\section{Sneutrino squared mass matrix and mixing matrix}\label{app:sneumix}

Extending the MSSM by right-handed neutrinos and giving them a Majorana
mass leads to a seesaw-like mechanism in the sneutrino sector. Similar
to the seesaw-extended Standard Model, where the neutrino spectrum gets
doubled, the sneutrino spectrum gets quadrupled. Why that? The MSSM
contains only three sneutrino states. Including right-handed fields, the
number of states get doubled, although half of them are singlets under
the SM gauge group. Moreover, due to Dirac and Majorana masses, the
physical spectrum gets even more enlarged. Effectively, we are left with
six light, more or less active states, and six heavy singlet-like
states. A priori, the sneutrino squared mass matrix is therefore a \(12
\times 12\) matrix, which can be perturbatively block-diagonalised
similar to the neutrino mass matrix. The complete procedure is described
in great detail by \cite{Dedes:2007ef}.

We choose the following basis: \(\tilde N = (\tilde\nu_\Ll,
\tilde\nu_\Ll^*, \tilde\nu_\Rr^*, \tilde\nu_\Rr)^\tp\) (such that
\(-\mathcal{L}^{\tilde\nul}_\upsh{mass} = N^\dag
(\mathcal{M}_{\mathbf{\tilde\nul}})^2 N\)) and classify chirality
conserving (\(LL, RR\)) and chirality changing blocks:
\begin{align*}
(\mathcal{M}_{\mathbf{\tilde\nu}})^2 =
\frac{1}{2}
        \begin{pmatrix}
                \mathcal{M}_{LL}^2 & \mathcal{M}_{LR}^2 \\
                \left(\mathcal{M}_{LR}^2\right)^\dag & \mathcal{M}_{RR}^2
        \end{pmatrix},
\end{align*}
with
\begin{subequations}
\begin{align}
  \mathcal{M}_{LL}^2 &= \begin{pmatrix}
      \mathcal{M}_{\tilde\ell}^2 + \frac{1}{2} M_Z^2 \cos2\beta
      \mathds{1}
      + \Mat{m}^\upsh{D}_\nul{\Mat{m}^\upsh{D}_\nul}^\dag & \Mat{0} \\
      \Mat{0} & (\mathcal{M}_{\tilde\ell}^2)^T
      + \frac{1}{2} M_Z^2 \cos2\beta \mathds{1}
      + {\Mat{m}^\upsh{D}_\nul}^* {\Mat{m}^\upsh{D}_\nul}^T
    \end{pmatrix},\\
  \mathcal{M}_{RL}^2 &= \begin{pmatrix}
      \Mat{m}^\upsh{D}_\nul\Mat{M}_\Rr &
      -\mu\cot\beta \Mat{m}^\upsh{D}_\nul + v_\uq \Mat{A}_\nul^* \\
      -\mu^* \cot\beta{\Mat{m}^\upsh{D}_\nul}^* + v_\uq \Mat{A}_\nul
      & {\Mat{m}^\upsh{D}_\nul}^* \Mat{M}_\Rr^*
    \end{pmatrix},\\
  \mathcal{M}_{RR}^2 &= \begin{pmatrix}
      (\mathcal{M}_{\tilde\nul}^2)^T +
      {\Mat{m}^\upsh{D}_\nul}^T{\Mat{m}^\upsh{D}_\nul}^*
      + \Mat{M}_\Rr^* \Mat{M}_\Rr & 2(\Mat{B}^2)^* \\
      2 \Mat{B}^2 & \mathcal{M}_{\tilde\nul}^2
      + {\Mat{m}^\upsh{D}_\nul}^\dag\Mat{m}^\upsh{D}_\nul
      + \Mat{M}_\Rr\Mat{M}^*_\Rr
    \end{pmatrix},
\end{align}
\end{subequations}
where bold face symbols as well as the soft slepton masses
\(\mathcal{M}^2_{\tilde \ell, \tilde\nul}\) denote \(3\times 3\)
matrices in flavour space and the singlet mass is symmetric:
\(\Mat{M}_\Rr = \Mat{M}_\Rr^T\). And \(\Mat{m}^\upsh{D}_\nul\)
is the Dirac neutrino mass matrix defined as
\(\Mat{m}^\upsh{D}_\nul=\frac{1}{\sqrt{2}} v_\uq \Mat{Y}_\nul\).
