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 production rate for $\eta'$ in $pp \rightarrow pp \eta'$ at rest is calculated in a covariant one boson exchange model, previously applied to study $\pi^0$ and $\eta$ production in NN collisions. The transition amplitudes for the elementary $BN \rightarrow \eta' N$ processes with $B$ being the meson exchanged ( $B = \pi , \sigma , \eta , \rho , \omega$ and $\delta$) are taken to be the sum of s and u channels with a nucleon in the intermediate states, and a $\delta$ meson pole in a t-channel. The couplings of the $\eta'$ to hadrons are a factor 0.437 weaker than the respective $\eta$-hadron couplings, as suggested by a quark model and a singlet-octet mixing angle $\theta = -23^o$. The model reproduces near threshold cross sections for the quasielastic processes $\pi^- p \rightarrow n \eta (\eta') $ and $pp \rightarrow pp \eta (\eta')$ reactions.\
     \
    PACS numbers : 25.40Ve, 13.75Cs, 14.40A9\
    Keywords : $\eta'$ meson production, Covariant OBE model.
author:
- 'E.Gedalin[^1], A.Moalem[^2] and L.Razdolskaja[^3]'
date: |
    Department of Physics\
    Ben-Gurion University of the Negev\
    Beer-Sheva, 84105, Israel\
title: 'On The $pp \rightarrow pp \eta (\eta'')$ Reactions Near Threshold'
---

[$\overline{p}$]{}[$\pi$ ]{} ø[$\omega$ ]{} 3[fm$^3$]{} 3[fm$^{-3}$]{} 11[$S11\ $]{}

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

The production of $\eta'$ mesons in proton-proton collisions near threshold has been reported recently by the SPESIII group at Saturne[@hibou98] and by COSY-J$\ddot u$lich[@moskal98]. The $\eta'$ is observed as a missing mass peak by detecting both final protons in a magnetic spectrometer. The total cross section of the $pp \rightarrow pp \eta'$ reaction is found to be a factor $\approx (m_{\pi}/m_{\eta '})^2$ smaller than for the corresponding $pp \rightarrow pp \pi^0$, indicating a similar production mechanisms for $\eta '$ and $\pi^0$. The observation that the $\eta$ production rate is nearly as large as $\pi^0$ production rate is attributed to a dominant contribution from resonant production via virtual excitation of the N$^*$ (1535 MeV) S11 nucleon isobar.

Various phenomenological one-boson-exchange (OBE) models for these processes have achieved impressive descriptions of extensive data from Saclay[@plouin90; @bergdolt93; @chiavassa94], Indiana[@meyer92], and Uppsala [@schuberth95; @bondar95; @calen96; @haggstrom97; @calen97]. Particularly, a relativistic covariant OBE model, reproduces consistently the cross section data (both scale and energy dependence) for the $pp \rightarrow pp \pi^0$ and $pp \rightarrow pp \eta$ reactions[@gedalin98; @gedalin981]. The main objective of the present note is to consider $\eta'$ production by applying a slightly generalized model. As in Ref.[@gedalin98; @gedalin981] we assume a reaction mechanism as depicted in Fig. 1, where a virtual boson B (B= $\pi , \eta , \sigma , \rho ,
\omega , \delta )$ created on one of the incoming protons is converted into a pseudoscalar meson P ($\pi ,\eta , \eta'$) on the second via a $BN \rightarrow PN$ conversion process. The half off mass-shell amplitudes for these conversion processes, hereafter denoted by $T_{BN \rightarrow PN}$, are taken to be the sum of three pole terms corresponding to s, u, and t-channels (see Fig. 2). The latter accounts for meson production occurring on internal lines such as $\sigma$ and $\delta$ meson lines, and requires knowledge about three meson legs vertices like $\sigma \pi \pi$, $\delta \eta \pi$ and $\delta \eta '\pi$ vertices. Such mechanisms are considered in Ref. [@gedalin981], where an effective $\sigma$-meson in a t channel, strongly enhanced due to offshellness, found to play a prominent role for $\pi^0$ production in the $pp \rightarrow pp \pi^0$ reaction at threshold. For $\eta '$ production most relevant is a $\delta \pi $ exchange. Generally, both nucleon and nucleon isobar excitations intermediate states may contribute to diagrams 2a and 2b. Such is the case for $\eta$ production, where production via exciation, propagation and subsequent decay of the N$^*$ (1535 MeV) S11 isobar into a $\eta$N pair play a significant role in the process. We shall demonstrate below that cross section data for $\eta '$ production can be explained without a resonant production term.

Much of the model success to explain meson production data in NN collisions depends on how well the half off mass shell amplitudes for the elementary conversion processes $BN \rightarrow PN$ are calculated. Unfortunately, there are no direct measurements which link directly to the off mass shell behavior of these amplitudes. In comparison with the $pp \rightarrow pp \eta$ reaction, the momentum transfer in $pp \rightarrow pp \eta'$ is nearly twice as large and it would be of interest to verify that the off shell behavior presumed reproduces $\eta'$ production rate as well. There are some near threshold cross section data for the $\pi N \rightarrow N \eta (\eta')$ reactions[@binnie73]. Here as for production in NN collisions the cross sections depart by more than an order of magnitude[@binnie73]. It is to be shown that taken on mass shell, the amplitudes $T_{\pi N \rightarrow  N\eta (\eta')}$ reproduce the cross sections for the quasielastic processes $\pi^0 p \rightarrow p \eta$ and $\pi^0 p \rightarrow p \eta'$ at their respective thresholds.

Theoretical perspective
=======================

To account for the production of $\eta'$ meson, we slightly generalize the Lagrangian of Refs. [@gedalin98; @gedalin981], by including a Lagrangian density, $$L_{\eta'NN} = \frac {f_{\eta' NN}}{m_{\eta'}}\bar{N}\gamma^5 
       \gamma^{\mu}\partial_{\mu}\eta'  N ~,
\label{lagran}$$ where N and $\eta'$ represent the fields of a nucleon and an $\eta'$ meson.

Further more, in order to allow $\eta'$ formation on an internal $\delta$-meson line we write the $\delta \eta' \pi$ vertex in a rather general form[@gedalin981], $$V_{\delta \eta' \pi} (k^2,q^2,(k-q)^2)= m_{\delta} \left( g_{0\eta'} 
    + g_{1\eta'} \frac {k^2}{m_{\delta}^2}
    + g_{2\eta'} \frac {q^2}{m_{\delta}^2}
    + g_{3\eta'} \frac {(k-q)^2}{m_{\delta}^2}\right)~.
\label{vertex}$$ This expression is similar to the $\delta \eta \pi$ vertex used in Ref. [@gedalin981] for $\pi^0$ (or $\eta$) production, where the vertex constants $g_{i\eta}; i=0 - 3$ were deduced from the partial decay width for $\delta \rightarrow \pi \eta$, and by applying the three Adler’s consistency conditions to the $T_{\pi^0 p \rightarrow \eta p}$ amplitude[@gedalin981].

There is no direct experimental information about the strength of the $\eta'$ couplings. However, being the heaviest member of the ground state pseudoscalar meson nonet, the $\eta'$ couplings can be related to those of the $\eta$. In the quark model the $\eta$NN and $\eta'$NN couplings are[@pdg], $$\begin{aligned}
 & & g_{\eta NN} = g_8 \cos {\theta} - g_1 \sin {\theta}~, \nonumber \\
 & & g_{\eta' NN} = g_8 \sin {\theta} + g_1 \cos {\theta}~,\end{aligned}$$ with $\theta, g_1 , g_8$ being the mixing angle, singlet and octet couplings, respectively. By making the simplifying assumption $g_8 = g_1$, and taking a linear mixing angle[@pdg], $\theta = -23^o$, one obtains, $$\frac {g_{\eta' NN}}{g_{\eta NN}} = 0.437~.
\label{ratio}$$ Similarly, the $\delta \eta \pi$ and $\delta \eta' \pi$ vertex constants scale also according to Eq. \[ratio\]. With the relevant $\eta$ couplings taken from Ref.[@gedalin981] and disregarding uncertainties in $\theta$ one obtains $$\begin{aligned}
& & g_{\eta' NN} = 2.68,~~~ g_{0\eta'} = 0.02 \pm 0.01,~~~ 
g_{1\eta'} = -2.6 \pm 0.20, \nonumber \\
& & g_{2\eta'} = -1.51 \pm 0.12,~~~ g_{3\eta'} = 1.44 \pm 0.11~.\end{aligned}$$

The $\eta$ meson couples rather strongly to the N$^*$ (1535 MeV) S11 and to a lesser extent to the N$^*$ (1710 MeV) P11 resonances. Particularly, the resonance mass almost coincides with the mass of a $\eta$ N pair so that contributions from graphs 2a and 2b with nucleon isobar excitations become prominent near the $\eta$ production threshold. There are no evidence for strong couplings of the $\eta'$ to baryon resonances. The decays of the N$^*$ (1535 MeV) S11 and N$^*$ (1710 MeV) P11 resonances into a free $\eta '$N pair are not accessible either. It is then reasonable to assume that resonant $\eta '$ production terms are small and can be neglected. With this in mind and following Refs.[@gedalin98; @gedalin981] the amplitudes for the $\pi^0 p \rightarrow \eta p$ and $\pi^0 p \rightarrow \eta' p$ are, $$\begin{aligned}
 & & T_{ \pi^0 p \rightarrow \eta p } = \nonumber\\
 & & -i g_{\eta NN^*} g_{\pi NN^*} f_{\eta} (k) f_{\pi} (q) 
\bar {u} (p') \left[ \frac {1}{M_R - \sqrt {s} + i \Gamma /2} + 
\frac {1}{M_R - \sqrt {u} + i \Gamma /2}\right] u(p)\nonumber \\
 & & i\frac {2M f_{\pi NN}}{m_{\pi}} \frac {2M f_{\eta NN}}{m_{\eta}}
     f_{\eta}(k) f_{\pi}(q)
{\bar u} (p') \left[k\! \! \! / \left(\frac {1}{s-M^2} - 
\frac {1}{u-M^2}\right) - \frac {1}{M}\right] u(p) + \nonumber \\
 & & i g_{\delta NN} \frac {f_{\delta}(q - k)} {(q - k)^2 - m_{\delta}^2}
V_{\delta \eta \pi}(k,q)
{\bar u}(p') u(p)~,
\label{vr:dep1}\end{aligned}$$ $$\begin{aligned}
 & & T_{\pi^0 p \rightarrow \eta' p} = \nonumber\\
 & & i\frac {2M f_{\pi NN}}{m_{\pi}} \frac {2M f_{\eta' NN}}{m_{\eta'}}
     f_{\eta'}(q) f_{\pi}(k)
{\bar u} (p') \left[k\! \! \! / \left(\frac {1}{s-M^2} - 
\frac {1}{u-M^2}\right) - \frac {1}{M}\right] u(p) + \nonumber \\
 & & i g_{\delta NN} \frac {f_{\delta}(q - k)} {(q - k)^2 - m_{\delta}^2}
V_{\delta \eta' \pi}(k,q)
{\bar u}(p') u(p)~,
\label{vr:dep2}\end{aligned}$$ where $M_R$, $\Gamma$ represent the mass and width of the N$^*$ (1535 MeV) S11 resonance; $M$ and $m_B$ the mass of a nucleon and a boson $B$; $f_B$ a source form factor parametrized in the usual form[@machleidt89], $$f_b = \frac {\Lambda_B^2 - m_b^2}{\Lambda_B^2 - q^2}~.$$ It is easy to trace in the expressions above the contribution from s, u and t channels. The first two terms in Eq. \[vr:dep1\] describe resonant production via nucleon isobar excitations, while the last term in both of these expressions stands for contributions from a $\delta$ meson t pole. The other meson conversion amplitudes and numerical details of the calculations are given in Refs.[@gedalin98; @gedalin981] and shall not be repeated here.

We may now use Eqns. \[vr:dep1\] and \[vr:dep2\] to evaluate the cross sections for the quasielastic processes $\pi^0 p \rightarrow \eta p$ and $\pi^0 p \rightarrow \eta' p$. At threshold these expressions predict for the amplitude squared : $|f(\pi^0 p \rightarrow \eta' p)|^2 = (15 \pm 7) \mu b/sr$ and $|f(\pi^0 p \rightarrow \eta p)|^2 = (346 \pm 40 )\mu b/sr$ which are consistent with the experimental values[@binnie73] $|f(\pi^0 p \rightarrow \eta' p)|^2 = 10\pm 1 \mu b/sr$ and $|f(\pi^0 p \rightarrow \eta p)|^2 = 365 \pm 30 \mu b/sr$, respectively. Predictions for $|f|^2$ are drawn in Figs. 3-4 versus the energy available in the center of mass (CM) system. Due to mutual cancellation, the contribution from both of the s and u nucleon pole terms (drawn separately as dot-dashed curve) is negligibly small for both processes. The t pole terms are also of the same size though playing a different role in the two cases. For the $\pi^0 p \rightarrow \eta p$ reaction the resonant production exceeds by far any of the other contributions. The t pole term is relatively weak, becoming noticeable only through interference with the strong resonant production term. In case of the $\pi^0 p \rightarrow \eta' p$ however, there is no strong resonance term and the t pole contribution determines the cross section almost solely.

We now turn to consider the cross sections for the $pp \rightarrow pp \eta$ and $p p \rightarrow pp \eta'$ reactions. Let us call $$\Pi_j = \frac {{\bf p}_j} {{E_j + M}}~,\\$$ where ${\bf p}_j$ and $E_j$ are three-momentum and total energy of the j-th nucleon. For the incoming particles in the CM system, $\Pi_1 = - \Pi_2 = \Pi$. The total energy square and the energy available in the CM system are respectively, $s = (p_1 + p_2)^2$ and $Q = \sqrt{s}- 2M - m_{\eta'} $. There is only one isovector amplitude which determines the reaction cross section at rest, corresponding to a ${}^{33}P_0 \rightarrow {}^{31}S_0$ transition in the two nucleon system. We write this amplitude in the form $$M_{11}(pp \rightarrow pp \eta') = M_{\pi} + M_{\eta} 
+ M_{\sigma} + M_{\delta} + M_{\rho} + M_{\omega}~,$$ where $M_B$ represents the contribution from the exchange of a boson B. Following Ref.[@gedalin98], $$\begin{aligned}
 & & M_{\pi}= i G_{\pi NN}  
g_{\pi NN}g_{\eta' NN}\Sigma_P \nonumber \\
 & & ~~-i f_{\pi NN} 
\left( \frac {2M}{m_{\pi}}\right)
\left( \frac {1}{m_{\pi}^2 - q^2}\right)f_{\pi }(q) g_{\delta NN}
\left( \frac {f_{\delta}(q)}{m_{\delta}-q^2}\right)
V_{\delta \eta' \pi} (k^2 = m_{\eta'}^2;q^2)~, \label{minpi} \\
 & & M_{\sigma} = i G_{\sigma NN} g_{\sigma NN}g_{\eta' NN}\Sigma_S~, 
\label{minsigma} \\
 & & M_{\eta} = i G_{\eta NN} 
g_{\eta NN}g_{\eta' NN}\Sigma_P~, \label{mineta} \\
 & & M_\rho = G_{\rho NN} 
g_{\rho NN}g_{\eta' NN}(\Sigma_{\rho}^{(1)} + 2 \Sigma_{\rho}^{(2)})~,~~
\label{minrhou} \\
 & & M_\omega = G_{\omega NN} 
g_{\omega NN}g_{\eta' NN}(\Sigma_{\omega}^{(1)} + 2 \Sigma_{\omega}^{(2)})~,
~~~~~\label{minomega} \\
 & &  M_{\delta} = i G_{\delta NN}  
g_{\delta NN}g_{\eta' NN}\Sigma_S~, \label{mindelta} \end{aligned}$$ where $$\begin{aligned}
 & & G_{BNN}= g_{B NN}\frac{E + M}{M}\frac{1}{M(m_{\eta'}+Q)
+ m_B^2}f_B^2 (-M[m_{\eta'}+Q]) \Pi~,\\  
 & & \Sigma_S =\frac{1}{M}\left(1 - \frac{5m_{\eta'}}
{2M+m_{\eta'}}\right)~,\\
 & & \Sigma_P = \left(\frac{m_{\eta'}}{2M}\right)^2
\left[1 + \frac{2(E +M)}{m_{\eta'}}{ 
\Pi \cdot \Pi}\right]\frac{1}{2M+m_\eta,}~.\\ 
 & & \Sigma_{\rho}^{(1)} = i \frac{m_{\eta'}}{4M^2}\nonumber\\
 & & ~~~~~~\left\{(1+\kappa \Pi \cdot \Pi)  \left[2-(1-
\frac{\kappa}{2})\frac{m_{\eta'}}{2M+m_{\eta'}} \right] +
\frac{\kappa m_{\eta'}^2}{4M(2M+m_{\eta'})} 
           \right\}~, \\
 & & \Sigma_{\rho}^{(2)} = i\kappa (1+\kappa)\frac{m_{\eta'}}{2M}
\nonumber \\
 & &  ~~~~~~\left\{-\frac{E+M}{M} {\Pi \cdot \Pi}
\left(1-\frac{m_{\eta'}}{2M+m_{\eta'}}\right) +
\frac{m_{\eta'}^2}{4M(2M+m_{\eta'})}\right\}~,\\
 & & \Sigma_{\omega}^{(1)}  
= i\frac{m_{\eta'}}{4M^2}\left(2-\frac{m_{\eta'}}{2M+m_{\eta'}}\right)~,\\
 & & \Sigma_{\omega}^{(2)} = 0~.
 \end{aligned}$$ All amplitudes are the sum of s and u nucleon pole terms, except $M_{\pi}$, Eqn. \[minpi\] which includes also a $\delta$ meson t pole term.

The various exchange contributions are shown in Fig. 5 versus the energy available in the CM system. The solid line is the transition amplitude obtained with the relative phases of all different exchange contributions set to be +1. Clearly, strong cancellations amongst these lower the transition amplitude to below $M_{\pi}$. Most important are the $\sigma$ and $\rho$ exchanges with ratios $M_{\sigma} : M_{\rho} : M_{\omega} : M_{\pi} : M_{\delta} : M_{\eta}
\\approx 14 : 10 : 8 : 7 : 2.5 : 1 $. The ratios quoted above differ considerably from the ones reported in Ref. [@gedalin98] for the $\eta$ production, where the reaction proceeds mainly via nucleon isobar excitations and the relative importance of various exchanges is determined by s+u nucleon isobar pole terms. Also the kinematic $\Sigma_B$ factors in Eqns. \[minsigma\]-\[minomega\] vary strongly with the mass of the meson produced, giving rise to quite different ratios for the s+u nucleon pole terms. At threshold the vertices $V_{\delta \eta \pi}, (k^2 = m_{\eta}^2)$ and $V_{\delta \eta' \pi}, (k^2 = m_{\eta'}^2)$ are small and practically do not affect the calculated cross sections.

Our predictions for the total cross sections are shown in Fig. 6 , along with the data from Refs.[@hibou98; @moskal98; @bergdolt93; @chiavassa94; @calen96; @schuberth95; @calen96; @haggstrom97]. The curves shown are corrected for final state interactions according to the procedure described in Ref.[@gedalin98]. To be consistent with the OBE picture of the NN interactions [@machleidt89] we have disregarded $\eta'$ exchange contributions. Such contributions scale like $(g_{\eta 'NN}/g_{\eta NN})^2$ and practically do not influence the cross sections presented here for the $pp \rightarrow pp \eta (\eta')$ reactions.

Summary
=======

In summary we have applied a covariant OBE model to calculate cross sections for the $\pi^0p  \rightarrow p \eta ' (\eta)$ and $pp \rightarrow pp \eta '(\eta)$ reactions. In marked difference with the $\eta$, we could reproduce near threshold cross sections for $\pi^0 p \rightarrow \eta'p$ and $pp \rightarrow pp \eta '$ without a resonant production via an intermediate baryon resonance. As in previous studies[@gedalin98; @gedalin981] the model is based on the OBE picture of the NN force[@machleidt89] and accounts for relativistic effects, energy dependence and nonlocality of the hadronic interactions. The calculations reported here and in Refs. [@gedalin98; @gedalin981] for $pp \rightarrow pp \pi^0 (\eta)$ provide a consistent description for pseudoscalar meson production in NN collisions. All calculations are carried out with the same formalism and meson-nucleon couplings as obtained by Machleidt[@machleidt89] from fitting NN scattering data. The model success to explain data for these processes depends on how well the half off mass shell amplitude for the conversion processes, $BN \rightarrow PN$ are calculated. As there are no data available which link directly to the off mass shell behavior of these amplitudes, this feature of the model needs still further verifications. Yet it is encouraging that such a simple model reproduce the production rate for as light as a pion and as heavy as the $\eta'$ meson.  \
 \
[**Acknowledgments**]{} This work was supported in part by the Israel ministry Of Immigration and Absorption. We would like to thank Dr. Z. Melamed for assistance in computation.

[999]{} F. Hibou et al., Phys. Lett. ${\bf B438}$ (1998) 41. P. Moskal et al., Phys. Lett. ${\bf B}$, in press. F. Plouin et al., Phys. Lett. ${\bf B276}$, (1990) 526; and references therein. A. M. Bergdolt et al., Phys. Rev. ${\bf D48}$, (1993) R2969. E. Chiavassa et al., Phys. Lett. ${\bf B322}$, (1994) 270; Phys. Lett. ${\bf B337}$, (1994) 192. H. O. Meyer et al., Nucl. Phys. ${\bf A539}$, (1992) 683. U. Schuberth, “The Reaction $pp \rightarrow pp\eta$ Close to Threshold”, Ph.D thesis, Uppsala University, 1995. A. Bondar et al., Phys. Lett. ${\bf B356}$, (1995) 8. H. Cal$\acute{e}$n et al., Phys. Lett. ${\bf B366}$, (1996) 39. S. H$\ddot{a}$ggstr$\ddot{o}$m, “ Production of $\eta$-mesons in Proton-Neutron Collisions”, Ph.D thesis, Uppsala University, 1997. H. Cal$\acute {e}$n et al., Phys. Rev. Lett. ${\bf 79}$, (1997) 2642. E. Gedalin, A. Moalem and L. Razdolskaja, Nucl. Phys. ${\bf A}$, (1998) in press. E. Gedalin, A. Moalem and L. Razdolskaja, nucl-th/9611005, and submitted for publication in Nucl. Phys. ${\bf A}$. D. M. Binnie et al., Phys. Rev. ${\bf D8}$, (1973) 2789. Particle Data Group, J. J. Hernandez et al. Phys. Rev. ${\bf D50}$ (1994) 1173. R. Machleidt, Adv. in Nucl. Phys. ${\bf 19}$ (1989) 189.

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[^1]: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>

[^2]: moalem @bgumail.bgu.ac.il

[^3]: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>