Convergent sum of EFT corrections to Schwarzschild metric requires UV locality

TL;DR

The paper studies perturbative corrections to the Schwarzschild solution within an effective field theory of gravity described by $C_{
2\u00a4\u00a8}{\cal F}_C(\Box/\Lambda^2)C^{\u00a8\u00a8\u0124\u0120}$, expanding in the perturbation parameter $\lambda$ and exploring resummation into closed-form expressions. By analyzing corrections far from the horizon, the authors derive explicit metric deformations and provide integral representations that depend on the form-factor ${\cal F}_C(\tau)$, linking the deformations to the UV properties of the graviton propagator. A key result is that the perturbative series converges only for UV-localizable theories; non-localizable choices with exponential growth in $\Box$ lead to divergent series, tying BH perturbativity to UV locality. In four dimensions, loop-induced logarithmic corrections ${\cal F}(\Box)=\gamma(\mu)\log(\Box/\mu^2)$ can dominate over tree-level EFT terms, highlighting the essential role of non-analytic loop effects in shaping the Schwarzschild geometry and motivating extensions to other BH solutions such as Kerr.

Abstract

Corrections to vacuum black hole solutions of general relativity (GR) are considered in an effective field theory (EFT) framework, perturbatively in EFT coefficients, focusing on the Schwarzschild solution of GR. We find dominant corrections to the Schwarzschild metric in all orders in the derivative expansion far away from the horizon. These corrections can be summed up in a closed form through EFT coefficients up to all orders in derivatives and to the second order in curvature. It occurs that such a summation is convergent only for localizable theories, making a direct connection between the graviton scattering amplitudes properties and the applicability of a perturbative treatment of an EFT of gravity. We further apply our results to logarithmic form-factors which appear in the 1-loop effective action for GR in four dimensions. We find out that the corresponding corrections to the Schwarzschild metric are stronger than those from the tree-level EFT operators. The developed framework can be extended to account for the corrections to the other BH solutions in GR, such as the Kerr metric.