- 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

Gauge choices, infrared pitfalls, and thermal effects in effective potentials

Debanjan Balui1,*, Tisa Biswas1,†, Joydeep Chakrabortty1,‡, Debmalya Dey1,§, Christoph Englert2,∥, and Subhendra Mohanty3,¶

  • *Contact author: debanjanb22@iitk.ac.in
  • †Contact author: tisab@iitk.ac.in
  • ‡Contact author: joydeep@iitk.ac.in
  • §Contact author: debmalyad23@iitk.ac.in
  • ∥Contact author: christoph.englert@glasgow.ac.uk
  • Contact author: subhendram@iiserb.ac.in
PDF

Phys. Rev. D 112, 056022 – Published 19 September, 2025

DOI: https://doi.org/10.1103/drsd-wfns

Abstract

The evaluation of effective potentials is critical for a range of phenomenological applications, including inflation, vacuum stability, and phase transitions. A drawback arises from the gauge dependence of the effective potential. Furthermore, in theories with spontaneous symmetry breaking, the effective potential exhibits infrared (IR) divergences in the limit of vanishing Goldstone masses. By considering the multiplicative anomaly that arises due to nonfactorization of elliptic operators in the Fermi gauge when computing the effective potential at one-loop order, we demonstrate that its gauge independence and IR behavior are improved to the corresponding findings of Landau gauge calculations simultaneously. The latter are straightforwardly and transparently reproduced using an approach that employs the heat kernel technique, thereby providing a shortcut to reflect anomaly-related cancellations from the outset. Our findings generalize to the treatment of the effective potential at finite temperature. In particular, the heat kernel extends gauge independence to any value of the expansion in mass over temperature.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (62)

  1. S. R. Coleman and E. J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7, 1888 (1973).
  2. M. Sher, Electroweak Higgs potentials and vacuum stability, Phys. Rep. 179, 273 (1989).
  3. R. Jackiw, Functional evaluation of the effective potential, Phys. Rev. D 9, 1686 (1974).
  4. L. Dolan and R. Jackiw, Gauge invariant signal for gauge symmetry breaking, Phys. Rev. D 9, 2904 (1974).
  5. H. Kluberg-Stern and J. B. Zuber, Ward identities and some clues to the renormalization of gauge invariant operators, Phys. Rev. D 12, 467 (1975).
  6. O. Piguet and K. Sibold, Gauge independence in ordinary Yang-Mills theories, Nucl. Phys. B253, 517 (1985).
  7. H. Kluberg-Stern and J. B. Zuber, Renormalization of non-Abelian gauge theories in a background field gauge. 1. Green functions, Phys. Rev. D 12, 482 (1975).
  8. N. K. Nielsen, On the gauge dependence of spontaneous symmetry breaking in gauge theories, Nucl. Phys. B101, 173 (1975).
  9. O. M. Del Cima, D. H. T. Franco, and O. Piguet, Gauge independence of the effective potential revisited, Nucl. Phys. B551, 813 (1999).
  10. P. Gambino and P. A. Grassi, The Nielsen identities of the SM and the definition of mass, Phys. Rev. D 62, 076002 (2000).
  11. P. A. Grassi, B. A. Kniehl, and A. Sirlin, Width and partial widths of unstable particles in the light of the Nielsen identities, Phys. Rev. D 65, 085001 (2002).
  12. S. Goria, G. Passarino, and D. Rosco, The Higgs boson lineshape, Nucl. Phys. B864, 530 (2012).
  13. N. K. Nielsen, Removing the gauge parameter dependence of the effective potential by a field redefinition, Phys. Rev. D 90, 036008 (2014).
  14. I. J. R. Aitchison and C. M. Fraser, Gauge invariance and the effective potential, Ann. Phys. (N.Y.) 156, 1 (1984).
  15. D. Metaxas and E. J. Weinberg, Gauge independence of the bubble nucleation rate in theories with radiative symmetry breaking, Phys. Rev. D 53, 836 (1996).
  16. H. H. Patel and M. J. Ramsey-Musolf, Baryon washout, electroweak phase transition, and perturbation theory, J. High Energy Phys. 07 (2011) 029.
  17. A. Andreassen, W. Frost, and M. D. Schwartz, Consistent use of effective potentials, Phys. Rev. D 91, 016009 (2015).
  18. L. P. Alexander and A. Pilaftsis, The one-loop effective potential in non-linear gauges, J. Phys. G 36, 045006 (2009).
  19. M. Garny and T. Konstandin, On the gauge dependence of vacuum transitions at finite temperature, J. High Energy Phys. 07 (2012) 189.
  20. J. R. Espinosa, G. F. Giudice, E. Morgante, A. Riotto, L. Senatore, A. Strumia, and N. Tetradis, The cosmological Higgstory of the vacuum instability, J. High Energy Phys. 09 (2015) 174.
  21. M. Quiros, Finite temperature field theory and phase transitions, in ICTP Summer School in High-Energy Physics and Cosmology (1999), pp. 187–259, arXiv:hep-ph/9901312.
  22. J. Hirvonen, J. Löfgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, Computing the gauge-invariant bubble nucleation rate in finite temperature effective field theory, J. High Energy Phys. 07 (2022) 135.
  23. J. Löfgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, Nucleation at finite temperature: A gauge-invariant perturbative framework, Phys. Rev. Lett. 130, 251801 (2023).
  24. F. Bernardo, P. Klose, P. Schicho, and T. V. I. Tenkanen, Higher-dimensional operators at finite-temperature affect gravitational-wave predictions, J. High Energy Phys. 08 (2025) 109.
  25. J. R. Espinosa, M. Garny, T. Konstandin, and A. Riotto, Gauge-independent scales related to the standard model vacuum instability, Phys. Rev. D 95, 056004 (2017).
  26. J. R. Espinosa, M. Garny, and T. Konstandin, Interplay of infrared divergences and gauge-dependence of the effective potential, Phys. Rev. D 94, 055026 (2016).
  27. J. Elias-Miro, J. R. Espinosa, and T. Konstandin, Taming infrared divergences in the effective potential, J. High Energy Phys. 08 (2014) 034.
  28. D. Balui, J. Chakrabortty, D. Dey, and S. Mohanty, Gauge invariant effective potential, Phys. Rev. D 111, 085032 (2025).
  29. S. W. Hawking, Zeta function regularization of path integrals in curved space-time, Commun. Math. Phys. 55, 133 (1977).
  30. E. Elizalde, Zeta-function regularization techniques for series summations and applications, in Proceedings Leipzig Workshop Quantum Field Theory Under the Influence of External Conditions (Leipzig) (, Naturwissenschaftlich-Theoretisches Zentrum, 1992), p. 10.
  31. E. Elizalde, Complete determination of the singularity structure of zeta functions, J. Phys. A 30, 2735 (1997).
  32. M. Wodzicki, Noncommutative Residue Chapter I. Fundamentals (Springer, Berlin, Heidelberg, 1987), pp. 320–399.
  33. J. Mickelsson, Wodzicki residue and anomalies of current algebras, Lect. Notes Phys. 436, 123 (1994).
  34. A. A. Bytsenko, G. Cognola, L. Vanzo, and S. Zerbini, Quantum fields and extended objects in space-times with constant curvature spatial section, Phys. Rep. 266, 1 (1996).
  35. E. Elizalde, L. Vanzo, and S. Zerbini, Zeta function regularization, the multiplicative anomaly and the Wodzicki residue, Commun. Math. Phys. 194, 613 (1998).
  36. E. Elizalde, A. Filippi, L. Vanzo, and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, arXiv:hep-th/9804071.
  37. R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math. 10, 288 (1967).
  38. A. A. Bel’kov, A. V. Lanyov, and A. Schaale, Calculation of heat kernel coefficients and usage of computer algebra, Comput. Phys. Commun. 95, 123 (1996).
  39. D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rep. 388, 279 (2003).
  40. I. G. Avramidi, Heat kernel approach in quantum field theory, Nucl. Phys. B, Proc. Suppl. 104, 3 (2002).
  41. M. Kontsevich and S. Vishik, Geometry of determinants of elliptic operators, arXiv:hep-th/9406140.
  42. I. G. Avramidi, The covariant technique for calculation of one loop effective action, Nucl. Phys. B355, 712 (1991); B509, 557(E) (1998).
  43. U. Banerjee, J. Chakrabortty, S. U. Rahaman, and K. Ramkumar, One-loop effective action up to dimension eight: Integrating out heavy scalar(s), Eur. Phys. J. Plus 139, 159 (2024).
  44. J. Chakrabortty, S. U. Rahaman, and K. Ramkumar, One-loop effective action up to dimension eight: Integrating out heavy fermion(s), Nucl. Phys. B1000, 116488 (2024).
  45. U. Banerjee, J. Chakrabortty, S. U. Rahaman, and K. Ramkumar, One-loop effective action up to any mass-dimension for non-degenerate scalars and fermions including light–heavy mixing, Eur. Phys. J. Plus 139, 169 (2024).
  46. U. Banerjee, J. Chakrabortty, and K. Ramkumar, Renormalization of scalar and fermion interacting field theory for arbitrary loop: Heat–Kernel approach, Eur. Phys. J. Plus 139, 714 (2024).
  47. N. Adhikary, J. Das, and D. Dey, Two-loop dimension six effective action: Integrating out heavy scalar, arXiv:2501.01313.
  48. N. Adhikary, T. Biswas, J. Chakrabortty, C. Englert, and M. Spannowsky, Electroweak scalar effects beyond dimension-6 in SMEFT, arXiv:2501.12160.
  49. A. Urbano, Inflation without gauge redundancy, J. Cosmol. Astropart. Phys. 04 (2020) 040.
  50. R. Fukuda and T. Kugo, Gauge invariance in the effective action and potential, Phys. Rev. D 13, 3469 (1976).
  51. A. Andreassen, W. Frost, and M. D. Schwartz, Consistent use of the standard model effective potential, Phys. Rev. Lett. 113, 241801 (2014).
  52. S. P. Martin, Taming the Goldstone contributions to the effective potential, Phys. Rev. D 90, 016013 (2014).
  53. A. O. Barvinsky and A. E. Kalugin, Notes on peculiarities of the Schwinger-DeWitt technique: One-loop double poles, total-derivative terms, and determinant anomalies, Phys. Rev. D 110, 105007 (2024).
  54. A. A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, and S. Zerbini, Analytic Aspects of Quantum Fields (World Scientific, Singapore, 2003).
  55. K. Fujikawa, B. W. Lee, and A. I. Sanda, Generalized renormalizable gauge formulation of spontaneously broken gauge theories, Phys. Rev. D 6, 2923 (1972).
  56. C. Becchi, A. Rouet, and R. Stora, Renormalization of gauge theories, Ann. Phys. (N.Y.) 98, 287 (1976).
  57. I. V. Tyutin, Gauge invariance in field theory and statistical physics in operator formalism, arXiv:0812.0580.
  58. L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys. Rev. D 9, 3320 (1974).
  59. E. Megias, E. Ruiz Arriola, and L. L. Salcedo, The thermal heat kernel expansion and the one loop effective action of QCD at finite temperature, Phys. Rev. D 69, 116003 (2004).
  60. J. Chakrabortty and S. Mohanty, One loop thermal effective action, arXiv:2411.14146.
  61. F. J. Moral-Gamez and L. L. Salcedo, Derivative expansion of the heat kernel at finite temperature, Phys. Rev. D 85, 045019 (2012).
  62. L. Di Luzio and L. Mihaila, On the gauge dependence of the standard model vacuum instability scale, J. High Energy Phys. 06 (2014) 079.

Outline

Information

Phys. Rev. D 112, 056022– Published 19 September, 2025
  • Received 1 August 2025
  • Accepted 4 September 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