On The Black Hole Weak Gravity Conjecture and Extremality in the Strong-Field Regime

We point out that the Weak Gravity Conjecture (WGC) implies that sufficiently small extremal black holes are necessarily in the strong-field regime of electrodynamics, and therefore probe the UV completion of the Maxwell sector. To investigate the WGC bounds arising from these small extremal black holes, we revisit black hole decay in generic field theories in asymptotically flat space. We derive a necessary and a sufficient condition for any black hole to decay, the latter amounting to a bound on the growth of charge relative to mass. We apply these conditions to extremal black holes derived in various UV completions of the Maxwell sector. We find that the Euler-Heisenberg and DBI effective actions satisfy the sufficient condition for decay, while the ModMax model fails the necessary one, rendering it incompatible with the WGC. Using the decay conditions, we show that the black hole WGC implies positivity of the $U(1)$ gauge coupling beta function. This provides an independent argument that classically stable (embedded-Abelian) colored black holes cannot exist. We also show that the black hole WGC constrains conformal hidden sector models, and is always satisfied in their AdS dual realizations.