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  • Open Access

Polyakov loops tame phase transitions

Lisa Biermann1,*, Joydeep Chakrabortty2,†, and Christoph Englert3,‡

  • *Contact author: lisa.biermann@psi.ch
  • †Contact author: joydeep@iitk.ac.in
  • ‡Contact author: christoph.englert@manchester.ac.uk
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Phys. Rev. D 114, 056031 – Published 29 September, 2026

DOI: https://doi.org/10.1103/5cbz-vt5j

Abstract

We estimate the impact of Polyakov loop (PL) contributions on electroweak phase transitions (PTs). We show that the PL, which is unavoidable in thermal gauge field theory, tends to tame thermal contributions, thereby softening electroweak PTs and affecting bubble dynamics, nucleation, and the related gravitational-wave spectrum. Including this contribution in perturbative approaches results in a thermal effective potential that disfavors first-order PTs over either second-order PTs or smooth crossovers. This feature is universal for both fermionic and bosonic contributions to the effective potential.

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References (121)

  1. A. D. Sakharov, Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe, Pis’ma Zh. Eksp. Teor. Fiz. 5, 32 (1967).
  2. C. L. Wainwright, CosmoTransitions: Computing cosmological phase transition temperatures and bubble profiles with multiple fields, Comput. Phys. Commun. 183, 2006 (2012).
  3. P. Basler and M. Mühlleitner, bsmpt (Beyond the Standard Model Phase Transitions): A tool for the electroweak phase transition in extended Higgs sectors, Comput. Phys. Commun. 237, 62 (2019).
  4. P. Athron, C. Balázs, A. Fowlie, and Y. Zhang, PhaseTracer: Tracing cosmological phases and calculating transition properties, Eur. Phys. J. C 80, 567 (2020).
  5. P. Basler, M. Mühlleitner, and J. Müller, bsmpt v2 a tool for the electroweak phase transition and the baryon asymmetry of the universe in extended Higgs sectors, Comput. Phys. Commun. 269, 108124 (2021).
  6. A. Ekstedt, P. Schicho, and T. V. I. Tenkanen, DRalgo: A package for effective field theory approach for thermal phase transitions, Comput. Phys. Commun. 288, 108725 (2023).
  7. P. Schicho and D. Schmitt, “DRansitions.” https://github.com/DMGW-Goethe/DRansitions.
  8. P. Basler, L. Biermann, M. Mühlleitner, J. Müller, R. Santos, and J. Viana, bsmpt v3 a tool for phase transitions and primordial gravitational waves in extended Higgs sectors, Comput. Phys. Commun. 316, 109766 (2025).
  9. P. Athron, C. Balazs, A. Fowlie, L. Morris, W. Searle, Y. Xiao, and Y. Zhang, PhaseTracer2: from the effective potential to gravitational waves, Eur. Phys. J. C 85, 559 (2025).
  10. V. Brdar, M. Finetti, M. Matteini, A. P. Morais, and M. Nemevšek, PT2 GWFinder: A package for cosmological first-order phase transitions and gravitational waves, Comput. Phys. Commun. 323, 110119 (2026).
  11. F. Costa, J. Hoefken Zink, M. Lucente, S. Pascoli, and S. Rosauro-Alcaraz, ELENA: a software for fast and precise computation of first order phase transitions and gravitational waves production in particle physics models, Eur. Phys. J. C 86, 851 (2026).
  12. V. Keus, L. Lewitt, and J. Thomson-Cooke, BLOOP https://github.com/MOREHIGGS/Bloop.
  13. J. Matuszak and C. Tasillo, TransitionListener v2.0—Robust gravitational wave predictions for cosmological phase transitions, arXiv:2605.15259.
  14. I. R. Wang, LeWRON: Agentic analysis of electroweak phase transitions, arXiv:2606.19425.
  15. K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, The electroweak phase transition: A nonperturbative analysis, Nucl. Phys. B466, 189 (1996).
  16. J. M. Cline and K. Kainulainen, Supersymmetric electroweak phase transition: Beyond perturbation theory, Nucl. Phys. B482, 73 (1996).
  17. M. Laine and K. Rummukainen, What’s new with the electroweak phase transition?, Nucl. Phys. B, Proc. Suppl. 73, 180 (1999).
  18. C. Ratti, M. A. Thaler, and W. Weise, Phases of QCD: Lattice thermodynamics and a field theoretical model, Phys. Rev. D 73, 014019 (2006).
  19. K. Kainulainen, V. Keus, L. Niemi, K. Rummukainen, T. V. I. Tenkanen, and V. Vaskonen, On the validity of perturbative studies of the electroweak phase transition in the two Higgs doublet model, J. High Energy Phys. 06 (2019) 075.
  20. O. Gould, J. Kozaczuk, L. Niemi, M. J. Ramsey-Musolf, T. V. I. Tenkanen, and D. J. Weir, Nonperturbative analysis of the gravitational waves from a first-order electroweak phase transition, Phys. Rev. D 100, 115024 (2019).
  21. O. Gould, S. Güyer, and K. Rummukainen, First-order electroweak phase transitions: A nonperturbative update, Phys. Rev. D 106, 114507 (2022).
  22. L. Niemi, M. J. Ramsey-Musolf, and G. Xia, Nonperturbative study of the electroweak phase transition in the real scalar singlet extended standard model, Phys. Rev. D 110, 115016 (2024).
  23. O. Gould, Real scalar phase transitions: a nonperturbative analysis, J. High Energy Phys. 04 (2021) 057.
  24. A. Ekstedt, O. Gould, and J. Löfgren, Radiative first-order phase transitions to next-to-next-to-leading order, Phys. Rev. D 106, 036012 (2022).
  25. O. Gould and T. V. I. Tenkanen, Perturbative effective field theory expansions for cosmological phase transitions, J. High Energy Phys. 01 (2024) 048.
  26. A. Ekstedt, P. Schicho, and T. V. I. Tenkanen, Cosmological phase transitions at three loops: The final verdict on perturbation theory, Phys. Rev. D 110, 096006 (2024).
  27. S. R. Coleman, The fate of the false vacuum. 1. Semiclassical theory, Phys. Rev. D 15, 2929 (1977); 16, 1248(E) (1977).
  28. C. G. Callan, Jr. and S. R. Coleman, The fate of the false vacuum. 2. First quantum corrections, Phys. Rev. D 16, 1762 (1977).
  29. A. D. Linde, On the vacuum instability, and the Higgs meson mass, Phys. Lett. 70B, 306 (1977).
  30. I. Affleck, Quantum statistical metastability, Phys. Rev. Lett. 46, 388 (1981).
  31. A. D. Linde, Decay of the false vacuum at finite temperature, Nucl. Phys. B216, 421 (1983); B223, 544(E) (1983).
  32. T. Konstandin and S. J. Huber, Numerical approach to multi dimensional phase transitions, J. Cosmol. Astropart. Phys. 06 (2006) 021.
  33. A. Ekstedt, Higher-order corrections to the bubble-nucleation rate at finite temperature, Eur. Phys. J. C 82, 173 (2022).
  34. O. Gould and J. Hirvonen, Effective field theory approach to thermal bubble nucleation, Phys. Rev. D 104, 096015 (2021).
  35. 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).
  36. 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.
  37. M. Kierkla, P. Schicho, B. Swiezewska, T. V. I. Tenkanen, and J. van de Vis, Finite-temperature bubble nucleation with shifting scale hierarchies, J. High Energy Phys. 07 (2025) 153.
  38. D. E. Morrissey and M. J. Ramsey-Musolf, Electroweak baryogenesis, New J. Phys. 14, 125003 (2012).
  39. J. van de Vis, J. de Vries, and M. Postma, Bubble trouble: A review on electroweak baryogenesis, Prog. Part. Nucl. Phys. 150, 104244 (2026).
  40. A. M. Polyakov, Compact gauge fields, and the infrared catastrophe, Phys. Lett. 59B, 82 (1975).
  41. A. M. Polyakov, Thermal properties of gauge fields, and quark liberation, Phys. Lett. 72B, 477 (1978).
  42. L. Susskind, Lattice models of quark confinement at high temperature, Phys. Rev. D 20, 2610 (1979).
  43. N. Weiss, The effective potential for the order parameter of gauge theories at finite temperature, Phys. Rev. D 24, 475 (1981).
  44. C. P. Korthals Altes, Constrained effective potential in hot QCD, Nucl. Phys. B420, 637 (1994).
  45. A. Dumitru, Y. Guo, and C. P. Korthals Altes, Two-loop perturbative corrections to the thermal effective potential in gluodynamics, Phys. Rev. D 89, 016009 (2014).
  46. H. Nishimura, C. Korthals-Altes, R. D. Pisarski, and V. Skokov, Thermal effective potential for the Polyakov loop to higher loop order, Proc. Sci. Confinement2018 (2019) 155 [arXiv:1811.11736].
  47. R. Fukuda and E. Kyriakopoulos, Derivation of the effective potential, Nucl. Phys. B85, 354 (1975).
  48. L. O’Raifeartaigh, A. Wipf, and H. Yoneyama, The constraint effective potential, Nucl. Phys. B271, 653 (1986).
  49. D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and instantons at finite temperature, Rev. Mod. Phys. 53, 43 (1981).
  50. R. D. Pisarski, Quark gluon plasma as a condensate of SU(3) Wilson lines, Phys. Rev. D 62, 111501 (2000).
  51. R. D. Pisarski, Notes on the deconfining phase transition, arXiv:hep-ph/0203271.
  52. V. N. Popov and S. A. Fedotov, The functional-integration method and diagram technique for spin systems, Sov. Phys. JETP 67, 535 (1988).
  53. O. Veits, R. Oppermann, M. Binderberger, and J. Stein, Extension of the Popov-Fedotov method to arbitrary spin, J. Phys. (Paris) I 4, 493 (1994).
  54. J. Chakrabortty and S. Mohanty, One loop thermal effective action, Nucl. Phys. B1020, 117165 (2025).
  55. S. Bandyopadhyay, J. Chakrabortty, D. Dey, P. Schicho, and Tushar, Higher-dimensional operators and Polyakov loop in hot Scalar QED from the heat kernel, arXiv:2606.09779.
  56. B. Svetitsky and L. G. Yaffe, Critical behavior at finite temperature confinement transitions, Nucl. Phys. B210, 423 (1982).
  57. J. Chakrabortty, S. Ghosh, and A. Kundu (to be published).
  58. T. Matsubara, A new approach to quantum statistical mechanics, Prog. Theor. Phys. 14, 351 (1955).
  59. A. J. Niemi and G. W. Semenoff, Finite temperature quantum field theory in Minkowski space, Ann. Phys. (N.Y.) 152, 105 (1984).
  60. N. P. Landsman and C. G. van Weert, Real and imaginary time field theory at finite temperature and density, Phys. Rep. 145, 141 (1987).
  61. R. Kubo, Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  62. P. C. Martin and J. S. Schwinger, Theory of many particle systems. 1, Phys. Rev. 115, 1342 (1959).
  63. 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).
  64. F. J. Moral-Gamez and L. L. Salcedo, Derivative expansion of the heat kernel at finite temperature, Phys. Rev. D 85, 045019 (2012).
  65. D. Balui, T. Biswas, J. Chakrabortty, D. Dey, C. Englert, and S. Mohanty, Gauge choices, infrared pitfalls, and thermal effects in effective potentials, Phys. Rev. D 112, 056022 (2025).
  66. L. L. Salcedo, Parity breaking in (2+1)-dimensions and finite temperature, Nucl. Phys. B549, 98 (1999).
  67. P. B. Arnold and L. G. Yaffe, The NonAbelian Debye screening length beyond leading order, Phys. Rev. D 52, 7208 (1995).
  68. M. Laine and O. Philipsen, The nonperturbative QCD Debye mass from a Wilson line operator, Phys. Lett. B 459, 259 (1999).
  69. S. R. Coleman and E. J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7, 1888 (1973).
  70. M. Quiros, Field theory at finite temperature, and phase transitions, Helv. Phys. Acta 67, 451 (1994).
  71. M. Quiros, Finite temperature field theory, and phase transitions, in ICTP Summer School in High-Energy Physics and Cosmology (World Scientific, Singapore, 1999), pp. 187–259.
  72. Y. Hidaka and R. D. Pisarski, Small shear viscosity in the semi quark gluon plasma, Phys. Rev. D 81, 076002 (2010).
  73. J. Braathen, S. Heinemeyer, C. P. Boatella, and A. Verduras Schaeidt, Complementarity of gravitational wave analyses and di-Higgs production in the exploration of the Electroweak Phase Transition dynamics in the RxSM, J. High Energy Phys. 06 (2026) 035.
  74. T. Biekötter, A. Dashko, M. Löschner, and G. Weiglein, Perturbative aspects of the electroweak phase transition with a complex singlet and implications for gravitational wave predictions, arXiv:2511.14831.
  75. D. Balui, J. Chakrabortty, D. Dey, and S. Mohanty, Gauge invariant effective potential, Phys. Rev. D 111, 085032 (2025).
  76. D. Balui, J. Chakrabortty, C. Englert, S. Mohanty, and Tushar, Background fields meet the heat kernel: Gauge invariance and RGEs without diagrams, Phys. Rev. D 113, 116040 (2026).
  77. J. Dumont, D. Gagnon et al., complex_bessel, 10.5281/zenodo.18220.
  78. D. E. Amos et al., SLATEC mathematical library, https://netlib.org/slatec/src/.
  79. D. Croon, TASI lectures on phase transitions, baryogenesis, and gravitational waves, Proc. Sci. TASI2022 (2024) 003. [arXiv:2307.00068].
  80. P. Athron, C. Balázs, A. Fowlie, L. Morris, and L. Wu, Cosmological phase transitions: From perturbative particle physics to gravitational waves, Prog. Part. Nucl. Phys. 135, 104094 (2024).
  81. H. H. Patel and M. J. Ramsey-Musolf, Baryon washout, electroweak phase transition, and perturbation theory, J. High Energy Phys. 07 (2011) 029.
  82. A. Ekstedt and J. Löfgren, A critical look at the electroweak phase transition, J. High Energy Phys. 12 (2020) 136.
  83. L. Di Luzio and L. Mihaila, On the gauge dependence of the standard model vacuum instability scale, J. High Energy Phys. 06 (2014) 079.
  84. 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.
  85. C. L. Hammer, J. E. Shrauner, and B. DeFacio, Alternate derivation of vacuum tunneling, Phys. Rev. D 19, 667 (1979).
  86. F. Devoto, S. Devoto, L. Di Luzio, and G. Ridolfi, False vacuum decay: An introductory review, J. Phys. G 49, 103001 (2022).
  87. V. K. Shante and S. Kirkpatrick, An introduction to percolation theory, Adv. Phys. 20, 325 (1971).
  88. C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: An update, J. Cosmol. Astropart. Phys. 03 (2020) 024.
  89. L. S. Friedrich, M. J. Ramsey-Musolf, T. V. I. Tenkanen, and V. Q. Tran, Addressing the gravitational wave—Collider inverse problem, arXiv:2203.05889.
  90. T. Biekötter, S. Heinemeyer, J. M. No, M. O. Olea-Romacho, and G. Weiglein, The trap in the early Universe: impact on the interplay between gravitational waves and LHC physics in the 2HDM, J. Cosmol. Astropart. Phys. 03 (2023) 031.
  91. M. J. Ramsey-Musolf, T. V. I. Tenkanen, and V. Q. Tran, Refining gravitational wave and collider physics dialogue via singlet scalar extension, arXiv:2409.17554.
  92. T. Biekötter, evortran: A modern Fortran package for genetic algorithms with applications from LHC data fitting to LISA signal reconstruction, SciPost Phys. Codebases 64, 1 (2026).
  93. A. Conaci, S. De Curtis, L. Delle Rose, A. Dey, A. Karan, S. Moretti, and M. Razzaq, Electro-weak phase transitions and collider signals in the aligned 2-Higgs doublet model, arXiv:2604.14099.
  94. M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Numerical simulations of acoustically generated gravitational waves at a first order phase transition, Phys. Rev. D 92, 123009 (2015).
  95. M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Shape of the acoustic gravitational wave power spectrum from a first order phase transition, Phys. Rev. D 96, 103520 (2017); 101, 089902(E) (2020).
  96. F. Giese, T. Konstandin, and J. van de Vis, Model-independent energy budget of cosmological first-order phase transitions—A sound argument to go beyond the bag model, J. Cosmol. Astropart. Phys. 07 (2020) 057.
  97. F. Giese, T. Konstandin, K. Schmitz, and J. van de Vis, Model-independent energy budget for LISA, J. Cosmol. Astropart. Phys. 01 (2021) 072.
  98. W. M. Folkner (LISA Collaboration), The LISA mission design, AIP Conf. Proc. 456, 11 (1998).
  99. K. Danzmann LISA Study Team), LISA mission overview, Adv. Space Res. 25, 1129 (2000).
  100. P. Amaro-Seoane et al. (LISA Collaboration), Laser Interferometer Space Antenna, arXiv:1702.00786.
  101. P. Auclair et al. (LISA Cosmology Working Group), Cosmology with the Laser Interferometer Space Antenna, Living Rev. Relativity 26, 5 (2023).
  102. C. Caprini, R. Jinno, M. Lewicki, E. Madge, M. Merchand, G. Nardini, M. Pieroni, A. Roper Pol, and V. Vaskonen (LISA Cosmology Working Group), Gravitational waves from first-order phase transitions in LISA: Reconstruction pipeline and physics interpretation, J. Cosmol. Astropart. Phys. 10 (2024) 020.
  103. N. Cabibbo, Unitary symmetry and leptonic decays, Phys. Rev. Lett. 10, 531 (1963).
  104. M. Kobayashi and T. Maskawa, CP violation in the renormalizable theory of weak interaction, Prog. Theor. Phys. 49, 652 (1973).
  105. E. Witten, An SU(2) anomaly, Phys. Lett. B 117, 324 (1982).
  106. R. Fok and G. D. Kribs, Four Generations, the electroweak phase transition, and supersymmetry, Phys. Rev. D 78, 075023 (2008).
  107. Q.-H. Cao, K. Hashino, X.-X. Li, Z. Ren, and J.-H. Yu, Electroweak phase transition triggered by fermion sector, J. High Energy Phys. 01 (2022) 001.
  108. L. Biermann, J. Chakrabortty, and C. Englert (to be published).
  109. M. Chala, J. C. Criado, L. Gil, and J. L. Miras, Higher-order-operator corrections to phase-transition parameters in dimensional reduction, J. High Energy Phys. 10 (2024) 025.
  110. A. Bhatnagar, D. Croon, and P. Schicho, Interpreting the 95 GeV resonance in the Two Higgs doublet model. Implications for the electroweak phase transition, J. High Energy Phys. 03 (2026) 014.
  111. M. Chala and G. Guedes, The high-temperature limit of the SM(EFT), J. High Energy Phys. 07 (2025) 085.
  112. M. Chala, L. Gil, and Z. Ren, Phase transitions in dimensional reduction up to three loops, Chin. Phys. C 49, 123105 (2025).
  113. M. Chala, M. C. Fiore, and L. Gil, Phase diagram of the standard model effective field theory, Phys. Rev. D 113, 115061 (2026).
  114. F. Bernardo, M. Chala, L. Gil, and P. Schicho, Hard thermal contributions to phase transition observables at NNLO, J. High Energy Phys. 06 (2026) 281.
  115. J. Chakrabortty, B. Siqueira Eduardo, S. Karmakar, and P. Schicho, Finite-temperature operator basis on R3×S1 for SMEFT, arXiv:2605.02878.
  116. F. Bernardo, R. G. Reinle, and P. Schicho, Matching higher-dimensional operators at finite temperature for general models, arXiv:2605.15176.
  117. R. Pasechnik, M. Reichert, F. Sannino, and Z.-W. Wang, Gravitational waves from composite dark sectors, J. High Energy Phys. 02 (2024) 159.
  118. R. Houtz, M. Ulloa, and M. West, Gravitational waves from confining dark sectors with self-consistent effective potentials, J. High Energy Phys. 08 (2026) 059.
  119. P. N. Meisinger and M. C. Ogilvie, Complete high temperature expansions for one loop finite temperature effects, Phys. Rev. D 65, 056013 (2002).
  120. A. Actor, Zeta function regularization of high temperature expansions in field theory, Nucl. Phys. B265, 689 (1986).
  121. A. Actor, More on Zeta function regularization of high temperature expansions, Fortschr. Phys. 35, 793 (1987).

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Phys. Rev. D 114, 056031– Published 29 September, 2026
  • Received 15 July 2026
  • Accepted 3 September 2026
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