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Weak Gravity Conjecture from Unitarity and Causality

Yuta Hamada, Toshifumi Noumi, Gary Shiu

TL;DR

This work provides an existence proof of the mild Weak Gravity Conjecture across broad classes of theories by linking unitarity and causality constraints to higher-derivative corrections in the photon-graviton EFT. By deriving positivity bounds on the relevant four-derivative couplings and analyzing causality, the authors show that heavy extremal black holes acquire a charge-to-mass ratio z>1 under two key UV scenarios: (i) dominance of tree-level effects from light neutral bosons, such as dilatons or moduli, and (ii) open-string-type UV completions where graviton Regge states are subdominant in photon scattering. They extend the analysis to higher dimensions and demonstrate positivity of black-hole entropy corrections, reinforcing the WGC linkage to fundamental consistency conditions. The results provide a robust, model-wide existence proof of the mild WGC in string-inspired settings and shed light on how UV completions shape low-energy gravity with quantum consistency.

Abstract

The weak gravity conjecture states that quantum gravity theories have to contain a charged state with a charge-to-mass ratio bigger than unity. By studying unitarity and causality constraints on higher derivative corrections to the charge-to-mass ratio of extremal back holes, we demonstrate that heavy extremal black holes can play the role of the required charged state under several assumptions. In particular, our argument is applicable when the higher spin states Reggeizing graviton exchange are subdominant in the photon scattering. It covers (1) theories with light neutral bosons such as dilaton and moduli, and (2) UV completion where the photon and the graviton are accompanied by different sets of Regge states just like open string theory. Our result provides an existence proof of the weak gravity conjecture in a wide class of theories, including generic string theory setups with the dilaton or other moduli stabilized below the string scale.

Weak Gravity Conjecture from Unitarity and Causality

TL;DR

This work provides an existence proof of the mild Weak Gravity Conjecture across broad classes of theories by linking unitarity and causality constraints to higher-derivative corrections in the photon-graviton EFT. By deriving positivity bounds on the relevant four-derivative couplings and analyzing causality, the authors show that heavy extremal black holes acquire a charge-to-mass ratio z>1 under two key UV scenarios: (i) dominance of tree-level effects from light neutral bosons, such as dilatons or moduli, and (ii) open-string-type UV completions where graviton Regge states are subdominant in photon scattering. They extend the analysis to higher dimensions and demonstrate positivity of black-hole entropy corrections, reinforcing the WGC linkage to fundamental consistency conditions. The results provide a robust, model-wide existence proof of the mild WGC in string-inspired settings and shed light on how UV completions shape low-energy gravity with quantum consistency.

Abstract

The weak gravity conjecture states that quantum gravity theories have to contain a charged state with a charge-to-mass ratio bigger than unity. By studying unitarity and causality constraints on higher derivative corrections to the charge-to-mass ratio of extremal back holes, we demonstrate that heavy extremal black holes can play the role of the required charged state under several assumptions. In particular, our argument is applicable when the higher spin states Reggeizing graviton exchange are subdominant in the photon scattering. It covers (1) theories with light neutral bosons such as dilaton and moduli, and (2) UV completion where the photon and the graviton are accompanied by different sets of Regge states just like open string theory. Our result provides an existence proof of the weak gravity conjecture in a wide class of theories, including generic string theory setups with the dilaton or other moduli stabilized below the string scale.

Paper Structure

This paper contains 22 sections, 69 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Strategy
  3. Unitarity constraints
  4. WGC from unitarity
  5. Causality constraints
  6. Case (1): theories with light neutral bosons
  7. Case (2): open string type UV completion
  8. Conclusion
  9. I. Derivation of the unitarity bounds
  10. A basis for forward scattering amplitudes
  11. Positivity bounds
  12. On graviton exchange and Regge states
  13. II. Extension to $D\geq5$
  14. Effective action and unitarity
  15. Causality constraints
  16. ...and 7 more sections

Figures (2)

  • Figure 1: A schematic picture of the particle spectrum: We assume that photon and graviton control the BH dynamics in the infrared. The ordinary QFT description breaks down at $\Lambda_{\rm QFT}$, which corresponds to the string scale $M_s$ in string theory. The spectrum may contain light particles below $\Lambda_{\rm QFT}$ (left), but it is also possible that there are no such light particles (right).
  • Figure 3: A flow chart for our derivation of the WGC from unitarity: Each step explains which conditions are necessary besides unitarity to show that heavy extremal BHs have the charge-to-mass ratio $z>1$ and thus the mild form of WGC is satisfied.