contribution

Let us firstly discuss the LL RR double insertion, ${\cal I}m(\d^{LL} \mu \eta_\ell m_\ell \d^{RR})_{ii}$, which turns out to be the quantitatively most interesting one. Since in general $\eta_\ell \approx \i3$ and the phase of $\mu$ - if any - is extremely small so that it can be safely neglected in the discussion, ${\cal I}m(\d^{LL}  \mu \eta_\ell m_\ell \d^{RR})_{ii}   \approx   \mu  {\cal I}m(\d^{LL}m_\ell  \d^{RR})_{ii}$. From eqs. (\[eq\_Rss\], \[eq\_Lss\]) it appears that $\d^{RR}_{ij}$, $\d^{LL}_{ij}$ are respectively of second and first order in the $t$’s. Thus, the lowest order is the cubic and the LL RR contribution reads: m(\^[LL]{} m\_\^[RR]{})\_[ii]{} = 8 m\_[i]{} \_[a&gt;b]{} t\_a t\_b (t\_a+t\_b) [I]{}m(N\_a E N\_b)\_[ii]{} . \[ss\_LLRR\] Notice that ${\cal I}m(N_a E N_b)_{ii} = \tan^2 \beta {\cal I}m(N_a m_\ell^2 N_b)_{ii} /m_t^2$, where $m_t$ is the top mass. Due to the hierarchy in $m_\ell$, a potentially important effect could only come from ${\cal I}m({N_a}_{i3}  {N_b}_{3i})$. Notice also that, on the contrary of the FC contribution, (\[ss\_LLRR\]) doesn’t strongly depend on $a_0 $.

The relative importance between this amplitude and the FC one can be easily appreciated by neglecting the terms proportional to $t_1$, which, as already mentioned, is justified when the lightest right-handed neutrino has the smallest 