importance to the tree-level.

We may also compute the value of $W_0$ required to cancel the cosmological constant: $$\begin{aligned}
 W_0=\left(\frac{e^{-K} V_{\mbox{log}} + \left(N_f'\Lambda'^2 +1 \right)h^2 N_c \mu^2}{3\left(N_f'\Lambda'^2+1\right)-(N_f'-2)\Lambda'^2} \right)^{1/2}\end{aligned}$$ Where $\Lambda'=\frac{\Lambda}{4\pi}$ and $N_f'=N_f^2+2 N_f N_c-1$.

The minima of the logarithmic potential and tree-level potential need not coincide, which is fortunate since the tree-level potential is minimised by Planck scale moduli vevs. Therefore the addition of the logarithmic potential to the tree-level will shift the minimum away from the tree-level, with the size of the shift being determined by the strength of the coupling constants and the relative importance of gravity. Inclusion of the quadratic, one-loop potential will shift the minimum closer to the tree-level result for all fields as discussed in appendix \[sec:appB\].

Remarks on methodology
----------------------

Since computation of $V_{\mbox{log}}$ requires diagonalisation of $\mathcal{M}^2$ it is significantly more involved than the calculation of the quadratic piece. It is possible to make some analytic progress by using the one-loop moduli masses derived in [@Intriligator:2006dd] and tree-level gravity corrections. However, more involved analytic calculations, such as computing the logarithmic piece of Eq. (\[eq:V\_one-loop\]) (with or without gravitational corrections to the masses) or computing corrections to the Kähler potential, as detailed in [@Grisaru:1996ve] 