acquires a VEV. Here this will be chosen, without loss of generality, to be the third generation. Electroweak symmetry breaking will induce VEVs in the remaining two right-handed sneutrino generations. However, these will be on the order of the neutrino masses and, therefore, are neglibible. Note that ${v_L}_i$ is in general complex.

Substituting the VEVs from Eq. (\[vevs\]) into the $F$-term, $D$-term and soft potentials yields $$\begin{aligned}
\label{F}
\begin{split}
\langle V_F\rangle &=
    \frac{1}{2}|\mu|^2 v^2+\frac{1}{4}|{Y_\nu}_{i3} {v_L}_i|^2v_R^2 + \frac{1}{4}|{Y_\nu}_{i3}|^2v_u^2v_R^2 + \frac{1}{4} |{Y_\nu}_{ij} {v_L}_i|^2v_u^2
\\
    & \quad  -\frac{1}{2\sqrt 2}(\mu {Y_\nu}_{i3} v_d {v_L}_i v_R+\text{h.c.}),
\end{split}
\\
\langle V_D\rangle&=\frac{g_2^2}{32}(v_u^2-v_d^2- |{v_L}_i|^2)^2 + \frac{g_{BL}^2}{32}(v_R^2- |{v_L}_i|^2)^2 + \frac{g_R^2}{32}(v_u^2-v_d^2-v_R^2)^2,
\\
\label{soft}
\langle V_{\mbox{soft}}\rangle&=\frac{1}{2}m^2_{\tilde L_{i}} \left| {v_L}_i \right|^2 + \frac{1}{2}m^2_{\tilde \nu_3^c}v_R^2 + \frac{1}{2}m_{H_u}^2v_u^2 + \frac{1}{2}m_{H_d}^2v_d^2 + b \, v_d v_u + \frac{1}{2\sqrt2}(a_{\nu_{i3}}v_u {v_L}_i v_R+\mbox{h.c.}),\end{aligned}$$ where repeated generational indices are summed and $g_R, g_2$ and $g_{BL}$ are the third component of right-handed isospin, left-handed isospin and $B-L$ gauge couplings respectively.

Equations (\[F\])-(\[soft\]) can be simplified by considering some general phenomenological features of this model. For example, neutrino masses are roughly proportional to the ${Y_\nu}_{ij}$ and ${v_L}_i$ parameters and, hence, ${Y_\nu}_{ij} \ll 1$ and ${v_L}_i\ll v_{u,d}, v_R$. With this in mind, the complete potential energy has the following minimization conditions: $$\begin{aligned}
v_R^2=&\frac{-8m^2_{\tilde 