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Fermi scale from quantum gravity scaling solution

Christof Wetterich

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Phys. Rev. D 114, 086002 – Published 5 October, 2026

DOI: https://doi.org/10.1103/3d3b-txny

Abstract

We propose that quantum gravity may predict the Fermi scale. Fundamental scale invariance implies the scale invariant standard model. Both the Fermi scale and the Planck mass are given by fields, and their ratio is dictated by a dimensionless cosmon-Higgs coupling. For an ultraviolet fixed point of quantum gravity, this coupling is an irrelevant parameter of the renormalization flow and becomes predictable. An analytic scaling solution for quantum gravity admits no free parameter for the mass term of the Higgs boson. We discuss a new asymptotically safe quantum gravity fixed point for which the scalar potential is not flat. If the largest intrinsic mass scale generated by the renormalization flow away from this fixed point is sufficiently below the Fermi scale, the couplings of the scale invariant standard model are determined by the scaling solution. For a given short distance model remaining valid to infinitely small distances, the ratio Fermi scale over Planck mass can then be predicted. With reasonable assumptions for the ultraviolet fixed point, a numerical solution finds a tiny value for the ratio between the Fermi and Planck scales, very close to a second order quantum electroweak phase transition. This could explain the observed gauge hierarchy.

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References (119)

  1. E. Gildener, Gauge symmetry hierarchies, Phys. Rev. D 14, 1667 (1976).
  2. S. Weinberg, Gauge hierarchies, Phys. Lett. 82B, 387 (1979).
  3. C. Wetterich, Fine tuning problem and the renormalization group, Phys. Lett. 140B, 215 (1984).
  4. W. A. Bardeen, On naturalness in the standard model, in Ontake Summer Institute on Particle Physics, 1995 (unpublished).
  5. R. Hempfling, The next-to-minimal Coleman-Weinberg model, Phys. Lett. B 379, 153 (1996).
  6. R. D. Peccei, J. Solà, and C. Wetterich, Adjusting the cosmological constant dynamically: Cosmons and a new force weaker than gravity, Phys. Lett. B 195, 183 (1987).
  7. K. A. Meissner and H. Nicolai, Conformal symmetry and the standard model, Phys. Lett. B 648, 312 (2007).
  8. K. A. Meissner and H. Nicolai, Effective action, conformal anomaly and the issue of quadratic divergences, Phys. Lett. B 660, 260 (2008).
  9. R. Foot, A. Kobakhidze, and R. R. Volkas, Electroweak Higgs as a pseudo-Goldstone boson of broken scale invariance, Phys. Lett. B 655, 156 (2007).
  10. M. Shaposhnikov and D. Zenhäusern, Quantum scale invariance, cosmological constant and hierarchy problem, Phys. Lett. B 671, 162 (2009).
  11. C. Wetterich, Where to look for solving the gauge hierarchy problem?, Phys. Lett. B 718, 573 (2012).
  12. M. Shaposhnikov and A. Shkerin, Gravity, scale invariance and the hierarchy problem, J. High Energy Phys. 10 (2018) 024.
  13. G. K. Karananas, M. Shaposhnikov, and S. Zell, Weyl-invariant Einstein-Cartan gravity: Unifying the strong CP and hierarchy puzzles, J. High Energy Phys. 11 (2024) 146.
  14. T. de Boer, M. Lindner, and A. Trautner, Hidden sector custodial naturalness, Phys. Rev. Lett. 137, 031801 (2026).
  15. C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).
  16. M. Reuter and C. Wetterich, Effective average action for gauge theories and exact evolution equations, Nucl. Phys. B417, 181 (1994).
  17. N. Tetradis and C. Wetterich, Critical exponents from the effective average action, Nucl. Phys. B422, 541 (1994).
  18. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  19. U. Ellwanger, Flow equations for N point functions and bound states, Z. Phys. C Part. Fields 62, 503 (1994).
  20. T. R. Morris, The exact renormalization group and approximate solutions, Int. J. Mod. Phys. A 09, 2411 (1994).
  21. C. Wetterich, Quadratic renormalization of the average potential and the naturalness of quadratic mass relations for the top quark, Z. Phys. C 48, 693 (1990).
  22. H. Gies, C. Gneiting, and R. Sondenheimer, Higgs mass bounds from renormalization flow for a simple Yukawa model, Phys. Rev. D 89, 045012 (2014).
  23. H. Gies and R. Sondenheimer, Higgs mass bounds from renormalization flow for a Higgs-top-bottom model, Eur. Phys. J. C 75, 68 (2015).
  24. T. Krajewski and Z. Lalak, Fine-tuning and vacuum stability in the Wilsonian effective action, Phys. Rev. D 92, 075009 (2015).
  25. T. Krajewski and Z. Lalak, Naturalness of effective theories in Wilsonian approach, Proc. Sci. PLANCK2015 (2015) 072. [arXiv:1510.02154].
  26. A. Eichhorn, H. Gies, J. Jaeckel, T. Plehn, M. M. Scherer, and R. Sondenheimer, The Higgs mass and the scale of new physics, J. High Energy Phys. 04 (2015) 022.
  27. H. Gies and R. Sondenheimer, Renormalization group flow of the Higgs potential, Phil. Trans. R. Soc. A 376, 20170120 (2018).
  28. M. Reichert, A. Eichhorn, H. Gies, J. M. Pawlowski, T. Plehn, and M. M. Scherer, Probing baryogenesis through the Higgs boson self-coupling, Phys. Rev. D 97, 075008 (2018).
  29. R. Sondenheimer, Nonpolynomial Higgs interactions and vacuum stability, Eur. Phys. J. C 79, 10 (2019).
  30. A. Held and R. Sondenheimer, Higgs stability-bound and fermionic dark matter, J. High Energy Phys. 02 (2019) 166.
  31. Á. Pastor-Gutiérrez, J. M. Pawlowski, and M. Reichert, The asymptotically safe standard model: From quantum gravity to dynamical chiral symmetry breaking, SciPost Phys. 15, 105 (2023).
  32. H. Gies, R. Schmieden, and L. Zambelli, Interplay of chiral transitions in the standard model, Eur. Phys. J. C 85, 56 (2025).
  33. J. P. Garcés, F. Goertz, M. Lindner, and Á. Pastor-Gutiérrez, The quantum criticality of the standard model and the hierarchy problem, J. High Energy Phys. 10 (2025) 134.
  34. S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in General Relativity: An Einstein Centenary Survey (Cambridge University Press, Cambridge, England, 1980), p. 790.
  35. M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57, 971 (1998).
  36. K. S. Stelle, Renormalization of higher-derivative quantum gravity, Phys. Rev. D 16, 953 (1977).
  37. E. S. Fradkin and A. A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B201, 469 (1982).
  38. I. G. Avramidy and A. O. Barvinsky, Asymptotic freedom in higher-derivative quantum gravity, Phys. Lett. 159B, 269 (1985).
  39. S. Sen, C. Wetterich, and M. Yamada, Asymptotic freedom and safety in quantum gravity, J. High Energy Phys. 03 (2022) 130.
  40. M. Shaposhnikov and C. Wetterich, Asymptotic safety of gravity and the Higgs boson mass, Phys. Lett. B 683, 196 (2010).
  41. A. Eichhorn, Y. Hamada, J. Lumma, and M. Yamada, Quantum gravity fluctuations flatten the Planck-scale Higgs potential, Phys. Rev. D 97, 086004 (2018).
  42. J. M. Pawlowski, M. Reichert, C. Wetterich, and M. Yamada, Higgs scalar potential in asymptotically safe quantum gravity, Phys. Rev. D 99, 086010 (2019).
  43. C. Wetterich, Quantum scale symmetry, arXiv:1901.04741.
  44. C. Wetterich and M. Yamada, Variable Planck mass from the gauge invariant flow equation, Phys. Rev. D 100, 066017 (2019).
  45. C. Wetterich, Effective scalar potential in asymptotically safe quantum gravity, Universe 7, 45 (2020).
  46. C. Wetterich, Fundamental scale invariance, Nucl. Phys. B964, 115326 (2021).
  47. T. Henz, J. Pawlowski, and C. Wetterich, Scaling solutions for dilaton quantum gravity, Phys. Lett. B 769, 105 (2017).
  48. T. Henz, J. Pawlowski, A. Rodigast, and C. Wetterich, Dilaton quantum gravity, Phys. Lett. B 727, 298 (2013).
  49. Y. Maitiniyazi, C. Wetterich, and M. Yamada, Scaling solutions for gauge invariant flow equations in dilaton quantum gravity, Phys. Rev. D 113, 106023 (2026).
  50. D. Dou and R. Percacci, The running gravitational couplings, Classical Quantum Gravity 15, 3449 (1998).
  51. W. Souma, Non-trivial ultraviolet fixed point in quantum gravity, Prog. Theor. Phys. 102, 181 (1999).
  52. O. Lauscher and M. Reuter, Ultraviolet fixed point and generalized flow equation of quantum gravity, Phys. Rev. D 65, 025013 (2001).
  53. M. Reuter and F. Saueressig, Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation, Phys. Rev. D 65, 065016 (2002).
  54. G. Narain and R. Percacci, Renormalization group flow in scalar-tensor theories: I, Classical Quantum Gravity 27, 075001 (2010).
  55. R. Percacci and G. P. Vacca, Search of scaling solutions in scalar–tensor gravity, Eur. Phys. J. C 75, 188 (2015).
  56. P. Donà, A. Eichhorn, P. Labus, and R. Percacci, Asymptotic safety in an interacting system of gravity and scalar matter, Phys. Rev. D 93, 044049 (2016).
  57. A. Eichhorn and M. Pauly, Constraining power of asymptotic safety for scalar fields, Phys. Rev. D 103, 026006 (2021).
  58. C. Laporte, A. D. Pereira, F. Saueressig, and J. Wang, Scalar-tensor theories within asymptotic safety, J. High Energy Phys. 12 (2021) 001.
  59. C. Wetterich and M. Yamada, Gauge hierarchy problem in asymptotically safe gravity—the resurgence mechanism, Phys. Lett. B 770, 268 (2017).
  60. S. Bornholdt and C. Wetterich, Selforganizing criticality, large anomalous mass dimension and the gauge hierarchy problem, Phys. Lett. B 282, 399 (1992).
  61. H. Gies and M. M. Scherer, Asymptotic safety of simple Yukawa systems, Eur. Phys. J. C 66, 387 (2010).
  62. H. Gies, S. Rechenberger, and M. M. Scherer, Towards an asymptotic-safety scenario for chiral Yukawa systems, Eur. Phys. J. C 66, 403 (2010).
  63. H. Gies and M. Picciau, Self-organized criticality in a relativistic Yukawa theory with luttinger fermions, Int. J. Theor. Phys. 64, 282 (2025).
  64. C. Wetterich, Cosmology and the fate of dilatation symmetry, Nucl. Phys. B302, 668 (1988).
  65. C. Wetterich, The quantum gravity connection between inflation and quintessence, Galaxies 10, 50 (2022).
  66. C. Wetterich, Quantum gravity and scale symmetry in cosmology, Handbook of Quantum Gravity (Springer, Singapore, 2023).
  67. C. Wetterich, Dark energy evolution from quantum gravity, arXiv:2407.03465.
  68. C. Wetterich, Cosmologies with variable Newton’s “constant”, Nucl. Phys. B302, 645 (1988).
  69. C. Wetterich, Probing quintessence with time variation of couplings, J. Cosmol. Astropart. Phys. 10 (2003) 002.
  70. C. M. Müller, G. Schäfer, and C. Wetterich, Nucleosynthesis and the variation of fundamental couplings, Phys. Rev. D 70, 083504 (2004).
  71. T. Dent, S. Stern, and C. Wetterich, Primordial nucleosynthesis as a probe of fundamental physics parameters, Phys. Rev. D 76, 063513 (2007).
  72. H. Meyer and U.-G. Meißner, Improved constraints on the variation of the weak scale from big bang nucleosynthesis, J. High Energy Phys. 06 (2024) 074.
  73. J.-P. Uzan, Fundamental constants: From measurement to the universe, a window on gravitation and cosmology, Living Rev. Relativity 28, 6 (2025).
  74. M. Shaposhnikov and D. Zenhäusern, Scale invariance, unimodular gravity and dark energy, Phys. Lett. B 671, 187 (2009).
  75. M. Shaposhnikov and D. Zenhäusern, Quantum scale invariance, cosmological constant and hierarchy problem, Phys. Lett. B 671, 162 (2009).
  76. M. Shaposhnikov and I. Tkachev, Quantum scale invariance on the lattice, Phys. Lett. B 675, 403 (2009).
  77. D. Blas, M. Shaposhnikov, and D. Zenhäusern, Scale-invariant alternatives to general relativity, Phys. Rev. D 84, 044001 (2011).
  78. G. K. Karananas and M. Shaposhnikov, Scale-invariant alternatives to general relativity. II. dilaton properties, Phys. Rev. D 93, 084052 (2016).
  79. J. García-Bellido, J. Rubio, M. Shaposhnikov, and D. Zenhäusern, Higgs-dilaton cosmology: From the early to the late universe, Phys. Rev. D 84, 123504 (2011).
  80. P. G. Ferreira, C. T. Hill, and G. G. Ross, Weyl current, scale-invariant inflation, and Planck scale generation, Phys. Rev. D 95, 043507 (2017).
  81. S. Casas, M. Pauly, and J. Rubio, Higgs-dilaton cosmology: An inflation–dark-energy connection and forecasts for future galaxy surveys, Phys. Rev. D 97, 043520 (2018).
  82. P. G. Ferreira, C. T. Hill, J. Noller, and G. G. Ross, Inflation in a scale-invariant universe, Phys. Rev. D 97, 123516 (2018).
  83. J. Rubio, Higgs inflation, Front. Astron. Space Sci. 5, 50 (2019).
  84. S. Casas, G. K. Karananas, M. Pauly, and J. Rubio, Scale-invariant alternatives to general relativity. III. The inflation-dark energy connection, Phys. Rev. D 99, 063512 (2019).
  85. Y. Fujii, Origin of the gravitational constant and particle masses in scale invariant scalar—tensor theory, Phys. Rev. D 26, 2580 (1982).
  86. U. Ellwanger and C. Wetterich, Evolution equations for the quark-meson transition, Nucl. Phys. B423, 137 (1994).
  87. S. Bornholdt and C. Wetterich, Average action for models with fermions, Z. Phys. C 58, 585 (1993).
  88. C. Wetterich, Gauge invariant flow equation, Nucl. Phys. B931, 262 (2018).
  89. C. Wetterich, Simplified functional flow equation, Phys. Lett. B 864, 139435 (2025).
  90. T. Papenbrock and C. Wetterich, Two-loop results from improved one loop computations, Z. Phys. C Part. Fields 65, 519 (1995).
  91. D. F. Litim, Optimized renormalization group flows, Phys. Rev. D 64, 105007 (2001).
  92. C. Wetterich, Field transformations in functional integral, effective action and functional flow equations, Nucl. Phys. B1008, 116707 (2024).
  93. B. Ratra and P. J. E. Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D 37, 3406 (1988).
  94. C. Wetterich, The cosmon model for an asymptotically vanishing time dependent cosmological ’constant’, Astron. Astrophys. 301, 321 (1995).
  95. J. A. Frieman, C. T. Hill, A. Stebbins, and I. Waga, Cosmology with ultralight pseudo Nambu-Goldstone bosons, Phys. Rev. Lett. 75, 2077 (1995).
  96. P. G. Ferreira and M. Joyce, Structure formation with a self-tuning scalar field, Phys. Rev. Lett. 79, 4740 (1997).
  97. P. T. P. Viana and A. R. Liddle, Perturbation evolution in cosmologies with a decaying cosmological constant, Phys. Rev. D 57, 674 (1998).
  98. E. J. Copeland, A. R. Liddle, and D. Wands, Exponential potentials and cosmological scaling solutions, Phys. Rev. D 57, 4686 (1998).
  99. R. R. Caldwell, R. Dave, and P. J. Steinhardt, Cosmological imprint of an energy component with general equation of state, Phys. Rev. Lett. 80, 1582 (1998).
  100. L. Amendola, Scaling solutions in general nonminimal coupling theories, Phys. Rev. D 60, 043501 (1999).
  101. L. Amendola, Coupled quintessence, Phys. Rev. D 62, 043511 (2000).
  102. E. V. Linder, The dynamics of quintessence, the quintessence of dynamics, Gen. Relativ. Gravit. 40, 329 (2007).
  103. C. Wetterich, Variable gravity universe, Phys. Rev. D 89, 024005 (2014).
  104. C. Wetterich, Growing neutrinos and cosmological selection, Phys. Lett. B 655, 201 (2007).
  105. L. Amendola, M. Baldi, and C. Wetterich, Quintessence cosmologies with a growing matter component, Phys. Rev. D 78, 023015 (2008).
  106. C. Wetterich, Graviton fluctuations erase the cosmological constant, Phys. Lett. B 773, 6 (2017).
  107. C. Wetterich, Scaling solution for field-dependent gauge couplings in quantum gravity, Nucl. Phys. B985, 116017 (2022).
  108. H. Gies and C. Wetterich, Renormalization flow of bound states, Phys. Rev. D 65, 065001 (2002).
  109. H. Gies and C. Wetterich, Universality of spontaneous chiral symmetry breaking in gauge theories, Phys. Rev. D 69, 025001 (2004).
  110. C. Wetterich, Spontaneously broken color, Phys. Rev. D 64, 036003 (2001).
  111. C. Wetterich, Higgs picture of the QCD-vacuum, AIP Conf. Proc. 739, 123 (2004).
  112. J.-E. Daum, U. Harst, and M. Reuter, Running gauge coupling in asymptotically safe quantum gravity, J. High Energy Phys. 01 (2010) 084.
  113. S. Folkerts, D. F. Litim, and J. M. Pawlowski, Asymptotic freedom of Yang-Mills theory with gravity, Phys. Lett. B 709, 234 (2012).
  114. U. Harst and M. Reuter, QED coupled to QEG, J. High Energy Phys. 05 (2011) 119.
  115. N. Christiansen and A. Eichhorn, An asymptotically safe solution to the U(1) triviality problem, Phys. Lett. B 770, 154 (2017).
  116. N. Christiansen, D. F. Litim, J. M. Pawlowski, and M. Reichert, Asymptotic safety of gravity with matter, Phys. Rev. D 97, 106012 (2018).
  117. A. Eichhorn and F. Versteegen, Upper bound on the Abelian gauge coupling from asymptotic safety, J. High Energy Phys. 01 (2018) 030.
  118. C. Wetterich, Gauge hierarchy due to strong interactions?, Phys. Lett. 104B, 269 (1981).
  119. B. Schrempp and M. Wimmer, Top quark and Higgs boson masses: Interplay between infrared and ultraviolet physics, Prog. Part. Nucl. Phys. 37, 1 (1996).

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Phys. Rev. D 114, 086002– Published 5 October, 2026
  • Received 27 March 2026
  • Accepted 14 September 2026
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