Quantum gravitational corrections to Reissner-Nordström black hole thermodynamics and their implications for the weak gravity conjecture

In this paper, we investigate the quantum gravitational corrections to the thermodynamical quantities of Reissner-Nordström black holes within the framework of effective field theory. The effective action originates from integrating out massless particles, including gravitons, at the one-loop level. We perform a complete thermodynamic analysis for both non-extremal and extremal black holes, and are mainly concerned about the shift in the charge-to-mass ratio $q/M$ that plays an important role in analyzing the weak gravity conjuecture. For non-extremal black holes, we identify a relationship between the shift in the charge-to-mass ratio and the thermodynamic stability of the black holes. For extremal black holes, we show that quantum gravity effects naturally lead to the super-extremality $q/M>1$ of charged black holes.

💡 Research Summary

In this work the authors investigate how one‑loop quantum‑gravity effects modify the thermodynamics of Reissner‑Nordström (RN) black holes and what implications these modifications have for the Weak Gravity Conjecture (WGC). Starting from the Einstein‑Maxwell action, they integrate out all massless fields—including gravitons—to obtain a non‑local effective action that captures the leading infrared quantum corrections. The non‑local part consists of curvature‑squared operators multiplied by logarithmic form factors, schematically
(L_{\rm hd}= -\alpha R\ln(\Box/\mu_^2)R + \beta R_{\mu\nu}\ln(\Box/\mu_^2)R^{\mu\nu} + \gamma R_{\mu\nu\rho\sigma}\ln(\Box/\mu_*^2)R^{\mu\nu\rho\sigma}).
Because the RN background is Ricci‑flat, the (\alpha) term does not contribute, leaving only the (\beta) and (\gamma) coefficients. These coefficients are universal: they are determined solely by the numbers of massless scalars ((n_s)), fermions ((n_f)) and vectors ((n_v)) in the low‑energy spectrum, as given in equations (4) and (5).

A direct variation of the action to obtain the corrected metric is technically prohibitive due to the presence of (\ln\Box). Instead the authors employ a recent theorem (Reall & Santos, 2020) stating that, to first order in the small parameter (\epsilon) that multiplies the higher‑derivative sector, the on‑shell Euclidean action—and therefore all thermodynamic quantities—can be evaluated on the unperturbed RN solution. The Einstein‑Hilbert part contributes only at order (\epsilon^2) because the RN metric already extremizes it, while the higher‑derivative part contributes linearly in (\epsilon) when evaluated on the classical background.

Carrying out the integration of the non‑local terms from the horizon to infinity (with appropriate regularisation of the logarithmic kernel) yields a corrected free energy (eq. 10):
(F(T,q)=F_{\rm RN}(T,q)-\frac{16\pi\epsilon(\beta+4\gamma)q^4-5\gamma q^2\bar r_h^2+5\gamma\bar r_h^4}{5\bar r_h^5},\ln(\bar r_h^2\mu_^2)).
Here (\bar r_h) is the horizon radius of the classical RN solution related to temperature and charge by the usual relation (T= ( \bar r_h - q^2/\bar r_h )/(4\pi\bar r_h^2)). The logarithm is assumed large, (\ln(\bar r_h^2\mu_^2)\gg1), so it dominates over subleading power‑suppressed contributions.

From the free energy the entropy (S=-\partial F/\partial T), electric potential (\Phi=\partial F/\partial q) and mass (M=F+TS+q\Phi) are derived. The key physical outcomes are:

1. Non‑extremal regime ((q\le \bar r_h<\sqrt{3},q)) – This interval coincides with the region where RN black holes have positive specific heat and are thermodynamically stable. The quantum correction reduces the mass at fixed charge, i.e. (\partial M_{\rm ext}(q,\epsilon)/\partial\epsilon<0). Consequently the charge‑to‑mass ratio (q/M) increases. For the standard particle content of the Standard Model (or any theory with a positive combination (\beta+4\gamma)), the shift is such that the corrected black hole moves closer to, and can even surpass, the extremality bound. This provides a concrete, model‑independent realization of the WGC condition that an object with (q/M>1) must exist.

2. Extremal limit ((\bar r_h\to q)) – In the zero‑temperature limit the logarithmic term becomes even more pronounced. The corrected extremal mass takes the form
   (M_{\rm ext}(q,\epsilon)=q\bigl