# Singlet neutrinos decaying to Higgs bosons during leptogenesis

(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.

• Q1: How can the $N$ have a U(1) lepton number charge? Wouldn't it be broken by the Majorana mass term $\bar{N^C} N$? – innisfree Jun 1 '15 at 16:27
• @innisfree, yes it does break U(1) lepton number by two units. A majorana mass term necessarily breaks any U1 it is charged under. – xi45 May 27 '16 at 17:55

Exactly as you mention in the second part of your post. Leptogenesis takes place at temperatures higher than the electroweak scale $T_{EW}\simeq100\,\mbox{GeV}$. Therefore, heavy neutrinos decay into lepton and higgs doublets, whose fields are still all physical.