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: 'We consider flavor changing effective neutrino interactions in the context of massive neutrinos in the issue of atmospheric neutrinos. Assuming as usual that this is an indication of the oscillation of muon neutrinos into tau neutrinos we show that there is a set of parameters which is consistent with the MSW resonance condition for the typical Earth density and atmospheric neutrino energies. In particular we show that even if the vacuum mixing angle vanishes it is still possible to have a resonance which is compatible with the atmospheric neutrino data. We also briefly consider the case of the solar neutrino problem.'
address: |
     Instituto de Física Teórica\
    Universidade Estadual Paulista\
    Rua Pamplona 145\
    01405-900– São Paulo, SP\
    Brazil
author:
- 'J. C. Montero [^1] and V. Pleitez [^2]'
title: MSW effect with flavor changing interactions and the atmospheric neutrino problem 
---

It is a well accepted fact that the solar [@solar] and atmospheric neutrino data [@sk1] strongly suggest that neutrinos are massive particles [@vb]. Other solutions like pure flavor changing neutrino interactions [@fcnit] or neutrino magnetic moments [@mg1; @mg2] are still possible for the solar neutrino problem, at least with the present data. Although the more favoured solution is the vacuum oscillation one, it is still possible that two or more of these mechanisms are in action simultaneously. For instance, the MSW [@lw1; @ms] effect together with effective flavor changing neutrino interactions was proposed in Ref.  [@er]. This is an interesting possibility since almost all extensions of the electroweak standard model implies new interactions for the neutrinos.

On the other hand, several years ago Bethe [@bethe] re-derived the resonance effect on neutrinos propagating through a medium [@lw1; @ms] based on a different view: his approach consists on considering the addition of a term $2EV$ upon the neutrino mass square due to the $W$-interactions of massive neutrinos propagating in a medium; $E$ is the neutrino energy and $V$ is the potential energy due to the matter effect. Here we will consider a similar effect when new interactions with matter (electrons or quarks) do exist and also if the neutrinos have arbitrary masses and for the sake of simplicity we will treat only two generations.

In this situation, the neutrino symmetry eigenstates $\nu_\alpha$ and $\nu_\beta$ are related to the neutrino mass eigenstates $\nu_1$ and $\nu_2$ as follows $$\vert \nu_\alpha\rangle=
\vert\nu_1\rangle\cos\theta+\vert\nu_2\rangle\sin\theta,
\quad
\vert \nu_\beta\rangle=
-\vert\nu_1\rangle\sin\theta+\vert\nu_2\rangle\cos\theta
\label{nutau}$$ and $\theta<45^o$; the masses of the states $\vert\nu_{1,2}\rangle$ are $m_{1,2}$ respectively and as we said before, we are assuming that they are arbitrary parameters. The square of the mass matrix in the $\vert\nu_{\alpha,\beta}\rangle$ flavor basis is given by

$${\cal M}^2=\frac{1}{2}(m^2_1+m^2_2)\left(
\begin{array}{cc}
1& 0\\
0&1
\end{array}
\right)+\frac{1}{2}(m^2_2-m^2_1)
\left(
\begin{array}{cc}
-\cos2\theta & \sin2\theta\\
\sin2\theta&\cos2\theta
\end{array}
\right).
\label{mm}$$

Assuming effective interactions of the form $(G_{\alpha f}G_{f\beta}/M^2_X)(\bar{\nu}_\alpha\nu_\beta)(\bar{f}f),
\;\alpha,\beta=e,\mu,\tau$; and $M_X$ is the mass of the interchanged particle and $f$ denotes a charged lepton or quark (we have omitted Lorentz indices) implies a potential energy that is given by (here we will consider only interactions with the electrons i.e., $f=e$): $$V^{NI}_{\alpha\beta}=
\frac{x}{M^2_W}
\,G_{\alpha e}G_{e \beta}N_e,\quad \alpha,\beta=e,\mu,\tau;
\label{inteff}$$ with $x=M^2_W/M^2_X$; $N_e$ is the number of electrons per unit volume. Constraints on $x$ and $G_{\alpha\beta}$ coming from several decays are model dependent so they will not be considered here at all. However, see Ref. [@fcnit].

In fact, our considerations are model independent but neutrinos must be massive particles with renormalizable (arbitrary) masses. This is a necessary condition in order to have neutrino masses that do not depend on the parameters of the flavor changing interactions $G_{\alpha e}$. For instance, in the context of the $R$-parity broken MSSM [@rpv] it means that one has to add right-handed neutrinos and its respective sneutrinos $\tilde{\nu}_R$. In the so called 3-3-1 model [@331] neutrinos can have arbitrary Majorana masses if a neutral component of the sextet gains a non vanishing vacuum expectation value [@ma1], or we can also add three right-handed neutrinos (and Dirac neutrinos), or both possibilities.

The momentum of a neutrino is related to its energy $E$ by $$k^2+m^2=(E-V)^2\approx E^2-2VE,
\label{mo}$$ and considering the atmospheric neutrino issue i.e., the muon and tau neutrino we see that $V_{\alpha\beta}$ is equivalent to an addition to $m^2$ given by $$m^2_{\alpha\beta}=2V_{\alpha\beta}E=2\sqrt{2}\,
\left(\frac{x}{\sqrt2}\,G_{\alpha e}G_{e \beta }
 +G_FM^2_W(1-4s^2_W)\,\delta_{\alpha\beta}\right)
\frac{Y_e}{M^2_Wm_n}\,\rho E
\equiv A_{\alpha\beta}
\label{m2}$$ where $\alpha,\beta=\mu,\tau$; $m_n$ is the mass of the nucleon and $Y_e$ the number of electrons per nucleon in the matter, usually $Y_e=1/2$ and we have introduced the $Z^0$ contribution as well. Taking the usual values for the parameters [@pdg] we have:

$$A_{\alpha\beta}=1.0\times10^{-3}\,\left(\frac{x}{\sqrt2}\,
G_{\alpha e}G_{e\beta }+0.006\,\delta_{\alpha\beta}
\right)\,\rho E,
\label{as}$$

where $\rho$ is in grams per cubic centimeter, $E$ is in GeV, and $A_{\alpha\beta}$ is in $\mbox{eV}^2$.

Thus, in a dense medium the original vacuum mass square matrix is modified by the $A_{\alpha\beta}$ term and it reads: $$\tilde{{\cal M}}^2=\frac{1}{2}\left(m^2_1+m^2_2+A_{\mu\mu}+A_{\tau\tau}\right)
\left(
\begin{array}{cc}
1& 0\\
0&1
\end{array}
\right)+\frac{1}{2}
\left(
\begin{array}{cc}
A_{\mu\mu}-A_{\tau\tau}-\Delta\cos2\theta & 2A_{\mu\tau}+\Delta\sin2\theta\\
2A_{\mu\tau}+\Delta\sin2\theta& -A_{\mu\mu}+A_{\tau\tau}+\Delta\cos2\theta
\end{array}
\right).
\label{mmi}$$ where $\Delta=m^2_2-m^2_1$ and we will assume $m^2_2>m^2_1$. The eigenvalues are

$$\tilde{m}^2_{1,2}=
\frac{1}{2}\left(m^2_1+m^2_2+A_{\mu\mu}+A_{\tau\tau}\right)\pm
\frac{1}{2}\left[(-A_{\mu\mu}+A_{\tau\tau}+
\Delta\cos2\theta)^2+\vert 2A_{\mu\tau}+ \Delta\sin2\theta\vert^2\right]^{1/2},
\label{eigenvalues}$$

and we see that the resonance condition is given by: $$A_{\mu\mu}-A_{\tau\tau}=
\Delta\cos2\theta.
\label{rc}$$

In Fig. \[fig1\] we show the neutrino energy $E$ as a function of the Earth density at the resonance given by Eq. (\[rc\]) for a given set of parameters, in particular with vanishing vacuum mixing angle. In Fig. \[fig2\] we show the two eigenvalues $\tilde{m}^2_{1,2}$ as a function of $E$ and not as a function of $\rho$ and we see that a conversion is possible for the typical Earth density and neutrino energies also with $\cos2\theta=1$. The occurrence of this resonance with $\theta=0$ does not occur in the context of the standard model. Both figures only illustrate that typical values for the Earth density and atmospheric neutrino energies are compatible with the resonance picture discussed above.

In fact, we can also consider the survival probability in a medium which is given by $$P(\nu_\mu\to\nu_\mu)=1-\sin^22\tilde\theta\sin^2\left(1.27\,
\tilde\Delta\,\frac{L}{E}\right),
\label{nunu}$$ where $$\sin^22\tilde\theta=\frac{(2A_{\mu\tau}+\Delta\sin2\theta)^2}
{(\Delta\cos2\theta-A_{\mu\mu}+A_{\tau\tau})^2+
(2A_{\mu\tau}+\Delta\sin2\theta)^2},\quad \mbox{and}
\quad \tilde\Delta=\tilde{m}^2_2-\tilde{m}^2_1,
\label{def}$$ with $\tilde\Delta$ in $\mbox{eV}^2$, $L$ in km and $E\approx \tilde{E}=
\vert\vec{p}\vert$ in GeV; and we have already used $\tilde{E}_2-\tilde{E}_1\approx 
\tilde\Delta/2\vert\vec{p}\vert$.

Notice that we can explain the atmospheric neutrino data since at the resonance $\sin^22\tilde\theta_r=1$ and if $\tilde\Delta_r=2.2\times10^{-3}\,\mbox{eV}^2$ (the subscript $r$ denotes the value at the resonance), which corresponds to the best fit of atmospheric neutrinos [@sk1; @sk2] leaving the vacuum parameters $\theta$ and $\Delta$ arbitrary. We have

$$\tilde\Delta^2=(\tilde{m}^2_2-\tilde{m}^2_1)^2=(-A_{\mu\mu}+A_{\tau\tau}+
\Delta\cos2\theta)^2+\vert 2A_{\mu\tau}+ \Delta\sin2\theta\vert^2,
\label{yupi}$$

and we see that, even at the resonance, if $\theta=0$, we can still have that $\tilde\Delta_r=2\vert A_{\mu\tau,r}\vert=2.2\times10^{-3}
\mbox{eV}^2$, which is accomplished, for instance, for neutrinos with $E\approx2$ GeV by taken $G_{\mu e}\approx G_{\tau e}\approx1,\,\rho=
2\,\mbox{g}/\mbox{cm}^3$. This also occurs if $\Delta=0$ (neutrinos mass degenerated or massless neutrinos) which also implies $A_{\mu\mu}=
A_{\tau\tau}$ at the resonance. This case corresponds to the pure flavor changing induced neutrino oscillations as in Ref. [@rpatm] but this situation may have problems with the up-going upward muon [@sk2] as has been pointed out in Ref. [@ln].

Likewise, we can consider the solar neutrino problem. In this case we have to change labels, $\mu\to e$, $\tau\to \mu$, in the expressions above and also to add the contribution of the $W$ boson. Then

$$m^2_{\alpha\beta}=2V_{\alpha\beta}E=2\sqrt{2}\,
\left(\frac{x}{\sqrt2}\,G_{\alpha e}G_{e\beta}
 +G_FM^2_W\,[\delta_{\alpha e}\delta_{\beta e}+(1-4s^2_W)
\delta_{\alpha\beta}]\right)
\frac{Y_e}{M^2_Wm_n}\,\rho E
\equiv A_{\alpha\beta},
\label{m3}$$

for $\alpha,\beta=e,\mu$. Putting values we have

$$A_{\alpha\beta}=1.0\times10^{-3}\,\left(\frac{x}{\sqrt2}\,
G_{\alpha e}G_{e\beta}+ 0.075\,\delta_{\alpha e}\delta_{\beta e}+
0.006\,\delta_{\alpha\beta}\right)\,\rho E,
\label{as2}$$

with the same units as in Eq. (\[as\]).

The eigenvalues are

$$\tilde{m}^2_{1,2}=
\frac{1}{2}\left(m^2_1+m^2_2+A_{ee}+A_{\mu\mu}\right)\pm
\frac{1}{2}\left[(-A_{ee}+A_{\mu\mu}+
\Delta\cos2\theta)^2+\vert 2A_{e\mu}+ \Delta\sin2\theta\vert^2\right]^{1/2}.
\label{eigenvalues2}$$

$\Delta$ has the same definition as before, however the masses $m_{1,2}$ may be different from that of the muon and tau neutrino case. In this case we have a similar resonance condition than that in Eq. (\[rc\]). In Fig. \[fig3\] we show neutrino energies as a function of typical Sun density at the resonance for a given set of parameters and using $\cos2\theta=1$ again. The occurrence of that resonance at $\theta=0$ does not happens if we consider the electron neutrino and the muon neutrino as in the Bethe’s paper [@bethe]. The latter case is got here if $A_{ee}=A$, $A_{\mu\mu}=A_{e\mu}=0$ and $\theta\not=0$.

Since in the present context the atmospheric neutrino data is almost independent of the value of vacuum mass square difference, it means that we could be able to explain solar [@solar], atmospheric [@sk1; @sk2] and LSND [@lsnd] neutrino data with three active neutrinos and without introducing a sterile neutrino. This will be the case if, in a truly three generation case, the $\Delta_{12}$ which dominates the $\nu_\mu\to\nu_\tau$ oscillation also dominates the $\nu_e\to\nu_\tau$ transition; while $\Delta_{13}$ is the one used for fitting the LSND results.

We have worked in the two-neutrino scheme because most of the data are presented in this situation. However a general three-neutrino scheme must be worked elsewhere; also more realistic fittings by considering the matter density profile of the Earth and Sun eventually have to be done [@mmg].

This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Conselho Nacional de Ciência e Tecnologia (CNPq) and by Programa de Apoio a Núcleos de Excelência (PRONEX).

K. Lande (Homestake Collaboration), in [*Neutrinos ’98*]{}, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Japan, 4-9 June 1998, edited by Y. Suzuki and T. Tosuka, to be published in Nucl. Phys. B (Proc. Suppl.); Y. Fukuda [*et al*]{}. (Kamiokande Collaboration), Phys. Rev. Lett. [**77**]{} 1683 (1996) 1683; T. Kirsten (GALLEX Collaboration) in [*Neutrinos ’98*]{}; V. Gavrin (SAGE Collaboration) in [*Neutrinos ’98*]{}; Y. Suzuki (SuperKamiokande Collaboration) in [*Neutrinos ’98*]{}. Y. Fukuda [*et al.*]{} (SuperKamiokande Collaboration), Phys. Rev. Lett. [**81**]{}, 1562 (1998). V. Barger, Talk presented at the [*5th International Conference on Physics Potential & Development of $\mu^+\mu^-$ Colliders*]{}, San Francisco, December 1999; hep-ph/0003212. M. M. Guzzo [*et al.*]{}, hep-ph/9908308. M. M. Guzzo, A. Masiero and S. Petcov, Phys. Lett. [**260**]{}, 154 (1991); S. Bergmann, M. M. Guzzo, P.C. de Holanda, P.I. Krastev and H. Nunokawa, hep-ph/0004049. M. M. Guzzo and H. Nunokawa, Astropart. Phys. [**12**]{} (1999) 87; hep-ph/9810408. L. Wolfenstein, Phys. Rev. D [**17**]{}, 2369 (1978). M. Mikheyev and A. Smirnov, Sov. J. Nucl. Phys. [**42**]{}, 913 (1986). E. Roulet, Phys. Rev. D [**44**]{}, 935 (1991). H. A. Bethe, Phys. Rev. Lett. [**56**]{}, 1305 (1986). L. Hall and M. Suzuki, Nucl. Phys. [**B231**]{}, 419 (1984). F. Pisano and V. Pleitez, Phys. Rev. [**D46**]{}, 410 (1992); P. Frampton, Phys. Rev. Lett. [**69**]{}, 2889 (1992). J. C. Montero, C. A. de S. Pires and V. Pleitez, Phy. Rev. D [**60**]{}, 115003 (1999); hep-ph/9903251. C. Caso [*et al.*]{}, The European Physical Journal, [**C3**]{}, 1 (1998). Y. Fukuda [*et al.*]{} (SuperKamiokande Collaboration), Phys. Rev. Lett. [**82**]{}, 2644 (1999). M.C. Gonzalez-Garcia [*et al.*]{}, Phys. Rev. Lett. [**82**]{}, 3202 (1999). P. Lipari and M. Lusignoli, Phys. Rev. D [**58**]{}, 073005 (2998). See however N. Fornengo, M. C. Gonzalez-Garcia and J. F. W. Valle, hep-ph/9906539. C. Athanassopoulos et al., Phys. Rev. Lett. 77, 3082 (1996); [*ibid*]{} 81, 1774 (1998). M. M. Guzzo [*et al.*]{}, work in progress.

0.01cm

= -0.009cm

0.01cm

-0.009cm

0.01cm

-0.009cm

[^1]: e-mail address:<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>

[^2]: e-mail address: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>