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

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

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

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

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

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

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

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

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

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

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

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

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

---

### *Example Input Text:*

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

---

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[end sys2]

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


---
abstract: 'The domain wall fermion (DWF) method, with its almost perfectly preserved chiral symmetry on the lattice, makes the calculation of the nucleon axial charge particularly easy. By maintaining chiral symmetry and using the Ward-Takahashi (WT) identity, one has $Z_A = Z_V$ and the bare lattice calculation yields the physical value without explicit renormalization. The DBW2 improved gauge action provides further enhancement of the symmetry and hence a more accurate WT identity at coarse lattice spacing. Taking advantage of these methods, we confirmed a significant volume dependence of the nucleon axial charge on $(1.2 {\rm fm})^3$ and $(2.4 {\rm fm})^3$ lattice volumes.'
address:
- 'Institute of Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki 305-0801, Japan'
- 'RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA'
author:
- 'Shigemi Ohta for the RBC collaboration[^1]'
title: |
    [Poster presented at “Lattice 2002” international symposium, June 24-29, 2002, Cambridge, MA, USA]{}\
    \
    \

    Nucleon axial charge from quenched lattice QCD with domain wall fermions and DBW2 gauge action
---

Domain wall fermions (DWF) [@DWF] preserve chiral symmetry on the lattice by introducing a fictitious fifth dimension in which the symmetry violation is exponentially suppressed. The DBW2 (“doubly blocked Wilson 2”) improved gauge action [@QCDTARO] improves the approach to the continuum by adding rectangular (2$\times$1) Wilson loops to the action. By combining the two, the “residual mass,” which controls low energy chiral behavior, is driven to as small as $am_{\rm res} < O(10^{-4}) $, or $\ll
{\rm 1\,MeV}$. Successful numerical lattice calculations investigating chiral symmetry and the ground-state hadron mass spectrum [@DWFspectroscopy], Kaon matrix elements [@DWFKaon], and negative-parity excited nucleon ($N^*$) mass [@DWFNstar] have been reported. Here we apply the method to the axial charge, $g_{_A}$, the simplest of nucleon electroweak matrix elements.

From neutron $\beta$ decay, we know $g_{_V} = G_F \cos \theta_c$ and $g_{_A}/g_{_V} = 1.2670(35)$ [@PDGgA]. These quantities are defined as $q^2\rightarrow 0$ limits of the form factors $g_{_V}(q^2)$ and $g_{_A}(q^2)$ in the relevant vector and axial current matrix elements, $\displaystyle \langle n| V^-_\mu(x) | p \rangle$, $$i\bar{u}_n [\gamma_\mu g_{_V}(q^2)
             +q_\lambda \sigma_{\lambda\mu} g_{_M}(q^2) ] u_p e^{-iqx},$$ and $\displaystyle \langle n| A^-_\mu(x) | p \rangle$, $$i\bar{u}_n \gamma_5
             [\gamma_\mu g_{_A}(q^2)
             +q_\mu g_{_P}(q^2) ] u_p e^{-iqx}.$$ On the lattice, in general, we calculate the relevant matrix elements of these currents with a lattice cutoff, $a^{-1}$ $\sim$ 1-2 GeV, and extrapolate to the continuum, $ a \rightarrow 0$, introducing lattice renormalization: $g_{_V, _A}^{\rm ren} = Z_{_V,
_A}^{\rm lat} g_{_V, _A}^{\rm lat}$. Also, unwanted lattice artefacts may result in unphysical mixing of chirally distinct operators. DWF make the calculation particularly easy, because the chiral symmetry is almost exact; the relation $Z_{_A} = Z_{_V}$ is maintained, so the lattice ratio $(g_{_{A}}/g_{_{V}})^{\rm lat}$ directly yields the renormalized ratio.

It is interesting to note the pre-QCD model estimations for this quantity. The non-relativistic quark model in its simplest form gives $5/3$, and the MIT bag model gives 1.07. The lattice QCD calculations with Wilson or clover fermions underestimate $g_A$ by 10 to 25 % [@WilsongA]. These calculation also suffer from systematic errors arising from $Z_{_A} \neq Z_{_V}$ and other renormalization complications.

Our formulation follows the standard one, and was summarized in our Lattice 2001 proceedings [@Lat01]. In addition to the wall sources, sequential sources are added [@Kostas].

Numerical calculations with Wilson (single plaquette) gauge action were performed at $\beta=6.0$ on a $16^3\times 32 \times 16$ lattice and $M_5
= 1.8$ with nucleon source at $t=5$, sink at 21, and current insertions in between. Results from 400 gauge configurations are summarized as follows. The local vector current renormalization $Z_{_V} = 1/g_{_V}^{\rm lattice}$ is well-behaved: the value $0.764(2)$ at $m_f = 0.02$ agrees well with $Z_{_A} = 0.7555(3)$ of the corresponding local axial current renormalization [@DWFspectroscopy; @DWFNPR]. A linear fit gives $Z_{_V} =
0.760(7)$ at $m_f = 0$, and a quadratic fit, 0.761(5). The axial to vector charge ratio, $g_{_A}/g_{_V}$, averaged in $10\le t \le 16$ time slices, yields a value of 0.81(11) in the linearly extrapolated chiral limit $m_f=0$. These are essentially the same as last year [@Lat01], but have been confirmed with much higher accuracy obtained through higher statistics.

Possible reasons for this small value of $g_{_A}/g_{_V}$ are finite lattice volume [@JaffeCohen], excited states (small separation between $t_{\rm source}$ and $t_{\rm sink}$), and quenching (zero modes, absent pion cloud, and so on). Here we concentrate on the first possibility. To investigate size-dependence, we need good chiral behavior or sufficient proximity to the continuum, and big enough lattice spatial volume at the same time. Improved gauge actions help both [@DWFDBW2]. In particular the DBW2 action [@QCDTARO] gives very small residual chiral symmetry breaking, $am_{\rm res} < 10^{-3}$, at a lattice spacing of $a$ $\sim$ 0.15 fm. At this lattice cutoff the hadron mass spectrum is well-behaved in the chiral limit too: $am_\rho = 0.592(9)$, $m_\rho/m_N \sim 0.8$, and $m_\pi (m_f=0.02) \sim 0.3 a^{-1}$.

DBW2 calculations are performed with both wall and sequential sources on two lattices: $8^3 \times 24 \times 16$ ($\sim$ $(1.2 {\rm fm})^3$), 405 configurations (wall and sequential), and $16^3 \times 32 \times 16$ ($\sim$ $(2.4 {\rm fm})^3$), 100 configurations (wall) and 204 (sequential). Nucleon source-sink separation is set at about 1.5 fm, and quark mass values of $m_f$ = 0.02, 0.04, 0.06, 0.08 and 0.10 are used. So the pion mass is as low as $m_\pi \ge 390 {\rm MeV}$ and satisfies $m_\pi L\ge 4.8$ and 2.4, respectively. The residual mass at this cutoff is about 0.8 MeV or $a m_{\rm res} \sim 6\times 10^{-4}$.

Again the vector and axial current renormalization factors are well-behaved (see Figure \[fig:DBW2Z\].)

![Local current renormalization factors, vector ($Z_{_V}$, circle) and axial ($Z_{_A}$, square).[]{data-label="fig:DBW2Z"}](FIG_Zv_Za_crop.eps){width="\columnwidth"}

The vector renormalization, $Z_{_V}$, calculated as $1/g_{_V}$, shows slight quadratic dependence on $m_f$ as expected: $V_\mu^{\rm conserved} = Z_{_V} V_\mu^{\rm local} + {\cal O}
(m_f^2 a^2)$, yielding a value $Z_{_V} = 0.791(17)$ in the chiral limit. This agrees well with the axial renormalization, $Z_{_A} = 0.77759(45)$, obtained from $\langle A_{_\mu}^{\rm conserved} (t) [\bar{q}\gamma_5 q](0) \rangle =
Z_{_A} \langle A_\mu^{\rm local} (t) [\bar{q}\gamma_5 q](0) \rangle$ [@DWFNPR]. This assures that the lattice calculation of the axial-to-vector ratio directly yields its continuum value without any further renormalization: $(g_{_A}/g_{_V})^{\rm lat} = (g_{_A}/g_{_V})^{\rm ren}$.

In Figure \[fig:ga\_o\_gv\] the quark-mass dependence of the ratio obtained from sequential source calculations is summarized for the two lattice volumes:

![$(g_{_A}/g_{_V})^{\rm ren}$: $m_f$ and volume dependence. Sequential source results on a large (cross) and small (circle) lattice. Wall source results are consistent with larger errors.[]{data-label="fig:ga_o_gv"}](ga_o_gv_crop.eps){width="\columnwidth"}

Clear volume dependence is seen between $(2.4 {\rm fm})^3$ and $(1.2 {\rm fm})^3$ volumes. The large volume results with the sequential source show a very mild $m_f$ dependence and extrapolate to a value of $(g_{_A}/g_{_V})^{\rm ren} = 1.21(3)(+4)$. The first error is statistical and the second is systematic from a preliminary finite-size analysis. At the lightest quark mass value of $m_f$=0.02, the volume dependence is no longer statistically significant. Indeed at this $m_f$ even the larger volume may not be big enough. In Figure \[fig:ga\_o\_gv\_phys\] the same data are presented in physical units together with the small volume Wilson gauge action results.

![$(g_{_A}/g_{_V})^{\rm ren}$: $m_f$ and volume dependence in physical scale set by $\rho$ meson mass. Wilson (diamond) and DBW2 results are consistent in the smaller volumes.[]{data-label="fig:ga_o_gv_phys"}](ga_o_gv_giusti_crop.eps){width="\columnwidth"}

The smaller volume results with Wilson ($16^3\times 32$ at $a$ $\sim$ 0.1 fm) and DBW2 ($8^3\times 24$ at $a$ $\sim$ 0.15 fm) actions are mutually consistent. In summary we confirmed a lattice-volume dependence of the nucleon axial charge, $g_{_A}/g_{_V}$. Similar dependence was seen in the bare tensor charge. Further investigation is under way.

[99]{} D.B. Kaplan, Phys. Lett. B288, 342 (1992); Y. Shamir, Nucl. Phys. B406, 90 (1993); V. Furman and Y. Shamir, Nucl. Phys. B439, 54 (1995). P. de Forcrand [*et al.*]{} (QCD-TARO collaboration), Nucl. Phys. B577, 263 (2000). T. Blum, [*et al.*]{} (RBC collaboration), to appear in Phys. Rev. D, hep-lat/0007038. T. Blum, [*et al.*]{} (RBC collaboration), RBRC Scientific Articles 4; hep-lat/0110075, submitted for publication in Phys. Rev. D. S. Sasaki, T. Blum and S. Ohta (for RBC collaboration), Phys. Rev. D65, 074503 (2002). The Particle Data Group. M. Fukugita, Y. Kuramashi, M. Okawa and A. Ukawa, Phys. Rev. Lett. 75, 2092 (1995); K.F. Liu, S.J. Dong, T.Draper and J.M. Wu, Phys. Rev. D49, 4755 (1994); M. Göckeler [*et al.*]{}, Phys.Rev. D53, 2317 (1996); D. Dolgov [*et al.*]{}, hep-lat/0201021; S. Capitani [*et al.*]{}, Nucl. Phys. B (Proc. Suppl.) 79, 548 (1999); R. Horsley [*et al.*]{}, Nucl. Phys. B (Proc. Suppl.) 94, 307 (2001); S. Güsken [*et al.*]{}, Phys. Rev. D59, 114502 (1999). S. Sasaki, Nucl. Phys. Proc. Suppl. 106, 30 (2002); and references cited there in. K. Orginos for the RBC collaboration, in these proceedings. T. Blum [*et al.*]{} (RBC Collaboration), Phys. Rev. D66, 014504 (2002). R.L. Jaffe, Phys. Lett. B529, 105 (2002); T.D. Cohen, Phys. Lett. B529, 50 (2002). RBC collaboration, in preparation.

[^1]: Current members are Y. Aoki, T. Blum, N. Christ, M. Creutz, C. Dawson, T. Izubuchi, L. Levkova, X. Liao, G. Liu, R. Mawhinney, Y. Nemoto, J. Noaki, S. Ohta, K. Orginos, S. Prelovsek, S. Sasaki and A. Soni. We thank RIKEN, BNL and the U.S. DOE for providing the facilities essential for the completion of this work.