\lambda^2_{22} / M_{\nu_2}$ and a heavy neutrino with mass $m_{\nu_3} \sim \lambda^2_{33} / M_{\nu_3}$.

Here we briefly summarize the parts of the Pati-Salam model [@PaSa] that are relevant for our analysis. For a more complete discussion see Ref. [@AnLe]. The SM fermions, together with the right-handed neutrinos, are conveniently accommodated in the following $F=(4,2,1)$ and $F^c=(\bar 4,1,\bar 2)$ representations : $$\qquad
F_A   = \left(\matrix{
            u & u & u & \nu \cr
            d & d & d & e \cr}
            \right)_A
\qquad
F^c_B = \left(\matrix{
            d^c & d^c & d^c & e^c \cr
            u^c & u^c & u^c & \nu^c \cr}
            \right)_B
\label{FcFields}$$ The MSSM Higgs bosons fields are contained in $h=(1,\bar 2,2)$ : $$h = \left(\matrix{h_d^- & h_u^0 \cr
                  h_d^0 & h_u^+ \cr} \right)
\label{SMHiggs}$$ whereas the heavy Higgs bosons $\bar H=(\bar 4,1,\bar 2)$ and $H=(4,1,2)$ are denoted by : $$\bar H = \left(\matrix{ \bar H_d & \bar H_d & \bar H_d & \bar H_e \cr
                        \bar H_u & \bar H_u & \bar H_u & \bar H_\nu
                        \cr}\right)
\qquad
H  = \left(\matrix{ H_d & H_d & H_d & H_e \cr
                    H_u & H_u & H_u & H_\nu  \cr}\right).
\label{RHiggs}$$ In addition to the Higgs fields in Eqs. , the 