(i) The Lagrangian of electroweak model extended with right-chiral singlet neutrinos $N_{iR}$ contains the Yukawa coupling term+ the bare Majorana mass term $$f_{\alpha i}\overline{l_{L\alpha}}\hat\phi N_{Ri}+M_{ij}\overline{N^c_{iR}}N_{jR}$$ where $\hat\phi=i\tau_2\phi^*$. To draw the Feynman diagram of tree-level lepton violating processes arising out of this we have to expand the doublets after spontaneous symmetry breaking. Right? Does $$N_{iR}\rightarrow \nu_{iL}+ h^0$$ give the only possible lepton-violating process at the tree-level? I use $h^0$ for the standard model (SM) higgs given by $$\phi=\begin{pmatrix}\phi^+\\\phi^0\end{pmatrix}\Rightarrow^{breaking}\begin{pmatrix}\phi^+\\\frac{1}{\sqrt{2}}(v+h^0+i\zeta)\end{pmatrix}$$ Is this correct and complete? If $N_R$ fields are assigned a lepton number 1, then why should the above decay be lepton number violating?
(ii) The confusion arose because of a notation in a paper by Yanagita and Fukugita where the possible processes were written as $$N_R\rightarrow l_L+\bar\phi$$ and $$N_R\rightarrow \bar{l}_L+\phi$$ I cannot understand this notation. Does this mean that this interaction can lead to a process in which neutrinos decay to charged leptons and charged higgs? But there is no charged Higgs in the SM. I suspect at high temperatures both the members of the Higgs doublet are physical particles and possible interactions are obtained by assuming $\phi=\begin{pmatrix}\phi^+\\\phi^0\end{pmatrix}$ and expanding the Yukawa coupling term.
