The Black Hole Weak Gravity Conjecture with Multiple Charges

TL;DR

This paper investigates how higher-derivative corrections affect extremal black holes with multiple charges, showing that the multi-charge Weak Gravity Conjecture is consistent with quantum corrections and not a Swampland criterion.

Contribution

It provides a detailed analysis of the extremality bound modifications due to higher-derivative operators and demonstrates the positivity of the extremality form at finite charge.

Findings

Higher-derivative operators are renormalized only if they produce invariant matrix elements.

The one-loop logarithmic running of Wilson coefficients is computed.

The multi-charge Weak Gravity Conjecture remains consistent with quantum corrections.

Abstract

We study the effect of higher-derivative corrections on asymptotically flat, four-dimensional, non-rotating dyonic black holes in low-energy models of gravity coupled to $N$ $U(1)$ gauge fields. For large extremal black holes, the leading $\mathcal{O}\left(1/Q^2\right)$ correction to the extremality bound is calculated from the most general low-energy effective action containing operators with up to four derivatives. Motivated by the multi-charge generalization of the Weak Gravity Conjecture, we analyze the necessary kinematic conditions for an asymptotically large extremal black hole to decay into a multi-particle state of finite charge extremal black holes. In the large black hole regime, we show that the convex hull condition degenerates to the requirement that a certain quartic form, constructed from the Wilson coefficients of the four-derivative effective operators, is everywhere positive. Using on-shell unitarity methods, we show that higher-derivative operators are renormalized at one-loop only if they generate local, on-shell matrix elements that are invariant tensors of the electromagnetic duality group $U(N)$. The one-loop logarithmic running of the four-derivative Wilson coefficients is calculated and shown to imply the positivity of the extremality form at some finite value of $Q^2$. This result generalizes a recently given argument by Charles, and shows that under the given assumptions the multi-charge Weak Gravity Conjecture is not a Swampland criterion.