can be read off from the matrix $M_e$ in Eq.(\[matrixMe\]), $\theta_{12}^e \approx \theta_C/3 \approx \la/3$: \_[13]{}\^l \^e\_[12]{} \^\_[23]{} \^e\_[12]{}/ /(3 ) 0.05 , for TB mixing in the neutrino sector. The other two lepton mixing angles are \^2 \^l\_[23]{} 1/2 , \^2 \^l\_[12]{} 1/3 + 2/9 \^l  , with $\delta^l$ being the leptonic Dirac CP phase. As has been shown in detail in [@HKL10] subleading corrections coming from operators with several flavons hardly alter the fermion mass matrices and the vacuum alignment, presented in Eqs.(\[VEVup\]-\[VEVnu\]). Thus all results achieved at LO are uncorrected to good approximation.

Extended SUSY $\bs{S_4 \times SU(5)}$ model {#sec3}
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Generation of large $\bs{\theta_{13}^l}$ {#largeth13}
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A simple way to enhance the value of $\theta_{13}^l$ is to add a new flavon $\eta$ which couples to right-handed neutrinos and gives rise to a contribution which partly breaks the $Z_2 \times Z_2$ subgroup preserved by the LO VEVs of the flavons $\Phi^\nu_{1}$, $\Phi^\nu_{2}$ and $\Phi^\nu_{3'}$. Recall that the $Z_2 \times Z_2$ symmetry is generated by $S$ and $U$, defined in the appendix of [@HKL10], and is responsible for TB mixing in the neutrino sector. Adding a field $\eta$ which transforms as ${\bf 1}$ under $S_4$ and carries $U(1)$ 