({\bf 10},{\bf 1}, 6)$, $\Delta \sim ({\bf 5},{\bf 1}, 7)$, $\Upsilon \sim ({\bf 5},{\bf 2}, 5)$, $\Omega \sim ({\bf 5},{\bf 3}, -6)$, $\Theta \sim ({\bf 5},{\bf 3}, -5)$ under $(SU(5),S_4,U(1))$ and their vector-like partners[^11] giving rise to the diagrams shown in figure 1 of [@HKL10].

In order to UV-complete the operator generating the mass of the bottom quark and the tau lepton, see Eq.(\[LOdown\]), we add the messengers $B$ and $\ol{B}$ with quantum numbers as found in table \[heavyfields\_new\]: \_1 T\_3 H\_ + \_2 B F \^d\_3 + M\_B B . The effective operators giving rise to up and charm quark masses, see Eq.(\[LOup\]), are promoted to renormalisable operators with the help of the messengers $A$ and $\Xi$ and their vector-like partners[^12] \_1 T A H\_5 + \_2 T \^u\_2 +\_1 \^u\_2 + \_2 T \^u\_2 + M\_A A + \_4 A \^u\_2 + M\_ . The operator, responsible for the largish value of the lepton mixing angle $\theta_{13}^l$, see Eq.(\[theta13op\]), arises in the UV completion from \_1 N \^d\_2 + \_2 N + M\_ +\_5 \^u\_2 . The diagram belonging to these messengers can be found in figure \[diagrameta\].

[diagrameta]{}

(50,20)

Integrating out the messengers, present in table \[heavyfields\_new\], 