content_image/343192.jpg content_image/343189.jpg content_image/343190.jpg
M ( a, 1 , b ) = ( M$_{1}$, M$_{2}$, M$_{3}$ ) respectively. Note that the overall scale of neutrino masses is determined by g 2 $^{$_{ν}$m}$− as will be shown in Eq. (32), while the magnitudes of function f ( α M ¯ $_{m$_{η}$}$) does not change a lot within the parameter region as we showed in Fig. 2. The neutrino spectrums are related to the ratio M ¯ m$_{η}$ and the value of α , and α = 1 corresponds to the mass of the second generation of light neutrinos m$_{2}$ . The locations of m$_{1}$ and m$_{3}$ are determined by the values of α ′ s (or a, b ) which are defined in Eq. (9), and the constraints come from the solar and atmospheric mass-squared differences. They are given by 

<img src='content_image/343192.jpg'>

<img src='content_image/343189.jpg'>

<img src='content_image/343190.jpg'>

in which, from the neutrino oscillation experiments we know that Δ m 2 21 is positive and dictates ω$_{2}$ > ω$_{1}$/a with the second term being sufficiently small x in Eq. (29). As will be shown later, in order for a leptogenesis to be successfully implemented at or around TeV scale in our scenario, we consider the case M 2 lightest ≃ ¯ m 2 η where the M$_{lightest}$ is the lightest of the heavy Majorana neutrino. Depending on the hierarchy of the heavy Majorana neutrino masses M$_{1}$ , M$_{2}$ and M$_{3}$ , the relative size of the parameter κ consistent with the possible mass ordering of light neutrinos and hierarchy of Δ m 2 32 and Δ m 2 21 can be classified as follows: 

