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  • Open Access

Convergent sum of EFT corrections to the Schwarzschild metric requires UV locality

Yang Liu1,*, Alexey S. Koshelev1,2,†, Anna Tokareva3,4,5,‡, and Ziyue Zhu3,§

  • *Contact author: liuyang2023@shanghaitech.edu.cn
  • †Contact author: askoshelev@shanghaitech.edu.cn
  • ‡Contact author: tokareva@ucas.ac.cn
  • §Contact author: zhuziyue25@mails.ucas.ac.cn
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Phys. Rev. D 114, 066002 – Published 1 September, 2026

DOI: https://doi.org/10.1103/6grg-s7zc

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 one-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 black hole solutions in GR, such as the Kerr metric.

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References (57)

  1. S. Deser, M. J. Duff, and C. J. Isham, Nonlocal conformal anomalies, Nucl. Phys. B111, 45 (1976).
  2. E. S. Fradkin and G. A. Vilkovisky, Conformal off mass shell extension and elimination of conformal anomalies in quantum gravity, Phys. Lett. B 73, 209 (1978).
  3. E. S. Fradkin and G. A. Vilkovisky, Conformal invariance and asymptotic freedom in quantum gravity, Phys. Lett. B 77, 262 (1978).
  4. A. O. Barvinsky and G. A. Vilkovisky, The generalized Schwinger-de Witt technique and the unique effective action in quantum gravity, Phys. Lett. 131B, 313 (1983).
  5. B. Zwiebach, Curvature squared terms and string theories, Phys. Lett. 156B, 315 (1985).
  6. S. Weinberg, Effective gauge theories, Phys. Lett. 91B, 51 (1980).
  7. J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50, 3874 (1994).
  8. M. Ruhdorfer, J. Serra, and A. Weiler, Effective field theory of gravity to all orders, J. High Energy Phys. 05 (2020) 083.
  9. D. Dou and R. Percacci, The running gravitational couplings, Classical Quantum Gravity 15, 3449 (1998).
  10. R. Percacci, Asymptotic safety, arXiv:0709.3851.
  11. M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57, 971 (1998).
  12. M. Reuter and F. Saueressig, Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation, Phys. Rev. D 65, 065016 (2002).
  13. B. Knorr, C. Ripken, and F. Saueressig, Form factors in asymptotic safety: Conceptual ideas and computational toolbox, Classical Quantum Gravity 36, 234001 (2019).
  14. K. S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16, 953 (1977).
  15. N. V. Krasnikov, Nonlocal gauge theories, Theor. Math. Phys. 73, 1184 (1987).
  16. N. V. Krasnikov, Nonlocal Gauge Theories Including Quantum Gravity (Springer, New York, 2024).
  17. Y. V. Kuzmin, The convergent nonlocal gravitation (in Russian), Sov. J. Nucl. Phys. 50, 1011 (1989).
  18. E. T. Tomboulis, Superrenormalizable gauge and gravitational theories, arXiv:hep-th/9702146.
  19. L. Modesto, Super-renormalizable quantum gravity, Phys. Rev. D 86, 044005 (2012).
  20. T. Biswas, A. S. Koshelev, and A. Mazumdar, Consistent higher derivative gravitational theories with stable de Sitter and anti–de Sitter backgrounds, Phys. Rev. D 95, 043533 (2017).
  21. D. G. Boulware and S. Deser, String generated gravity models, Phys. Rev. Lett. 55, 2656 (1985).
  22. C. Garraffo and G. Giribet, The Lovelock black holes, Mod. Phys. Lett. A 23, 1801 (2008).
  23. T. Padmanabhan and D. Kothawala, Lanczos-Lovelock models of gravity, Phys. Rep. 531, 115 (2013).
  24. H. Lu, A. Perkins, C. N. Pope, and K. S. Stelle, Black holes in higher-derivative gravity, Phys. Rev. Lett. 114, 171601 (2015).
  25. H. Lü, A. Perkins, C. N. Pope, and K. S. Stelle, Spherically symmetric solutions in higher-derivative gravity, Phys. Rev. D 92, 124019 (2015).
  26. Y. Kats, L. Motl, and M. Padi, Higher-order corrections to mass-charge relation of extremal black holes, J. High Energy Phys. 12 (2007) 068.
  27. C. Cheung and G. N. Remmen, Positive signs in massive gravity, J. High Energy Phys. 04 (2016) 002.
  28. S. Melville, Causality and quasi-normal modes in the GREFT, Eur. Phys. J. Plus 139, 725 (2024).
  29. N. Arkani-Hamed, Y.-t. Huang, J.-Y. Liu, and G. N. Remmen, Causality, unitarity, and the weak gravity conjecture, J. High Energy Phys. 03 (2022) 083.
  30. P. Bueno, P. A. Cano, and R. A. Hennigar, Regular black holes from pure gravity, Phys. Lett. B 861, 139260 (2025).
  31. P. Bueno, R. A. Hennigar, and Á. J. Murcia, Birkhoff implies quasi-topological, Classical Quantum Gravity 43, 095020 (2026).
  32. Y.-P. Wang, L. Ma, and Y. Pang, Quantum corrections to pair production of charged black holes in de Sitter space, J. Cosmol. Astropart. Phys. 01 (2023) 007.
  33. W. L. Boyce and J. E. Santos, EFT corrections to charged black hole quasinormal modes, J. High Energy Phys. 12 (2025) 017.
  34. F. S. Miguel, EFT corrections to scalar and vector quasinormal modes of rapidly rotating black holes, Phys. Rev. D 109, 104016 (2024).
  35. G. T. Horowitz, M. Kolanowski, G. N. Remmen, and J. E. Santos, Extremal Kerr black holes as amplifiers of new physics, Phys. Rev. Lett. 131, 091402 (2023).
  36. G. T. Horowitz, M. Kolanowski, G. N. Remmen, and J. E. Santos, Sudden breakdown of effective field theory near cool Kerr-Newman black holes, J. High Energy Phys. 05 (2024) 122.
  37. G. T. Horowitz, M. Kolanowski, and J. E. Santos, Almost all extremal black holes in AdS are singular, J. High Energy Phys. 01 (2023) 162.
  38. G. T. Horowitz and J. E. Santos, Smooth extremal horizons are the exception, not the rule, J. High Energy Phys. 02 (2025) 169.
  39. C. Y. R. Chen, C. de Rham, and A. J. Tolley, Deformations of extremal black holes and the UV, Phys. Rev. D 111, 024056 (2025).
  40. F. Del Porro, F. Ferrarin, and A. Platania, Impact of quantum gravity on the UV sensitivity of extremal black holes, Fortschr. Phys. 74, e70101 (2026).
  41. A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, J. High Energy Phys. 10 (2006) 014.
  42. D.-Y. Hong, Z.-H. Wang, and S.-Y. Zhou, Causality bounds on scalar-tensor EFTs, J. High Energy Phys. 10 (2023) 135.
  43. T. Biswas, A. S. Koshelev, and A. Mazumdar, Gravitational theories with stable (anti-)de Sitter backgrounds, Fundam. Theor. Phys. 183, 97 (2016).
  44. J. Borissova, B. L. Giacchini, and A. Held, Spherically symmetric solutions in quasi-local Einstein-Weyl gravity, J. Cosmol. Astropart. Phys. 07 (2026) 080.
  45. A. S. Koshelev, O. Melichev, and L. Rachwal, Cancellation of UV divergences in ghost-free infinite derivative gravity, arXiv:2512.18006.
  46. B. Zwiebach, A First Course in String Theory (Cambridge University Press, Cambridge, England, 2006), 7.
  47. S. P. Robinson, Normalization conventions for Newton’s constant and the Planck scale in arbitrary spacetime dimension, arXiv:gr-qc/0609060.
  48. C. de Rham, S. Jaitly, and A. J. Tolley, Constraints on Regge behavior from IR physics, Phys. Rev. D 108, 046011 (2023).
  49. J. Tokuda, Extension of positivity bounds to non-local theories: IR obstructions to Lorentz invariant UV completions, J. High Energy Phys. 05 (2019) 216.
  50. L. Buoninfante, J. Tokuda, and M. Yamaguchi, New lower bounds on scattering amplitudes: Non-locality constraints, J. High Energy Phys. 01 (2024) 082.
  51. L. Buoninfante, L.-Q. Shao, and A. Tokareva, Nonlocal positivity bounds: Islands in terra incognita, Phys. Rev. D 112, L021904 (2025).
  52. X. Calmet, A. Giusti, and M. Sebastianutti, Black-hole solutions in quantum gravity with Vilkovisky-DeWitt effective action, Europhys. Lett. 150, 69001 (2025).
  53. M. Shaposhnikov and A. Tokareva, Exact quantum conformal symmetry, its spontaneous breakdown, and gravitational Weyl anomaly, Phys. Rev. D 107, 065015 (2023).
  54. M. Shaposhnikov and A. Tokareva, Anomaly-free scale symmetry and gravity, Phys. Lett. B 840, 137898 (2023).
  55. https://xact.es/index.html.
  56. N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, The string landscape, black holes and gravity as the weakest force, J. High Energy Phys. 06 (2007) 060.
  57. I. S. Gradshteyn, I. M. Ryzhik, D. Zwillinger, and V. Moll, Table of Integrals, Series, and Products (Academic Press, Amsterdam, 2015).

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Phys. Rev. D 114, 066002– Published 1 September, 2026
  • Received 19 February 2026
  • Accepted 7 July 2026
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