be possible to chose a cut-off small enough that $W_0$ will not change dramatically, and our results will be qualitatively unchanged with respect to the case with the quadratically divergent term omitted. We have assumed, and prove in appendix \[sec:appB\], that the derivatives of $V_{\mbox{quad.}}$ are similar in form and magnitude to $V_{\mbox{tree}}$’s. It is interesting to note that $V_{\mbox{quad.}}$ can play an important role in the potential despite being generated by gravity, in close analogy to the role played by $-3
e^K |W|^2$. This is in contrast to the gravitational corrections to the logarithmic potential, which are negligible in comparison to the globally supersymmetric terms. Naturally, we still have to re-tune to get $V=0$, but the loop corrections do not increase the degree of tuning required. Finally, if $Z\gtrsim 3$ it is clear that these models break down and the cosmological constant cannot be tuned to zero. This will not be the case unless the cut-off is close to the Planck scale. In these models, the relevant cut-off is the scale at which the magnetic description is no longer valid as supersymmetry is best described by the low energy variables in the magnetic theory.

We have implicitly assumed, in 