\sim ({\bf 1},-8)$.[^10] Up to the order $\la^8$ the following terms can contribute M\_[V\_1]{} V\_1 \^\_1 + V\_1 \^u\_2 \^\_2 + V\_1 (\^d\_[2]{})\^8/M\^6 . Plugging in the LO vacuum alignment of the flavons renders the last term irrelevant and thus we get a correlation between $\wt\varphi^u_2$, $\varphi^\nu_1$ and $M_{V_1}$. We can determine $\wt\varphi^u_2$ and it has order $\la^4 M$, if we choose the mass scale $M_{V_1}$ to be of the order $\la^4 M$. Corrections to the relation obtained from the $F$-term of $V_1$ arise from terms at order $\la^9$ and higher and are not relevant because they only slightly change the value of the VEVs given in terms of parameters of the flavon superpotential.

Fixing the VEV of ${\bs \eta}$
------------------------------

The simplest possibility to relate the VEV of the field $\eta$ to an explicit mass scale is to introduce a driving field $V_{\eta}$ transforming as ${\bf
  1}$ under $S_4$ and carrying $U(1)$ charge $-7$. Then the term \[VetaLO\] M\_ V\_is allowed. Furthermore, up to order $\la^9$ the structures V\_(\^d\_2)\^7/M\^5 + V\_(\^d\_2)\^3 \^d\_3 \^\_[3’]{}/M\^3 are compatible with all symmetries of the model. Plugging in the LO vacuum, only the latter is relevant and leads together with the operator in Eq.(\[VetaLO\]) 