Brdar, Helmboldt and Lindner (JHEP 12 (2019) 158) showed that in the minimal classically scale-invariant extension of the Standard Model, demanding renormalization-group consistency — the absence of Landau poles below the Planck scale — caps the scalar portal couplings at O(10−1)\mathcal{O}(10^{-1}), and that for every fully consistent parameter point the electroweak phase transition cannot complete by thermal bubble nucleation: the Universe supercools through a vacuum-dominated epoch until the QCD chiral phase transition triggers electroweak symmetry breaking at temperatures of order 100 MeV. They conjectured this link between RG consistency and strong supercooling to hold generally in purely scalar classically scale-invariant extensions of the Standard Model, while suggesting that scale-invariant gauge extensions with larger gauge couplings might avoid strong supercooling. Both parts remain open: no general proof or counterexample of the relation in purely scalar extensions is known, and no systematic analysis of whether larger gauge couplings can prevent strong supercooling exists. The problem posed is to establish or refute the general relation in purely scalar extensions and to determine whether RG-consistent gauge extensions can complete their scale-generating transition without strong supercooling.

Background

Classical scale invariance is one of the most minimal proposed solutions to the gauge hierarchy problem — the unexplained stability of the electroweak scale (v≃246v \simeq 246 GeV) against the Planck scale. A classically scale-invariant (CSI) theory contains no dimensionful parameters in its tree-level Lagrangian; all scales, including the electroweak scale, arise by dimensional transmutation through radiative (Coleman–Weinberg) symmetry breaking, in which loop corrections to the effective potential generate a vacuum expectation value away from the origin (Coleman and Weinberg, Phys. Rev. D 7 (1973) 1888). For such a model to genuinely address the hierarchy problem it must remain perturbative up to the Planck scale: if any coupling develops a Landau pole — a finite energy where the coupling diverges under renormalization-group (RG) running — the theory loses predictivity precisely in the range where the hierarchy problem is posed. This requirement of RG consistency strongly constrains the scalar portal couplings, the quartic couplings that mix the Higgs field with new scalar fields.

The minimal phenomenologically consistent CSI realization (Brdar, Helmboldt and Lindner, JHEP 12 (2019) 158) extends the Standard Model by two real gauge-singlet scalars, ss and rr, with rr odd under a Z2\mathbb{Z}_2 symmetry so that it is stable and can serve as a dark matter candidate. Requiring the absence of sub-Planckian Landau poles caps the portal couplings at O(10−1)\mathcal{O}(10^{-1}) at the TeV scale. With such weak portals, the authors computed the finite-temperature effective potential and the thermal bubble nucleation rate and found that for every fully consistent parameter point the electroweak phase transition (EWPT) cannot complete by nucleation of critical bubbles: the nucleation rate never competes with the Hubble expansion. Instead the Universe enters a vacuum-dominated (quasi-de Sitter) supercooling epoch while the Higgs remains in the symmetric phase, until the QCD chiral phase transition — which proceeds with six effectively massless quarks, since electroweak symmetry breaking has not yet generated their masses — produces a quark condensate that, through the top-quark Yukawa interaction, destabilizes the Higgs potential and triggers electroweak symmetry breaking at temperatures of order 100 MeV. The transition is therefore strongly supercooled, in the standard sense that the nucleation (or percolation) temperature lies orders of magnitude below the critical temperature and the strength parameter α\alpha (latent heat relative to the radiation energy density at completion) is much larger than one. The QCD-induced completion scenario builds on earlier work by Iso, Serpico and Shimada (Phys. Rev. Lett. 119 (2017) 141301) and von Harling and Servant (JHEP 01 (2018) 159). The same paper also found that the associated gravitational-wave background is not testable (relying on the external computation of Helmboldt, Kubo and van der Woude, Phys. Rev. D 100 (2019) 055025, for the speed of the chiral transition), that the 'supercool' dark matter mechanism of Hambye, Strumia and Teresi (JHEP 08 (2018) 188) — in which particles present during the supercooled epoch are diluted and their abundance re-set at reheating — is negligible in this model, and that ordinary thermal freeze-out of rr reproduces the observed relic abundance only at the level of a few benchmark points, without a full dark matter parameter-space analysis.

In their conclusions the authors state the general open issue: they conjectured that the relation between RG consistency and strongly supercooled scale-generating phase transitions holds generally in purely scalar CSI extensions of the Standard Model, while in scale-invariant gauge extensions the dynamics may differ, because gauge couplings can be larger without inducing Landau poles and strong supercooling may thereby be prevented. Neither statement was proven; both are the subject of the present problem.

Problem Statement

Determine the general relation between renormalization-group consistency and supercooling in classically scale-invariant extensions of the Standard Model.

(i) Prove or refute the conjecture of Brdar, Helmboldt and Lindner that in any purely scalar classically scale-invariant extension of the Standard Model — no extra gauge groups, arbitrary scalar field content and portal structure — that generates the electroweak scale radiatively and is RG-consistent (perturbative, with no Landau pole below the Planck scale), the scale-generating electroweak phase transition is necessarily strongly supercooled: thermal bubble nucleation cannot complete the transition at temperatures of order the critical temperature, so the Universe undergoes a vacuum-dominated epoch before completion, and the nucleation or percolation temperature lies parametrically below the critical temperature. A refutation may take the form of one explicit, fully specified RG-consistent purely scalar classically scale-invariant model and parameter point in which the transition completes without strong supercooling.

(ii) Determine whether scale-invariant gauge extensions of the Standard Model — for example with an extra U(1) or SU(2) gauge group whose couplings can be sizable without violating RG consistency — admit RG-consistent parameter points in which the electroweak or scale-generating transition completes without strong supercooling, or establish that strong supercooling persists throughout the RG-consistent region of this class as well.

Both parts serve a single objective: establish whether Landau-pole avoidance — the consistency condition a classically scale-invariant theory must satisfy in order to address the hierarchy problem — generically fixes the strength and duration of its cosmological phase transition. An answer should state precisely the class of models covered, the order of the RG and finite-temperature analysis used, and the criterion by which a transition is classified as strongly supercooled.

References

  1. V. Brdar, A. J. Helmboldt, M. Lindner, 'Strong supercooling as a consequence of renormalization group consistency', JHEP 12 (2019) 158, doi:10.1007/JHEP12(2019)158, arXiv:1910.13460, https://arxiv.org/abs/1910.13460
  2. S. Coleman and E. Weinberg, 'Radiative corrections as the origin of spontaneous symmetry breaking', Phys. Rev. D 7 (1973) 1888, doi:10.1103/PhysRevD.7.1888, https://journals.aps.org/prd/abstract/10.1103/PhysRevD.7.1888
  3. S. Iso, P. D. Serpico, K. Shimada, 'QCD-Electroweak First-Order Phase Transition in a Supercooled Universe', Phys. Rev. Lett. 119 (2017) 141301, doi:10.1103/PhysRevLett.119.141301, arXiv:1704.04955, https://arxiv.org/abs/1704.04955
  4. B. von Harling, G. Servant, 'QCD-induced Electroweak Phase Transition', JHEP 01 (2018) 159, doi:10.1007/JHEP01(2018)159, arXiv:1711.11554, https://arxiv.org/abs/1711.11554
  5. T. Hambye, A. Strumia, D. Teresi, 'Super-cool Dark Matter', JHEP 08 (2018) 188, doi:10.1007/JHEP08(2018)188, arXiv:1805.01473, https://arxiv.org/abs/1805.01473
  6. A. J. Helmboldt, J. Kubo, S. van der Woude, 'Observational prospects for gravitational waves from hidden or dark chiral phase transitions', Phys. Rev. D 100 (2019) 055025, doi:10.1103/PhysRevD.100.055025, arXiv:1904.07891, https://arxiv.org/abs/1904.07891
  7. D. Bödeker, 'Remarks on the QCD-electroweak phase transition in a supercooled universe', Phys. Rev. D 104 (2021) L111501, doi:10.1103/PhysRevD.104.L111501, arXiv:2108.11966, https://arxiv.org/abs/2108.11966
  8. K. Kawana, 'Cosmology of a supercooled universe', Phys. Rev. D 105 (2022) 103515, doi:10.1103/PhysRevD.105.103515, arXiv:2201.00560, https://arxiv.org/abs/2201.00560
  9. M. Kierkla, A. Karam, B. Świeżewska, 'Conformal model for gravitational waves and dark matter: a status update', JHEP 03 (2023) 007, doi:10.1007/JHEP03(2023)007, arXiv:2210.07075, https://arxiv.org/abs/2210.07075
  10. A. Salvio, 'Model-independent radiative symmetry breaking and gravitational waves', JCAP 04 (2023) 051, doi:10.1088/1475-7516/2023/04/051, arXiv:2302.10212, https://arxiv.org/abs/2302.10212
  11. A. Salvio, 'Supercooling in Radiative Symmetry Breaking: Theory Extensions, Gravitational Wave Detection and Primordial Black Holes', JCAP 12 (2023) 046, doi:10.1088/1475-7516/2023/12/046, arXiv:2307.04694, https://arxiv.org/abs/2307.04694
  12. Y. Guan, S. Matsuzaki, 'Ladder top-quark condensation imprints in supercooled electroweak phase transition', JHEP 09 (2024) 140, doi:10.1007/JHEP09(2024)140, arXiv:2405.03265, https://arxiv.org/abs/2405.03265
  13. M. Arteaga, A. Ghoshal, A. Strumia, 'Gravitational waves and black holes from the phase transition in models of dynamical symmetry breaking', JCAP 05 (2025) 029, doi:10.1088/1475-7516/2025/05/029, arXiv:2409.04545, https://arxiv.org/abs/2409.04545
  14. K.-P. Xie, C.-H. Zhan, 'Dark matter in classically conformal theories: WIMP and supercooling', arXiv:2603.09126 (2026), https://arxiv.org/abs/2603.09126
  15. J. Cesca, A. Kobakhidze, 'QCD and electroweak phase transitions with hidden scale invariance: implications for primordial black holes, quark-lepton nuggets and gravitational waves', arXiv:2603.18406 (2026), https://arxiv.org/abs/2603.18406
  16. F. Cutrona, F. Rescigno, A. Salvio, 'ACT-Planck data and phase transitions from a viable no-scale Standard Model completion', Phys. Rev. D 114 (2026) 043513, doi:10.1103/38mz-pw8t, arXiv:2603.24664, https://arxiv.org/abs/2603.24664
  17. M. Kierkla, M. Lewicki, P. Schicho, D. Schmitt, B. Świeżewska, 'Theoretical uncertainties in reconstructing model parameters with gravitational waves from supercooled phase transitions', arXiv:2607.18233 (2026), https://arxiv.org/abs/2607.18233