confirmed by Eqs.(\[thetanu\],\[asr\]). The requirement that $\theta_{13}^l \sim \la$ originates from the operator in Eq.(\[theta13op\]) implies that the order of the VEV of $\eta$ has to be \^4 M , since the leading terms giving rise to right-handed neutrino masses are of the order $\la^4 \, M$, see Eq.(\[matrixMR\]). The explicit form of the contribution due to the term in Eq.(\[theta13op\]) is \^d\_2 (N\_1 N\_2 + N\_2 N\_1 + N\_3 N\_3 )/M , using the LO vacuum in Eq.(\[VEVdown\]). Note that we can also write down an operator similar to the one in Eq.(\[theta13op\]) for the choice $\eta\sim
{\bf 1'}$. We comment on this possibility and the differences between the two choices, $\eta \sim {\bf 1}$ and $\eta \sim {\bf 1'}$, in section \[eta1pr\]. The light neutrino mass matrix, arising from the type I see-saw mechanism, can be cast into the form \[mnuLO\] m\_\^[eff]{} =(

[ccc]{} b\_+c\_-a\_& a\_& a\_+ d\_\
a\_& b\_+ d\_ & c\_\
a\_+ d\_ & c\_& b\_

) ( ) . The matrix $m^{eff}_\nu$ clearly shows that the tri-maximal vector $(1,1,1)^T$ is still an eigenvector of $m_\nu^{eff}$, even for $d_\nu \neq 0$, which is traced back to the invariance of the neutrino mass matrix under $S$ \[Strimax\] 