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  • Open Access

Infinite heat order in 3+1 dimensions

Borut Bajc1,*, Giulia Muco2,†, Francesco Sannino2,3,‡, and Sophie Wagner2,§

  • *Contact author: borut.bajc@ijs.si
  • †Contact author: giulia@qtc.sdu.dk
  • ‡Contact author: sannino@cp3.sdu.dk
  • §Contact author: sowag21@student.sdu.dk
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Phys. Rev. D 113, 125029 – Published 22 June, 2026

DOI: https://doi.org/10.1103/tfz8-yf7t

Abstract

We investigate whether spontaneous symmetry breaking can persist up to arbitrarily high temperature in ultraviolet-complete quantum field theories in four spacetime dimensions. We focus on completely asymptotically free models with gauge group SU(Nc1)×SU(Nc2) and two complex scalar fields, each transforming in the fundamental representation of one gauge factor and singlet under the other. The scalar potential contains quartic self-interactions together with a negative portal coupling between the two sectors. In the Veneziano limit, this class of theories was previously shown to admit fixed-flow trajectories for which one scalar acquires a negative thermal mass at asymptotically large temperature, leading to symmetry nonrestoration. Here we extend that analysis to finite numbers of colors and flavors. We derive the finite-N fixed-flow equations, compute the leading 1/N corrections to the large-N solutions, and solve the full finite-N system numerically. We find explicit finite-N benchmark theories for which the scalar potential remains bounded from below, the gauge sector is asymptotically free, and one scalar thermal mass stays negative at arbitrarily high temperature. This provides an explicit perturbative example of infinite heat order in a four-dimensional ultraviolet-complete quantum field theory with a finite field content.

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References (73)

  1. S. Weinberg, Gauge and global symmetries at high temperature, Phys. Rev. D 9, 3357 (1974).
  2. Yiqiu Han, Xiaoyang Huang, Zohar Komargodski, Andrew Lucas, and Fedor K. Popov, Entropic order, Nat. Commun. 17, 87 (2026).
  3. R. N. Mohapatra and G. Senjanović, Soft CP violation at high temperature, Phys. Rev. Lett. 42, 1651 (1979).
  4. R. N. Mohapatra and G. Senjanović, Broken symmetries at high temperature, Phys. Rev. D 20, 3390 (1979).
  5. P. Langacker and S. Y. Pi, Magnetic monopoles in grand unified theories, Phys. Rev. Lett. 45, 1 (1980).
  6. P. Salomonson, B. S. Skagerstam, and A. Stern, On the primordial monopole problem in grand unified theories, Phys. Lett. 151B, 243 (1985).
  7. G. R. Dvali, A. Melfo, and G. Senjanović, Is there a monopole problem?, Phys. Rev. Lett. 75, 4559 (1995).
  8. H. Lew and R. R. Volkas, Discrete quark—lepton symmetry need not pose a cosmological domain wall problem, Phys. Rev. D 47, 1356 (1993).
  9. G. R. Dvali and G. Senjanović, Is there a domain wall problem?, Phys. Rev. Lett. 74, 5178 (1995).
  10. R. N. Mohapatra and G. Senjanović, Cosmological baryon production in a ‘superconducting’ early universe, Phys. Rev. D 21, 3470 (1980).
  11. V. A. Kuzmin, M. E. Shaposhnikov, and I. I. Tkachev, Gauge hierarchies and unusual symmetry behavior at high temperatures, Phys. Lett. 105B, 159 (1981).
  12. V. A. Kuzmin, M. E. Shaposhnikov, and I. I. Tkachev, Matter—antimatter domains in the universe: A solution of the vacuum walls problem, Phys. Lett. 105B, 167 (1981).
  13. V. A. Kuzmin, M. E. Shaposhnikov, and I. I. Tkachev, Baryon generation and unusual symmetry behavior at high temperatures, Nucl. Phys. B196, 29 (1982); B202, 543(E) (1982).
  14. S. Dodelson and L. M. Widrow, Baryon symmetric baryogenesis, Phys. Rev. Lett. 64, 340 (1990).
  15. S. Dodelson, B. R. Greene, and L. M. Widrow, Baryogenesis, dark matter and the width of the Z, Nucl. Phys. B372, 467 (1992).
  16. J. W. Lee and I. G. Koh, Inflation and inverse symmetry breaking, Phys. Rev. D 54, 7153 (1996).
  17. A. D. Linde, High density and high temperature symmetry behavior in gauge theories, Phys. Rev. D 14, 3345 (1976).
  18. H. E. Haber and H. A. Weldon, Finite temperature symmetry breaking as Bose- Einstein condensation, Phys. Rev. D 25, 502 (1982).
  19. K. M. Benson, J. Bernstein, and S. Dodelson, Phase structure and the effective potential at fixed charge, Phys. Rev. D 44, 2480 (1991).
  20. A. Riotto and G. Senjanović, Supersymmetry and broken symmetries at high temperature, Phys. Rev. Lett. 79, 349 (1997).
  21. J. Liu and G. Segre, Baryon asymmetry of the universe and large lepton asymmetries, Phys. Lett. B 338, 259 (1994).
  22. B. Bajc, A. Riotto, and G. Senjanović, Large lepton number of the universe and the fate of topological defects, Phys. Rev. Lett. 81, 1355 (1998).
  23. B. Bajc, A. Riotto, and G. Senjanović, R—charge kills monopoles, Mod. Phys. Lett. A 13, 2955 (1998).
  24. B. Bajc and G. Senjanović, Large lepton number and high temperature symmetry breaking in MSSM, Phys. Lett. B 472, 373 (2000).
  25. H. S. Kang and G. Steigman, Cosmological constraints on neutrino degeneracy, Nucl. Phys. B372, 494 (1992).
  26. W. H. Kinney and A. Riotto, Measuring the cosmological lepton asymmetry through the CMB anisotropy, Phys. Rev. Lett. 83, 3366 (1999).
  27. J. Lesgourgues and S. Pastor, Cosmological implications of a relic neutrino asymmetry, Phys. Rev. D 60, 103521 (1999).
  28. G. Barenboim, W. H. Kinney, and W. I. Park, Resurrection of large lepton number asymmetries from neutrino flavor oscillations, Phys. Rev. D 95, 043506 (2017).
  29. G. Barenboim, W. H. Kinney, and W. I. Park, Flavor versus mass eigenstates in neutrino asymmetries: Implications for cosmology, Eur. Phys. J. C 77, 590 (2017).
  30. G. Barenboim and W. I. Park, A full picture of large lepton number asymmetries of the universe, J. Cosmol. Astropart. Phys. 04 (2017) 048.
  31. H. E. Haber, Baryon asymmetry and the scale of supersymmetry breaking, Phys. Rev. D 26, 1317 (1982).
  32. M. L. Mangano, Global and gauge symmetries in finite temperature supersymmetric theories, Phys. Lett. B 147, 307 (1984).
  33. B. Bajc, A. Melfo, and G. Senjanović, On supersymmetry at high temperature, Phys. Lett. B 387, 796 (1996).
  34. G. R. Dvali and Lawrence M. Krauss, High temperature symmetry breaking, SUSY flat directions, and the monopole problem, arXiv:hep-ph/9811298.
  35. B. Bajc and G. Senjanović, High temperature symmetry breaking via flat directions, Phys. Rev. D 61, 103506 (2000).
  36. G. Bimonte and G. Lozano, On symmetry nonrestoration at high temperature, Phys. Lett. B 366, 248 (1996).
  37. J. Orloff, The UV price for symmetry nonrestoration, Phys. Lett. B 403, 309 (1997).
  38. T. G. Roos, Wilson renormalization group study of inverse symmetry breaking, Phys. Rev. D 54, 2944 (1996).
  39. M. Pietroni, N. Rius, and N. Tetradis, Inverse symmetry breaking and the exact renormalization group, Phys. Lett. B 397, 119 (1997).
  40. G. Amelino-Camelia, Rayleigh-Ritz variational approximation and symmetry nonrestoration, Phys. Lett. B 388, 776 (1996).
  41. M. B. Pinto and R. O. Ramos, A nonperturbative study of inverse symmetry breaking at high temperatures, Phys. Rev. D 61, 125016 (2000).
  42. G. Bimonte and G. Lozano, Can symmetry nonrestoration solve the monopole problem?, Nucl. Phys. B460, 155 (1996).
  43. Y. Fujimoto and S. Sakakibara, On symmetry nonrestoration at high temperature, Phys. Lett. 151B, 260 (1985).
  44. K. G. Klimenko, 1/N expansion in the O(N) x O(N) scalar theory and the problem of symmetry restoration at high temperature, Theor. Math. Phys. 80, 929 (1989).
  45. M. P. Grabowski, The effective potential and symmetry breaking in the O(N) x O(N) model, Z. Phys. C 48, 505 (1990).
  46. M. B. Gavela, O. P‘ene, N. Rius, and S. Vargas-Castrillón, Fading of symmetry nonrestoration at finite temperature, Phys. Rev. D 59, 025008 (1998).
  47. G. Bimonte, D. Iniguez, A. Tarancon, and C. L. Ullod, A Monte Carlo study of inverse symmetry breaking, Phys. Rev. Lett. 81, 750 (1998).
  48. S. I. Hong and J. B. Kogut, Symmetry nonrestoration in a Gross-Neveu model with random chemical potential, Phys. Rev. D 63, 085014 (2001).
  49. N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici, and M. Smolkin, Thermal order in conformal theories, Phys. Rev. D 102, 065014 (2020).
  50. Alex Buchel, Thermal order in holographic CFTs and no-hair theorem violation in black branes, Nucl. Phys. B967, 115425 (2021).
  51. Alex Buchel, Fate of the conformal order, Phys. Rev. D 103, 026008 (2021).
  52. N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici, and M. Smolkin, Symmetry breaking at all temperatures, Phys. Rev. Lett. 125, 131603 (2020).
  53. Goran Senjanovic, Rochelle salt: A prototype of particle physics, 1st International Conference on Particle Physics and the Early Universe (1998), pp. 437–445, 10.1142/9789814447263_0062.
  54. Borut Bajc, High temperature symmetry nonrestoration, 3rd International Conference on Particle Physics and the Early Universe (2000), pp. 247–253, 10.1142/9789812792129_0039.
  55. Alex Buchel, Compactified holographic conformal order, Nucl. Phys. B973, 115605 (2021).
  56. Alex Buchel, A bestiary of black holes on the conifold with fluxes, J. High Energy Phys. 06 (2021) 102.
  57. Alex Buchel, The quest for a conifold conformal order, J. High Energy Phys. 08 (2022) 080.
  58. Alex Buchel, Holographic conformal order with higher derivatives, Nucl. Phys. B1004, 116578 (2024).
  59. Noam Chai, Anatoly Dymarsky, Mikhail Goykhman, Ritam Sinha, and Michael Smolkin, A model of persistent breaking of continuous symmetry, SciPost Phys. 12, 181 (2022).
  60. Noam Chai, Anatoly Dymarsky, and Michael Smolkin, Model of persistent breaking of discrete symmetry, Phys. Rev. Lett. 128, 011601 (2022).
  61. Noam Chai, Eliezer Rabinovici, Ritam Sinha, and Michael Smolkin, The bi-conical vector model at 1/N, J. High Energy Phys. 05 (2021) 192.
  62. Borut Bajc, Adrián Lugo, and Francesco Sannino, Asymptotically free and safe fate of symmetry nonrestoration, Phys. Rev. D 103, 096014 (2021).
  63. Yu Nakayama, On the trace anomaly of the Chaudhuri–Choi–Rabinovici model, Symmetry 13, 276 (2021).
  64. Soumyadeep Chaudhuri and Eliezer Rabinovici, Symmetry breaking at high temperatures in large N gauge theories, J. High Energy Phys. 08 (2021) 148.
  65. Johan Carlström and Egor Babaev, Spontaneous breakdown of time-reversal symmetry induced by thermal fluctuations, Phys. Rev. B 91, 140504 (2015).
  66. K. G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4, 3174 (1971).
  67. K. G. Wilson, Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior, Phys. Rev. B 4, 3184 (1971).
  68. David J. Gross and Frank Wilczek, Ultraviolet behavior of nonAbelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973).
  69. H. David Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30, 1346 (1973).
  70. Steven Weinberg, Critical phenomena for field theorists, 14th International School of Subnuclear Physics: Understanding the Fundamental Constitutents of Matter (1976), .
  71. Zohar Komargodski and Fedor K. Popov, Temperature-resistant order in 2+1 dimensions, Phys. Rev. Lett. 135, 091602 (2025).
  72. Oleg Antipin, Jens Krog, Matin Mojaza, and Francesco Sannino, Stable extensions with(out) gravity, Nucl. Phys. B886, 125 (2014).
  73. Gian F. Giudice, Gino Isidori, Alberto Salvio, and Alessandro Strumia, Softened gravity and the extension of the standard model up to infinite energy, J. High Energy Phys. 02 (2015) 137.

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Phys. Rev. D 113, 125029– Published 22 June, 2026
  • Received 29 April 2026
  • Accepted 21 May 2026
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