the operators directly relevant for fermion masses and mixing in this model. Note that the way of generating large $\theta_{13}^l$ discussed here is very similar to the one proposed in [@KL11] where it has been realised in the context of a non-unified SUSY $S_4$ model.

The structure of the paper is as follows: in section \[recap\] we recapitulate the main features and results of the original model; in section \[sec3\] we discuss the operators which arise from introducing the new flavon $\eta$ and show the results for the lepton mixing; section \[flavons\] contains a brief discussion of the relevant parts of the flavon superpotential, while we present the messengers of a simple UV completion in section \[messengers\]. We comment on the differences between the choices $\eta \sim {\bf 1}$ and $\eta \sim {\bf 1'}$ under $S_4$ in section \[eta1pr\] and conclude in section \[conclusions\].

Original SUSY $\bs{S_4 \times SU(5)}$ model {#recap}
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We consider a SUSY $SU(5)$ model in four dimensions. The full flavour symmetry of the model is $S_4 \times U(1)$. For details of the group theory of $S_4$ we refer to the appendix of [@HKL10]. The $U(1)$ symmetry is necessary to control the operators with several flavons.[^4] The 