in the Goldstone directions. We expect there to be both logarithmic and quadratic corrections stemming from this. The quadratic terms we can ignore if $\Lambda \ll 1$, but the logarithmic terms we have to consider more carefully. We note that all the operators in [@Gaillard:1993es] that contain $V$ are dimension 8 and so the largest possible linear contribution would be $\mathcal{O}(W_0^2 \mu X)\sim \mu^5X$ where $X$ is a generic field. This should be compared to the largest contribution from gravity at tree-level, $X F W_0\sim \mu^4 X$, and so we expect these effects to be negligible.

As noted in [@Intriligator:2006dd] and [@Grisaru:1996ve] one can capture some information about the effective potential purely by integrating out fields and calculating the correction to the Kähler potential. However, as described in the appendix of [@Intriligator:2006dd], this is an approximation only valid to 2nd order in $F$, we also note that it is harder to work with numerically. Hence we opt to calculate the full one-loop potential. It is nonetheless interesting to compare these two approaches and we see that the (somewhat arbitrary) corrections to the Kähler potential introduced in [@Haba:2008vj], created an explicit cut-off dependence into the effective potential for $\Phi_0$ and hence 