Structure of right-handed neutrino mass matrix

Recently, Nishiura and the author proposed a unified quark-lepton mass matrix model under a family symmetry $\mathrm{U}(3)\times \mathrm{U}(3{)}^{\prime }$. The model can give excellent parameter fitting to the observed quark and neutrino data. The model has a reasonable basis as far as the quark sector, but, in the neutrino sector, the form of the right-handed neutrino mass matrix $M_R$ does not have a theoretical basis; that is, it was nothing but a phenomenological assumption. In this paper, it is pointed out that the form of $M_R$ is originated in the structure of Majorana mass matrix ($4\times 4$ matrix) for the left-handed fields $(({\nu }_L{)}_i,({\nu }_R^c{)}^i,(N_L{)}_{\alpha },(N_R^c{)}^{\alpha })$ where ${\nu }_i$ ($i=1$, 2, 3) and $N_{\alpha }$ ($\alpha =1$, 2, 3) are U(3)-family and ${\mathrm{U}(3)}^{\prime }$-family triplets, respectively.