Causality, Unitarity, and the Weak Gravity Conjecture

TL;DR

This paper investigates how unitarity and causality constraints influence extremal black hole charge-to-mass ratios within effective field theories, providing evidence supporting the Weak Gravity Conjecture through threshold correction analysis.

Contribution

It demonstrates that unitarity and causality impose positivity constraints on threshold corrections, ensuring the WGC is satisfied in a broad class of theories, including supersymmetric and minimally coupled models.

Findings

Logarithmic running at one loop increases the Wilson coefficient, supporting large black holes with Q>M.

Causality and unitarity imply positive threshold corrections from integrating out massive states.

Shift in extremal Q/M ratio correlates with on-shell action shift, favoring the WGC.

Abstract

We consider the shift of charge-to-mass ratio for extremal black holes in the context of effective field theory, motivated by the Weak Gravity Conjecture. We constrain extremality corrections in different regimes subject to unitarity and causality constraints. In the asymptotic IR, we demonstrate that for any supersymmetric theory in flat space, and for all minimally coupled theories, logarithmic running at one loop pushes the Wilson coefficient of certain four-derivative operators to be larger at lower energies, guaranteeing the existence of sufficiently large black holes with $Q>M$. We identify two exceptional cases of nonsupersymmetric theories involving large numbers of light states and Planck-scale nonminimal couplings, in which the sign of the running is reversed, leading to black holes with negative corrections to $Q/M$ in the deep IR, but argue that these do not rule out extremal black holes as the requisite charged states for the WGC. We separately show that causality and unitarity imply that the leading threshold corrections to the effective action from integrating out massive states, in any weakly coupled theory, can be written as a sum of squares and is manifestly positive for black hole backgrounds. Quite beautifully, the shift in the extremal $Q/M$ ratio is directly proportional to the shift in the on-shell action, guaranteeing that these threshold corrections push $Q>M$ in compliance with the WGC. Our results apply for black holes with or without dilatonic coupling and charged under any number of ${\rm U}(1)$s.