The gauge hierarchy problem is traditionally stated as the fine-tuning of the Standard Model Higgs mass parameter against quadratically cutoff-sensitive radiative corrections. A persistent line of work — C. Wetterich (1981, 1992, 2012), C. Wetterich and M. Yamada (2017), and F. Jegerlehner (2015) — argues that this tuning is an artifact of ordinary perturbative expansion and of scheme-dependent bookkeeping: in the electroweak broken phase the Higgs mass is set by the order parameter vv through the Higgs mechanism (mH2=λv2/3m_H^2 = \lambda v^2/3 in the source's conventions), and the only regulator-independent object is the universal deviation δμ2(k)\delta\mu^2(k) of the running mass parameter from the critical trajectory of the electroweak phase transition, in terms of which 'no fine-tuning problem exists from the point of view of the renormalization group'. Recent quantitative work contests exactly this: Branchina, Branchina, Contino and Darvishi (2022) and Branchina, Branchina and Contino (2023) find that the absence of quadratic divergences in dimensional regularization does not dissolve a physical tuning, and Garcés, Goertz, Lindner and Pastor-Gutiérrez (2025), implementing the first full-Standard-Model Wilsonian functional renormalization group, agree that the distance to the critical surface is the scheme-independent quantity but conclude that the near-critical Standard Model exhibits a scheme-independent physical tuning that is enhanced near the quantum phase transition. The problem, kept at the sources' generality: formulate electroweak naturalness entirely in scheme-independent terms, determine whether the Standard Model Higgs mass so formulated exhibits a physical fine-tuning that demands an explanatory mechanism or whether the tuning is a scheme artifact, and state which classes of proposed solutions of the hierarchy problem the resulting verdict renders necessary, unnecessary, or unaffected. Literature status as of August 2026: open and actively contested, with no consensus resolution.

Background

In the Standard Model of particle physics, electroweak symmetry is broken spontaneously: the Higgs field acquires a vacuum expectation value v≈246v \approx 246 GeV, and the Higgs boson, the WW and ZZ bosons, and fermions acquire masses proportional to vv. The gauge hierarchy problem is the observation that, in a cutoff regularization, the loop corrections to the Higgs mass-squared parameter μ2\mu^2 contain terms proportional to the square of the ultraviolet (UV) cutoff Λ\Lambda — 'quadratic divergences' — so keeping the physical Higgs mass near 125 GeV while the theory is extended to the Planck scale MPl∼1019M_{\rm Pl} \sim 10^{19} GeV appears to require canceling contributions to roughly thirty-five significant digits. Within effective field theory (EFT), the framework in which a field theory is treated as the low-energy limit of an unknown UV completion, this is usually phrased as the instability of a light scalar mass under heavy physical thresholds, and it motivated most beyond-Standard-Model model building (low-energy supersymmetry, compositeness) of the past four decades.

A dissenting line argues the problem, so stated, is not physical. Its key notions come from the Wilsonian renormalization group (RG), the description of a field theory by a flow of couplings as fluctuations are integrated out scale by scale, implemented non-perturbatively by the functional renormalization group (FRG), an exact flow equation for the scale-dependent effective action. The electroweak phase transition defines a critical surface (equivalently critical hypersurface) in theory space: the set of couplings for which the theory sits exactly at a second-order phase transition between the symmetric and broken phases. The precise location of this surface depends on the parametrization and regularization scheme — dimensional regularization, for example, displays no quadratic divergences at all, a point emphasized by Bardeen (1995). What is scheme-independent, on this view, is the deviation δμ2(k)=μ2(k)−μ∗2(k)\delta\mu^2(k) = \mu^2(k) - \mu^2_*(k) of the running mass parameter from the critical trajectory μ∗2(k)\mu^2_*(k) (the scaling solution lying on the surface), or equivalently the dimensionless combination γ=δμ2/k2\gamma = \delta\mu^2/k^2, whose flow ∂tγ=Aγγ\partial_t \gamma = A_\gamma \gamma (with t=ln⁡kt = \ln k the logarithmic RG scale) has a stability coefficient AγA_\gamma; γ=0\gamma = 0 is a fixed point because RG flows cannot cross the critical surface. 'Self-tuned criticality' (Wetterich 1981; Bornholdt and Wetterich 1992) denotes the situation in which γ\gamma is driven toward zero automatically, i.e. all deviations from criticality are irrelevant; a proposed realization is 'high scale attraction', where a UV fixed point with anomalous mass dimension AμA_\mu large enough renders γ\gamma irrelevant, a candidate being asymptotically safe quantum gravity (a quantum gravity scenario in which the gravitational couplings approach a non-trivial UV fixed point, making the theory non-perturbatively renormalizable). Separately, Jegerlehner (2015) argues that in the broken phase the Higgs mass is fixed by the emergent order parameter vv itself — in his cutoff conventions mH2=(1/3)λv2m_H^2 = (1/3)\lambda v^2 — so tuning a bare mass against cutoff contributions addresses a quantity that is not the physical input, and the hierarchy problem in the broken phase is a 'pseudo problem'.

Wetterich and Yamada (2017) combine these threads: the fine-tuning debate concerns only the scheme-dependent location of the critical surface, which 'is not a quantity of interest', whereas δμ2(k)\delta\mu^2(k) is 'universal and physically meaningful', and 'no fine-tuning problem exists from the point of view of the renormalization group' — though they retain as the surviving core the question 'why is the world so close to criticality', answered by their resurgence mechanism built on asymptotic safety. Against this, an active counter-line (Branchina et al. 2022, 2023; Garcés et al. 2025) uses direct Wilsonian calculations to argue that a scheme-independent physical tuning survives and is quantifiable, and mainstream assessments (Giudice 2017; Craig 2023) treat the hierarchy problem as real but unresolved after the LHC found no naturalness-motivated new physics. The question of whether the hierarchy problem is a physical problem, and what exactly the scheme-independent content of 'electroweak fine-tuning' is, therefore remains genuinely open.

Problem Statement

At the natural generality of the source claims, with no restriction to specific models, schemes, or regulators: (1) Give a formulation of electroweak naturalness — the smallness of the Higgs mass-squared parameter relative to the ultraviolet scale of the theory (the Planck scale, or any physically realized heavy threshold such as a grand-unification scale) — in which every quantity entering the statement is regulator- and scheme-independent. (2) Determine, within this scheme-independent formulation, whether the observed Standard Model exhibits a physical fine-tuning — whether its proximity to the critical surface, or the smallness of γ\gamma in the infrared, is a fact about the theory that demands an explanatory mechanism beyond the Standard Model — or whether the appearance of tuning arises only from scheme-dependent objects (quadratic divergences, the bare mass parameter, the location of the critical surface in a given parametrization), so that the hierarchy problem in this form is not a physical problem. The determination must adjudicate the conflicting published conclusions: Jegerlehner's broken-phase pseudo-problem claim; Wetterich–Yamada's 'no fine-tuning problem exists from the point of view of the renormalization group' with its surviving core 'why is the world so close to criticality'; and the counter-results of Branchina et al. (2022, 2023) and Garcés et al. (2025) that a scheme-independent physical tuning exists and is enhanced near criticality — including the treatment of finite threshold corrections from heavy physics, which the sources claim reduce to the same scheme-dependent location issue. (3) State which classes of proposed solutions of the gauge hierarchy problem the resulting verdict renders necessary, unnecessary, or unaffected (for example: low-energy supersymmetry, compositeness or other symmetry-protection dynamics, anthropic selection in a landscape, and scale-invariant or asymptotically safe ultraviolet completions).

References

  1. F. Jegerlehner, 'The hierarchy problem and the cosmological constant problem in the Standard Model', 2015, arXiv:1503.00809, https://arxiv.org/abs/1503.00809
  2. C. Wetterich, 'Where to look for solving the gauge hierarchy problem?', Physics Letters B 718 (2012) 573, 2011, arXiv:1112.2910, DOI 10.1016/j.physletb.2012.11.020, https://arxiv.org/abs/1112.2910
  3. C. Wetterich and M. Yamada, 'Gauge hierarchy problem in asymptotically safe gravity — the resurgence mechanism', Physics Letters B 770 (2017) 268, 2016, arXiv:1612.03069, DOI 10.1016/j.physletb.2017.04.049, https://arxiv.org/abs/1612.03069
  4. C. Wetterich, 'Gauge hierarchy due to strong interactions?', Physics Letters B 104 (1981) 269, 1981, DOI 10.1016/0370-2693(81)90124-6, https://doi.org/10.1016/0370-2693(81)90124-6
  5. S. Bornholdt and C. Wetterich, 'Self-organizing criticality, large anomalous mass dimension and the gauge hierarchy problem', Physics Letters B 282 (1992) 399, 1992, DOI 10.1016/0370-2693(92)90659-R, https://doi.org/10.1016/0370-2693(92)90659-R
  6. W. A. Bardeen, 'On naturalness in the standard model', 1995, FERMILAB-CONF-95/391-T, INSPIRE https://inspirehep.net/literature/404517, full text https://lss.fnal.gov/archive/1995/conf/Conf-95-391-T.pdf
  7. C. Branchina, V. Branchina, F. Contino and N. Darvishi, 'Dimensional regularization, Wilsonian RG, and the naturalness and hierarchy problem', Physical Review D 106 (2022) 065007, 2022, arXiv:2204.10582, DOI 10.1103/PhysRevD.106.065007, https://arxiv.org/abs/2204.10582
  8. C. Branchina, V. Branchina and F. Contino, 'Physical tuning and naturalness', Physical Review D 107 (2023) 096012, 2022, arXiv:2208.05431, DOI 10.1103/PhysRevD.107.096012, https://arxiv.org/abs/2208.05431
  9. J. P. Garcés, F. Goertz, M. Lindner and Á. Pastor-Gutiérrez, 'The quantum criticality of the Standard Model and the hierarchy problem', JHEP 10 (2025) 134, 2025, arXiv:2506.15919, DOI 10.1007/JHEP10(2025)134, https://arxiv.org/abs/2506.15919
  10. G. F. Giudice, 'The Dawn of the Post-Naturalness Era', 2017, arXiv:1710.07663 (published in 'From My Vast Repertoire...', World Scientific, 2018, pp. 267-292, DOI 10.1142/9789813238053_0013), https://arxiv.org/abs/1710.07663
  11. N. Craig, 'Naturalness: past, present, and future', European Physical Journal C 83 (2023) 825, 2022, arXiv:2205.05708, DOI 10.1140/epjc/s10052-023-11928-7, https://arxiv.org/abs/2205.05708
  12. C. Wetterich, 'Fermi scale from quantum gravity scaling solution', 2026, arXiv:2601.16731, https://arxiv.org/abs/2601.16731