A Note on the Magnitude of the Flux Superpotential

The magnitude of the flux superpotential $W_{flux}$ plays a crucial role in determining the scales of IIB string compactifications after moduli stabilisation. It has been argued that values of $W_{flux}$ much less than one are preferred, and even required for physical and consistency reasons. This note revisits these arguments. We establish that the coupling (g) of heavy Kaluza-Klein modes to light states scales as ${M_{KK} / M_{Pl}}$ (hence is suppressed by two third powers of the inverse volume of compactification) and argue that consistency of the superspace derivative expansion requires $gF/M^2 \sim m_{3/2}/ M_{KK} << 1$, where $F$ is the auxiliary field of the light fields and $M$ the ultraviolet cutoff. This gives only a mild constraint on the flux superpotential, $W_{flux} << V^{1/3}$ (where V is the volume of the compactification), which can be easily satisfied for order one values of $W_{flux}$. This regime is also statistically favoured and makes the Bousso-Polchinski mechanism for the vacuum energy hierarchically more efficient.

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