structures (up to order $\la^{12}$, $D^\nu=Y^\nu_2, Z^\nu_{3'}$, $a, b=1,2,3'$): D\^\^\_a \^\_b , D\^\^d\_2 \^\_a/M , D\^\^2 (\^d\_2)\^2/M\^2 , D\^\^\_a \^\_b \^u\_2/M , D\^(\^d\_2)\^8 \^\_a/M\^7 . As one can check, they only involve flavons whose VEV alignment preserves the generator $S$ and, hence, the shifts $\Delta^\nu_{i,j}$ are of the form in Eq.(\[nushifts\]).[^9] The shifts $\Delta^\nu_{i,j}$ are enhanced by a factor $\la^{-3}$ compared to the original model, cf. Eq.(5.3) in [@HKL10]. Thus, they contribute at the same level, if plugged into the leading operators in Eq.(\[LOnu\]), as the operator in Eq.(\[theta13op\]), to the neutrino mass matrix. In particular, they also induce a non-zero value of $\theta_{13}^\nu$ in the neutrino sector. Due to the preservation of the generator $S$ all such contributions can be cast into the form of $m^{eff}_\nu$ shown in Eq.(\[mnuLO\]). Furthermore, these enhanced shifts do not generate corrections to the charged fermion mass matrices which are of order $\la^8$ or larger.

There are several operators involving the driving fields and the field $\eta$ which arise at the level $\la^{11}$ or smaller and are irrelevant for the discussion of the vacuum alignment as well as for the size of the leading VEV shifts. Thus, we get the same results for the 