- 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

High-temperature massless scalar partition function on general stationary backgrounds

Manuel Valle1,* and Miguel Á. Vázquez-Mozo2,†

  • *Contact author: manuel.valle@ehu.es
  • †Contact author: vazquez@usal.es
PDF

Phys. Rev. D 112, 105008 – Published 10 November, 2025

DOI: https://doi.org/10.1103/m5qv-q8hq

Abstract

The high-temperature equilibrium partition function of a massless real scalar field nonminimally coupled to the scalar curvature is computed at second order in the derivative expansion on a generic stationary background. Using covariant perturbation theory, the expression of the thermal partition function at second order in powers of curvatures is also obtained, including its nonlocal contributions. For conformal coupling, the Weyl anomaly at fourth order in derivatives and second order in curvatures is evaluated using both expansions and the results found to be consistent.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (43)

  1. S. Deser, M. J. Duff, and C. J. Isham, Nonlocal conformal anomalies, Nucl. Phys. B111, 45 (1976).
  2. M. J. Duff, Observations on conformal anomalies, Nucl. Phys. B125, 334 (1977).
  3. M. J. Duff, Twenty years of the Weyl anomaly, Classical Quantum Gravity 11, 1387 (1994).
  4. S. Deser, Conformal anomalies: Recent progress, Helv. Phys. Acta 69, 570 (1996).
  5. V. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity (Cambridge University Press, Cambridge, England, 2007).
  6. Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rep. 569, 1 (2015).
  7. R. A. Bertlmann, Anomalies in Quantum Field Theory (Oxford University Press, New York, 1996).
  8. L. Álvarez-Gaumé and M. Á. Vázquez-Mozo, Anomalies and the Green-Schwarz Mechanism, in Handbook of Quantum Gravity, edited by C. Bambi, L. Modesto, and I. Shapiro (Springer, New York, 2024), 10.1007/978-981-19-3079-9_68-1.
  9. C. Eling, Y. Oz, S. Theisen, and S. Yankielowicz, Conformal anomalies in hydrodynamics, J. High Energy Phys. 05 (2013) 037.
  10. M. N. Chernodub, Anomalous transport due to the conformal anomaly, Phys. Rev. Lett. 117, 141601 (2016).
  11. S. Diles, The role of Weyl symmetry in hydrodynamics, Phys. Lett. B 779, 331 (2018).
  12. P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium (Cambridge University Press, Cambridge, England, 2019).
  13. M. N. Chernodub and E. Kilinçarslan, Conformal anomaly and helicity effects in kinetic theory via scale-dependent coupling, Phys. Rev. D 103, 036019 (2021).
  14. K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz, and A. Yarom, Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109, 101601 (2012).
  15. N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla, and T. Sharma, Constraints on fluid dynamics from equilibrium partition functions, J. High Energy Phys. 09 (2012) 046.
  16. J. S. Dowker and G. Kennedy, Finite temperature and boundary effects in static space-times, J. Phys. A 11, 895 (1978).
  17. J. S. Dowker and J. P. Schofield, High temperature expansion of the free energy of a massive scalar field in a curved space, Phys. Rev. D 38, 3327 (1988).
  18. Y. Gusev and A. Zelnikov, Finite temperature non-local effective action for scalar fields, Classical Quantum Gravity 15, L13 (1998).
  19. Y. V. Gusev and A. I. Zelnikov, Finite temperature nonlocal effective action for quantum fields in curved space, Phys. Rev. D 59, 024002 (1999).
  20. I. S. Kalinichenko and P. O. Kazinski, High-temperature expansion of the one-loop free energy of a scalar field on a curved background, Phys. Rev. D 87, 084036 (2013).
  21. J. I. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications (Cambridge University Press, Cambridge, England, 2011).
  22. A. O. Barvinsky and G. A. Vilkovisky, Covariant perturbation theory. 2: Second order in the curvature. General algorithms, Nucl. Phys. B333, 471 (1990).
  23. P. B. Gilkey, The spectral geometry of a Riemannian manifold, J. Diff. Geom. 10, 601 (1975).
  24. I. G. Avramidi, The covariant technique for calculation of one loop effective action, Nucl. Phys. B355, 712 (1991).
  25. I. G. Avramidi, Heat Kernel and Quantum Gravity (Springer, New York, 2000).
  26. D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rep. 388, 279 (2003).
  27. I. G. Avramidi, Heat Kernel Method and Its Applications (Springer, New York, 2015).
  28. L. Parker and D. J. Toms, New form for the coincidence limit of the Feynman propagator, or heat kernel, in curved space-time, Phys. Rev. D 31, 953 (1985).
  29. L. Parker and D. J. Toms, Explicit curvature dependence of coupling constants, Phys. Rev. D 31, 2424 (1985).
  30. L. E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity (Cambridge University Press, Cambridge, England, 2009).
  31. D. V. Fursaev, Kaluza-Klein method in theory of rotating quantum fields, Nucl. Phys. B596, 365 (2001).
  32. I. G. Avramidi, The covariant technique for the calculation of the heat kernel asymptotic expansion, Phys. Lett. B 238, 92 (1990).
  33. S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309, 279 (1993).
  34. D. Karateev, Z. Komargodski, J. Penedones, and B. Sahoo, Trace anomalies and the graviton-dilaton amplitude, J. High Energy Phys. 11 (2024) 067.
  35. A. O. Barvinsky and G. A. Vilkovisky, Covariant perturbation theory. 3: Spectral representations of the third order form-factors, Nucl. Phys. B333, 512 (1990).
  36. A. O. Barvinsky, Y. V. Gusev, G. A. Vilkovisky, and V. V. Zhytnikov, Asymptotic behaviors of the heat kernel in covariant perturbation theory, J. Math. Phys. (N.Y.) 35, 3543 (1994).
  37. A. O. Barvinsky, Y. V. Gusev, G. A. Vilkovisky, and V. V. Zhytnikov, The One loop effective action and trace anomaly in four-dimensions, Nucl. Phys. B439, 561 (1995).
  38. A. O. Barvinsky and Y. V. Gusev, Covariant algorithms for one loop radiation currents in gauge theories and quantum gravity, Classical Quantum Gravity 9, 383 (1992).
  39. K. Yamada, Quantum corrections to the stress tensor in perturbation theory, Phys. Rev. D 10, 599 (1974).
  40. P. d. M. Teixeira, I. L. Shapiro, and T. G. Ribeiro, One-loop effective action: Nonlocal form factors and renormalization group, Gravitation Cosmol. 26, 185 (2020).
  41. A. Schwimmer and S. Theisen, Spontaneous breaking of conformal invariance and trace anomaly matching, Nucl. Phys. B847, 590 (2011).
  42. A. Schwimmer and S. Theisen, Comments on trace anomaly matching, J. Phys. A 56, 465402 (2023).
  43. J. L. Mañes, E. Megías, M. Valle, and M. Á. Vázquez-Mozo, Non-Abelian anomalous (super)fluids in thermal equilibrium from differential geometry, J. High Energy Phys. 11 (2018) 076.

Outline

Information

Phys. Rev. D 112, 105008– Published 10 November, 2025
  • Received 13 September 2025
  • Accepted 16 October 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