content_image/332941.jpg content_image/332951.jpg content_image/332950.jpg content_image/332949.jpg
<img src='content_image/332941.jpg'>

duced, both of which carry the lepton number of two-unit, and their interactions are given by 

<img src='content_image/332951.jpg'>

where L$_{L}$ is the left-handed lepton doublet and ℓ$_{R}$ is the right-handed lepton singlet. The matrices f$_{ij}$ and g ′ ab are respectively an anti-symmetric and a symmetric couplings and the lepton number is violated by the interaction with the parameter µ . 

The neutrino mass matrix is generated at the two-loop level via the diagram in Fig. 1 (left); 

where y$_{i}$ [= √$_{2}$$_{m}$$_{i}$$_{/v}$ ( i = e , µ , τ )] are the SM Yukawa coupling constants of charged leptons with the masses m$_{i}$ and the VEV v ( ≃ 246 GeV), g$_{ij}$ are defined as g$_{ii}$ = g ′ ii and g$_{ij}$ = 2 g ′ ij ( i ̸ = j ), m$_{ω}$ and m$_{k}$ are masses of ω ± and k $^{±±}$, and 

<img src='content_image/332950.jpg'>

<img src='content_image/332949.jpg'>

where I$_{1}$ ( r ) takes the value of around 3 - 0.2 for 10 − 2 < ∼ r < ∼ 10 $^{2}$. The universal scale of neutrino masses is determined by the two-loop suppression factor 1 / (16 π $^{2}$)2 and the lepton number violating parameter µ . The charged lepton Yukawa coupling constants y$_{ℓ}$$_{i}$ ( y$_{e}$ ≪ y$_{µ}$ ≪ y$_{τ}$ < ∼ 10 − $^{2}$) give an additional suppression factor. Thus, any of f$_{ij}$ or g$_{ij}$ can be of O (1) when m$_{ω}$ and m$_{k}$ are at the TeV scale. The flavor structure of the mass matrix is determined by the combination of the coupling constants f$_{ij}$ and y$_{i}$g$_{ij}$y$_{j}$ . 

