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: 'Dark and baryonic matter contribute comparable energy density to the present Universe. The dark matter may also be responsible for the cosmic positron/electron excesses. We connect these phenomena with Dirac seesaw for neutrino masses. In our model (i) the dark matter relic density is a dark matter asymmetry generated simultaneously with the baryon asymmetry so that we can naturally understand the coincidence between the dark and baryonic matter; (ii) the dark matter mostly decays into the leptons so that its decay can interpret the anomalous cosmic rays with positron/electron excesses.'
author:
- 'Pei-Hong Gu$^{1}_{}$'
- 'Utpal Sarkar$^{2,3}_{}$'
- 'Xinmin Zhang$^{4}_{}$'
title: 'Visible and Dark Matter Genesis and Cosmic Positron/Electron Excesses '
---

Precise cosmological observations indicate that dark and baryonic matter have different properties but contribute comparable energy density to the present Universe. This intriguing coincidence inspires us to propose a common origin for the creation and evolution of the dark and baryonic matter. Since the baryonic matter currently exists because of a matter-antimatter asymmetry, the dark matter relic density may also be an asymmetry between the dark matter and dark antimatter [@kuzmin1997; @kl2004; @cht2005; @klz2009]. The dark matter asymmetry and the baryon asymmetry may originate from decays of the same heavy particles, so that their coincidence is not surprising at all. On the other hand, recently cosmic-ray data [@chang2008; @torii2008; @adriani2008; @aharonian2008; @abdo2009] suggest [@mpsv2009] that (i) the dark matter should dominantly annihilate or decay into leptons with a large cross section or a long life time; (ii) the dark matter annihilation or decay should be consistent with the constraints from the observations on the gamma and neutrino fluxes. In this paper we explain these phenomena in a unified scenario where the neutrino masses originate through the Dirac seesaw mechanism [@gh2006].

We extend the standard model (SM) with a global $U(1)^{}_{lepton}
\times U(1)^{}_{dark}$ symmetry and include additional particles: singlet right-handed neutrino $\nu_{R}^{}(\mathbf{1},\mathbf{1},0)(1,1)$, heavy doublet scalar $\eta(\mathbf{1},\mathbf{2},-1/2)(0,-1)$, charged singlet scalar $\xi(\mathbf{1},\mathbf{1},1)(-2,0)$, and neutral singlet scalars $\sigma(\mathbf{1},\mathbf{1},0)(0,-1)$ and $\chi(\mathbf{1},\mathbf{1},0)(-2,-1)$, where the transformations are given under the SM gauge group $SU(3)_{c}^{}\times SU(2)_L^{}
\times U(1)_{Y}^{}$ and the global symmetry $U(1)_{lepton}^{}\times
U(1)_{dark}^{}$. For simplicity, we only present the relevant part of the Lagrangian for purpose of demonstration, $$\begin{aligned}
\mathcal{L} &\supset&
-y\overline{\psi_{L}^{}}\eta\nu_{R}^{}-f\xi\overline{\psi_{L}^{c}}
i\tau_2^{}\psi_{L}^{}-\mu\sigma\eta^\dagger_{}\phi-\kappa\chi^\dagger_{}\xi\eta^{T}_{}i\tau_2^{}\phi\nonumber\\
[2mm]
&&+\textrm{H.c}-M_\eta^2\left(\eta^\dagger_{}\eta\right)-m_\xi^2\left(\xi^\dagger_{}\xi\right)-m_\chi^2\left(\chi^\dagger\chi\right)\nonumber\\
[2mm]
&&-\left(\chi^\dagger_{}\chi\right)\left[\alpha\left(\xi^\dagger_{}\xi\right)
+\beta\left(\phi^\dagger_{}\phi\right)+\gamma\left(\sigma^\dagger_{}\sigma\right)\right]\nonumber\\
[2mm]
&&-\left(\sigma^\dagger_{}\sigma\right)\left[\zeta\left(\xi^\dagger_{}\xi\right)
+\epsilon\left(\phi^\dagger_{}\phi\right)+\vartheta\left(\sigma^\dagger_{}\sigma\right)\right]\,.\end{aligned}$$ Here $\psi_{L}^{}(\mathbf{1},\mathbf{2},-1/2)(1,0)$ and $\phi(\mathbf{1},\mathbf{2},-1/2)(0,0)$, respectively, are the SM lepton and Higgs doublets. The right-handed neutrinos neither have Yukawa couplings with the SM Higgs doublet nor have Majorana masses as a result of the $U(1)_{lepton}^{}\times U(1)_{dark}^{}$ conservation. The global symmetry $U(1)_{lepton}^{}$ will be exactly conserved at any energy scales while the global symmetry $U(1)_{dark}^{}$ will be spontaneously broken above the weak scale.

The singlet scalar $\sigma$ acquires a vacuum expectation value (VEV) to break $U(1)_{dark}^{}$ and then induces a small VEV of the heavy doublet scalars $\eta$ after the electroweak symmetry breaking by $\langle\phi\rangle\simeq 174\,\textrm{GeV}$, $$\begin{aligned}
\langle\eta\rangle \simeq
-\frac{\mu\langle\sigma\rangle\langle\phi\rangle}{M_{\eta}^2}~~\textrm{for}~~M_{\eta}^{}\gtrsim
\mu \gg \langle\sigma\rangle > \langle\phi\rangle\,.\end{aligned}$$ Therefore through the Dirac seesaw mechanism [@gh2006], the neutrinos obtain very small Dirac masses naturally through their Yukawa couplings to the heavy doublet, $$\begin{aligned}
m_{\nu}^{}= y\langle\eta\rangle\,.\end{aligned}$$

In our model, the heavy doublet scalar $\eta$ has three lepton number conserving decay channels: $$\begin{aligned}
\eta\,\rightarrow\, \psi_{L}^{}\nu_{R}^{c} \,,\quad
\eta\,\rightarrow\, \phi^\ast_{}\xi^\ast_{}\chi\,,\quad
\eta\,\rightarrow\, \phi\,\sigma \,.\end{aligned}$$ We consider two heavy scalars $\eta_{1,2}^{}$ to incorporate CP violation. These decays of $\eta_{1,2}^{}$ at loop orders (as shown in Fig. \[asymmetry\]) can interfere to generate three types of lepton asymmetries after $\eta_{1,2}^{}$ go out of equilibrium: the first one (including that transferred from the charged singlet scalar $\xi$) is stored in the left-handed lepton doublets $\psi_{L}^{}$; the second one is stored in the singlet right-handed neutrinos $\nu_{R}^{}$; and the third one is stored in the neutral singlet $\chi$, although the total lepton asymmetry vanishes as the lepton number is exactly conserved.

The effective Yukawa couplings of the left-handed lepton doublets to the SM Higgs doublet and the right-handed neutrinos are extremely weak, so that they can not go into equilibrium until the temperature falls well below the electroweak scale. This will prevent the $\nu_{R}^{}$ asymmetry to cancel the $\psi_{L}^{}$ asymmetry before the sphaleron transitions are over. Thus, before the electroweak phase transition, the sphaleron [@krs1985] action will partially convert the lepton asymmetry stored in the left-handed lepton doublets to the baryon asymmetry in the Universe. Therefore, we can make use of the leptogenesis [@fy1986] via neutrinogenesis [@dlrw1999] mechanism to understand the baryon asymmetry in the Universe. On the other hand, the asymmetry between the neutral singlet $\chi$ and its CP-conjugate will always survive after it is produced because there are no other processes breaking the lepton number stored in $\chi$. We will show later that this $\chi$ asymmetry corresponds to an amount of energy density, equal to the dark matter relic density, so that $\chi$ becomes the dark matter candidate.

For an estimate of the amount of CP asymmetry, we work in the basis, in which the $\eta_{1,2}^{}$ mass matrix is real and diagonal: $M_{\eta}^{2}=\textrm{diag}\left(M_{\eta_1^{}}^{2},
M_{\eta_2^{}}^{2}\right)$. We also assume $M_{\eta_2^{}}^{} \gg
M_{\eta_1^{}}^{}$, so that the final lepton asymmetry stored in the left-handed lepton doublets $\psi_{L}^{}$ and the dark matter asymmetry stored in the neutral singlet $\chi$ should mainly come from the decays of the lighter $\eta_{1}^{}$: $$\begin{aligned}
\hskip -.4cm\varepsilon_{\eta_1^{}}^{\psi_{L}^{}}&\equiv&
\frac{\Gamma(\eta_{1}^{} \rightarrow\psi_L\nu^c_R
)-\Gamma(\eta_{1}^{\ast}\rightarrow\psi^c_L\nu_R)}{\Gamma_{\eta_1^{}}^{}}\nonumber\\
[1mm] &&+2\frac{\Gamma(\eta_{1}^{} \rightarrow
\phi^\ast_{}\xi^\ast_{}\chi)-\Gamma(\eta_{1}^{\ast}\rightarrow\phi\xi\chi^\ast_{})}{\Gamma_{\eta_1^{}}^{}}\nonumber\\
[2mm]
&=&\frac{1}{4\pi}\frac{\textrm{Im}\left[\textrm{Tr}\left(y_{1}^{\dagger}y_{2}^{}\right)\left(\frac{\mu_1^\ast\mu_2^{}}{M_{\eta_2^{}}^2}
-\frac{1}{32\pi^2_{}}\kappa_1^{}\kappa_2^\ast
\frac{M_{\eta_1^{}}^2}{M_{\eta_2^{}}^2}\right)\right]}
{\textrm{Tr}\left(y_{1}^{\dagger}y_{1}^{}\right)+\frac{1}{32\pi^2_{}}|\kappa_1^{}|^2_{}+\frac{|\mu_1^{}|^2_{}}{M_{\eta_1^{}}^2}}\nonumber\\
[1mm] &&
-\frac{1}{64\pi^3}\frac{\textrm{Im}\left\{\kappa_1^{\ast}\kappa_2^{}\left[\frac{\mu_1^\ast\mu_2^{}}{M_{\eta_2^{}}^2}
+\textrm{Tr}\left(y_{2}^{\dagger}y_{1}^{}\right)\frac{M_{\eta_1^{}}^2}{M_{\eta_2^{}}^2}\right]\right\}}
{\textrm{Tr}\left(y_{1}^{\dagger}y_{1}^{}\right)+\frac{1}{32\pi^2_{}}|\kappa_1^{}|^2_{}+\frac{|\mu_1^{}|^2_{}}{M_{\eta_1^{}}^2}}\,,
\\
[4mm]\varepsilon_{\eta_1^{}}^{\chi}&\equiv& \frac{\Gamma(\eta_{1}^{}
\rightarrow
\phi^\ast_{}\xi^\ast_{}\chi)-\Gamma(\eta_{1}^{\ast}\rightarrow\phi\xi\chi^\ast_{})}{\Gamma_{\eta_{1}^{}}^{}}\nonumber\\
[1mm]
&=&-\frac{1}{128\pi^3}\frac{\textrm{Im}\left\{\kappa_1^{\ast}\kappa_2^{}\left[\frac{\mu_1^\ast\mu_2^{}}{M_{\eta_2^{}}^2}
+\textrm{Tr}\left(y_{2}^{\dagger}y_{1}^{}\right)\frac{M_{\eta_1^{}}^2}{M_{\eta_2^{}}^2}\right]\right\}}
{\textrm{Tr}\left(y_{1}^{\dagger}y_{1}^{}\right)+\frac{1}{32\pi^2_{}}|\kappa_1^{}|^2_{}+\frac{|\mu_1^{}|^2_{}}{M_{\eta_1^{}}^2}}\,.
\phantom{xxx}\end{aligned}$$ As the decays of $\xi$ into two leptons $\psi_L$ are in equilibrium, we included the lepton asymmetry in $\xi$ for estimating the total lepton asymmetry involved in the sphaleron process. Here $\Gamma_{\eta_i^{}}^{}$ denotes the total decay width of $\eta_{i}^{}$ or $\eta_{i}^{\ast}$, $$\begin{aligned}
\Gamma_{\eta_i^{}}&\equiv&\Gamma\left(\eta_{i}^{}\rightarrow
\psi_{L}^{}\nu_{R}^{c}\right)+\Gamma\left(\eta_{i}^{}\rightarrow
\phi^\ast_{}\xi^\ast_{}\chi^{}_{}\right) +\Gamma\left(\eta_{i}^{}\,
\rightarrow
\phi^{}_{}\sigma^{}_{}\right)\nonumber\\[2mm] &=&
\frac{1}{16\pi}\left[\textrm{Tr}\left(y_{i}^{\dagger}y_i^{}\right)
+\frac{1}{32\pi^{2}_{}}|\kappa_{i}^{}|^{2}_{}+\frac{|\mu_{i}^{}|^{2}_{}}{M_{\eta_{i}^{}}^{2}}\right]M_{\eta_{i}^{}}^{}\,.\end{aligned}$$ The unitarity and the CPT conservation imply the total decay width of $\eta_1$ and $\eta^\ast$ to be the same $\Gamma_{\eta_i}
= \Gamma_{\eta_i^\ast}$. For simplicity, we assume $y_{2}^{}=y_{1}^{}e^{i\varphi}_{},\,\kappa_{2}^{}=\kappa_{1}^{}e^{i\varphi}_{},\,\mu_{2}^{}=\mu_{1}^{}e^{i(\delta-\varphi)}_{}$ and then derive $$\begin{aligned}
\varepsilon_{\eta_1^{}}^{\psi_{L}^{}}&=&\frac{\sin\delta}{4\pi}\frac{\left[\textrm{Tr}\left(y_{1}^{\dagger}y_{1}^{}\right)
-\frac{1}{16\pi^2_{}}|\kappa_1^{}|^2_{}\right]\frac{|\mu_1^{}|^2_{}}{M_{\eta_2^{}}^2}}
{\textrm{Tr}\left(y_{1}^{\dagger}y_{1}^{}\right)+\frac{1}{32\pi^2_{}}|\kappa_1^{}|^2_{}+\frac{|\mu_1^{}|^2_{}}{M_{\eta_1^{}}^2}}\,,\\
[1mm]
\varepsilon_{\eta_1^{}}^{\chi}&=&-\frac{\sin\delta}{4\pi}\frac{\frac{1}{32\pi^2_{}}|\kappa_1^{}|^2_{}\frac{|\mu_1^{}|^2_{}}{M_{\eta_2^{}}^2}}
{\textrm{Tr}\left(y_{1}^{\dagger}y_{1}^{}\right)+\frac{1}{32\pi^2_{}}|\kappa_1^{}|^2_{}+\frac{|\mu_1^{}|^2_{}}{M_{\eta_1^{}}^2}}\,.\end{aligned}$$ The ratio between $\varepsilon_{\eta_1^{}}^{\psi_L^{}}$ and $\varepsilon_{\eta_1^{}}^{\chi}$ then becomes, $$\begin{aligned}
\varepsilon_{\eta_1^{}}^{\psi_{L}^{}}:\varepsilon_{\eta_1^{}}^{\chi}=\textrm{Tr}\left(y_{1}^{\dagger}y_{1}^{}\right)
-\frac{1}{16\pi^2_{}}|\kappa_1^{}|^2_{}:-\frac{1}{32\pi^2_{}}|\kappa_1^{}|^2_{}\,.\end{aligned}$$

The final baryon asymmetry and dark matter asymmetry would contribute energy density to the present Universe as below [@kt1990], $$\begin{aligned}
\label{baryon-asymmetry}
\rho_{B}^{0}&=&n_{B}^{0}m_{N}^{}=\frac{n_{B}^{0}}{s_0^{}}m_{N}^{}s_0^{}
=-\frac{28}{79}\frac{n_{L_{SM}^{}}^{}}{s}\left|_{T\simeq M_{\eta_1^{}}^{}}^{}m_{N}^{}s_0^{}\right.\nonumber\\
&\simeq&
-\frac{28}{79}\varepsilon_{\eta_1^{}}^{\psi_{L}^{}}\frac{n_{\eta_1^{}}^{eq}}{s}\left|_{T\simeq
M_{\eta_1^{}}^{}}^{}m_{N}^{}s_0^{}\right.\,,\\
[1mm] \label{dark-matter-asymmetry}
\rho_{\chi}^{0}&=&n_{\chi}^{0}m_{\chi}^{}=\frac{n_{\chi}^{0}}{s_0^{}}m_{\chi}^{}s_0^{}
=\frac{n_{\chi}^{}}{s}\left|_{T\simeq M_{\eta_1^{}}^{}}^{}m_{\chi}^{}s_0^{}\right.\nonumber\\
&\simeq&
\varepsilon_{\eta_1^{}}^{\chi}\frac{n_{\eta_1^{}}^{eq}}{s}\left|_{T\simeq
M_{\eta_1^{}}^{}}^{}m_{\chi}^{}s_0^{}\right.\,,\end{aligned}$$ where $m_{N}^{}\simeq 1\,\textrm{GeV}$ is the masses of the nucleons, $s$ is the entropy density, $n_{B}^{}$ and $n_{\chi}^{}$, respectively, are the number density of baryon and dark matter, $n_{\eta_{1}^{}}^{eq}$ is the equilibrium distribution of the heavy singlet $\eta_{1}^{}$. The solutions (\[baryon-asymmetry\]) and (\[dark-matter-asymmetry\]) for the baryon and dark matter density are obtained, assuming that the decays of $\eta_1$ satisfies the out-of-equilibrium condition, $$\begin{aligned}
\Gamma_{\eta_1^{}}^{}\lesssim H(T)\left.\right|_{T=
M_{\eta_1^{}}^{}}^{}=
\left(\frac{8\pi^{3}_{}g_{\ast}^{}}{90}\right)^{\frac{1}{2}}_{}\frac{M^{2}_{\eta_1^{}}}{M_{\textrm{Pl}}^{}}
\,,\end{aligned}$$ where $H(T)$ is the Hubble constant with relativistic degrees of freedom $g_{\ast}^{}\simeq 100$ and the Planck mass $M_{\textrm{Pl}}^{}\simeq 10^{19}_{}\,\textrm{GeV}$. In the presence of the fast annihilation between the dark matter and dark antimatter, the dark matter asymmetry should be equivalent to the dark matter relic density. In this scenario, the contributions from the baryonic and dark matter to the present Universe should have the following ratio, $$\begin{aligned}
\Omega_{B}^{}:\Omega_{\chi}^{}\equiv\rho_{B}^{0}:\rho_{\chi}^{0}=-\frac{28}{79}\varepsilon_{\eta_1^{}}^{\psi_{L}^{}}
m_{N}^{}: \varepsilon_{\eta_1^{}}^{\chi} m_{\chi}^{}\,.\end{aligned}$$ Conventionally, we define $$\begin{aligned}
\eta_{B}^{0}&=&\frac{n_B^{0}}{n_{\gamma}^{0}} \simeq
7.04\times\left[-\frac{28}{79}\varepsilon_{\eta_1^{}}^{\psi_{L}^{}}\frac{n_{\eta_1^{}}^{eq}}{s}\left|_{T\simeq
M_{\eta_1^{}}^{}}^{}\right.\right]\nonumber\\
[1mm]
&\simeq&\frac{7.04\,\varepsilon_{\eta_1^{}}^{\psi_{L}^{}}}{15\,g_\ast^{}}\,,\end{aligned}$$ to describe the current baryon asymmetry. Here $n_{\gamma}^{}$ is the number density of photon.

For giving a numerical example, we take $M_{\eta_1^{}}
=0.1M_{\eta_2^{}}^{}=5\times 10^{13}_{}\,\textrm{GeV},\, |\mu_1^{}|
=|\mu_2^{}|=10^{12}_{}\,\textrm{GeV}$ and obtain $\langle\eta_{1}^{}\rangle =100\langle\eta_{2}^{}\rangle\simeq
1\,\textrm{eV}$ for $\langle\sigma\rangle=1.5\times
10^4_{}\,\textrm{GeV}$. Consequently the neutrino masses would be, $$\begin{aligned}
\label{parameter-3} m_{\nu}^{}\sim \mathcal{O}(0.01\,\textrm{eV}) ~~
\textrm{for} ~~y_1^{}=y_2^{}e^{-i\varphi}=\mathcal{O}(0.01)\,.\end{aligned}$$ We further input $m_\chi^{}=4\,\textrm{TeV}$, $|\kappa_{1}^{}|=|\kappa_{2}^{}|=0.02$, $\sin\delta=-0.5$ and $\textrm{Tr}\left(m_{\nu}^\dagger
m_\nu^{}\right)=\textrm{Tr}\left(y_{1}^\dagger
y_1^{}\right)\langle\eta_{1}^{}\rangle^2_{}=3\times
10^{-3}_{}\,\textrm{eV}^2_{}$ to obtain a lepton asymmetry $$\begin{aligned}
\label{parameter-6}
\varepsilon_{\eta_1^{}}^{\psi_L^{}}\simeq-2.3\times 10^3_{}\,
\varepsilon_{\eta_1^{}}^{\chi}\simeq -1.4\times 10^{-7}_{}\,.\end{aligned}$$ Eventually, we obtain the baryon asymmetry and the dark matter asymmetry as below $$\begin{aligned}
\label{parameter-7} \eta_{B}^{0}\simeq 6.2\times
10^{-10}\,,~~\Omega_{\chi}^{}:\Omega_{B}^{}\simeq 5\,,\end{aligned}$$ which are fully consistent with the experimental observations [@amsler2008].

In the non-relativistic limit, the annihilation cross-section of dark matter and dark antimatter reads, $$\begin{aligned}
\hskip -.5cm\langle\sigma v\rangle =
\frac{1}{32\pi}\left[\left(\alpha-\frac{\gamma\zeta}{2\vartheta}\right)^{2}_{}+2\left(\beta-\frac{\epsilon\zeta}{2\vartheta}\right)^2_{}
+2\gamma^2_{}\right]\frac{1}{m_\chi^2}\,,\end{aligned}$$ which could be very high as $\alpha,\beta,\gamma,\zeta,\epsilon,\vartheta<\sqrt{4\pi}$. For example, we obtain $\langle\sigma
v\rangle=18\,\textrm{pb}\,\left({4\,\textrm{TeV}}/{m_\chi^{}}\right)^2_{}$ for $\alpha,\beta,\vartheta=0.1$ and $\gamma,\zeta,\epsilon=1$. The thermally produced dark matter, with the mass within the range of a few GeV to a few TeV, should have an annihilation cross section slightly smaller than $1\,\textrm{pb}$ to give the desired relic density (\[parameter-7\]). If the cross section is much bigger than $1\,\textrm{pb}$, the thermally produced relic density should be negligible. Therefore the dark matter relic density can be well approximated by the dark matter asymmetry.

The recent cosmic-ray experiments [@chang2008; @torii2008; @adriani2008; @aharonian2008; @abdo2009] suggest [@mpsv2009] that (i) the TeV-scale dark matter should mostly annihilate or decay into the leptons with a large cross section or a long life time; (ii) the dark matter annihilation or decay should not result in overabundant gamma and neutrino fluxes. For demonstration, we take the rotation, $$\begin{aligned}
&&Z_1^{}\left[\langle\phi\rangle\phi+\langle\eta_1^{}\rangle\eta_1^{}+\langle\eta_2^{}\rangle\eta_2^{}\right]
\rightarrow\phi =\left(
\begin{array}{c}
\frac{1}{\sqrt{2}}h+\langle\phi\rangle\\
0\end{array} \right)\,,\nonumber\\
[1mm]
&&Z_1^{}Z_2^{}\left[\langle\phi\rangle\left(\langle\eta_1^{}\rangle\eta_1^{}+\langle\eta_2^{}\rangle\eta_2^{}\right)
-\left(\langle\eta_1^{}\rangle^2_{}+\langle\eta_2^{}\rangle^2_{}\right)\phi\right]\rightarrow\eta_1^{}
\,,\nonumber\\
[2mm] &&Z_2^{}\left[
\langle\eta_1^{}\rangle\eta_2^{}-\langle\eta_2^{}\rangle\eta_1^{}\right]
\rightarrow\eta_2^{}\,,\end{aligned}$$ where $Z_1^{} =
[\langle\phi\rangle^2_{}+\langle\eta_1^{}\rangle^2_{}
+\langle\eta_2^{}\rangle^2_{}]^{-\frac{1}{2}}_{}$ and $Z_2^{} =
[\langle\eta_1^{}\rangle^2_{}
+\langle\eta_2^{}\rangle^2_{}]^{-\frac{1}{2}}_{}$. Clearly, the Yukawa couplings of the heavy doublet scalars to the quarks are highly suppressed by $\displaystyle{\mathcal{O}(\langle\eta\rangle/\langle\phi\rangle)}$. Therefore, the dark matter $\chi$ should mostly decay into the charged singlet scalar $\xi$, the left-handed charged leptons $\ell_L^{}$, the right-handed neutrinos $\nu_R^{}$ and the physical Higgs boson $h$. We show the dominant decay channels in Fig. \[dm\]. Here we have taken into account that (1) $\xi$ can uniquely and rapidly decay into the left-handed leptons $\ell_L^{}$ and $\nu_L^{}$ through the Yukawa coupling $f\xi\overline{\psi_{L}^c}i\tau_2^{}\psi_{L}^{}$; (2) $h$ can dominantly decay into the bottom quarks. The decay width is given by $$\begin{aligned}
\Gamma_{\chi}^{}&=&\frac{1}{192\cdot
(2\pi)^3}\Sigma_{i,j=1}^{2}\kappa_i^{\ast}\kappa_j^{}\textrm{Tr}\left(y_i^{}y^\dagger_j\right)
\frac{\langle\phi\rangle^2_{}m_\chi^3}{M_{\eta_i^{}}^2M_{\eta_j^{}}^2}\nonumber\\
[1mm]
&&\times\left[1+\frac{1}{96\cdot\left(2\pi\right)^2_{}}\frac{m_\chi^2}{\langle\phi\rangle^2_{}}\right]\,.\end{aligned}$$ For giving a numerical example, we choose the parameters considered before and then determine the life time, $$\begin{aligned}
\tau_\chi^{} = \frac{1}{\Gamma_\chi^{}}&\simeq & 0.74\times
10^{26}\,\textrm{sec}\,.\end{aligned}$$ We notice that the SM Higgs boson and then the bottom quark appear in the final states of the dark matter decay. However, the branching ratio is small (less than $4\%$) for the present choice of the parameters. This means the dark matter mostly decays into the leptons. Compared with the good fitting in [@mpsv2009], we find that the dark matter decay in our model can induce the desired positron/electron excesses but avoid the overabundant gamma and neutrino fluxes.

The quartic coupling of the dark matter scalar $\chi$ to the SM Higgs doublet $\phi$, i.e. $\beta\left(\chi^\dagger_{}\chi\right)\left(\phi^\dagger_{}\phi\right)$, will result in an elastic scattering of dark matter on nuclei and hence a nuclear recoil [@aht2008]. The spin-independent cross section of the dark-matter-nucleon elastic scattering would be, $$\begin{aligned}
\label{dama3} &&\sigma\left(\chi N \rightarrow \chi N\right)=
\frac{\beta^2_{}}{4\pi}\frac{\mu_r^2}{m_h^4
m_\chi^2}f^2_{}m_N^2\nonumber\\
[2mm] && \hskip -.5cm\simeq 10^{-44}_{}\,\textrm{cm}^2_{}\,\times
\frac{\beta^2_{}}{4\pi}\left(\frac{f}{0.3}\right)^2_{}\left(\frac{120\,\textrm{GeV}}{m_h^{}}\right)^4_{}
\left(\frac{4\,\textrm{TeV}}{m_\chi^{}}\right)^2_{}\,,\phantom{xxx}\end{aligned}$$ which is below the current experimental limit [@angle2007; @ahmed2008], however, reachable for the future experiments. Here $\mu_{r}^{}=m_\chi^{} m_N^{}/(m_\chi^{} + m_N^{})$ is the nucleon-dark-matter reduced mass, $m_h^{}$ is the mass of the physical Higgs boson $h$ (The mixing between $h$ and $h'$ (from $\sigma$) is negligible for $\langle\sigma\rangle\gg\langle\phi\rangle$.), the factor $f$ with a central value $f=0.30$ [@aht2008] parameterizes the Higgs to nucleons coupling from the trace anomaly, $fm_{N}^{}\equiv\langle
N|\sum_q^{}m_q^{}\bar{q}q|N\rangle$.

In this paper we connect the neutrino masses with the origin of the dark matter relic density and the baryon asymmetry in the present Universe. In our model, the dark matter relic density is a dark matter asymmetry because the thermal relic density of dark matter is negligible in the presence of the fast annihilation between the dark matter and dark antimatter. This dark matter asymmetry originate simultaneously with a lepton asymmetry, which can explain the baryon asymmetry via the sphaleron process, and given by the amount of CP violation in out-of-equilibrium decays of heavy scalars. The dark matter mostly decays into the leptons to generate the positron/electron excesses without the overabundant gamma and neutrino fluxes so that we can explain the results from the recent cosmic-ray experiments.

**Acknowledgement**: We thank Xiao-Jun Bi and Goran Senjanovi$\rm\acute{c}$ for helpful discussions. US thanks the Department of Physics and the McDonnell Center for the Space Sciences at Washington University in St. Louis for inviting him as Clark Way Harrison visiting professor. XZ is supported in part by the National Natural Sciences Foundation of China under grants 10533010 and 10675136, and by the Chinese Academy of Sciences under grant KJCX3-SYW-N2.

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