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The black hole weak gravity conjecture (WGC) is a set of linear inequalitieson the four-derivative corrections to Einstein--Maxwell theory. Remarkably, infour dimensions, these combinations appear in the $2 \to 2$ photon amplitudes,leading to the hope that the conjecture might be supported using dispersionrelations. However, the presence of a pole arising in the forward limit due tograviton exchange greatly complicates the use of such arguments. In this paper,we apply recently developed numerical techniques to handle the graviton pole,and we find that standard dispersive arguments are not strong enough to implythe black hole WGC. Specifically, under a fairly typical set of assumptions,including weak coupling of the EFT and Regge boundedness, a small violation ofthe black hole WGC is consistent with unitarity and causality. We quantify thesize of this violation, which vanishes in the limit where gravity decouples andalso depends logarithmically on an infrared cutoff. We discuss the meaning ofthese bounds in various scenarios. We also implement a method for boundingamplitudes without manifestly positive spectral densities, which could beapplied to any system of non-identical states, and we use it to improve boundson the EFT of pure photons in absence of gravity.

Bounding Violations of the Weak Gravity Conjecture