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]


---
author:
- 'B. Pire[^1]'
- 'L. Szymanowski[^2]'
- 'J. Wagner[^3]'
title: 'Probing the transversity spin structure of a nucleon in neutrino-production of a charmed meson'
---

Introduction {#intro}
============

The now well established framework of collinear QCD factorization [@fact1; @fact2; @fact3] for exclusive reactions mediated by a highly virtual photon in the generalized Bjorken regime describes hadronic amplitudes using generalized parton distributions (GPDs) which give access to a 3-dimensional analysis [@3d] of the internal structure of hadrons. Neutrino production is another way to access (generalized) parton distributions [@weakGPD]. Although neutrino induced cross sections are orders of magnitudes smaller than those for electroproduction and neutrino beams are much more difficult to handle than charged lepton beams, they have been very important to scrutinize the flavor content of the nucleon and the advent of new generations of neutrino experiments opens new possibilities. In particular, the flavor changing character of the electroweak current allows charmed quark to be produced in processes involving light quark partonic distributions [@PS]. This in turn allows helicity flip hard amplitudes to occur at the $O(\frac{m_c}{Q})$ level where $Q$ is the typical large scale allowing QCD collinear factorization. Such a coefficient function has to be attached to a chiral-odd generalized parton distribution, the elusive transversity GPDs [@transGPDdef; @transGPDno; @transGPDacc]. The transverse character of these GPDs select the transverse polarization of the $W-$boson, which phenomenologically allows a separation of this interesting amplitude through the azimuthal distribution of the final state particles [@PS].

Kinematics {#sec-1}
==========

For definiteness, we consider the exclusive production of a pseudoscalar $D-$meson through the reaction (see Fig. 1): $$\begin{aligned}
\nu_l (k)\,N(p_1) &\to& l^- (k')\,D^+ (p_D)\,N(p_2) \,,\end{aligned}$$ where N is a proton or a neutron, in the kinematical domain where collinear factorization leads to a description of the scattering amplitude in terms of nucleon GPDs and the $D-$meson distribution amplitude, with the hard subprocesses: $$\begin{aligned}
W^+ (\varepsilon, q)\,d &\to& D^+ (p_D)\,d\,.\end{aligned}$$ Our kinematical notations are as follows ($m$ and $M_D$ are the nucleon and $D-$meson masses, $m_c$ will denote the charmed quark mass): $$\begin{aligned}
&&q=k-k'~~~~~; ~~~~~Q^2 = -q^2~~~~~; ~~~~~\Delta = p_2-p_1 ~~~~~; ~~~~~\Delta^2=t \,;\nonumber\\
&&p_1^\mu=(1+\xi)p^\mu +\frac{1}{2}  \frac{m^2-\Delta_T^2/4}{1+\xi} n^\mu -\frac{\Delta_T^\mu}{2}~~~~;~~~~ p_2^\mu=(1-\xi)p^\mu +\frac{1}{2}  \frac{m^2-\Delta_T^2/4}{1-\xi} n^\mu +\frac{\Delta_T^\mu}{2}\,; \\
&&q^\mu= -2\xi' p^\mu +\frac{Q^2}{4\xi'} n^\mu ~~~~;~~~p_D^\mu=  2(\xi-\xi') p^\mu +\frac{M_D^2-\Delta_T^2}{4(\xi-\xi')}  n^\mu -\Delta_T^\mu \,,\nonumber\end{aligned}$$ with $p^2 = n^2 = 0$ and $p.n=1$. As in the double deeply virtual Compton scattering case [@DDVCS], it is meaningful to introduce two distinct momentum fractions: $$\begin{aligned}
\xi = - \frac{(p_2-p_1).n}{2} ~~~~~,~~~~~\xi' = - \frac{q.n}{2} \,.\end{aligned}$$ Neglecting the nucleon mass and $\Delta_T$, the approximate values of $\xi$ and $\xi'$ are $$\begin{aligned}
\xi \approx \frac{Q^2+M_D^2}{4p_1.q-Q^2-M_D^2} ~~~~,~~~  \xi' \approx \frac{Q^2}{4p_1.q-Q^2-M_D^2} \,.\end{aligned}$$ To unify the description of the scaling amplitude, we define a modified Bjorken variable $
x_B^D \equiv \frac {Q^2+M_D^2}{2p_1.q} 
$ which allows to express $\xi$ and $\xi'$ in a compact form: $$\begin{aligned}
\xi \approx \frac{x_B^D }{2-x_B^D } ~~~~~,~~~~~\xi' \approx \frac{x_B }{2-x_B^D } \,.\end{aligned}$$ If the meson mass is the relevant large scale (for instance in the limiting case where $Q^2$ vanishes as in the timelike Compton scattering kinematics [@TCS]) : $$\begin{aligned}
Q^2\to0 ~~;~~\xi' \to 0 ~;~ \xi \approx \frac{\tau}{2-\tau} ~~;~~\tau = \frac{M_D^2}{s_{WN}-m^2}\,.\end{aligned}$$

The transverse amplitude
========================

![Feynman diagrams for the factorized amplitude for the $ \nu_\mu N \to \mu^-  D^+ N'$ or the $ \nu_\mu N \to \mu^-  D^0 N'$ process involving the quark GPDs; the thick line represents the heavy quark. In the Feynman gauge, diagram (a) involves convolution with both the transversity GPDs and the chiral even ones, whereas diagram (b) involves only chiral even GPDs.[]{data-label="fig-1"}](fig1neutrinoPoetic.pdf){width="10cm"}

In the Feynman gauge, the non-vanishing $m_c-$dependent part of the Dirac trace in the hard scattering part depicted in Fig. 1a reads: $$\begin{aligned}
Tr[\sigma^{pi}\gamma^\nu \hat p_D \gamma^5 \gamma^{\nu'} \frac{m_c}{D_1}\hat \varepsilon (1- \gamma^5)\frac{-g_{\nu \nu'}}{D_2} ]
= \frac{2 (Q^2+M_D^2)}{\xi}  \varepsilon_\mu[ \epsilon^{\mu p i n} + i g^{\mu i}_\perp]   \frac{m_c}{D_1} \frac{1}{D_2} \, ,
 \end{aligned}$$ where $\varepsilon$ is the polarization vector of the $W^\pm$boson (we denote $\hat p = p_\mu \gamma^\mu$ for any vector $p$). The fermionic trace vanishes for the diagram shown on Fig. 1b thanks to the identity $\gamma^\rho \sigma^{\alpha \beta}\gamma_\rho = 0$. The denominators of the propagators read: $$\begin{aligned}
&&D_1 = k_c^2 - m_c^2 +i\epsilon= \frac{Q^2 }{2 \xi'} (x+\xi-2\xi') - m_c^2 +i\epsilon =  \frac{Q^2+M_D^2 }{2 \xi} (x+\xi) - Q^2   - m_c^2 +i\epsilon \,,\\
 && D_2 = k_g^2 +i\epsilon= \bar z [\bar z m_D^2 + \frac{Q^2 + m_D^2}{2 \xi} (x-\xi) +i\epsilon]\, , \nonumber
 \end{aligned}$$ where $k_c$ ($k_g$) is the heavy quark (gluon) momentum. The transverse amplitude is then written as ($\tau = 1-i2$): $$\begin{aligned}
T_{T}  = \frac{iC \xi (m_c-2M_D)}{\sqrt 2 (Q^2+M_D^2)}  \bar{N}(p_{2}) \left[  {\mathcal{H}}_{T}^\phi i\sigma^{n\tau} +\tilde {\mathcal{H}}_{T}^\phi \frac{\Delta^{\tau}}{m_N^2}  
+ {\mathcal E}_{T}^\phi \frac{\hat n \Delta ^{\tau}+2\xi  \gamma ^{\tau}}{2m_N} - \tilde {\mathcal E}_{T}^\phi \frac{\gamma ^{\tau}}{m_N}\right] N(p_{1}), 
\label{tramp}\end{aligned}$$ with $C= \frac{2\pi}{3}C_F \alpha_s V_{dc}$, in terms of transverse form factors that we define as : $$\begin{aligned}
{\cal F }_T^\phi=f_{D}\int \frac{\phi(z)dz}{\bar z}\hspace{-.1cm}\int \frac{F^d_T(x,\xi,t) dx }{(x-\xi+\beta \xi+i\epsilon) (x-\xi +\alpha \bar z+i\epsilon)},
\label{TFF}
 \end{aligned}$$ where $F^d_T$ is any d-quark transversity GPD, $\alpha = \frac {2 \xi M_D^2}{Q^2+M_D^2}$, $\beta =  \frac {2 (M_D^2-m_c^2)}{Q^2+M_D^2}$ and we shall denote ${\bar {\mathcal{E}}_T^\phi}=\xi{\mathcal{E}}_T^\phi-{\tilde {\mathcal{E}}_T^\phi}$ . In the following, we shall put $\beta$ to $0$.

The prefactor in Eq.(\[tramp\]) shows the two sources of the transverse amplitude : $m_c$ signals the contribution from the helicity changing part of the heavy quark propagator, while $M_D$ signals the contribution from the twist 3 heavy meson distribution amplitude which we parametrize (omitting the Wilson lines) as : $$\begin{aligned}
  \langle 0 | \bar c(y)~\gamma^5~d(-y) |D^- (P_D) \rangle  =
  - i  \frac{f_D}{4} M_D \int^1_0 dz~ e^{i(2z-1) P_D.y}~ \phi^{(s)}_D(z)\,, 
         \end{aligned}$$ and for simplicity we identify $\phi^s$ with the leading twist 2 pseudoscalar charmed meson DA $\phi_D$ defined as: $$\langle 0 | \bar c(y)~ \gamma^\mu \gamma^5~d(-y) |D^-(P_D) \rangle  =
 - i  \frac{f_D}{4}  P_D^\mu \int^1_0 dz~e^{i(2z-1) P_D.y}   \phi_D(z)\,.$$

The azimuthal dependence of neutrinoproduction.
===============================================

The dependence of a leptoproduction cross section on azimuthal angles is a widely used way to analyze the scattering mechanism. This procedure is helpful as soon as one can define an angle $\varphi$ between a leptonic and a hadronic plane, as for deeply virtual Compton scattering [@DGPR] and related processes. In the neutrino case, it reads : $$\begin{aligned}
\label{cs}
\frac{d^4\sigma(\nu N\to l^- N'D)}{dx_B\, dQ^2\, dt\,  d\varphi}&=&
%-\varepsilon\cos(2\varphi)\sigma_{+-}
\\
 \tilde\Gamma
\Bigl\{ ~\frac{1+ \sqrt{1-\varepsilon^2}}{2} \sigma_{- -}&+&\varepsilon\sigma_{00}+  \sqrt{\varepsilon}(\sqrt{1+\varepsilon}+\sqrt{1-\varepsilon} )(\cos\varphi\
{\rm Re}\sigma_{- 0} + \sin\varphi\
 {\rm Im}\sigma_{- 0} )\ \Bigr\},\nonumber\end{aligned}$$ with $$\tilde \Gamma = \frac{G_F^2}{(2 \pi)^4} \frac{1}{16x_B} \frac{1}{\sqrt{ 1+4x_B^2m_N^2/Q^2}}\frac{1}{(s-m_N^2)^2} \frac{Q^2}{1-\epsilon}\,, \nonumber$$ and the “cross-sections” $\sigma_{lm}=\epsilon^{* \mu}_l W_{\mu \nu} \epsilon^\nu_m$ are product of amplitudes for the process $ W(\epsilon_l) N\to D N' $, averaged (summed) over the initial (final) hadron polarizations. In the anti-neutrino case, one gets a similar expression with $\sigma_{--} \to \sigma_{++}$ , $\sigma_{-0}\to \sigma_{+0}$, $1+ \sqrt{1-\varepsilon^2} \to 1- \sqrt{1-\varepsilon^2}$ and $\sqrt{1+\varepsilon}+\sqrt{1-\varepsilon} \to \sqrt{1+\varepsilon}-\sqrt{1-\varepsilon}$. We use the standard notations of deep exclusive leptoproduction, namely $y=p_1.q/p_1.k$ and $\epsilon \simeq 2(1-y)/[1+(1-y)^2]$. The azimuthal angle $\varphi$ is defined in the initial nucleon rest frame as: $$sin ~\varphi = \frac {\vec q \cdot[(\vec q \times \vec p_D) \times (\vec q \times \vec k)]}{|\vec q||\vec q\times \vec p_D||
\vec q\times \vec k|}\,,$$ while the final nucleon momentum lies in the $xz$ plane ($\Delta^y = 0$).

The quantity $\sigma_{- 0} $ is directly related to the observables $<cos~\varphi>$ and $<sin~\varphi>$ through $$\begin{aligned}
  <cos ~\varphi>&=&\frac{\int cos ~\varphi ~d\varphi ~d^4\sigma}{\int d\varphi ~d^4\sigma}= \frac{\sqrt \varepsilon (\sqrt{1+\varepsilon}+\sqrt{1-\varepsilon} )\, {\cal R}e \sigma_{- 0}}{2\epsilon \sigma_{0 0} + (1+ \sqrt{1-\varepsilon^2}) \sigma_{--}}  \,, \nonumber \\
   <sin~ \varphi>&=&  \frac{\int sin ~\varphi ~d\varphi ~d^4\sigma}{\int d\varphi ~d^4\sigma}= \frac{\sqrt \varepsilon (\sqrt{1+\varepsilon}+\sqrt{1-\varepsilon} )\, {\cal I}m \sigma_{- 0}}{2\epsilon \sigma_{0 0} + (1+ \sqrt{1-\varepsilon^2}) \sigma_{--}}      \,.
   \end{aligned}$$

![Contribution of $\sigma_{ - -}$ to the differential cross section $\frac{d\sigma}{dx_B dQ^2 d t}$ of neutrino-production of a $D^+$ meson on a proton (dashed line) or a neutron (solid line) at $s = 20$ GeV$^2$ , $y=0.7$ and $t  =t_0$, as a function of $Q^2$. []{data-label="figresult"}](neutrino_proton_neutron_2.pdf){width="10cm"}

Estimating the counting rates and the angular observables defined above is in progress [@prog]. As a first step, we calculate $\sigma_{--} $ which is bilinear in transversity quark GPDs. At zeroth order in $\Delta_T$, $\sigma_{--}$ reads: $$\begin{aligned}
\sigma_{--} =   \frac{4\xi^2 C^2 (m_c+2M_D)^2}{(Q^2+M_D^2)^2}\biggl\{(1-\xi^2)|{\mathcal{H}_T^\phi}|^2  + \frac{\xi^2}{1-\xi^2} | {\bar {\mathcal{E}}_T^\phi}|^2 -2\xi \mathcal{R}e [ \mathcal{H}_T^\phi {\bar {\mathcal{E}}_T^{\phi *}}]\biggr\} . \end{aligned}$$ Using the model of Ref [@heavyDA2] for the $D^+$ meson distribution amplitude and the parametrization of the dominant transversity GPD $H_T(x,\xi,t)$ from Ref [@ElB] (and neglecting for the time being other chiral-odd GPDs contributions), we compute the contribution to the differential cross section given in Eq.(\[cs\]) integrated over $\varphi$. The result is shown in Fig. \[figresult\] as a function of $Q^2$ for $s=20$ GeV$^2$, $y=0.7$ and $t=t_{min}$. Since the process selects the $d-$ quark contribution, the proton and neutron target cases allow to access $H_T^d$ and $H_T^u$ respectively. Although small, the cross-sections are of the same order of magnitude as those for the neutrino production of $\pi$ or $D_s$ mesons estimated in [@weakGPD]. This shows that these processes should be measurable in intense neutrino beam facilities.

Let us remind the reader that we allow $Q^2$ to be quite small since the hard scale governing our process is $M_D^2+Q^2$.

Conclusion.
===========

Collinear QCD factorization has allowed us to calculate neutrino production of $D-$mesons in terms of GPDs. Gluon and both chiral-odd and chiral-even quark GPDs contribute to the amplitude for different polarization states of the $W^\pm$ boson. The azimuthal dependence of the cross section allows to separate different contributions. Planned high energy neutrino facilities [@NOVA] which have their scientific program oriented toward the understanding of neutrino oscillations or elusive inert neutrinos may thus allow - without much additional equipment - some important progress in the realm of hadronic physics.

This work was partially supported by the COPIN-IN2P3 Agreement and by the french grant ANR PARTONS (Grant No. ANR-12-MONU-0008-01); L. Sz was partially supported by grant of National Science Center, Poland, No. 2015/17/B/ST2/01838.

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