superpotential {#flavons}
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A crucial ingredient for the construction of the flavon superpotential is the assumption of a continuous $R$-symmetry under which matter superfields carry charge $+1$, flavons and Higgs fields are uncharged and fields driving the alignment of the flavon VEVs have charge $+2$. As a consequence, the equations determining the vacuum alignment are given by the $F$-terms of the driving fields which only appear linearly in the superpotential. The latter also do not have any direct couplings to matter superfields.

Impact of ${\bs \eta}$ on the vacuum alignment
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As one can check the field $\eta$ does not lead to any operator which strongly perturbs the vacuum alignment of the flavons achieved with the help of the driving fields $X^d_1$, $Y^d_2$, $Z^\nu_{3'} $, $Y^\nu_2$, $\ol X^d_1$, $X^{\nu d}_{1'}$, $Y^{du}_2$ , $X^u_1$ and $X^{\mathrm{new}}_1$, $\wt
X^{\mathrm{new}}_{1'}$, found in table 3 and in Eq.(5.1) of [@HKL10]. At the subleading level the most important new operators are \[etaflavonleading\] Y\^\_2 \^d\_2 \^\_1/M + Y\^\_2 \^d\_2 \^\_2/M + Z\^\_[3’]{} \^d\_2 \^\_[3’]{}/M , which are responsible for shifts in the vacuum of the fields $\Phi^\nu_2$ and $\Phi^\nu_{3'}$ and thus give rise to additional contributions to the light neutrino mass matrix which lead to further 