High-scale attraction at ultraviolet fixed points: can a large anomalous mass dimension in asymptotically safe gravity explain the gauge hierarchy?

The observed ratio between the Fermi scale and the Planck scale, about $10^{-16}$ , could be explained without new physics near the TeV scale if an ultraviolet fixed point of the renormalization-group flow rendered the Higgs mass parameter irrelevant — 'high scale attraction'. Whether this is possible hinges on the anomalous mass dimension $A_{\mu}$ at candidate fixed points, above all the asymptotically safe fixed point of gravity coupled to Standard-Model matter: a reliable computation of $A_{\mu}$ , and the decision whether it is large enough ( $A_{\mu}>2$ for fixed scale $\chi$ ; $A_{\mu}>0$ when $\chi$ runs proportionally to the renormalization scale $k$ ), was posed as the chief open problem by Wetterich (2012) and remains unresolved. Functional-renormalization-group studies since then have made the question precise — the mass term becomes irrelevant only if gravity is sufficiently strong at short distances, i.e. the fixed-point value of the dimensionless Planck mass is small enough — but different truncations currently disagree: some find conditional irrelevance of the mass term, a comprehensive asymptotically safe Standard-Model study finds it relevant with a flat Higgs potential, and a 2026 review concludes that the hierarchy problem is expected to persist because the fixed point is near-perturbative. Whether any ultraviolet fixed point with the required characteristics exists, and whether it would predict the observed rather than an overpredicted hierarchy (a Fermi scale near the QCD scale), is open.

Background

The gauge hierarchy problem asks why the Fermi scale $\varphi_{0}\approx 175$ GeV — the vacuum expectation value of the Standard Model (SM) Higgs field, which sets the $W$ - and $Z$ -boson masses — is sixteen orders of magnitude smaller than the reduced Planck mass $M_{\mathrm{p}}=(8\pi G_{N})^{-1/2}\approx 2.4\times 10^{18}$ GeV that governs gravity. In the Wilsonian renormalization-group (RG) picture the problem is one of relevance: the dimensionless Higgs mass parameter — in the convention of the source paper, $\gamma\sim m_{H}^{2}/\chi^{2}$ , the coefficient of $(\varphi^{\dagger}\varphi)\chi^{2}$ in the effective scalar potential, while the deviation from electroweak criticality that runs under the RG flow is $\gamma\chi^{2}/k^{2}\sim m_{H}^{2}/k^{2}$ , with $k$ the RG scale — is controlled by a relevant coupling, so $\gamma$ must be tuned to $|\gamma|\sim 10^{-32}$ at the Planck scale to reproduce the observed electroweak scale. This tuning is equivalent to placing the RG trajectory exponentially close to the critical surface — the set of trajectories ending at a vanishing Higgs mass — of the zero-temperature electroweak phase transition, which is a second-order (continuous) quantum phase transition. A 'high-scale solution' of the hierarchy problem, advocated in the source paper (Wetterich 2012), instead seeks an ultraviolet (UV) fixed point of the RG flow — a scale at which all dimensionless couplings cease to run — above some scale $\chi$ (associated with the Planck mass), at which the deviation from electroweak criticality is itself an irrelevant parameter. Writing the flow of the mass coupling as $\partial_{t}\gamma=A_{\mu}\gamma$ with $t=\ln(k/\chi)$ , the anomalous mass dimension $A_{\mu}$ measures how strongly the fixed point attracts $\gamma$ toward zero: for fixed $\chi$ , the deviation from criticality $\gamma\chi^{2}/k^{2}$ becomes irrelevant only if $A_{\mu}>2$ ; if the scale $\chi$ runs proportionally to $k$ , as happens when the Planck mass is itself generated by a dynamical field, $A_{\mu}>0$ suffices. This 'high scale attraction' is the RG analogue of how inflation drives the cosmic density to criticality. The leading candidate for such a fixed point is asymptotic safety — the conjecture (Weinberg 1979) that quantum gravity is non-perturbatively renormalizable because its RG flow approaches a non-Gaussian, interacting fixed point (NGFP) at trans-Planckian scales — studied in practice with the functional renormalization group (FRG), an exact Wilsonian flow equation for the scale-dependent effective action (Reuter 1998). A precedent for predictive power exists: imposing the Planck-scale boundary condition $\lambda(M_{\mathrm{p}})=0$ on the Higgs quartic coupling $\lambda$ , motivated by such a fixed point, Shaposhnikov and Wetterich (2010) predicted a Higgs boson mass near 126 GeV, close to the observed value of about 125 GeV, together with a top pole mass near 171.5 GeV; Wetterich (2012) sharpened this to the pair of conditions $\lambda(\chi)=0$ and $\beta_{\lambda}(\chi)=0$ . The boundary condition on $\lambda$ , however, is an assumption about the unknown UV fixed point, and the paper's central unresolved issue concerns the mass term: no reliable computation of $A_{\mu}$ at any candidate UV fixed point existed — fixed points known from studies of four-fermion interactions or large Yukawa couplings had not exhibited large $A_{\mu}$ — so it was unknown whether high-scale attraction is realizable at all. Two linked issues compound it: the properties of the fixed point itself (its spectrum; the fixed-point values of $\lambda$ , the top Yukawa coupling $h$ , and the gauge couplings; and whether the heavy states responsible for the fixed point decouple smoothly at $\chi$ , since threshold corrections to $\beta_{\lambda}$ could invalidate the imposed low-energy conditions), and the phenomenological precision — top pole mass, Higgs mass, RG and matching uncertainties, and possible intermediate scales such as grand-unification or $B-L$ breaking scales — needed to confirm or falsify the predicted numerical relations.

Problem Statement

Determine whether high-scale (ultraviolet-fixed-point) solutions of the gauge hierarchy problem are realized or falsified, with the anomalous mass dimension $A_{\mu}$ of the Higgs mass parameter as the decisive quantity. Concretely: (1) compute reliably — with controlled truncation, regulator, and gauge dependence — the anomalous mass dimension $A_{\mu}$ (related to the critical exponent $\theta_{m}$ of the Higgs mass parameter near the fixed point by $\theta_{m}\simeq 2-A_{\mu}$ up to field anomalous dimensions, so that the irrelevance criterion $A_{\mu}>2$ is equivalent to a negative critical exponent, $\theta_{m}<0$ ) at candidate ultraviolet fixed points, most importantly the asymptotically safe fixed point of gravity coupled to Standard-Model-like matter, and decide whether $A_{\mu}$ is large enough — $A_{\mu}>2$ for a fixed decoupling scale $\chi$ (the scale, of order the Planck mass, at which the fixed-point physics decouples from the low-energy effective theory), or $A_{\mu}>0$ where $\chi(k)$ runs proportionally to the renormalization scale $k$ — to render the deviation from electroweak criticality (distance from the critical surface of the electroweak phase transition) irrelevant, i.e. naturally small at low energies ('high scale attraction'); (2) establish the fixed-point properties this requires or excludes — the spectrum of relevant and irrelevant directions and the fixed-point values of the Higgs quartic coupling $\lambda$ , the top Yukawa coupling $h$ , and the gauge couplings — together with whether the heavy states responsible for the fixed point decouple smoothly at the scale $\chi$ , so that the low-energy boundary conditions $\lambda(\chi)=0$ and $\beta_{\lambda}(\chi)=0$ (vanishing quartic coupling and vanishing running of $\lambda$ at $\chi$ ) remain valid; (3) decide whether, if high scale attraction is realized, the resulting renormalization-group flow quantitatively predicts the observed ratio $\varphi_{0}/M_{\mathrm{p}}\approx 10^{-16}$ of the Fermi scale $\varphi_{0}$ to the reduced Planck mass $M_{\mathrm{p}}$ — including the associated predictions for the Higgs and top pole masses — rather than tying the electroweak scale to the largest intrinsic scale generated away from the fixed point (which, for Standard-Model matter only, would lie near the QCD scale and is excluded). These three parts are components of the single determination requested above: parts (1) and (2) together decide whether high-scale attraction is realized or excluded, while the quantitative prediction in part (3) is required only in the case that it is realized.

References

C. Wetterich, 'Where to look for solving the gauge hierarchy problem?', Phys. Lett. B 718 (2012) 573-576, doi:10.1016/j.physletb.2012.11.020, arXiv:1112.2910, https://arxiv.org/abs/1112.2910
S. Weinberg, 'Ultraviolet divergences in quantum theories of gravitation', in General Relativity: An Einstein Centenary Survey (S. W. Hawking and W. Israel, eds.), Cambridge University Press, 1979, pp. 790-831, https://inspirehep.net/literature/159043
M. Reuter, 'Nonperturbative evolution equation for quantum gravity', Phys. Rev. D 57 (1998) 971-985, doi:10.1103/PhysRevD.57.971, arXiv:hep-th/9605030, https://arxiv.org/abs/hep-th/9605030
M. Shaposhnikov, C. Wetterich, 'Asymptotic safety of gravity and the Higgs boson mass', Phys. Lett. B 683 (2010) 196-200, doi:10.1016/j.physletb.2009.12.022, arXiv:0912.0208, https://arxiv.org/abs/0912.0208
A. Eichhorn, A. Held, J. M. Pawlowski, 'Quantum-gravity effects on a Higgs-Yukawa model', Phys. Rev. D 94 (2016) 104027, doi:10.1103/PhysRevD.94.104027, arXiv:1604.02041, https://arxiv.org/abs/1604.02041
C. Wetterich, M. Yamada, 'Gauge hierarchy problem in asymptotically safe gravity — the resurgence mechanism', Phys. Lett. B 770 (2017) 268-271, doi:10.1016/j.physletb.2017.04.049, arXiv:1612.03069, https://arxiv.org/abs/1612.03069
Y. Hamada, M. Yamada, 'Asymptotic safety of higher derivative quantum gravity non-minimally coupled with a matter system', JHEP 08 (2017) 070, doi:10.1007/JHEP08(2017)070, arXiv:1703.09033, https://arxiv.org/abs/1703.09033
A. Eichhorn, Y. Hamada, J. Lumma, M. Yamada, 'Quantum gravity fluctuations flatten the Planck-scale Higgs potential', Phys. Rev. D 97 (2018) 086004, doi:10.1103/PhysRevD.97.086004, arXiv:1712.00319, https://arxiv.org/abs/1712.00319
M. Shaposhnikov, A. Shkerin, 'Gravity, Scale Invariance and the Hierarchy Problem', JHEP 10 (2018) 024, doi:10.1007/JHEP10(2018)024, arXiv:1804.06376, https://arxiv.org/abs/1804.06376
J. M. Pawlowski, M. Reichert, C. Wetterich, M. Yamada, 'Higgs scalar potential in asymptotically safe quantum gravity', Phys. Rev. D 99 (2019) 086010, doi:10.1103/PhysRevD.99.086010, arXiv:1811.11706, https://arxiv.org/abs/1811.11706
C. Wetterich, 'Effective scalar potential in asymptotically safe quantum gravity', Universe 7 (2021) 045, doi:10.3390/universe7020045, arXiv:1911.06100, https://arxiv.org/abs/1911.06100
Á. Pastor-Gutiérrez, J. M. Pawlowski, M. Reichert, 'The Asymptotically Safe Standard Model: From quantum gravity to dynamical chiral symmetry breaking', SciPost Phys. 15 (2023) 105, doi:10.21468/SciPostPhys.15.3.105, arXiv:2207.09817, https://arxiv.org/abs/2207.09817
J. P. Garcés, F. Goertz, M. Lindner, Á. Pastor-Gutiérrez, 'The quantum criticality of the Standard Model and the hierarchy problem', JHEP 10 (2025) 134, doi:10.1007/JHEP10(2025)134, arXiv:2506.15919, https://arxiv.org/abs/2506.15919
[CMS and ATLAS Collaborations], 'Combination of Measurements of the Top Quark Mass from Data Collected by the ATLAS and CMS Experiments at $\sqrt{s}=7$ and 8 TeV', Phys. Rev. Lett. 132 (2024) 261902, doi:10.1103/PhysRevLett.132.261902, arXiv:2402.08713, https://arxiv.org/abs/2402.08713
C. Wetterich, 'Fermi scale from quantum gravity scaling solution', arXiv:2601.16731 (2026), https://arxiv.org/abs/2601.16731
A. Eichhorn, 'Asymptotically safe quantum gravity and its phenomenology — a review', arXiv:2606.21522 (2026), https://arxiv.org/abs/2606.21522