Naturalness and the Weak Gravity Conjecture

The weak gravity conjecture (WGC) is an ultraviolet consistency condition asserting that an Abelian force requires a state of charge q and mass m with q>m/mPl. We generalize the WGC to product gauge groups and study its tension with the naturalness principle for a charged scalar coupled to gravity. Reconciling naturalness with the WGC either requires a Higgs phase or a low cutoff at Λ∼qmPl. If neither applies, one can construct simple models that forbid a natural electroweak scale and whose observation would rule out the naturalness principle.