_{\,e}$;  $\eta/\eta^{\,\prime\,} \rightarrow \gamma\gamma^*$          $38.0\pm 2.8$              $-16.7\pm 2.8$            [@R9]
                                                                                                   $41.2\pm 4.7$              $-13.5\pm 4.7$           [@R11]
  $\pi^-p \rightarrow (\eta/\eta^{\,\prime\,})\,n$                                                 $36.5\pm 1.4$              $-18.2\pm 1.4$            [@R6]
                                                                                                   $39.3\pm 1.2$              $-15.4\pm 1.2$            [@R7]
  $\bar{p}p \rightarrow (\eta/\eta^{\,\prime\,})\,(\pi^0/\eta/\omega)$                             $37.4\pm 1.8$              $-17.3\pm 1.8$            [@R8]
  [*average*]{}                                                                                    $39.2\pm 1.3$              $-15.5\pm 1.3$         [@R1; @R10]
  ${\rm Z}^0 \rightarrow hadrons$                                                                  $42.3\pm 3.5$              $-12.4\pm 3.5$        this analysis

  : Compilation of the values of the $\eta$–$\eta^{\,\prime}$ mixing angles $\varphi_P$ and $\theta_P$ obtained from different processes (with $\theta_P = \varphi_P - \arctan\sqrt{2} \simeq 
  \varphi_P - 54.7^{\circ}$).

\[tab1\]

The theoretical analyses of the mixing angles are usually model dependent, and the predictions for $\theta_P$ range from $-23^{\circ}$ to $-10^{\circ}$. Classic examples are the quadratic and linear Gell-Mann–Okubo (GMO) mass formulae which yield the values $\theta_P^{\,quad} \simeq -10^{\circ}$ and $\theta_P^{\,lin} \simeq -23^{\circ}$ [@PDG]. For the vector ($\varphi_V$) and tensor ($\varphi_T$) mixing angles the most of theoretical and phenomenological analyses (see, for example, Refs. [@PDG; @R1; @R2; @R12; @R13; @R14; @R15]) predict the values which are very close to the “ideal” mixing: $\varphi_V = \theta_V - \theta^{\,ideal} \simeq +3.4^{\circ}$ and $\varphi_T = \theta_T - \theta^{\,ideal} \simeq -7.3^{\circ}$ ($\theta^{\,ideal} = \arctan{1/\sqrt{2}} \simeq 35.3^{\circ}$).

The interpretation of the $f_0$(980) scalar meson is one of the most controversial in meson spectroscopy [@I10]. The question is whether the $f_0$(980) consists mostly of nonstrange or of strange quarks. The recent phenomenological analyses of the experimental data on the following decay processes $\phi \rightarrow \pi^0\pi^0\gamma$, $J/\psi \rightarrow \omega\pi\pi$ [@I11], $D_s^+ \rightarrow f_0(980)\pi^+$ [@I9] and $f_0(980) \rightarrow \pi\pi/K\bar{K}/\gamma\gamma$ [@I8] favour the $s\bar{s}$ dominance of the $f_0$(980). The phenomenological study of low-energy $\pi\pi$ scattering [@I12] and the phase shift analysis of $\pi K$ scattering [@I13] have led to the same conclusion.

In the present paper the strangeness contents of light-flavour isoscalars are obtained for the first time from its total production rates per hadronic Z decay measured at LEP. The new decay process (${\rm Z}^0 \rightarrow hadrons$) is added to the above-listed processes used to determine the mixing angles.

Recently it has been shown [@U1]-[@U4] that the total production rates per hadronic Z decay ($\langle n \rangle$) of all light-flavour mesons ($M$) and baryons ($B$) measured so far at LEP follow phenomenological laws related to the spin ($J$), isospin ($I$), strangeness content and mass ($m$) of the particles. These regularities can be combined into one empirical formula: $$\label{eq7}
{\langle n \rangle} \,=\, 
A\cdot\beta_H\cdot(2J+1)\cdot\gamma^{\,k}\cdot\exp{[-b_H(m/m_0)^{N_H}]},$$ where $H$ = $M$ or $B$, $m_0=1$ GeV/$c^2$, $\gamma$ is the strangeness suppression factor with a value of $\gamma \simeq 0.5$ for all hadrons, $k$ is the number of $s$ and $\bar{s}$ quarks in the hadron and $b_H$ is the slope of the mass dependence. The values of the degree $N_H$ and of the coefficient $\beta_H$ are different for mesons and baryons: $$\label{eq8}
N_M = 1,~~ N_B = 2 ~~~{\rm and}~~~
\beta_M = 1,~~ \beta_B = {4\over{C_{\pi/p}\cdot\lambda_{\,QS}}},$$ where $\lambda_{QS} = (2J+1)(2I+1)$ can be interpreted as a fermion suppression factor originating from quantum statistics properties of bosons and fermions and $C_{\pi/p}$ is the $\pi$/p ratio at the zero mass limit with a value of $C_{\pi/p} \simeq 3$ which could be expected from quark combinatorics.

According to the results of Refs. [@U2; @U3; @U4] the normalization parameter $A$ in Eq. (\[eq7\]) is the same for all mesons, but the meson slope $b_M$ is split into two: one for vector, tensor and scalar mesons ($b_{V,T,S}$) and another for pseudoscalar mesons ($b_P$). The slope splitting of the mass dependence of meson production rates can probably be explained by the influence of the spin-spin interaction between the quarks of the meson (the spins of quarks are parallel for vector, tensor and scalar mesons and anti-parallel for pseudoscalar mesons). However, there is no influence of the value and orientation (with respect to the net spin) of the orbital angular momentum of the quarks, i.e. of the spin-orbital interaction of the quarks.

In the analyses [@U1]-[@U4] the strangeness contents $k$ of baryons and $I$$\neq$0 mesons were taken from the prediction of the SU(3) quark model. For isoscalars the following values for $k$ were used: $k(\omega)=k(f_2^{~})=k(f_0)=0$, $k(\eta)=k(\eta^{\,\prime})=1$ and $k(\phi)=k(f_2^{\prime})=2$. The [*purpose of the present paper*]{} is to obtain the values of the “effective” numbers $k$ of the $\eta$–$\eta^{\,\prime}$, $\omega$–$\phi$, $f_2^{~}$–$f_2^{\prime}$ and $f_0$(980) isoscalars from the fit of Eq. (\[eq7\]) to the total production rates per hadronic Z decay of all light-flavour hadrons measured so far at LEP and then to determine the corresponding mixing angles. The [*basic idea*]{} of this analysis is the following relations between the numbers $k$ and the nonstrange-strange mixing angles $\varphi$: $$\label{eq9}
k(\eta)\, \,\equiv\, 2 - k(\eta^{\,\prime}) \,=\, 2 \sin^2\varphi_P,$$ $$\label{eq10}
k(\phi)\, \,\equiv\, 2 - k(\omega)\,\, \,=\, 2 \cos^2\varphi_V,$$ $$\label{eq11}
k(f_2^{\prime}) \,\equiv\, 2 - k(f_2^{~}) \,=\, 2 \cos^2\varphi_T,$$ $$\label{eq12}
k(f_0) \,=\, 2 \sin^2\varphi_S,$$ which assume the orthogonality of the physical isoscalar partners and no mixing with other isoscalars and glueballs.

The total[^1] production rates per hadronic Z decay of light-flavour hadrons, used in this analysis, were obtained for at least one state of a given isomultiplet as a weighted-average[^2] of the measurements of the four LEP experiments: ALEPH [@A1]-[@A3], DELPHI [@D9],[@D1]-[@D8], L3 [@L1]-[@L4] and OPAL [@O1]-[@O9]. Then, in the fits of our analysis, we use the [*production rates per isospin state*]{} ($\langle n \rangle$) which were obtained by averaging the above-obtained production rates of particles belonging to the same isomultiplet (${\langle n \rangle}_i$): $$\label{eq13}
{\langle n \rangle} \,=\, 
{1\over X}\,\sum_{i\,=\,1}^{i\,=\,X}{\langle n \rangle}_i,$$ where $X \leq (2I+1)$ is the number of [*measured*]{} isospin states. The exact definitions (\[eq13\]) and the experimental values of the data points $\langle n \rangle$ are given in Table \[tab2\] together with the corresponding references.

  hadron                                       definition                         $\langle n \rangle$  References
  ---------------------- ------------------------------------------------------- --------------------- -------------------------------------
  $\pi$                            [$1\over3$]{}($\pi^0$+$\pi^{\pm}$)                8.83$\pm$0.15     [@A1; @A2; @D4; @D7; @L1; @O2; @O9]
  K                          [$1\over4$]{}(K$^0$+$\bar{\rm K}^0$+K$^{\pm}$)         1.075$\pm$0.016    [@A2; @A3; @D2; @D7; @L3; @O2; @O4]
  $\eta$                                                                             0.94$\pm$0.08     [@L1; @O9]
  $\eta^{\prime}$                                                                    0.17$\pm$0.05     [@L2; @O9]
  $\rho$                          [$1\over3$]{}($\rho^0$+$\rho^{\pm}$)               1.21$\pm$0.15     [@A1; @D8; @O9]
  K$^*$                   [$1\over4$]{}(K$^{*0}$+$\bar{\rm K}^{*0}$+K$^{*\pm}$)     0.367$\pm$0.014    [@A1; @D2; @D6; @O1; @O5]
  $\omega$                                                                          1.084$\pm$0.086    [@A1; @L2; @O9]
  $\phi$                                                                           0.0966$\pm$0.0073   [@A1; @D6; @O8]
  N                                                 p                               1.037$\pm$0.040    [@A2; @D7; @O2]
  $\Lambda$                                                                         0.388$\pm$0.011    [@A3; @D3; @L3; @O6]
  $\Sigma$                   [$1\over3$]{}($\Sigma^-$+$\Sigma^0$+$\Sigma^+$)        0.089$\pm$0.005    [@A1; @D5; @D9; @L4; @O7]
  $\Xi$                                          $\Xi^-$                           0.0265$\pm$0.0011   [@A1; @D3; @O6]
  $\Delta$                                    $\Delta^{++}$                         0.088$\pm$0.035    [@D1; @O3]
  $\Sigma^*$                          [$1\over2$]{}$\Sigma^{*\pm}$                 0.0234$\pm$0.0022   [@A1; @D3; @O6]
  $\Xi^*$                                      $\Xi^{*0}$                          0.0058$\pm$0.0010   [@A1; @D3; @O6]
  $\Omega^-$                                                                      0.0013 $\pm$0.00024  [@A1; @D5; @O6]
  $a_0$(980)                            [$1\over2$]{}$a_0^{\pm}$                    0.135$\pm$0.055    [@O9]
  $f_0$(980)                                                                        0.147$\pm$0.011    [@D8; @O8]
  K$_2^*$(1430)              [$1\over2$]{}(K$_2^{*0}$+$\bar{\rm K}_2^{*0}$)         0.042$\pm$0.020    [@D8; @O5]
  $f_2$(1270)                                                                       0.169$\pm$0.025    [@D8; @O8]
  $f_2^{\prime}$(1525)                                                              0.012$\pm$0.006    [@D8]
  $\Lambda$(1520)                                                                  0.0225$\pm$0.0028   [@D9; @O6]

  : Definitions and values of the total production rates per isospin state of light-flavour hadrons in hadronic Z decays obtained from the weighted-average values of the four LEP experiments. The baryon rates include charge conjugated states.

\[tab2\]

In comparison with the analyses of Refs. [@U1]-[@U4], in the present analysis Eq. (\[eq7\]) is simultaneously fitted to all experimental data points given in Table \[tab2\] and it assumes the validity of the relation $V/P = (2J+1) = 3$ for the vector-to-pseudoscalar ratio at the zero mass limit. In the first fit (fit 1) the numbers $k$ for the $\eta$–$\eta^{\,\prime}$, $\omega$–$\phi$ and $f_2^{~}$–$f_2^{\prime}$ isoscalars are fixed and given by Eqs. (\[eq9\])-(\[eq11\]) with the values of the mixing angles predicted by the quadratic GMO mass formula: $\varphi_P \simeq 44.7^{\circ}$, $\varphi_V \simeq 3.7^{\circ}$ and $\varphi_T \simeq -7.3^{\circ}$ [@PDG]. The fixed value of $k=0$ is used for the $f_0$(980) scalar. The result of the fit 1 is shown in Fig. \[fig1\] and in Table \[tab3\]. Fig. \[fig1\] shows the mass dependence of the total production rates per spin and isospin state for hadrons in hadronic Z decays weighted by a factor $\lambda_{QS}\,\gamma^{-k}$ where $\lambda_{QS}=1$ for mesons and $\lambda_{QS}=(2J+1)(2I+1)$ for baryons. Three curves on this figure are the result of the fit 1 for baryons, for mesons with the net spin S=0 (pseudoscalars) and for mesons with the net spin S=1 (vectors, tensors and scalars). The values for $\gamma$ and $C_{\pi/p}$ are found to be $\gamma = 0.51\pm0.02$ and $C_{\pi/p} = 2.8\pm0.2$. The latter is consistent with the quark combinatorics prediction $\pi$/p = 3 for the direct production rates. This coincidence can probably be explained by the absence of decay processes at zero masses.

                           fit 1           fit 2           fit 3           fit 4
  ------------------- --------------- --------------- --------------- ---------------
  $A$                  15.4$\pm$0.4    14.7$\pm$0.9    15.4$\pm$0.4    15.2$\pm$0.4
  $C_{\pi/p}$           2.8$\pm$0.2      3    fixed      3    fixed     2.8$\pm$0.2
  $\gamma$             0.51$\pm$0.02   0.51$\pm$0.02   0.50$\pm$0.02   0.50$\pm$0.02
  $b_P$                4.01$\pm$0.11   4.03$\pm$0.11   3.97$\pm$0.12   3.94$\pm$0.14
  $b_{V,T,S}$          4.71$\pm$0.06   4.67$\pm$0.07   4.69$\pm$0.06   4.67$\pm$0.08
  $b_B$                2.70$\pm$0.07   2.57$\pm$0.07   2.62$\pm$0.04   2.66$\pm$0.08
  $N_M$                  1    fixed    1.04$\pm$0.06     1    fixed      1    fixed
  $N_B$                  2    fixed    2.07$\pm$0.05     2    fixed      2    fixed
  $k(\eta)$              0.99 fixed      0.99 fixed    0.90$\pm$0.12   0.91$\pm$0.12
  $k(\phi)$              1.99 fixed      1.99 fixed    1.94$\pm$0.09   1.94$\pm$0.09
  $k(f_2^{\prime})$      1.97 fixed      1.97 fixed    1.86$\pm$0.20   1.84$\pm$0.21
  $k(f_0)$               0    fixed      0    fixed      0    fixed    0.09$\pm$0.13
  $\chi^2/ndf$            9.4/16          8.6/15          9.8/14          8.1/12

  : Values of the parameters in Eq. (\[eq7\]) obtained in the four fits of this analysis.

\[tab3\]

In the second fit (fit 2) of our analysis we test the sensitivity of the empirical formula (\[eq7\]) to the values of the degree $N_H$. This was not discussed in the previous analyses [@U1]-[@U4]. In the fit 2 the numbers $k$ for the isoscalars are still fixed with the same values as in the fit 1, the ratio $C_{\pi/p}$ is fixed with a value of $C_{\pi/p}=3$ and the degrees $N_M$ and $N_B$ are free parameters. The values for $N_M$ and $N_B$ are found to be (see Table \[tab3\]): $$\label{eq14}
N_M \,=\, 1.04\pm0.06 ~~~{\rm and}~~~ N_B \,=\, 2.07\pm0.05.$$ These values are close within the relatively small errors to the values of 1 and 2 used previously. So, we can conclude that this test (fit 2) strongly suggests the use of $N_M=1$ and $N_B=2$ in our phenomenological analysis. These values are fixed in the next fits.

In the third fit (fit 3) the strangeness contents $k(\eta)$, $k(\phi)$ and $k(f_2^{\prime})$ are free parameters. The values for $k(\eta^{\,\prime})$, $k(\omega)$ and $k(f_2^{~})$ are given by the constraints (\[eq9\])-(\[eq11\]). The fixed parameters in the fit 3 are: $C_{\pi/p}=3$, $N_M=1$, $N_B=2$ and $k(f_0)=0$. The values for $k(\eta)$, $k(\phi)$ and $k(f_2^{\prime})$ are found to be (see Table \[tab3\]): $k(\eta)=0.90\pm0.12$, $k(\phi)=1.94\pm0.09$ and $k(f_2^{\prime})=1.86\pm0.20$. These values agree within the errors with theoretical and phenomenological analyses. Only one question in our analysis is still open. The strangeness content of the $f_0$(980) scalar is fixed with the ad hoc value of $k(f_0)=0$ which was suggested in Refs. [@U1; @U2; @U3; @U4]. However, the recent phenomenological analyses of different physical processes suggest the $s\bar{s}$ dominance of the $f_0$(980).

In our final fit (fit 4) only the values of $N_M=1$ and $N_B=2$ are fixed. All other parameters are free. Our final values for $\gamma$ and $C_{\pi/p}$ (see Table \[tab3\]) are: $$\label{eq15}
\gamma \,=\, 0.50\pm0.02 ~~~{\rm and}~~~ C_{\pi/p} = 2.8\pm0.2,$$ in good agreement with our previous results [@U2; @U3; @U4]. The strangeness contents of the isoscalars, the main purpose of this paper, are found to be: $$\label{eq16}
k(\eta)\, \,\equiv\, 2 - k(\eta^{\,\prime}) \,=\, 0.91 \pm 0.12,$$ $$\label{eq17}
k(\phi)\, \,\equiv\, 2 - k(\omega)\, \,=\, 1.94 \pm 0.09,$$ $$\label{eq18}
k(f_2^{\prime}) \,\equiv\, 2 - k(f_2^{~}) \,=\, 1.84 \pm 0.21,$$ $$\label{eq19}
k(f_0) \,=\, 0.09 \pm 0.13.$$ The corresponding values of the mixing angles are given by Eqs. (\[eq9\])-(\[eq12\]) with the values (\[eq16\])-(\[eq19\]) for the numbers $k$.

Our value (\[eq16\]) for the number $k(\eta)$ corresponds to the following values of the singlet-octet and nonstrange-strange mixing angles: $$\label{eq20}
\theta_P \,=\, -12.4^{\circ} \pm 3.5^{\circ},~~~
\varphi_P \,=\, 42.3^{\circ} \pm 3.5^{\circ},$$ which are compared in Table \[tab1\] with the mixing angle values obtained from the well established and accepted phenomenology and from the experimental data available at present. Quite remarkably, our values (\[eq20\]) are compatible within the errors with the values for $\theta_P$ and $\varphi_P$ obtained from very different physical processes (Table \[tab1\]).

Our values (\[eq17\])-(\[eq18\]) for the numbers $k(\phi)$ and $k(f_2^{\prime})$ correspond to the following values of the vector ($\varphi_V$) and tensor ($\varphi_T$) nonstrange-strange mixing angles (our analysis is not sensitive to the sign of $\varphi$): $$\label{eq21}
\vert\varphi_V\vert \,=\, 10^{\circ} \pm 8^{\circ},~~~
\vert\varphi_T\vert \,=\, 16^{\circ} \pm 11^{\circ},$$ which are very close to the predictions $\vert\varphi_V\vert \simeq 3.4^{\circ}$ and $\vert\varphi_T\vert \simeq 7.3^{\circ}$ of most theoretical and phenomenological analyses (see, for example, Refs. [@R1; @R2; @R12; @R13; @R14; @R15]) including the quadratic and linear GMO mass formulae [@PDG]. The values (\[eq21\]) are also compatible with the “ideal” mixing angle $\varphi^{ideal}=0$.

The value of the mixing angle $\varphi_S$ for the $f_0$(980) scalar meson is found from Eq. (\[eq12\]) with the value (\[eq19\]) for the number $k(f_0)$: $$\label{eq22}
\vert\varphi_S\vert \,=\, 13^{\circ} \pm 9^{\circ}.$$ This value of the mixing angle $\varphi_S$ is not consistent with the results of recent phenomenological studies. For example, the present data for decays of $f_0$(980) into the channels $\pi\pi$, $K\bar{K}$ and $\gamma\gamma$ allow to interpret the $f_0$(980) as either the quasi-singlet state with $\varphi_S \simeq  55^{\circ}$ or the quasi-octet one with $\varphi_S \simeq  -43^{\circ}$ [@I8]. Also the studies of the decay processes $\phi \rightarrow \pi^0\pi^0\gamma$, $J/\psi \rightarrow \omega\pi\pi$ [@I11] and $D_s^+ \rightarrow f_0(980)\pi^+$ [@I9] have led (in terms of $\varphi_S$ defined by Eq. (\[eq6\])) to the values of $\varphi_S \simeq 70^{\circ}$ and $\varphi_S \simeq 76^{\circ}$, respectively.

The obtained disagreement for the $f_0$(980) can probably be related with a question which is still open. This is whether the $f_0$(980) belongs to the scalar $q\bar{q}$ nonet $1{\,}^3P_0$ or whether it should be considered as an exotic state. The scalar nonet classification can be performed in terms of so-called “bare states” (the $K$-matrix poles) corresponding to $q\bar{q}$ states “before” the mixing which is caused by the transitions $q\bar{q}~state \rightarrow real~mesons$ [@K1; @K2; @K3; @I7]. In this way [@I7] the scalar $q\bar{q}$ nonet $1{\,}^3P_0$ has been found to be: $f_0^{bare}(720\pm100)$, $f_0^{bare}(1260\pm30)$, $a_0^{bare}(960\pm30)$ and $K_0^{bare}(1220_{-150}^{+~50})$. The flavour wave functions of the orthogonal states $f_0^{bare}(720)$ and $f_0^{bare}(1260)$ are given in the form (\[eq6\]) by the mixing angle $\varphi_S[f_0^{bare}(720)] = {-70^{\circ\,}}_{-16^{\circ}}^{+~5^{\circ}}$ [@I7]. The transitions $q\bar{q}~state \rightarrow real~mesons$ mix these bare states with each other as well as with nearly states. The real $f_0$(980) meson is a result of this mixing and is related both to the $f_0^{bare}(720\pm100)$ and $f_0^{bare}(1260\pm30)$ states.

There is some interesting relation between our phenomenology and the $K$-matrix analysis which can be illustrated by Fig. \[fig2\], where the total production rates per spin and isospin state for vector, tensor and scalar mesons are plotted as a function of $m$ and three curves with $k=0$, $k=1$ and $k=2$ are the result of our final fit (fit 4).

According to (\[eq22\]) the data point for the $f_0$(980) is close to the curve with $k=0$. As was observed by L. Montanet [@MPC] using the figure from Ref. [@U3], the $f_0$(980) data point shifted from the real mass ($m$ = 980 MeV/$c^2$) to the bare mass ($m^{bare}$ = $720\pm100$ MeV/$c^2$) is close to the curve with $k=2$ (Fig. \[fig2\]). This fact can probably be considered as a speculative argument to re-determine the strangeness content of the $f_0$(980) using our phenomenology, but replacing the real mass of the $f_0$(980) to the bare one. Therefore the fit 4 is repeated with this replacement and gives the following value of the scalar mixing angle $\varphi_S^{\,bare}$: $$\label{eq23}
\vert\varphi_S^{\,bare}\vert \,=\, 73^{\circ} \pm 7^{\circ} \pm 24^{\circ},$$ where the second error is due to the uncertainty $\pm100$ MeV/$c^2$ of the bare mass. This value is well consistent with the predictions of the $K$-matrix analysis [@K2; @I7] and of the recent phenomenological studies [@I8; @I9; @I11]. It can be noted also that there is only one more scalar meson in our analysis, i.e. $a_0$(980), but the real mass of this scalar is very close to the bare mass of the $a_0^{bare}(960\pm30)$ state. So, if the $f_0$(980) is mostly $s\bar{s}$ state, our results for the scalar mesons suggest that its total production rates are probably given by the bare masses.

In conclusion, the new phenomenological approach has been suggested to determine the strangeness contents of light-flavour isoscalars. For the first time for this purpose the total production rates per hadronic Z decay of all light-flavour hadrons measured so far at LEP were used. So, the new physical process (${\rm Z}^0 \rightarrow hadrons$) is added to the list of ones used in the world available analyses on the mixing angle determination. Assuming the one-mixing-angle scheme, the orthogonality of the isoscalar partners and no mixing with other states and glueballs, our approach gives us the following values of the pseudoscalar, vector and tensor mixing angles: $\varphi_P = 42.3^{\circ} \pm 3.5^{\circ}$ ($\theta_P = -12.4^{\circ} \pm 3.5^{\circ}$), $\vert\varphi_V\vert = 10^{\circ} \pm 8^{\circ}$ and $\vert\varphi_T\vert = 16^{\circ} \pm 11^{\circ}$, in good agreement with the present experimental evidence. The same analysis gives us two values of the scalar mixing angle: $\vert\varphi_S\vert = 13^{\circ} \pm 9^{\circ}$ if the $f_0$(980) is taken with the real mass, but $\vert\varphi_S^{\,bare}\vert = 73^{\circ} \pm 7^{\circ} \pm 24^{\circ}$ if the $f_0$(980) is taken with the bare mass ($K$-matrix pole) of the $f_0^{bare}(720\pm100)$ state. Only the second value is consistent with recent phenomenological analyses. [*If their conclusions are correct*]{}, it means that in the framework of our approach the total production rates of scalar mesons are probably given by the bare masses.

[*Acknowledgments.*]{} I am grateful to L. Montanet for interesting and inspiring discussions and for useful comments.

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[^1]:  The quoted rates include decay products from resonances and particles with $c\tau < 10$ cm.

[^2]:  In calculating the errors of averages, the standard weighted least-squares procedure suggested by the PDG [@PDG] was applied: if the quantity $[\chi^2/(N-1)]^{1/2}$ was greater than 1, the error of the average was multiplied by this scale factor.