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  • Open Access

Thermal damping of mass-modulating scalars

Abhishek Banerjee1,*, Ngan H. Nguyen2,†, and Erwin H. Tanin3,‡

  • *Contact author: abanerj4@umd.edu
  • †Contact author: nnguye53@jhu.edu
  • ‡Contact author: ehtanin@stanford.edu
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Phys. Rev. D 114, 015032 – Published 22 July, 2026

DOI: https://doi.org/10.1103/rqq9-1dc2

Abstract

The cosmological evolution of a scalar field is shaped by Hubble damping. Any nongravitational couplings of the scalar with the primordial thermal bath generically contribute additional damping. Although often neglected, such thermal damping could be the dominant dissipative effect on the scalar. We derive approximate yet highly general thermal damping rates of scalar fields that modulate the masses of thermally populated particles. We extend previous results to cover cases of particular phenomenological interest where the scalar background oscillates sinusoidally but not necessarily slowly compared to the thermalization rates of the primordial bath. As applications, we estimate the thermal damping of scalars coupled linearly to neutrinos, quadratically to gluons, and linearly to Weakly Interacting Massive Particles, and demonstrate the importance of this effect in certain parameter spaces of these models. We also estimate the thermal damping rates in models of QCD axions.

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References (107)

  1. A. Berera, Warm inflation, Phys. Rev. Lett. 75, 3218 (1995).
  2. L. D. McLerran, E. Mottola, and M. E. Shaposhnikov, Sphalerons and axion dynamics in high temperature QCD, Phys. Rev. D 43, 2027 (1991).
  3. G. D. Moore and M. Tassler, The sphaleron rate in SU(N) gauge theory, J. High Energy Phys. 02 (2010) 105.
  4. G. D. Moore, Computing the strong sphaleron rate, Phys. Lett. B 412, 359 (1997).
  5. K. V. Berghaus, P. W. Graham, and D. E. Kaplan, Minimal warm inflation, J. Cosmol. Astropart. Phys. 03 (2020) 034; 10 (2023) E02.
  6. K. Minami, K. Mukaida, and K. Nakayama, Reheating with thermal dissipation and primordial gravitational waves, J. Cosmol. Astropart. Phys. 03 (2026) 016.
  7. K. Mukaida and K. Nakayama, Dissipative effects on reheating after inflation, J. Cosmol. Astropart. Phys. 03 (2012) 002.
  8. M. Drewes and J. U. Kang, The kinematics of cosmic reheating, Nucl. Phys. B875, 315 (2013); B888, 284(E) (2014).
  9. P. W. Graham, D. E. Kaplan, and S. Rajendran, Relaxation of the cosmological constant, Phys. Rev. D 100, 015048 (2019).
  10. L. Ji, D. E. Kaplan, S. Rajendran, and E. H. Tanin, Thermal perturbations from cosmological constant relaxation, Phys. Rev. D 105, 015025 (2022).
  11. K. V. Berghaus and T. Karwal, Thermal friction as a solution to the Hubble tension, Phys. Rev. D 101, 083537 (2020).
  12. K. V. Berghaus, P. W. Graham, D. E. Kaplan, G. D. Moore, and S. Rajendran, Dark energy radiation, Phys. Rev. D 104, 083520 (2021).
  13. K. Choi, S. H. Im, H. J. Kim, and H. Seong, Axion dark matter with thermal friction, J. High Energy Phys. 02 (2022) 180.
  14. A. Papageorgiou, P. Quílez, and K. Schmitz, Axion dark matter from frictional misalignment, J. High Energy Phys. 01 (2022) 169.
  15. G. N. Felder, H. Kim, W.-I. Park, and E. D. Stewart, Preheating and Affleck-Dine leptogenesis after thermal inflation, J. Cosmol. Astropart. Phys. 06 (2007) 005.
  16. E. H. Tanin and E. D. Stewart, Damping of an oscillating scalar field indirectly coupled to a thermal bath, J. Cosmol. Astropart. Phys. 11 (2017) 019.
  17. A. Banerjee, N. H. Nguyen, and E. H. Tanin, Thermal damping of neutrino-coupled scalar dark matter, arXiv:2509.25308.
  18. K. V. Berghaus, M. Drewes, and S. Zell, Warm inflation with the standard model, Phys. Rev. Lett. 135, 171002 (2025).
  19. J. Yokoyama and A. D. Linde, Is warm inflation possible?, Phys. Rev. D 60, 083509 (1999).
  20. K. Mukaida, K. Nakayama, and M. Takimoto, Fate of Z2 Symmetric Scalar Field, J. High Energy Phys. 12 (2013) 053.
  21. K. Mukaida and K. Nakayama, Dynamics of oscillating scalar field in thermal environment, J. Cosmol. Astropart. Phys. 01 (2012) 017.
  22. M. Bastero-Gil, A. Berera, and R. O. Ramos, Dissipation coefficients from scalar and fermion quantum field interactions, J. Cosmol. Astropart. Phys. 09 (2010) 033.
  23. J. Yokoyama, Fate of oscillating scalar fields in the thermal bath and their cosmological implications, Phys. Rev. D 70, 103511 (2004).
  24. J. Yokoyama, Can oscillating scalar fields decay into particles with a large thermal mass?, Phys. Lett. B 635, 66 (2006).
  25. M. Morikawa, Classical fluctuations in dissipative quantum systems, Phys. Rev. D 33, 3607 (1986).
  26. W.-Y. Ai, A. Beniwal, A. Maggi, and D. J. E. Marsh, From QFT to Boltzmann: Freeze-in in the presence of oscillating condensates, J. High Energy Phys. 02 (2023) 122.
  27. W.-Y. Ai, M. Drewes, D. Glavan, and J. Hajer, Oscillating scalar dissipating in a medium, J. High Energy Phys. 11 (2021) 160.
  28. A. Berera, M. Gleiser, and R. O. Ramos, Strong dissipative behavior in quantum field theory, Phys. Rev. D 58, 123508 (1998).
  29. M. Gleiser and R. O. Ramos, Microphysical approach to nonequilibrium dynamics of quantum fields, Phys. Rev. D 50, 2441 (1994).
  30. L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys. Rev. D 9, 3320 (1974).
  31. G. Aarts, D. Ahrensmeier, R. Baier, J. Berges, and J. Serreau, Far from equilibrium dynamics with broken symmetries from the 2PI—1/N expansion, Phys. Rev. D 66, 045008 (2002).
  32. J. M. Cornwall, R. Jackiw, and E. Tomboulis, Effective action for composite operators, Phys. Rev. D 10, 2428 (1974).
  33. E. A. Calzetta and B.-L. B. Hu, Nonequilibrium Quantum Field Theory (Oxford University Press, New York, 2009).
  34. D. Bödeker and J. Nienaber, Scalar field damping at high temperatures, Phys. Rev. D 106, 056016 (2022).
  35. A. Hosoya and M.-a. Sakagami, Time development of Higgs field at finite temperature, Phys. Rev. D 29, 2228 (1984).
  36. S. Jeon, Hydrodynamic transport coefficients in relativistic scalar field theory, Phys. Rev. D 52, 3591 (1995).
  37. M. Drewes, S. Mendizabal, and C. Weniger, The Boltzmann equation from quantum field theory, Phys. Lett. B 718, 1119 (2013).
  38. B. Garbrecht and M. Garny, Finite width in out-of-Equilibrium propagators and kinetic theory, Ann. Phys. (Amsterdam) 327, 914 (2012).
  39. M. Laine and S. Procacci, Minimal warm inflation with complete medium response, J. Cosmol. Astropart. Phys. 06 (2021) 031.
  40. G. N. Felder, L. Kofman, and A. D. Linde, Instant preheating, Phys. Rev. D 59, 123523 (1999).
  41. M. L. Bellac, Thermal Field Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2011).
  42. M. Quiros, Finite temperature field theory and phase transitions, in ICTP Summer School in High-Energy Physics and Cosmology (1999), pp. 187–259, .
  43. A. Berlin, Neutrino oscillations as a probe of light scalar dark matter, Phys. Rev. Lett. 117, 231801 (2016).
  44. G. Krnjaic, P. A. N. Machado, and L. Necib, Distorted neutrino oscillations from time varying cosmic fields, Phys. Rev. D 97, 075017 (2018).
  45. V. Brdar, J. Kopp, J. Liu, P. Prass, and X.-P. Wang, Fuzzy dark matter and nonstandard neutrino interactions, Phys. Rev. D 97, 043001 (2018).
  46. J. Liao, D. Marfatia, and K. Whisnant, Light scalar dark matter at neutrino oscillation experiments, J. High Energy Phys. 04 (2018) 136.
  47. F. Capozzi, I. M. Shoemaker, and L. Vecchi, Neutrino oscillations in dark backgrounds, J. Cosmol. Astropart. Phys. 07 (2018) 004.
  48. G.-Y. Huang and N. Nath, Neutrinophilic Axion-Like dark matter, Eur. Phys. J. C 78, 922 (2018).
  49. J. M. Cline, Viable secret neutrino interactions with ultralight dark matter, Phys. Lett. B 802, 135182 (2020).
  50. A. Dev, P. A. N. Machado, and P. Martínez-Miravé, Signatures of ultralight dark matter in neutrino oscillation experiments, J. High Energy Phys. 01 (2020) 094.
  51. G.-y. Huang and N. Nath, Neutrino meets ultralight dark matter: 0νββ decay and cosmology, J. Cosmol. Astropart. Phys. 05 (05) 034,
  52. M. Losada, Y. Nir, G. Perez, and Y. Shpilman, Probing scalar dark matter oscillations with neutrino oscillations, J. High Energy Phys. 04 (2021) 030.
  53. E. J. Chun, Neutrino transition in dark matter, arXiv:2112.05057.
  54. A. Dev, G. Krnjaic, P. Machado, and H. Ramani, Constraining feeble neutrino interactions with ultralight dark matter, Phys. Rev. D 107, 035006 (2023).
  55. G.-y. Huang, M. Lindner, P. Martínez-Miravé, and M. Sen, Cosmology-friendly time-varying neutrino masses via the sterile neutrino portal, Phys. Rev. D 106, 033004 (2022).
  56. M. Losada, Y. Nir, G. Perez, I. Savoray, and Y. Shpilman, Parametric resonance in neutrino oscillations induced by ultra-light dark matter and implications for KamLAND and JUNO, J. High Energy Phys. 03 (2022) 032.
  57. R. Plestid and S. Tevosyan, The cosmology of ultralight scalar dark matter coupled to right-handed neutrinos, J. High Energy Phys. 07 (2024) 012.
  58. T. Gherghetta and A. Shkerin, Probing a local dark matter halo with neutrino oscillations, Phys. Rev. D 108, 095009 (2023).
  59. Z. Chacko and R. K. Mishra, Effective theory of a light dilaton, Phys. Rev. D 87, 115006 (2013).
  60. A. Banerjee, C. Csáki, M. Geller, Z. Heller-Algazi, and A. Ismail, Ultralight dilatonic dark matter, J. High Energy Phys. 04 (2026) 158.
  61. P. W. Graham, D. E. Kaplan, and S. Rajendran, Cosmological relaxation of the electroweak scale, Phys. Rev. Lett. 115, 221801 (2015).
  62. A. Banerjee, H. Kim, and G. Perez, Coherent relaxion dark matter, Phys. Rev. D 100, 115026 (2019).
  63. F. Piazza and M. Pospelov, Sub-eV scalar dark matter through the super-renormalizable Higgs portal, Phys. Rev. D 82, 043533 (2010).
  64. L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, The landscape of QCD axion models, Phys. Rep. 870, 1 (2020).
  65. H. Kim and G. Perez, Oscillations of atomic energy levels induced by QCD axion dark matter, Phys. Rev. D 109, 015005 (2024).
  66. A. Hook and J. Huang, Probing axions with neutron star inspirals and other stellar processes, J. High Energy Phys. 06 (2017) 036.
  67. A. Hook, Solving the hierarchy problem discretely, Phys. Rev. Lett. 120, 261802 (2018).
  68. A. Banerjee, M. A. Buen-Abad, and A. Hook, Constructing a light QCD axion without 1/N tuning, Phys. Rev. D 112, 075027 (2025).
  69. Y. V. Stadnik and V. V. Flambaum, Can dark matter induce cosmological evolution of the fundamental constants of nature?, Phys. Rev. Lett. 115, 201301 (2015).
  70. K. A. Olive and M. Pospelov, Environmental dependence of masses and coupling constants, Phys. Rev. D 77, 043524 (2008).
  71. A. Hees, O. Minazzoli, E. Savalle, Y. V. Stadnik, and P. Wolf, Violation of the equivalence principle from light scalar dark matter, Phys. Rev. D 98, 064051 (2018).
  72. A. Banerjee, G. Perez, M. Safronova, I. Savoray, and A. Shalit, The phenomenology of quadratically coupled ultra light dark matter, J. High Energy Phys. 10 (2022) 042.
  73. K. Bartnick, K. Springmann, S. Stelzl, and A. Weiler, ϕ-dwarfs: White dwarfs probe quadratically coupled scalars, J. High Energy Phys. 04 (2026) 061.
  74. E. G. Adelberger, B. R. Heckel, and A. E. Nelson, Tests of the gravitational inverse square law, Annu. Rev. Nucl. Part. Sci. 53, 77 (2003).
  75. P. Touboul et al. (MICROSCOPE Collaboration), MICROSCOPE mission: Final results of the test of the equivalence principle, Phys. Rev. Lett. 129, 121102 (2022).
  76. L. Hui, A. Nicolis, and C. Stubbs, Equivalence principle implications of modified gravity models, Phys. Rev. D 80, 104002 (2009).
  77. T. Damour and J. F. Donoghue, Equivalence principle violations and couplings of a light dilaton, Phys. Rev. D 82, 084033 (2010).
  78. S. Ghosh, K. K. Boddy, and T.-T. Yu, Early universe constraints on variations in fundamental constants induced by ultralight scalar dark matter, arXiv:2511.14532.
  79. S. Sibiryakov, P. Sørensen, and T.-T. Yu, BBN constraints on universally-coupled ultralight scalar dark matter, J. High Energy Phys. 12 (2020) 075.
  80. V. V. Flambaum and E. V. Shuryak, Limits on cosmological variation of strong interaction and quark masses from big bang nucleosynthesis, cosmic, laboratory and oklo data, Phys. Rev. D 65, 103503 (2002).
  81. V. V. Flambaum and A. J. Mansour, Constraints on the variation of the QCD interaction scale ΛQCD, J. High Energy Phys. 11 (2025) 086.
  82. C. Delaunay, M. Geller, Z. Heller-Algazi, G. Perez, and K. Springmann, Natural ultralight dark matter: The quadratic twin, Phys. Rev. D 113, 035011 (2026).
  83. G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, The QCD axion, precisely, J. High Energy Phys. 01 (2015) 034.
  84. M. P. Hertzberg, M. Tegmark, and F. Wilczek, Axion cosmology and the energy scale of inflation, Phys. Rev. D 78, 083507 (2008).
  85. M. Kawasaki, N. Kitajima, and F. Takahashi, Relaxing isocurvature bounds on string axion dark matter, Phys. Lett. B 737, 178 (2014).
  86. I. J. Allali, M. P. Hertzberg, and Y. Lyu, Altered axion abundance from a dynamical Peccei-Quinn scale, Phys. Rev. D 105, 123517 (2022).
  87. P. W. Graham and D. Racco, Revisiting isocurvature bounds on the minimal QCD axion, J. High Energy Phys. 12 (2025) 028.
  88. L. Heurtier, F. Huang, and T. M. P. Tait, Resurrecting low-mass axion dark matter via a dynamical QCD scale, J. High Energy Phys. 12 (2021) 216.
  89. S. Ipek and T. M. P. Tait, Early cosmological period of QCD confinement, Phys. Rev. Lett. 122, 112001 (2019).
  90. V. Vovchenko, B. B. Brandt, F. Cuteri, G. Endrődi, F. Hajkarim, and J. Schaffner-Bielich, Pion condensation in the early universe at nonvanishing lepton flavor asymmetry and its gravitational wave signatures, Phys. Rev. Lett. 126, 012701 (2021).
  91. M. M. Middeldorf-Wygas, I. M. Oldengott, D. Bödeker, and D. J. Schwarz, Cosmic QCD transition for large lepton flavor asymmetries, Phys. Rev. D 105, 123533 (2022).
  92. M. M. Wygas, I. M. Oldengott, D. Bödeker, and D. J. Schwarz, Cosmic QCD epoch at nonvanishing lepton asymmetry, Phys. Rev. Lett. 121, 201302 (2018).
  93. L. Di Luzio, B. Gavela, P. Quilez, and A. Ringwald, An even lighter QCD axion, J. High Energy Phys. 05 (2021) 184.
  94. L. Di Luzio, B. Gavela, P. Quilez, and A. Ringwald, Dark matter from an even lighter QCD axion: Trapped misalignment, J. Cosmol. Astropart. Phys. 10 (2021) 001.
  95. R. T. Co, E. Gonzalez, and K. Harigaya, Axion misalignment driven to the hilltop, J. High Energy Phys. 05 (2018) 163.
  96. R. T. Co, T. Lee, and O. P. Leonard, (Non-)perturbative dynamics of a light QCD axion: Dark matter and the strong CP problem, Phys. Rev. D 112, 115007 (2025).
  97. C. García-Cely, G. Landini, and Ó. Zapata, Dark matter in QCD-like theories with a theta vacuum: Cosmological and astrophysical implications, Phys. Rev. D 111, 063044 (2025).
  98. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can confinement ensure natural CP invariance of strong interactions?, Nucl. Phys. B166, 493 (1980).
  99. R. J. Crewther, P. Di Vecchia, G. Veneziano, and E. Witten, Chiral estimate of the electric dipole moment of the neutron in quantum chromodynamics, Phys. Lett. B 88, 123 (1979); 91B, 487(E) (1980).
  100. P. W. Graham, H. Ramani, O. Simon, and E. H. Tanin, Cosmological limits on strong dark forces, arXiv:2511.09614.
  101. R. K. Leane, T. R. Slatyer, J. F. Beacom, and K. C. Y. Ng, GeV-scale thermal WIMPs: Not even slightly ruled out, Phys. Rev. D 98, 023016 (2018).
  102. I. Kuznetsova, D. Habs, and J. Rafelski, Pion and muon production in e−, e+, gamma plasma, Phys. Rev. D 78, 014027 (2008).
  103. A. L. Mota, M. C. Nemes, B. Hiller, and H. Walliser, Meson properties in a renormalizable version of the NJL model, Nucl. Phys. A652, 73 (1999).
  104. A. M. Bernstein and B. R. Holstein, Neutral pion lifetime measurements and the QCD chiral anomaly, Rev. Mod. Phys. 85, 49 (2013).
  105. H. Terazawa, Pion pair production by two photons, Phys. Rev. D 51, R954 (1995).
  106. J. Gasser, M. A. Ivanov, and M. E. Sainio, Revisiting gamma gamma—> pi+ pi- at low energies, Nucl. Phys. B745, 84 (2006).
  107. V. F. Mukhanov, Nucleosynthesis without a computer, Int. J. Theor. Phys. 43, 669 (2004).

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Phys. Rev. D 114, 015032– Published 22 July, 2026
  • Received 13 February 2026
  • Accepted 23 June 2026
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