content_image/433816.jpg content_image/433817.jpg content_image/433823.jpg content_image/433830.jpg content_image/433834.jpg content_image/433826.jpg content_image/433821.jpg content_image/433829.jpg content_image/433824.jpg content_image/433818.jpg content_image/433819.jpg content_image/433820.jpg content_image/433822.jpg content_image/433827.jpg content_image/433831.jpg content_image/433832.jpg content_image/433833.jpg content_image/433825.jpg
where Γ$_{R}$$_{j}$ and Γ$_{L}$$_{j}$ are the decay width at tree level, 

<img src='content_image/433816.jpg'>

<img src='content_image/433817.jpg'>

Although the Yukawa couplings of L are much smaller than those of R [cf. (25)], the lepton asymmetries (27a) and (27b) can arrive at a same magnitude, 

because the Yukawa couplings of R dominate the loop corrections in the decays of both R and L . For generating the observed baryon asymmetry [1] 

<img src='content_image/433823.jpg'>

the Yukawa couplings y$_{N}$ and the masses M should fulfill 

The 3-body decays of L don’t have significant contribu- tions to the total decay width and the lepton asymmetry, but it is sufficient to produce abundant right-handed neu- trinos for the dark matter relic density. We now clarify this interesting scenario in details. It is easy to read the relic density, 

where κ ′ $_{L$_{j}$}$( y$_{N}$, M ) is a wash out factor and 

<img src='content_image/433830.jpg'>

is the branching ratio. From the baryon asymmetry (26) and the relic density (32), we can perform the following ratio, 

<img src='content_image/433834.jpg'>

<img src='content_image/433826.jpg'>

<img src='content_image/433821.jpg'>

## IV. DARK MATTER 

<img src='content_image/433829.jpg'>

For a given relic density, we can determine the masses of the right-handed neutrinos by the observed baryon asym- metry, 

Here m$_{N}$ ∼ 940 MeV is the nucleon mass. On the other hand, as shown in Eq. (23), the mass matrix of the right- handed neutrinos has a same texture with that of the left-handed neutrinos. For example, let’s consider the case with the degenerate neutrino masses, 

which immediately yields 

With further assumptions, 

which is true in the weak washout region, and 

<img src='content_image/433824.jpg'>

we can obtain the dark matter masses 

Accordingly, the left-right symmetry breaking scale can be given by 

The above breaking scale points to the necessity of the resonant [12, 13] leptogenesis because of the parameter choice (39), i.e. M$_{j}$ ≃ 186 GeV. It also indicates that the gauge interactions of the right-handed neutrinos decouple at a high temperature ∼ ( v$_{R}$ v$_{L}$ ) 4 3 O (MeV) = O (100 GeV) so that the non-thermally [23] produced right-handed neutrinos can successfully explain the dark matter relic density. 

We should keep in mind that the right-handed neutri- nos mix with their left-handed partners due to the Dirac mass term in Eq. (16) so that they will decay at tree level and loop orders. In absence of the mixing between W ± L and W ± R (cf. (8)), the decay width should be [24, 25] 

<img src='content_image/433818.jpg'>

<img src='content_image/433819.jpg'>

<img src='content_image/433820.jpg'>

<img src='content_image/433822.jpg'>

<img src='content_image/433827.jpg'>

<img src='content_image/433831.jpg'>

<img src='content_image/433832.jpg'>

<img src='content_image/433833.jpg'>

<img src='content_image/433825.jpg'>

