content_image/371275.jpg content_image/371277.jpg content_image/371278.jpg content_image/371279.jpg content_image/371280.jpg content_image/371281.jpg
associated with N$_{2}$ and the lepton flavors α = e, µ, τ are given as 

<img src='content_image/371275.jpg'>

where the common factor g 2 $_{ν}$m ∗ M ≃ 8 π $^{$^{2}$m}$∗ m$_{0}$/ ( ¯ m$_{η}$$^{m}$− ), all K -factors are evaluated at tempera- ture T = M$_{2}$ , and M$_{2}$ is the lightest of the heavy Majorana neutrinos. And the corresponding CP asymmetries ε α 2 are approximately given, for x ≪ 1, by 

<img src='content_image/371277.jpg'>

in which ε µ 2 + ε τ 2 = − ε e 2 is satisfied due to Im[ H$_{2}$$_{j}$ ] = 0 for ϕ$_{1}$$_{,}$$_{2}$ = 0. From Eq. (50) we see that all K -factors are almost equal, so this case can be classified as K α 2 ≥ 1 and K α 2 < 1. In strong wash-out regime K α 2 ≥ 1 ( α = e, µ, τ ), given the initial thermal abundance of N$_{2}$ and the condition for K α $_{2}$, the resulting baryon-to-photon ratio η$_{B}$ approximately given as 

<img src='content_image/371278.jpg'>

where x sin φ is from the common factor in K µ,τ 2 . In weak wash-out regime K α 2 < 1 ( α = e, µ, τ ), the resulting baryon-to-photon ratio η$_{B}$ can be simply given as 

<img src='content_image/371279.jpg'>

where x sin φ comes from the common factor in K µ,τ 2 , K µ 2 ≃ K τ 2 and K$_{2}$ = K e 2 + K µ 2 + K τ $_{2}$. If we take, for example, δ$_{N}$$_{3}$$_{η}$ = 1 . 5 × 10 − $^{6}$, the magnitude of washout factors are given as 

<img src='content_image/371280.jpg'>

and Eq. (52) can be simplified as 

<img src='content_image/371281.jpg'>

where g$_{ν}$ ≃ 0 . 4 , b ≃ 2 . 7 and ( K µ $_{2}$) − 1 . 16 ≃ 0 . 2. Fig. 10 shows the behavior of η$_{B}$ as functions of φ and x , representing only in the limit of φ = 3 π/ 2 the value of x can become order of O (0 . 1). However, Fig. (10) shows that η$_{B}$ is not enough for BAU to be satisfied. 

