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Lorentz violating quadratic gravity

R. B. Alfaia1,*, Willian Carvalho1,†, A. C. Lehum1,‡, J. R. Nascimento2,§, A. Yu. Petrov2,∥, and P. J. Porfírio2,¶

  • *Contact author: ramon.alfaia@icen.ufpa.br
  • †Contact author: willian.carvalho@icen.ufpa.br
  • ‡Contact author: lehum@ufpa.br
  • §Contact author: jroberto@fisica.ufpb.br
  • ∥Contact author: petrov@fisica.ufpb.br
  • Contact author: pporfirio@fisica.ufpb.br
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Phys. Rev. D 113, 125035 – Published 30 June, 2026

DOI: https://doi.org/10.1103/vyn6-l63g

Abstract

In this paper, we explore the perturbative renormalization and study the classical dynamics of the bumblebee model coupled to quadratic gravity, a theoretical setting that allows the violation of Lorentz symmetry. Such a violation arises from a vector field whose potential is engineered to induce a nonzero vacuum expectation value, thereby leading to the emergence of a preferred direction in spacetime and, consequently, to the spontaneous breaking of Lorentz symmetry. Working in dimensional regularization and expanding the metric around flat space, we compute the one-loop divergent parts of the two-point functions of the bumblebee and graviton fields, with special emphasis on the role of Lorentz violating insertions in internal lines. These results determine the counterterms required to renormalize the gravitational and bumblebee sectors in the presence of a preferred background direction, and make explicit how Lorentz violating interactions feed back into the UV structure of quadratic gravity. On the classical side, we derive the field equations and identify exact solutions supported by bumblebee backgrounds. In particular, we show that the Schwarzschild geometry remains an exact solution for an appropriate bumblebee configuration, even in the presence of nonminimal couplings. We close with a discussion of the operator content suggested by the one-loop structure and of prospective extensions to cosmological and less symmetric backgrounds.

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References (53)

  1. K. S. Stelle, Phys. Rev. D 16, 953 (1977).
  2. E. Tomboulis, Phys. Lett. 70B, 361 (1977).
  3. S. D. Odintsov and I. L. Shapiro, Classical Quantum Gravity 9, 873 (1992).
  4. G. Narain and R. Anishetty, J. High Energy Phys. 10 (2013) 203.
  5. A. Salvio and A. Strumia, J. High Energy Phys. 06 (2014) 080.
  6. A. Salvio and A. Strumia, Eur. Phys. J. C 78, 124 (2018).
  7. M. B. Einhorn and D. R. T. Jones, J. High Energy Phys. 03 (2015) 047.
  8. I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum Gravity (IOP Publishing, Abingdon, 2017).
  9. J. F. Donoghue and G. Menezes, Nuovo Cimento Soc. Ital. Fis. 45C, 26 (2022).
  10. G. ’t Hooft and M. J. G. Veltman, Ann. Inst. Henri Poincare Phys. Theor. A 20, 69 (1974).
  11. S. Deser and P. van Nieuwenhuizen, Phys. Rev. Lett. 32, 245 (1974).
  12. S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 10, 411 (1974).
  13. J. F. Donoghue, Phys. Rev. D 50, 3874 (1994).
  14. C. P. Burgess, Living Rev. Relativity 7, 5 (2004).
  15. J. F. Donoghue, M. M. Ivanov, and A. Shkerin, arXiv:1702.00319.
  16. L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. Lüst, and A. Riotto, Fortschr. Phys. 64, 176 (2016).
  17. B. Holdom and J. Ren, Phys. Rev. D 93, 124030 (2016).
  18. D. Anselmi and M. Piva, J. High Energy Phys. 05 (2018) 027.
  19. A. Salvio, Front. Phys. 6, 77 (2018).
  20. D. Buccio, J. F. Donoghue, G. Menezes, and R. Percacci, Phys. Rev. Lett. 133, 021604 (2024).
  21. M. Aoki, J. Kubo, and J. Yang, J. Cosmol. Astropart. Phys. 01 (2022) 005.
  22. M. Aoki, J. Kubo, and J. Yang, J. Cosmol. Astropart. Phys. 05 (2024) 096.
  23. C. Álvarez-Luna, S. de la Calle-Leal, J. A. R. Cembranos, and J. J. Sanz-Cillero, J. High Energy Phys. 02 (2023) 232.
  24. L. Buoninfante, J. High Energy Phys. 12 (2023) 111.
  25. I. D. Gialamas, A. Karam, and A. Racioppi, J. Cosmol. Astropart. Phys. 11 (2020) 014.
  26. V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989).
  27. V. A. Kostelecký and Z. Li, Phys. Rev. D 103, 024059 (2021).
  28. V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004).
  29. R. Bluhm and V. A. Kostelecky, Phys. Rev. D 71, 065008 (2005).
  30. O. Bertolami and J. Paramos, Phys. Rev. D 72, 044001 (2005).
  31. R. Casana, A. Cavalcante, F. P. Poulis, and E. B. Santos, Phys. Rev. D 97, 104001 (2018).
  32. A. A. A. Filho, J. R. Nascimento, A. Y. Petrov, and P. J. Porfírio, Phys. Rev. D 108, 085010 (2023).
  33. A. A. Araújo Filho, J. R. Nascimento, A. Y. Petrov, and P. J. Porfírio, J. Cosmol. Astropart. Phys. 07 (2024) 004.
  34. D. Capelo and J. Páramos, Phys. Rev. D 91, 104007 (2015).
  35. R. V. Maluf and J. C. S. Neves, J. Cosmol. Astropart. Phys. 10 (2021) 038.
  36. A. F. Santos, A. Y. Petrov, W. D. R. Jesus, and J. R. Nascimento, Mod. Phys. Lett. A 30, 1550011 (2015).
  37. W. D. R. Jesus and A. F. Santos, Int. J. Mod. Phys. A 35, 2050050 (2020).
  38. A. Övgün, K. Jusufi, and İ. Sakall𝚤, Phys. Rev. D 99, 024042 (2019).
  39. R. V. Maluf, C. A. S. Almeida, R. Casana, and M. M. Ferreira, Jr., Phys. Rev. D 90, 025007 (2014).
  40. A. Delhom, J. R. Nascimento, G. J. Olmo, A. Y. Petrov, and P. J. Porfírio, Phys. Lett. B 826, 136932 (2022).
  41. A. Delhom, T. Mariz, J. R. Nascimento, G. J. Olmo, A. Y. Petrov, and P. J. Porfírio, J. Cosmol. Astropart. Phys. 07 (2022) 018.
  42. A. C. Lehum, J. R. Nascimento, A. Y. Petrov, and P. J. Porfirio, Gen. Relativ. Gravit. 57, 66 (2025).
  43. A. A. Araújo Filho, J. Cosmol. Astropart. Phys. 06 (2025) 026; 02 (2026) E01.
  44. A. C. Lehum, J. R. Nascimento, A. Y. Petrov, and P. J. Porfírio, Phys. Rev. D 110, 125007 (2024).
  45. S. R. Coleman and E. J. Weinberg, Phys. Rev. D 7, 1888 (1973).
  46. A. Delhom, J. R. Nascimento, G. J. Olmo, A. Y. Petrov, and P. J. Porfírio, Eur. Phys. J. C 81, 287 (2021).
  47. J. L. Chkareuli, J. Jejelava, and Z. Kepuladze, Bled Workshops Phys. 19, 74 (2018), arXiv:1811.09578.
  48. T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001).
  49. Q. G. Bailey and V. A. Kostelecky, Phys. Rev. D 74, 045001 (2006).
  50. R. V. Maluf, J. E. G. Silva, and C. A. S. Almeida, Phys. Lett. B 749, 304 (2015).
  51. S. M. Carroll and H. Tam, Phys. Rev. D 78, 044047 (2008).
  52. B. Racskó and L. Á. Gergely, Eur. Phys. J. Plus 138, 413 (2023).
  53. C. van de Bruck, M. A. Gorji, N. A. Nilsson, M. C. Pookkillath, and M. Yamaguchi, arXiv:2509.11647.

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Phys. Rev. D 113, 125035– Published 30 June, 2026
  • Received 13 March 2026
  • Accepted 9 June 2026
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