\[table:lam dep\] we can see that for $\mu=\sqrt{2}~10^{-7}$, $\<\Phi_0\>= -4.295~10^{-12} + 6.886~10^{-14}
 \mbox{Log}_{10} (\Lambda)$. Table \[table:Mu dep\] shows that there are no significant changes as one varies $\mu$. In table \[table:Non-pert\] $\Lambda_{\mbox{non-pert.}}$ is varied and the non-perturbative term becomes relevant for $\Lambda_{\mbox{non-pert.}}>4~10^{-2}$. Since we expect the non-perturbative term to be suppressed by an extra power of $\Phi$ when $N_f=4$, corresponding to $\Lambda_{\mbox{non-pert.}}\sim \Phi
 \sim \mu^2 = 2~10^{-14}$ it is clearly well under control.

Analytic Approximation
----------------------

In addition we can compare our numerical results to analytical approximations. Because of the expected smallness of the fields it is sufficient to take the leading order of $\mu$ when searching for the minimum. The gravitational corrections to the logarithmic one loop potential appear with higher powers of $\mu$ than the SUSY effective potential and can be neglected in the analytical approximation, while the quadratic corrections are both relevant and readily calculable. The solution to $\frac{\partial V}{\partial \Phi_0}=0$, derived in appendix \[sec:appB\], under the assumption[^8] that $V_{\mbox{log}}$ is described by Eq. , is given by:

$     \begin{array}[b]{|c|ccccccc|}
       \hline
       N_f & 2 & 2 & 2 & 2 & 2 & 2 & 2\\
       N   & 1 & 1 & 1 & 1 & 1 