On the Search for Low W0

The magnitude of the vacuum expectation value of the Gukov-Vafa-Witten superpotential (Formula presented.) plays a central role in the phenomenology of type IIB flux compactifications. Recent analytical constructions have shown that perturbatively flat vacua can be used to obtain very low values of (Formula presented.). We present systematic algorithms to carry out exhaustive numerical searches for such vacua. We also analyse them in the statistical context, as part of the entire ensemble of type IIB flux vacua at low (Formula presented.). Our preliminary analysis indicates that these perturbatively flat vacua are statistically sparse in the whole set of vacua at low (Formula presented.) as calculated by Denef and Douglas. Two-moduli examples are used to illustrate these more general findings in specific settings. We find that these simple (two moduli) cases are good examples for existence proofs but they do not feature a large statistical tuning freedom for phenomenological applications.