or could be identified with $\wh{S}_4$ or $\wh{S}_5$ of (\[Wfull\]). To the extent that string models seldom give self-couplings at such a low order in the superpotential, we might expect $\lambda_{11}  =
\lambda_{22} = \lambda_{33} =0$, thereby generating (\[perturb\]) at roughly the correct order of magnitude.

Now let us consider the leptonic (PMNS) mixing matrix defined by $U_{\rm PMNS} = U_e^{\dagger}U_{\nu}$, where $U_{\nu}$ is the matrix in (\[Unu\]) and $U_e$ is the analogous matrix for the charged leptons. Most of the earlier studies [@DeGouvea:2005gd; @Langacker:2004xy; @Altarelli:2004za; @Mohapatra:2004vr; @Barger:1998ta; @Barbieri:1998mq; @Altarelli:1998nx; @Jezabek:1998du; @Mohapatra:1999zr; @Babu:2002ex] of the texture in (\[lambdaT1\]) assumed that this form holds in the basis for which $U_e=1$. In that case, one has an inverted hierarchy and $U_{\rm PMNS} = U_{\nu}$ is bimaximal for $a=b$, i.e., $\theta_{12}=\theta_{23}=\pi/4$, while $\theta_{13}=0$. For $|a|\ne |b|$ the solar mixing remains maximal while the atmospheric mixing angle is $|\tan \theta_{23}|=|b|/|a|$. It is now well established, however, that the solar mixing is not maximal, i.e., $\pi/4-\theta_{12}=0.19^{+0.05}_{-0.06}$ [@Fogli:2005cq], where the quoted errors are 2$\sigma$. It is well-known that reasonable perturbations on this texture (still with $U_e=1$) have difficulty yielding a realistic solar mixing and mass splitting. To see this, let us add a perturbation $\delta$ to 