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Revisiting the sphaleron and axion production rates in QCD at high temperatures

Sayak Guin* and Sayantan Sharma

  • *Contact author: sayakg@imsc.res.in
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Phys. Rev. D 114, 014520 – Published 27 July, 2026

DOI: https://doi.org/10.1103/6t1x-trw7

Abstract

We report our new lattice results for the sphaleron rate calculated within a thermal effective field theory of soft SU(N) gluons whose momenta are below the magnetic scale, where N=2, 3, for a wide range of temperatures spanning from 0.6 to 1015  GeV at sufficiently large volumes. Comparing these results with sphaleron rates in a nonthermal SU(N) plasma where the infrared gluons are overoccupied, we estimate the typical thermalization time for these ultrasoft gluons during the early stages of reheating after inflation. We also calculate the thermal production rate of relativistic axions due to these nonperturbatively interacting soft gluons which shows a significant deviation from its perturbative estimate even at the electroweak scale.

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References (66)

  1. A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. 59B, 85 (1975).
  2. G. ’t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D 14, 3432 (1976); 18, 2199(E) (1978).
  3. F. R. Klinkhamer and N. S. Manton, A saddle point solution in the Weinberg-Salam theory, Phys. Rev. D 30, 2212 (1984).
  4. L. D. McLerran, E. Mottola, and M. E. Shaposhnikov, Sphalerons and axion dynamics in high temperature QCD, Phys. Rev. D 43, 2027 (1991).
  5. D. E. Kharzeev, J. Liao, and P. Tribedy, Chiral magnetic effect in heavy ion collisions: The present and future, Int. J. Mod. Phys. E 33, 2430007 (2024).
  6. R. N. Mohapatra and X.-m. Zhang, QCD sphalerons at high temperature and baryogenesis at electroweak scale, Phys. Rev. D 45, 2699 (1992).
  7. D. Y. Grigoriev and V. A. Rubakov, Soliton pair creation at finite temperatures. Numerical study in (1+1)-dimensions, Nucl. Phys. B299, 67 (1988).
  8. P. B. Arnold and L. G. Yaffe, High temperature color conductivity at next-to-leading log order, Phys. Rev. D 62, 125014 (2000).
  9. D. Bodeker, On the effective dynamics of soft non-Abelian gauge fields at finite temperature, Phys. Lett. B 426, 351 (1998).
  10. D. Bodeker, L. D. McLerran, and A. V. Smilga, Really computing nonperturbative real time correlation functions, Phys. Rev. D 52, 4675 (1995).
  11. G. D. Moore and M. Tassler, The sphaleron rate in SU(N) gauge theory, J. High Energy Phys. 02 (2010) 105.
  12. M. Mace, S. Schlichting, and R. Venugopalan, Off-equilibrium sphaleron transitions in the glasma, Phys. Rev. D 93, 074036 (2016).
  13. S. Schlichting, Turbulent thermalization of weakly coupled non-abelian plasmas, Phys. Rev. D 86, 065008 (2012).
  14. J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan, Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies, Phys. Rev. D 89, 074011 (2014).
  15. J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan, Universal attractor in a highly occupied non-Abelian plasma, Phys. Rev. D 89, 114007 (2014).
  16. J. Berges, K. Boguslavski, L. de Bruin, T. Butler, and J. M. Pawlowski, Order parameters for gauge invariant condensation far from equilibrium, Phys. Rev. D 109, 114011 (2024).
  17. M. Bresciani, M. D. Brida, L. Giusti, and M. Pepe, QCD equation of state at very high temperature: Computational strategy, simulations, and data analysis, Phys. Rev. D 113, 034506 (2026).
  18. T. Kunihiro, B. Muller, A. Ohnishi, A. Schafer, T. T. Takahashi, and A. Yamamoto, Chaotic behavior in classical Yang-Mills dynamics, Phys. Rev. D 82, 114015 (2010).
  19. J. Engels, F. Karsch, and K. Redlich, Scaling properties of the energy density in SU(2) lattice gauge theory, Nucl. Phys. B435, 295 (1995).
  20. G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier, and B. Petersson, Thermodynamics of SU(3) lattice gauge theory, Nucl. Phys. B469, 419 (1996).
  21. S. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, Precision SU(3) lattice thermodynamics for a large temperature range, J. High Energy Phys. 07 (2012) 056.
  22. F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, The color glass condensate, Annu. Rev. Nucl. Part. Sci. 60, 463 (2010).
  23. J. Ambjorn and A. Krasnitz, Improved determination of the classical sphaleron transition rate, Nucl. Phys. B506, 387 (1997).
  24. G. D. Moore, Measuring the broken phase sphaleron rate nonperturbatively, Phys. Rev. D 59, 014503 (1999).
  25. M. Barroso Mancha and G. D. Moore, The sphaleron rate from 4D Euclidean lattices, J. High Energy Phys. 01 (2022) 155.
  26. C. Bonanno, F. D’Angelo, M. D’Elia, L. Maio, and M. Naviglio, Sphaleron rate of Nf=2+1 QCD, Phys. Rev. Lett. 132, 051903 (2024).
  27. A. H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23, 347 (1981).
  28. A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. 91B, 99 (1980).
  29. Y. Akrami et al. (Planck Collaboration), Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641, A10 (2020).
  30. L. Kofman, A. D. Linde, and A. A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73, 3195 (1994).
  31. L. Kofman, A. D. Linde, and A. A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D 56, 3258 (1997).
  32. K. Harigaya and K. Mukaida, Thermalization after/during reheating, J. High Energy Phys. 05 (2013) 006.
  33. S. Passaglia, W. Hu, A. J. Long, and D. Zegeye, Achieving the highest temperature during reheating with the Higgs condensate, Phys. Rev. D 104, 083540 (2021).
  34. M. Drees and B. Najjari, Energy spectrum of thermalizing high energy decay products in the early universe, J. Cosmol. Astropart. Phys. 10 (2021) 009.
  35. K. Mukaida and M. Yamada, Perturbative reheating and thermalization of pure Yang-Mills plasma, J. High Energy Phys. 05 (2024) 174.
  36. S. Guin, H. Pandey, and S. Sharma, Understanding thermalization in a non-Abelian gauge theory in terms of its soft modes, Phys. Lett. B 866, 139490 (2025).
  37. R. Baier, A. H. Mueller, D. Schiff, and D. T. Son, “Bottom up” thermalization in heavy ion collisions, Phys. Lett. B 502, 51 (2001).
  38. A. Kurkela and G. D. Moore, Bjorken flow, plasma instabilities, and thermalization, J. High Energy Phys. 11 (2011) 120.
  39. Y. Fu, J. Ghiglieri, S. Iqbal, and A. Kurkela, Thermalization of non-Abelian gauge theories at next-to-leading order, Phys. Rev. D 105, 054031 (2022).
  40. B. Barman, N. Bernal, and J. Rubio, Two or three things particle physicists (mis)understand about (pre)heating, Nucl. Phys. B1018, 116996 (2025).
  41. I. Rudenok, Y. Shtanov, and S. Vilchinskii, Post-inflationary preheating with weak coupling, Phys. Lett. B 733, 193 (2014).
  42. Y. Ema, R. Jinno, K. Mukaida, and K. Nakayama, Gravitational particle production in oscillating backgrounds and its cosmological implications, Phys. Rev. D 94, 063517 (2016).
  43. A. Caravano, E. Komatsu, K. D. Lozanov, and J. Weller, Lattice simulations of inflation, J. Cosmol. Astropart. Phys. 12 (2021) 010.
  44. F. L. Bezrukov and M. Shaposhnikov, The standard model Higgs boson as the inflaton, Phys. Lett. B 659, 703 (2008).
  45. R. D. Peccei and H. R. Quinn, CP conservation in the presence of instantons, Phys. Rev. Lett. 38, 1440 (1977).
  46. R. D. Peccei and H. R. Quinn, Constraints imposed by CP conservation in the presence of instantons, Phys. Rev. D 16, 1791 (1977).
  47. G. G. Raffelt, Astrophysical axion bounds, Lect. Notes Phys. 741, 51 (2008).
  48. J. Preskill, M. B. Wise, and F. Wilczek, Cosmology of the invisible axion, Phys. Lett. 120B, 127 (1983).
  49. L. F. Abbott and P. Sikivie, A cosmological bound on the invisible axion, Phys. Lett. 120B, 133 (1983).
  50. M. Dine and W. Fischler, The not so harmless axion, Phys. Lett. 120B, 137 (1983).
  51. G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, The QCD axion, precisely, J. High Energy Phys. 01 (2016) 034.
  52. P. Petreczky, H.-P. Schadler, and S. Sharma, The topological susceptibility in finite temperature QCD and axion cosmology, Phys. Lett. B 762, 498 (2016).
  53. S. Borsanyi et al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature (London) 539, 69 (2016).
  54. P. Graf and F. D. Steffen, Thermal axion production in the primordial quark-gluon plasma, Phys. Rev. D 83, 075011 (2011).
  55. K. Akita and M. Yamaguchi, A precision calculation of relic neutrino decoupling, J. Cosmol. Astropart. Phys. 08 (2020) 012.
  56. J. Froustey, C. Pitrou, and M. C. Volpe, Neutrino decoupling including flavour oscillations and primordial nucleosynthesis, J. Cosmol. Astropart. Phys. 12 (2020) 015.
  57. J. J. Bennett, G. Buldgen, P. F. De Salas, M. Drewes, S. Gariazzo, S. Pastor, and Y. Y. Y. Wong, Towards a precision calculation of Neff in the standard model II: Neutrino decoupling in the presence of flavour oscillations and finite-temperature QED, J. Cosmol. Astropart. Phys. 04 (2020) 073.
  58. K. Bouzoud and J. Ghiglieri, Thermal axion production at hard and soft momenta, J. High Energy Phys. 01 (2024) 163.
  59. A. D. Linde, Infrared problem in thermodynamics of the Yang-Mills gas, Phys. Lett. 96B, 289 (1980).
  60. K. Bouzoud, J. Ghiglieri, M. Laine, and G. S. S. Sakoda, Energy and momentum dependence of the soft-axion interaction rate, J. High Energy Phys. 05 (2026) 034.
  61. E. Masso, F. Rota, and G. Zsembinszki, On axion thermalization in the early universe, Phys. Rev. D 66, 023004 (2002).
  62. J. Bernstein, L. S. Brown, and G. Feinberg, The cosmological heavy neutrino problem revisited, Phys. Rev. D 32, 3261 (1985).
  63. M. S. Turner, Thermal production of not so invisible axions in the early universe, Phys. Rev. Lett. 59, 2489 (1987); 60, 1101(E) (1987).
  64. A. Notari, F. Rompineve, and G. Villadoro, Improved hot dark matter bound on the QCD axion, Phys. Rev. Lett. 131, 011004 (2023).
  65. M. Escudero, G. Jackson, M. Laine, and S. Sandner, Fast and flexible neutrino decoupling. Part I. The standard model, J. Cosmol. Astropart. Phys. 02 (2025) 046.
  66. N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).

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Phys. Rev. D 114, 014520– Published 27 July, 2026
  • Received 8 May 2026
  • Accepted 1 July 2026
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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.

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