- 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

Heat kernel of nonminimal second-order operators

Dario Sauro*

  • *Contact author: dario.sauro@uni-jena.de
PDF

Phys. Rev. D 112, 125012 – Published 5 December, 2025

DOI: https://doi.org/10.1103/qg8b-fkjw

Abstract

We analyze the spectra of general nonminimal second-order operators. To do this, we derive the local part of the trace of the second Seeley-DeWitt heat kernel coefficient for such operators in a completely model-independent way. Afterwards, we provide three examples to show how our result can be applied in practical scenarios. In particular, we emphasize this discussion when dealing with a toy model of dynamical torsion, which is viewed as a simple instance of higher-spin fields. All our results are compatible with the literature, and we provide a Mathematica notebook with the model-independent results that are written in the paper.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (56)

  1. D. V. Vassilevich, Phys. Rep. 388, 279 (2003).
  2. J. S. Schwinger, Phys. Rev. 82, 664 (1951).
  3. B. S. DeWitt, Conf. Proc. C 630701, 585 (1964); Les Houches Lect. Notes 13, 585 (1964).
  4. L. F. Abbott, Nucl. Phys. B185, 189 (1981).
  5. R. Seeley, Am. J. Math. 91, 889 (1969).
  6. M. Atiyah, R. Bott, and V. K. Patodi, Inventiones Math. 19, 279 (1973).
  7. P. B. Gilkey, J. Diff. Geom. 10, 601 (1975).
  8. I. G. Avramidi, Lect. Notes Phys. Monogr. 64, 1 (2000).
  9. A. O. Barvinsky and G. A. Vilkovisky, Phys. Rep. 119, 1 (1985).
  10. R. Percacci, An Introduction to Covariant Quantum Gravity and Asymptotic Safety (World Scientific, Hackensack, NJ, 2017), ISBN [Amazon][WorldCat], [Amazon][WorldCat].
  11. I. L. Buchbinder and I. Shapiro, Introduction to Quantum Field Theory with Applications to Quantum Gravity (Oxford University Press, Oxford, 2021), ISBN [Amazon][WorldCat], [Amazon][WorldCat].
  12. J. Daas, W. Oosters, F. Saueressig, and J. Wang, Phys. Lett. B 809, 135775 (2020).
  13. D. V. Vassilevich, J. High Energy Phys. 08 (2005) 085.
  14. A. O. Barvinsky and W. Wachowski, Phys. Rev. D 108, 045014 (2023).
  15. D. V. Fursaev and S. N. Solodukhin, Phys. Lett. B 365, 51 (1996).
  16. H. Gies and A. S. Salek, Phys. Rev. D 103, 125027 (2021).
  17. I. Jack and H. Osborn, Nucl. Phys. B207, 474 (1982).
  18. A. V. Ivanov and N. V. Kharuk, J. Phys. G 48, 015002 (2020).
  19. A. V. Ivanov, Nucl. Phys. B1006, 116647 (2024).
  20. I. Carneiro and G. von Gersdorff, J. High Energy Phys. 12 (2024) 140.
  21. J. Fuentes-Martín, A. Moreno-Sánchez, A. Palavrić, and A. E. Thomsen, J. High Energy Phys. 08 (2025) 099.
  22. I. Carneiro and G. von Gersdorff, J. High Energy Phys. 10 (2025) 078.
  23. Y. n. Obukhov, Nucl. Phys. B212, 237 (1983).
  24. V. P. Gusynin, Nucl. Phys. B333, 296 (1990).
  25. V. P. Gusynin, E. V. Gorbar, and V. V. Romankov, Nucl. Phys. B362, 449 (1991).
  26. K. Groh, F. Saueressig, and O. Zanusso, arXiv:1112.4856.
  27. D. T. Grasso and S. M. Kuzenko, J. High Energy Phys. 06 (2023) 120.
  28. L. Farolfi and F. Fecit, Eur. Phys. J. C 85, 356 (2025).
  29. O. Melichev, J. High Energy Phys. 08 (2025) 130.
  30. T. M. Sangy, T. de Paula Netto, and I. L. Shapiro, arXiv:2505.00214.
  31. R. Martini, G. Paci, and D. Sauro, J. High Energy Phys. 12 (2024) 138.
  32. R. Endo, Prog. Theor. Phys. 71, 1366 (1984).
  33. R. Endo, Classical Quantum Gravity 12, 1157 (1995).
  34. A. O. Barvinsky, A. E. Kalugin, and W. Wachowski, Phys. Rev. D 112, 076032 (2025).
  35. J. M. Martín-García, xact: Efficient tensor computer algebra for Mathematica, http://www.xact.es/.
  36. J. M. Martín-García, xtensor: Fast abstract tensor computer algebra, http://www.xact.es/xTensor.
  37. J. M. Martín-García, Comput. Phys. Commun. 179, 597 (2008).
  38. T. Nutma, Comput. Phys. Commun. 185, 1719 (2014).
  39. M. B. Fröb, arXiv:2008.12422.
  40. See Supplemental Material at http://link.aps.org/supplemental/10.1103/qg8b-fkjw for the Mathematica code version of the model-independent trace of the second Seeley-DeWitt coefficient.
  41. D. Sauro, https://arxiv.org/src/2508.09017v1/anc.
  42. T. S. Evans, Phys. Lett. B 457, 127 (1999).
  43. D. Sauro and O. Zanusso, Classical Quantum Gravity 39, 185001 (2022).
  44. D. Sauro, R. Martini, and O. Zanusso, Int. J. Geom. Methods Mod. Phys. 20, 2350237 (2023).
  45. G. Paci and O. Zanusso, Ann. Phys. (Amsterdam) 473, 169877 (2025).
  46. D. M. Ghilencea, J. High Energy Phys. 03 (2019) 049.
  47. I. L. Buchbinder, P. R. B. R. d. Vale, G. Y. Oyadomari, and I. L. Shapiro, Phys. Rev. D 110, 125015 (2024).
  48. M. Piva, Phys. Rev. D 108, 045011 (2023).
  49. M. Kalb and P. Ramond, Phys. Rev. D 9, 2273 (1974).
  50. I. L. Shapiro, Eur. Phys. J. Plus 140, 534 (2025).
  51. I. L. Shapiro, Phys. Rep. 357, 113 (2002).
  52. G. A. Vilkovisky, Nucl. Phys. B234, 125 (1984).
  53. T. Curtright, Phys. Lett. 165B, 304 (1985).
  54. M. Piva, Phys. Rev. D 105, 045006 (2022).
  55. C. Marzo, Phys. Rev. D 106, 024045 (2022).
  56. R. Martini, G. Paci, and D. Sauro, Classical Quantum Gravity 40, 195005 (2023).

Outline

Information

Phys. Rev. D 112, 125012– Published 5 December, 2025
  • Received 16 August 2025
  • Accepted 13 November 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