\times U(1)_{B-L}$ into $U(1)_Y$ at some high scale to obtain the SM gauge symmetry at the electroweak (EW) scale. The $H_L$ doublet is required for preservation of the left-right symmetry. The bidoublet field $\Phi$ is responsible for generation of quark and charged lepton masses and Cabibbo-Kobayashi-Maskawa (CKM) mixing angles. The charged singlet field $\delta^{\pm}$ is needed for generation of neutrino masses through one-loop diagrams as will be discussed later in this section.

The Yukawa Lagrangian is given as: $$\begin{aligned}
\mathcal{L}_Y&=&Y_{ij}^{q1}\overline{Q}_{Li}\Phi Q_{Rj}+Y_{ij}^{q2}\overline{Q}_{Li}\widetilde{\Phi}Q_{Rj}+Y_{ij}^{l1}\overline{l}_{Li}\Phi l_{Rj} + Y_{ij}^{l2}\overline{l}_{Li}\widetilde{\Phi}l_{Rj} + \l_{L_{ij}} l^T_{Li} i \tau_2 l_{Lj} \delta^+ + \l_{R_{ij}} l^T_{Ri} i \tau_2 l_{Rj} \delta^+ + H.C.~,~~
\label{eq:yuk}\end{aligned}$$ where $Y$ and $\l$ are the Yukawa couplings and $$\widetilde{\Phi}=\tau_2\Phi^\ast\tau_2,$$ The structure of $\l_{L/R_{ij}}$ term is such that the only terms that will survive are the ones with $i \neq j$. This is exactly the same as in the Zee mechanism of neutrino mass generation. If we expand out any one of the terms involving $\d^+$ in the Yukawa Lagrangian we will get: $$\mathcal{L} \supset \sum_{i \neq j} \nu_i e_j(\l_{ij}-\l_{ji}),$$ where $\nu_i$ and $e_j$ are both in the flavor basis. Thus if we redefine the $\l$ matrix to $\l'_{ij}=\l_{ij}-\l_{ji}$, then this new $\l'$ matrix is completely anti-symmetric and the 