Bounding Violations of the Weak Gravity Conjecture

TL;DR

We address bounding four-derivative corrections in Einstein–Maxwell EFT via the black hole WGC by employing dispersion relations for 2→2 photon scattering. The graviton pole in the forward limit complicates standard positivity arguments, so we implement a numerical, functionally-based SDP approach, including integral sum rules and a generalized optical theorem to bound amplitudes even when spectral densities are not manifestly positive. In the gravity-full theory, we demonstrate that the WGC can be violated by amounts suppressed by $\frac{M^2}{M_{\mathrm P}^2}$ and enhanced by a logarithmic IR cutoff, with explicit bounds such as $g_2 \ge -\frac{c}{M^2 M_{\mathrm P}^2}\log\left(\frac{M}{m_{\mathrm IR}}\right) + \frac{c_0}{M^2 M_{\mathrm P}^2}$ (numerical constants given), while gravity decoupling restores the standard WGC positivity. The work also develops methods to bound amplitudes without manifestly positive spectral densities, improving photon EFT bounds in the absence of gravity. Overall, gravity weakens the rigidity of dispersion-based EFT bounds and provides a concrete, quantitative window for BH-WGC violations, with clear implications for UV completions and the Swampland program.

Abstract

The black hole weak gravity conjecture (WGC) is a set of linear inequalities on the four-derivative corrections to Einstein--Maxwell theory. Remarkably, in four dimensions, these combinations appear in the $2 \to 2$ photon amplitudes, leading to the hope that the conjecture might be supported using dispersion relations. However, the presence of a pole arising in the forward limit due to graviton 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 imply the black hole WGC. Specifically, under a fairly typical set of assumptions, including weak coupling of the EFT and Regge boundedness, a small violation of the black hole WGC is consistent with unitarity and causality. We quantify the size of this violation, which vanishes in the limit where gravity decouples and also depends logarithmically on an infrared cutoff. We discuss the meaning of these bounds in various scenarios. We also implement a method for bounding amplitudes without manifestly positive spectral densities, which could be applied to any system of non-identical states, and we use it to improve bounds on the EFT of pure photons in absence of gravity.

Figures (15)

• Figure 1: Schematic representation of our bounds: the weak gravity conjectures is satisfied at leading order (dark shaded region), but violations are still admissible at sub-leading order in $M/M_\mathrm{P}$ (light shaded region).
• Figure 2: The starting point for our dispersive argument is the dashed contour at infinity in the $s'$ plane. It is deformed inwards to pick up contributions from three low-energy poles and two infinite cuts.
• Figure 3: Some bounds on the six-derivative term $M^2 h_3/g_2$. The dots refer to the partial UV completions as above.
• Figure 4: Allowed region in the space $(M^4 g_{4,1}/g_2,M^4 g_{4,2}/g_2)$. In the left figure we show in orange the one sided bound (allowed above, disallowed below) from Arkani-Hamed:2020blm, in light blue the result from Henriksson:2021ymi and in dark blue our new result. In the other figure we zoom in the new allowed region, adding some partial completions \ref{['tab:EFTcoefValues_trees']}.
• Figure 5: Exclusion plot in the plane $(g_2, \beta^2)$ properly normalized and divided by $\log(M/m_\mathrm{IR})$. The shaded regions represent the allowed values obtained by using various combinations of dispersion relations. In particular the light blue only uses the $\mathcal{I}_g$ dispersion relation, while the darker blue uses the $\mathcal{I}_g$,$\mathcal{I}_0$ and the $\mathcal{I}_g$, $\mathcal{I}_0$, $\mathcal{I}_{\beta^2}$ dispersion relations. The bounds have been obtained in the parametric limit $\log(M/m_\mathrm{IR})\gg 1$.