- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Gravitational instantons and the quality problem of the QCD axion: Facts, speculations, and statements in between

Pier Giuseppe Catinari* and Alfredo Urbano†

  • *Contact author: piergiuseppe.catinari@uniroma1.it
  • †Contact author: alfredo.urbano@uniroma1.it
PDF

Phys. Rev. D 111, 125007 – Published 12 June, 2025

DOI: https://doi.org/10.1103/n5nz-8232

Abstract

In this work, we critically reanalyze the explicit breaking of the Peccei-Quinn global symmetry—and the corresponding corrections to the QCD axion potential—induced by gravity. Specifically, we examine the role of gravitational instantons, which are nonperturbative, finite-action solutions to the Euclidean Einstein equations. These instantons represent topologically nontrivial configurations of spacetime and are analogous to instantons in gauge theory. The amount of symmetry breaking induced by gravitational instantons can be computed in a controlled way within the framework of semiclassical gravity, using ’t Hooft operators, in full analogy to the computation of the axion potential arising from QCD small instanton effects. Contrary to previous results in the literature, we find that the effects of gravitational instantons are extremely small and therefore do not give rise to a significant quality problem for the axion solution to the strong CP problem, both within the Standard Model and in beyond-the-Standard-Model scenarios that involve multiple copies of the Standard Model. In conclusion, we argue that, assuming the ultraviolet completion of gravity is weakly coupled, the axion solution to the strong CP problem remains free from any quality issues due to gravity. Along the way, we derive the effective Lagrangian of the QCD axion, including its gravitational coupling.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (87)

  1. M. Kamionkowski and J. March-Russell, Phys. Lett. B 282, 137 (1992).
  2. T. Banks and N. Seiberg, Phys. Rev. D 83, 084019 (2011).
  3. L. Susskind, arXiv:hep-th/9501106.
  4. R. Bousso, J. High Energy Phys. 06 (1999) 028.
  5. R. Bousso, Rev. Mod. Phys. 74, 825 (2002).
  6. T. Banks and M. Dine, Nucl. Phys. B505, 445 (1997).
  7. P. Svrcek and E. Witten, J. High Energy Phys. 06 (2006) 051.
  8. T. Banks and L. J. Dixon, Nucl. Phys. B307, 93 (1988).
  9. D. Harlow and H. Ooguri, Commun. Math. Phys. 383, 1669 (2021).
  10. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 436, 257 (1998).
  11. S. H. Shenker, The strength of nonperturbative effects in string theory, in Random Surfaces and Quantum Gravity, edited by O. Alvarez, E. Marinari, and P. Windey (Springer US, Boston, MA, 1991), pp. 191–200.
  12. F. Larsen and F. Wilczek, Nucl. Phys. B458, 249 (1996).
  13. X. Calmet, S. D. H. Hsu, and D. Reeb, Phys. Rev. D 77, 125015 (2008).
  14. G. Dvali, Fortschr. Phys. 58, 528 (2010).
  15. G. Dvali and M. Redi, Phys. Rev. D 77, 045027 (2008).
  16. D. Baumann and L. McAllister, Inflation and String Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2015).
  17. S. B. Giddings and A. Strominger, Nucl. Phys. B306, 890 (1988).
  18. L. F. Abbott and M. B. Wise, Nucl. Phys. B325, 687 (1989).
  19. R. Kallosh, A. D. Linde, D. A. Linde, and L. Susskind, Phys. Rev. D 52, 912 (1995).
  20. R. Alonso and A. Urbano, J. High Energy Phys. 02 (2019) 136.
  21. P.-S. Hsin, L. V. Iliesiu, and Z. Yang, Classical Quantum Gravity 38, 194004 (2021).
  22. J. Alvey and M. Escudero, J. High Energy Phys. 01 (2021) 032; 11 (2023) 223(E).
  23. A. Hebecker, T. Mikhail, and P. Soler, Front. Astron. Space Sci. 5, 35 (2018).
  24. T. Hertog, B. Truijen, and T. Van Riet, Phys. Rev. Lett. 123, 081302 (2019).
  25. R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37, 172 (1976).
  26. C. G. Callan, R. Dashen, and D. J. Gross, Phys. Rev. D 17, 2717 (1978).
  27. S. W. Hawking, Phys. Lett. A 60, 81 (1977).
  28. G. W. Gibbons and S. W. Hawking, Commun. Math. Phys. 66, 291 (1979).
  29. T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. Rep. 66, 213 (1980).
  30. J. W. York, Jr., Phys. Rev. Lett. 28, 1082 (1972).
  31. G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
  32. R. Schon and S.-T. Yau, Commun. Math. Phys. 65, 45 (1979).
  33. E. Witten, Commun. Math. Phys. 80, 381 (1981).
  34. G. W. Gibbons and C. N. Pope, Commun. Math. Phys. 66, 267 (1979).
  35. T. Eguchi and A. J. Hanson, Ann. Phys. (N.Y.) 120, 82 (1979).
  36. G. W. Gibbons and S. W. Hawking, Phys. Lett. 78B, 430 (1978).
  37. A. H. Taub, Ann. Math. 53, 472 (1951).
  38. E. Newman, L. Tamburino, and T. Unti, J. Math. Phys. (N.Y.) 4, 915 (1963).
  39. A. A. Belavin and D. E. Burlankov, Phys. Lett. 58A, 7 (1976).
  40. W. J. Marciano, H. Pagels, and Z. Parsa, Phys. Rev. D 15, 1044 (1977).
  41. T. Eguchi and P. G. O. Freund, Phys. Rev. Lett. 37, 1251 (1976).
  42. A. Hebecker and P. Henkenjohann, J. High Energy Phys. 09 (2019) 038.
  43. S. Deser, M. J. Duff, and C. J. Isham, Phys. Lett. 93B, 419 (1980).
  44. R. Holman, T. W. Kephart, and S.-J. Rey, Phys. Rev. Lett. 71, 320 (1993).
  45. S.-J. Rey, in Proceedings of the 7th Meeting of the APS Division of Particles Fields (1992), pp. 1565–1567, https://inspirehep.net/literature/342182.
  46. Z. Chen and A. Kobakhidze, Eur. Phys. J. C 82, 596 (2022).
  47. Z. Chen, A. Kobakhidze, C. A. J. O’Hare, Z. S. C. Picker, and G. Pierobon, Eur. Phys. J. C 82, 940 (2022).
  48. Z. Chen, A. Kobakhidze, C. A. J. O’Hare, Z. S. C. Picker, and G. Pierobon, arXiv:2110.11014.
  49. H. Boutaleb-Joutei, A. Chakrabarti, and A. Comtet, Phys. Rev. D 21, 979 (1980).
  50. A. Chakrabarti, Fortschr. Phys. 35, 1 (1987).
  51. L. Alvarez-Gaume and E. Witten, Nucl. Phys. B234, 269 (1984).
  52. L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, Phys. Rep. 870, 1 (2020).
  53. L. Di Luzio, F. Mescia, and E. Nardi, Phys. Rev. Lett. 118, 031801 (2017).
  54. P. Sikivie, Phys. Rev. Lett. 48, 1156 (1982).
  55. K. Choi and S. H. Im, J. High Energy Phys. 01 (2016) 149.
  56. D. E. Kaplan and R. Rattazzi, Phys. Rev. D 93, 085007 (2016).
  57. M. Farina, D. Pappadopulo, F. Rompineve, and A. Tesi, J. High Energy Phys. 01 (2017) 095.
  58. G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, J. High Energy Phys. 01 (2016) 034.
  59. R. Jackiw, Rev. Mod. Phys. 52, 661 (1980).
  60. K. Huang, Quarks, Leptons & Gauge Fields (World Scientific, Singapore, 1992).
  61. S. Hawking, Nucl. Phys. B144, 349 (1978).
  62. G. W. Gibbons, S. W. Hawking, and M. J. Perry, Nucl. Phys. B138, 141 (1978).
  63. D. N. Page, Phys. Rev. D 18, 2733 (1978).
  64. J. W. York, Jr., Phys. Rev. Lett. 26, 1656 (1971).
  65. M. Shifman, A. Vainshtein, and V. Zakharov, Nucl. Phys. B163, 46 (1980).
  66. G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976).
  67. A. Belavin, A. Polyakov, A. Schwartz, and Y. Tyupkin, Phys. Lett. 59B, 85 (1975).
  68. S. Weinberg, The Quantum Theory of Fields. Vol. 2: Modern Applications (Cambridge University Press, Cambridge, England, 2013).
  69. R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
  70. G. ’t Hooft, Nucl. Phys. B315, 517 (1989).
  71. G. Franchetti, Nonlinearity 31, 2419 (2018).
  72. C. Csáki, R. T. D’Agnolo, E. Kuflik, and M. Ruhdorfer, J. High Energy Phys. 04 (2024) 074.
  73. M. Dine and N. Seiberg, Nucl. Phys. B273, 109 (1986).
  74. J. M. Flynn and L. Randall, Nucl. Phys. B293, 731 (1987).
  75. M. Bianchi, F. Fucito, G. Rossi, and M. Martellini, Nucl. Phys. B440, 129 (1995).
  76. C. Csáki, R. Tito D’Agnolo, R. S. Gupta, E. Kuflik, T. S. Roy, and M. Ruhdorfer, J. High Energy Phys. 10 (2023) 139.
  77. N. Arkani-Hamed, T. Cohen, R. T. D’Agnolo, A. Hook, H. D. Kim, and D. Pinner, Phys. Rev. Lett. 117, 251801 (2016).
  78. S.-J. Rey, Phys. Rev. D 47, R2652 (1993).
  79. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 1984).
  80. T. Ortin, Gravity and Strings, 2nd ed., Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2015).
  81. T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. Rep. 66, 213 (1980).
  82. Z. Xiao, Commun. Theor. Phys. 42, 235 (2004).
  83. S. Hawking and C. Pope, Phys. Lett. 73B, 42 (1978).
  84. C. N. Pope, J. Phys. A 14, L133 (1981).
  85. M. F. Atiyah, V. K. Patodi, and I. M. Singer, Math. Proc. Cambridge Philos. Soc. 77, 43 (1975).
  86. M. Shifman, Advanced Topics in Quantum Field Theory: A Lecture Course, 2nd ed. (Cambridge University Press, Cambridge, England, 2022).
  87. G. ’t Hooft, Phys. Rev. D 14, 3432 (1976).

Outline

Information

Phys. Rev. D 111, 125007– Published 12 June, 2025
  • Received 15 November 2024
  • Accepted 21 May 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation