\nu_{3}^c}  + g_R^2\left(v_u^2 - v_d^2 \right)}{g_R^2+g_{BL}^2}\\
\frac{1}{8}(g_2^2+g_R^2)v^2=&-|\mu|^2+\frac{M_{H_u}^2\tan^2\beta-M_{H_d}^2}{1-\tan^2\beta}\\
\frac{2b}{\sin2\beta}=&2|\mu|^2+m_{H_u}^2+m_{H_d}^2
$$ $$\begin{aligned}
{v_L}_i=&\frac{\frac{v_R}{\sqrt 2}(Y_{\nu_{i3}}^* \mu v_d-a_{\nu_{i3}}^* v_u)}{m_{\tilde L_{i}}^2-\frac{g_2^2}{8}(v_u^2-v_d^2)-\frac{g_{BL}^2}{8}v_R^2}\end{aligned}$$ where $v^2 = v_d^2 + v_u^2$ and $$\begin{aligned}
    M_{H_u}^2&\equiv&m_{H_u}^2-\frac{1}{8}g_R^2v_R^2\\
    M_{H_d}^2&\equiv&m_{H_d}^2+\frac{1}{8}g_R^2v_R^2 \ .\end{aligned}$$ These conditions necessarily mean that the soft mass of the sneutrino that acquires a VEV, the third generation here, must have a tachyonic soft mass. Radiative mechanisms for achieving such a mass have been discussed in references [@Ambroso:2009jd; @Ambroso:2009sc; @Ambroso:2010pe].

Prior to electroweak symmetry breaking, $B-L$ breaking leaves one linear combination of the third component of right-handed isospin and $B-L$ gauge bosons massless– the hypercharge gauge boson. The other linear combination, $Z_R$, becomes massive. Including electroweak symmetry breaking effects, the mass of $Z_R$ is $$M_{Z_R}^2 \simeq \frac{1}{4} \left(g_R^2 + g_{BL}^2\right) v_R^2
    \left(
        1 + \frac{g_R^4}{\left(g_R^2+g_{BL}^2\right)^2} \frac{v^2}{v_R^2}
    \right).$$ See reference [@Everett:2009vy] for more details. Current bounds on $M_{Z_R}$ are at around 2.5 TeV [@ATLAS:2013jma; @CMS:2013qca].

R-parity Violation {#RPV}
==================

$R$-parity violation in this model is best parameterized by the two flavorful parameters– ${v_L}_i$ and $$\epsilon_i  \equiv \frac{1}{\sqrt 2} {Y_\nu}_{i3} v_R \ .$$ The superpotential expanded around the vacuum now contains the $R$-parity violating terms $$\label{W.brpv}
    W \supset \epsilon_i \,  L_i \,  H_u - \frac{1}{\sqrt 2 }{Y_e}_i \, {v_L}_i \,  H_d^- \,   e^c_i \ 