A Note on the Magnitude of the Flux Superpotential

TL;DR

This note reevaluates the traditional requirement of a tiny flux superpotential $W_0$ in IIB compactifications. It shows, via three independent analyses, that heavy–light couplings between Kaluza–Klein modes and light fields are suppressed as $g \sim \mathcal{V}^{-2/3}$, so the EFT consistency condition becomes $W_0/\mathcal{V}^{1/3} \ll 1$, a mild constraint easily satisfied at large volume. Consequently, order-one $W_0$ values are natural, especially in the Large Volume Scenario (LVS), and actually enhance the efficiency of Bousso–Polchinski-type tuning for the cosmological constant; small $W_0$ is not a general necessity for phenomenology or EFT validity. The work also contrasts KKLT and LVS, highlighting that LVS does not require $W_0$ to be small to yield viable moduli stabilization or TeV-scale soft terms, with the gravitino mass set by $m_{3/2} \sim e^{K/2}W_0 \sim W_0 M_P/\mathcal{V}$.

Abstract

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.