the Weinberg operator is at most of the relative order $\la^8$ with respect to the leading contribution from the operators in Eq.(\[LOnu\]), if we assume the Weinberg operator to be suppressed by the generic messenger scale $M$, which is supposed to be of the order of the scale of grand unification.[^8]

Subleading operators involving ${\bs \eta}$ {#eta_charged_sector}
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Since the additional flavon $\eta$ transforms as a trivial singlet under $S_4$, we find the following operator (up to $\la^8$ assuming the sizes of the flavon VEVs as given above) \[opcheta\] F T\_3 H\_ \^d\_3 /M\^2  , which contributes to the down quark and the charged lepton mass matrix. Using the LO VEV of $\wt\Phi^d_3$, see Eq.(\[VEVdown\]), the operator leads to a contribution to the (32) ((23)) and (33) entries of $M_d$ (and $M_e$) which is of order $\la^7$. Such contributions can be absorbed into the parameters present in the down quark and charged lepton mass matrices found in [@HKL10] and thus do not change the results for fermion masses and mixing presented there. Apart from the operator in Eq.(\[opcheta\]) no additional operators involving $\eta$ and contributing directly to the charged fermion mass matrices up to the order $\la^8$ are generated.

Flavon 