are gauge singlets from the point of view of the [[Standard Model ]{}]{}gauge symmetries, they need not be added “sequentially”, so as to match in number the left-handed ones. For the general case, the relevant part of the Lagrangian is written as, $$\begin{aligned}
- \mathcal{L} = \sum_{a,i}Y_{\nu}^{ai} \bar{\ell}_{L}^{a} \tilde{H} \nu_{R_i} + \frac{1}{2}\sum_{i,j}M^{ij}_{R}\overline{\nu_{R_i}^{c}}\nu_{R_j} + \text{H.c.}
\label{seesaw-lag}\end{aligned}$$ where $\ell_{L}^{a} = (\nu^a_L, l^a_L)^T$ with $a=1,\,2,\,3$ denotes the three families of left lepton doublets, while $i,j=1,2,\cdots n$ labels the right-handed singlet neutrinos, and $H$ is the [[Standard Model ]{}]{}Higgs doublet. After the electroweak symmetry breaking, the Higgs gets vacuum expectation value (vev) $\vev{H} = \frac{v}{\sqrt{2}}$ with $v = 246\,\text{GeV}$. Then the full neutrino mass matrix is expressed as, $$\begin{aligned}
\mathcal{M}_{\nu}=
\begin{pmatrix}
 0 & m_{D} \\
 m_{D}^{T} & M_{R}  \\
\end{pmatrix}
\label{mass-matrix}\end{aligned}$$ where $m_{D}=\frac{Y_{\nu}}{\sqrt{2}}v$ is the “Dirac mass matrix” generated after spontaneous electroweak symmetry breaking.

Being invariant under the gauge symmetry, the n$\times$n right-handed neutrinos “Majorana mass matrix” $M^{ij}_R$ is independent of the electroweak scale and can be large, $|M^{ij}_R| >> v$ implying $|\frac{m^{aj}_{D}}{M^{ij}_{R}}| << 1$. Hence the mass matrix in Eq. can be block-diagonalized perturbatively [@Schechter:1981cv]. To leading order the mass matrix elements for light neutrinos $m^{ab}_{\nu}$ are given as [@Schechter:1981cv] $$\begin{aligned}
m^{ab}_{\nu}\simeq -m^{ai}_{D} 