The Hierarchy Problem and the Cosmological Constant Problem Revisited

Higgs Inflation and a New View on the SM of Particle Physics
  • Fred JegerlehnerEmail author
Part of the following topical collections:
  1. Naturalness, Hierarchy, and Fine-Tuning


We argue that the Standard Model (SM) in the Higgs phase does not suffer from a “hierarchy problem” and that similarly the “cosmological constant problem” resolves itself if we understand the SM as a low energy effective theory emerging from a cutoff-medium at the Planck scale. We actually take serious Veltman’s “The Infrared–Ultraviolet Connection” addressing the issue of quadratic divergences and the related huge radiative correction predicted by the SM in the relationship between the bare and the renormalized theory, usually called “the hierarchy problem” and claimed that this has to be cured. We discuss these issues under the condition of a stable Higgs vacuum, which allows extending the SM up to the Planck cutoff. The bare Higgs boson mass then changes sign below the Planck scale, such that the SM in the early universe is in the symmetric phase. The cutoff enhanced Higgs mass term as well as the quartically enhanced cosmological constant term provide a large positive dark energy that triggers the inflation of the early universe. Reheating follows via the decays of the four unstable heavy Higgs particles, predominantly into top–antitop pairs, which at this stage are massless. Preheating is suppressed in SM inflation since in the symmetric phase bosonic decay channels are absent at tree level. The coefficients of the shift between bare and renormalized Higgs mass as well as of the shift between bare and renormalized vacuum energy density exhibit close-by zeros at about \(7.7 \times 10^{14}\ \hbox {GeV}\) and \(3.1 \times 10^{15}\ \hbox {GeV}\), respectively. The zero of the Higgs mass counter term triggers the electroweak phase transition, from the low energy Higgs phase and to the symmetric phase above the transition point. Since inflation tunes the total energy density to take the critical value of a flat universe and all contributing components are positive, it is obvious that the cosmological constant today is naturally a substantial fraction of the total critical density. Thus taking cutoff enhanced corrections seriously the Higgs system provides besides the masses of particles in the Higgs phase also dark energy, inflation and reheating in the early universe. The main unsolved problem in our context remains the origin of dark matter. Higgs inflation is possible and likely even unavoidable provided new physics does not disturb the known relevant SM properties substantially. The scenario highly favors to understand the SM and its main properties as a natural structure emerging at long distance. This in particular concerns the SM symmetry patterns and their consequences.


Higgs vacuum stability Hierarchy problem Cosmological constant problem Inflation 



I thank the organizers of the Naturalness, Hierarchy and Fine Tuning Workshop, at the RWTH Aachen, for the kind invitation and the support.


  1. 1.
    Jegerlehner, F.: The standard model as a low-energy effective theory: what is triggering the Higgs mechanism? Acta Phys. Polon. B 45, 1167 (2014). [arXiv:1304.7813 [hep-ph]]ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Jegerlehner, F.: Higgs inflation and the cosmological constant. Acta Phys. Polon. B 45, 1215 (2014). [arXiv:1402.3738 [hep-ph]]ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    ’t Hooft, G.: Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking. NATO Adv. Stud. Inst. Ser. B Phys. 59, 135 (1980)Google Scholar
  4. 4.
    Veltman, M.J.G.: The infrared—ultraviolet connection. Acta Phys. Polon. B 12, 437 (1981)Google Scholar
  5. 5.
    Decker, R., Pestieau, J.: Lepton self-mass, Higgs scalar and heavy quark masses, arXiv:hep-ph/0512126
  6. 6.
    Al-sarhi, M.S., Jack, I., Jones, D.R.T.: Quadratic divergences in gauge theories. Z. Phys. C 55, 283 (1992)ADSCrossRefGoogle Scholar
  7. 7.
    Hamada, Y., Kawai, H., Oda, K.Y.: Bare Higgs mass at Planck scale. Phys. Rev. D 87, 053009 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    Jones, D.R.T.: The quadratic divergence in the Higgs mass revisited. Phys. Rev. D 88, 098301 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    Wetterich, C.: Fine tuning problem and the renormalization group. Phys. Lett. 140B, 215 (1984)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Glashow, S.L.: Partial symmetries of weak interactions. Nucl. Phys. 22, 579 (1961)CrossRefGoogle Scholar
  11. 11.
    Weinberg, S.: A model of leptons. Phys. Rev. Lett. 19, 1264 (1967)ADSCrossRefGoogle Scholar
  12. 12.
    Fritzsch, H., Gell-Mann, M., Leutwyler, H.: Advantages of the color octet gluon picture. Phys. Lett. 47, 365 (1973)CrossRefGoogle Scholar
  13. 13.
    Weinberg, S.: Nonabelian gauge theories of the strong interactions. Phys. Rev. Lett. 31, 494 (1973)ADSCrossRefGoogle Scholar
  14. 14.
    Mather, J.C., et al.: A preliminary measurement of the cosmic microwave background spectrum by the Cosmic Background Explorer (COBE) satellite. Astrophys. J. (Letter) 354, 37 (1990)ADSCrossRefGoogle Scholar
  15. 15.
    Smoot, G., et al.: Structure in the COBE differential microwave radiometer first year maps. Astrophys. J. (Letters) 396, 1 (1992)ADSCrossRefGoogle Scholar
  16. 16.
    Bennett, C.L., et al.: [WMAP Collaboration], Nine-year Wilkinson microwave anisotropy probe (WMAP) observations: final maps and results. Astrophys. J. Suppl. 208, 20 (2013)Google Scholar
  17. 17.
    Ade, P.A.R., et al.: [Planck Collaboration], Planck 2013 results. I. Overview of products and scientific results. Astron. Astrophys. 571, A1 (2014); Planck 2013 results. XVI. Cosmological parameters. Astron. Astrophys. 571, A16 (2014)Google Scholar
  18. 18.
    Adam, R., et al.: [Planck Collaboration], Planck 2015 results. I. Overview of products and scientific results. Astron. Astrophys. 594, A1 (2016)Google Scholar
  19. 19.
    Riess, A.G., et al.: [Supernova Search Team], Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998)Google Scholar
  20. 20.
    Perlmutter, S., et al.: [Supernova Cosmology Project Collaboration], Measurements of Omega and Lambda from 42 high redshift supernovae. Astrophys. J. 517, 565 (1999)Google Scholar
  21. 21.
    Bass, S.D.: The cosmological constant puzzle: vacuum energies from QCD to dark energy. Acta Phys. Polon. B 45, 1269 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ade, P.A.R., et al.: [Planck Collaboration], Planck 2013 results. XXIV. Constraints on primordial non-gaussianity. Astron. Astrophys. 571, A24 (2014)Google Scholar
  23. 23.
    Dreitlein, J.: Broken symmetry and the cosmologial constant. Phys. Rev. Lett. 33, 1243 (1974)ADSCrossRefGoogle Scholar
  24. 24.
    Felten, J.E., Isaacman, R.: Scale factors \(R(t)\) and critical values of the cosmological constant Lambda in Friedmann universes. Rev. Mod. Phys. 58, 689 (1986)ADSCrossRefGoogle Scholar
  25. 25.
    Sahni, V., Starobinsky, A.A.: The case for a positive cosmological Lambda term. Int. J. Mod. Phys. D 9, 373 (2000)ADSGoogle Scholar
  26. 26.
    Guth, A.H.: The inflationary universe: a possible solution to the horizon and flatness problems. Phys. Rev. D 23, 347 (1981)ADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Starobinsky, A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B 91, 99 (1980)ADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Albrecht, A., Steinhardt, P.J.: Cosmology for grand unified theories with radiatively induced symmetry breaking. Phys. Rev. Lett. 48, 1220 (1982)ADSCrossRefGoogle Scholar
  29. 29.
    Linde, A.D.: A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Phys. Lett. B 108, 389 (1982)ADSCrossRefGoogle Scholar
  30. 30.
    Linde, A.D.: Chaotic inflation. Phys. Lett. B 129, 177 (1983)ADSCrossRefGoogle Scholar
  31. 31.
    Kolb, E.W., Turner, M.S.: The early universe. Front. Phys. 69, 1 (1990)ADSMathSciNetzbMATHGoogle Scholar
  32. 32.
    Weinberg, S.: Cosmology, p. 593. Oxford Univ. Press, Oxford (2008)zbMATHGoogle Scholar
  33. 33.
    Minkowski, P.: On the spontaneous origin of Newton’s constant. Phys. Lett. 71B, 419 (1977)ADSCrossRefGoogle Scholar
  34. 34.
    Zee, A.: A broken symmetric theory of gravity. Phys. Rev. Lett. 42, 417 (1979)ADSCrossRefGoogle Scholar
  35. 35.
    Bezrukov, F., Shaposhnikov, M.: The standard model Higgs boson as the inflaton. Phys. Lett. B 659, 703 (2008)ADSCrossRefGoogle Scholar
  36. 36.
    Barbon, J.L.F., Espinosa, J.R.: On the naturalness of Higgs inflation. Phys. Rev. D 79, 081302 (2009)ADSCrossRefGoogle Scholar
  37. 37.
    Bezrukov, F., Magnin, A., Shaposhnikov, M., Sibiryakov, S.: Higgs inflation: consistency and generalisations. JHEP 1101, 016 (2011)ADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Bezrukov, F., Shaposhnikov, M.: Higgs inflation at the critical point. Phys. Lett. B 734, 249 (2014)ADSCrossRefGoogle Scholar
  39. 39.
    Bezrukov, F., Rubio, J., Shaposhnikov, M.: Living beyond the edge: Higgs inflation and vacuum metastability. Phys. Rev. D 92, 083512 (2015)ADSCrossRefGoogle Scholar
  40. 40.
    Hamada, Y., Kawai, H., Oda, K.Y., Park, S.C.: Higgs inflation is still alive after the results from BICEP2. Phys. Rev. Lett. 112, 241301 (2014)ADSCrossRefGoogle Scholar
  41. 41.
    Hamada, Y., Kawai, H., Oda, K.Y.: Eternal Higgs inflation and the cosmological constant problem. Phys. Rev. D 92, 045009 (2015)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Peccei, R., Sola, J., Wetterich, C.: Adjusting the cosmological constant dynamically: cosmons and a new force weaker than gravity. Phys. Lett. B 195, 183 (1987)ADSCrossRefGoogle Scholar
  43. 43.
    Wetterich, C.: Cosmologies with variable Newton’s constant. Nucl. Phys. B 302, 645 (1988)ADSCrossRefGoogle Scholar
  44. 44.
    Wetterich, C.: Cosmology and the fate of dilatation symmetry. Nucl. Phys. B 302, 668 (1988)ADSCrossRefGoogle Scholar
  45. 45.
    Wetterich, C.: Inflation, quintessence, and the origin of mass. Nucl. Phys. B 897, 111 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Aad, G., et al.: [ATLAS Collab.], Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B 716, 1 (2012); [ATLAS Collaboration], A particle consistent with the Higgs Boson observed with the ATLAS Detector at the Large Hadron Collider. Science 338, 1576 (2012)Google Scholar
  47. 47.
    Chatrchyan, S., et al.: [CMS Collab.], Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B 716, 30 (2012); [CMS Collaboration], A new boson with a mass of 125-GeV observed with the CMS experiment at the Large Hadron Collider. Science 338, 1569 (2012)Google Scholar
  48. 48.
    Englert, F., Brout, R.: Broken symmetry and the mass of gauge vector mesons. Phys. Rev. Lett. 13, 321 (1964)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Higgs, P.W.: Broken symmetries, massless particles and gauge fields. Phys. Lett. 12, 132 (1964)ADSCrossRefGoogle Scholar
  50. 50.
    Cabibbo, N., Maiani, L., Parisi, G., Petronzio, R.: Bounds on the fermions and higgs boson masses in grand unified theories. Nucl. Phys. B 158, 295 (1979)ADSCrossRefGoogle Scholar
  51. 51.
    Hung, P.Q.: Vacuum instability and new constraints on fermion masses. Phys. Rev. Lett. 42, 873 (1979)ADSCrossRefGoogle Scholar
  52. 52.
    Lindner, M.: Implications of triviality for the standard model. Z. Phys. C 31, 295 (1986)ADSCrossRefGoogle Scholar
  53. 53.
    Grzadkowski, B., Lindner, M.: Stability of triviality mass bounds in the standard model. Phys. Lett. B 178, 81 (1986)ADSCrossRefGoogle Scholar
  54. 54.
    Lindner, M., Sher, M., Zaglauer, H.W.: Probing vacuum stability bounds at the Fermilab collider. Phys. Lett. B 228, 139 (1989)ADSCrossRefGoogle Scholar
  55. 55.
    Sher, M.: Electroweak Higgs potentials and vacuum stability. Phys. Rept. 179, 273 (1989)ADSCrossRefGoogle Scholar
  56. 56.
    Hambye, T., Riesselmann, K.: Matching conditions and Higgs mass upper bounds revisited. Phys. Rev. D 55, 7255 (1997)ADSCrossRefGoogle Scholar
  57. 57.
    Casas, J.A., Espinosa, J.R., Quiros, M.: Improved Higgs mass stability bound in the standard model and implications for supersymmetry. Phys. Lett. B 342, 171 (1995)ADSCrossRefGoogle Scholar
  58. 58.
    Casas, J.A., Espinosa, J.R., Quiros, M.: Standard model stability bounds for new physics within LHC reach. Phys. Lett. B 382, 374 (1996)ADSCrossRefGoogle Scholar
  59. 59.
    Espinosa, J.R., Quiros, M.: Improved metastability bounds on the standard model Higgs mass. Phys. Lett. B 353, 257 (1995)ADSCrossRefGoogle Scholar
  60. 60.
    Schrempp, B., Wimmer, M.: Top quark and Higgs boson masses: interplay between infrared and ultraviolet physics. Prog. Part. Nucl. Phys. 37, 1 (1996)ADSCrossRefGoogle Scholar
  61. 61.
    Isidori, G., Ridolfi, G., Strumia, A.: On the metastability of the standard model vacuum. Nucl. Phys. B 609, 387 (2001)ADSzbMATHCrossRefGoogle Scholar
  62. 62.
    Espinosa, J.R., Giudice, G.F., Riotto, A.: Cosmological implications of the Higgs mass measurement. JCAP 0805, 002 (2008). [arXiv:0710.2484]ADSCrossRefGoogle Scholar
  63. 63.
    Ellis, J., Espinosa, J.R., Giudice, G.F., Höcker, A., Riotto, A.: The probable fate of the standard model. Phys. Lett. B 679, 369 (2009)ADSCrossRefGoogle Scholar
  64. 64.
    Feldstein, B., Hall, L.J., Watari, T.: Landscape prediction for the Higgs boson and top quark masses. Phys. Rev. D 74, 095011 (2006)ADSCrossRefGoogle Scholar
  65. 65.
    Degrassi, G., Di Vita, S., Elias-Miro, J., Espinosa, J.R., Giudice, G.F., Isidori, G., Strumia, A.: Higgs mass and vacuum stability in the standard model at NNLO. JHEP 1208, 098 (2012)ADSCrossRefGoogle Scholar
  66. 66.
    Shaposhnikov, M., Wetterich, C.: Asymptotic safety of gravity and the Higgs boson mass. Phys. Lett. B 683, 196 (2010)ADSCrossRefGoogle Scholar
  67. 67.
    Holthausen, M., Lim, K.S., Lindner, M.: Planck scale boundary conditions and the Higgs mass. JHEP 1202, 037 (2012)ADSzbMATHCrossRefGoogle Scholar
  68. 68.
    Bezrukov, F., Kalmykov, M.Y., Kniehl, B.A., Shaposhnikov, M.: Higgs Boson mass and new physics. JHEP 1210, 140 (2012)ADSCrossRefGoogle Scholar
  69. 69.
    Alekhin, S., Djouadi, A., Moch, S.: The top quark and Higgs boson masses and the stability of the electroweak vacuum. Phys. Lett. B 716, 214 (2012)ADSCrossRefGoogle Scholar
  70. 70.
    Mihaila, L.N., Salomon, J., Steinhauser, M.: Gauge coupling beta functions in the standard model to three loops. Phys. Rev. Lett. 108, 151602 (2012)ADSCrossRefGoogle Scholar
  71. 71.
    Bednyakov, A.V., Pikelner, A.F., Velizhanin, V.N.: Anomalous dimensions of gauge fields and gauge coupling \(\beta \)-functions in the standard model at three loops. JHEP 1301, 017 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Bednyakov, A.V., Pikelner, A.F., Velizhanin, V.N.: Yukawa coupling \(\beta \)-functions in the standard model at three loops. Phys. Lett. B 722, 336 (2013)ADSzbMATHCrossRefGoogle Scholar
  73. 73.
    Pikelner, A.F., Velizhanin, V.N.: Higgs self-coupling \(\beta \)-function in the standard model at three loops. Nucl. Phys. B 875, 552 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Pikelner, A.F., Velizhanin, V.N.: Three-loop Higgs self-coupling \(\beta \)-function in the standard model with complex Yukawa matrices. Nucl. Phys. B 879, 256 (2014)ADSzbMATHCrossRefGoogle Scholar
  75. 75.
    Pikelner, A.F., Velizhanin, V.N.: Three-loop SM \(\beta \)-functions for matrix Yukawa couplings. Phys. Lett. B 737, 129 (2014)ADSzbMATHCrossRefGoogle Scholar
  76. 76.
    Chetyrkin, K.G., Zoller, M.F.: Three-loop \(\beta \)-functions for top-Yukawa and the Higgs self-interaction in the standard model. JHEP 1206, 033 (2012)ADSCrossRefGoogle Scholar
  77. 77.
    Chetyrkin, K.G., Zoller, M.F.: \(\beta \)-function for the Higgs self-interaction in the standard model at three-loop level. JHEP 1304, 091 (2013)ADSCrossRefGoogle Scholar
  78. 78.
    Fleischer, J., Jegerlehner, F.: Radiative corrections to Higgs decays in the extended Weinberg-Salam model. Phys. Rev. D 23, 2001 (1981)ADSCrossRefGoogle Scholar
  79. 79.
    Sirlin, A., Zucchini, R.: Dependence of the quartic coupling \(\overline{h}_{\overline{{\rm MS}}}(M)\) on \(m_H\) and the possible onset of new physics in the Higgs sector of the standard model. Nucl. Phys. B 266, 389 (1986)ADSCrossRefGoogle Scholar
  80. 80.
    Jegerlehner, F., Kalmykov, M.Y., Veretin, O.: \(\overline{{\rm MS}}\) versus pole masses of gauge bosons: electroweak bosonic two loop corrections. Nucl. Phys. B 641, 285 (2002)ADSCrossRefGoogle Scholar
  81. 81.
    Kalmykov, M.Y., Veretin, O.: Full two loop electroweak corrections to the pole masses of gauge bosons. Nucl. Phys. Proc. Suppl. 116, 382 (2003)ADSzbMATHCrossRefGoogle Scholar
  82. 82.
    Jegerlehner, F., Kalmykov, M.Y., Veretin, O.: \(\overline{{\rm MS}}\) versus pole masses of gauge bosons. 2. Two loop electroweak fermion corrections. Nucl. Phys. B 658, 49 (2003)ADSzbMATHCrossRefGoogle Scholar
  83. 83.
    Jegerlehner, F., Kalmykov, M.Y., Kniehl, B.A.: On the difference between the pole and the \(\overline{{\rm MS}}\) masses of the top quark at the electroweak scale. Phys. Lett. B 722, 123 (2013)ADSzbMATHCrossRefGoogle Scholar
  84. 84.
    Jegerlehner, F., Kalmykov, M.Y., Kniehl, B.A.: Self-consistence of the standard model via the renormalization group analysis. J. Phys. Conf. Ser. 608, 012074 (2015)CrossRefGoogle Scholar
  85. 85.
    Buttazzo, D., Degrassi, G., Giardino, P.P., Giudice, G.F., Sala, F., Salvio, A., Strumia, A.: Investigating the near-criticality of the Higgs boson. JHEP 1312, 089 (2013)ADSCrossRefGoogle Scholar
  86. 86.
    Bednyakov, A.V., Kniehl, B.A., Pikelner, A.F., Veretin, O.L.: Stability of the electroweak vacuum: gauge independence and advanced precision. Phys. Rev. Lett. 115, 201802 (2015)ADSCrossRefGoogle Scholar
  87. 87.
    Kniehl, B.A., Pikelner, A.F., Veretin, O.L.: Two-loop electroweak threshold corrections in the standard model. Nucl. Phys. B 896, 19 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Martin, S.P.: Matching relations for decoupling in the standard model at two loops and beyond. Phys. Rev. D 99, 033007 (2019). [arXiv:1812.04100 [hep-ph]]ADSCrossRefGoogle Scholar
  89. 89.
    Martin, S.P., Robertson, D.G.: Higgs boson mass in the standard model at two-loop order and beyond. Phys. Rev. D 90, 073010 (2014). [arXiv:1407.4336 [hep-ph]]ADSCrossRefGoogle Scholar
  90. 90.
    Martin, S.P.: Pole mass of the W Boson at two-loop order in the pure \(\overline{MS}\) scheme. Phys. Rev. D 91, 114003 (2015). [arXiv:1503.03782 [hep-ph]]ADSCrossRefGoogle Scholar
  91. 91.
    Martin, S.P.: \(Z\)-Boson pole mass at two-loop order in the pure \(\overline{MS}\) scheme. Phys. Rev. D 92, 014026 (2015)ADSCrossRefGoogle Scholar
  92. 92.
    Awramik, M., Czakon, M.: Complete two loop electroweak contributions to the muon lifetime in the standard model. Phys. Lett. B 568, 48 (2003). in the standard model,ADSCrossRefGoogle Scholar
  93. 93.
    Awramik, M., Czakon, M., Freitas, A., Weiglein, G.: Complete two-loop electroweak fermionic corrections to \(\sin ^{2} \theta ^{{\rm lept}}_{{\rm eff}}\) and indirect determination of the Higgs boson mass. Phys. Rev. Lett. 93, 201805 (2004)ADSCrossRefGoogle Scholar
  94. 94.
    Awramik, M., Czakon, M., Freitas, A.: Bosonic corrections to the effective weak mixing angle at \(O(\alpha ^2)\). Phys. Lett. B 642, 563 (2006a)Google Scholar
  95. 95.
    Awramik, M., Czakon, M., Freitas, A.: Electroweak two-loop corrections to the effective weak mixing angle. JHEP 0611, 048 (2006b)Google Scholar
  96. 96.
    Jadach, S., Płaczek, W., Skrzypek, M., Ward, B.F.L., Yost, S.A.: The path to 0.01% theoretical luminosity precision for the FCC-ee. Phys. Lett. B 790, 314 (2019)ADSCrossRefGoogle Scholar
  97. 97.
    Accomando, E., et al.: [ECFA/DESY LC Physics Working Group], Physics with \(e^{+} e^{-}\) linear colliders. Phys. Rept. 299, 1 (1998); Aguilar-Saavedra, J.A., et al. [ECFA/DESY LC Physics Working Group], TESLA: The Superconducting electron positron linear collider with an integrated X-ray laser laboratory. Technical design report. Part 3. Physics at an e+ e- linear collider, arXiv:hep-ph/0106315
  98. 98.
    Azzi, P., et al.: Physics Behind Precision, arXiv:1703.01626 [hep-ph]; Theory report on the 11th FCC-ee workshop, 8–11 (January 2019), CERN, Geneva, to appear
  99. 99.
    Beneke, M., Marquard, P., Nason, P., Steinhauser, M.: On the ultimate uncertainty of the top quark pole mass. Phys. Lett. B 775, 63 (2017)ADSCrossRefGoogle Scholar
  100. 100.
    Appelquist, T., Carazzone, J.: Infrared singularities and massive fields. Phys. Rev. D 11, 2856 (1975)ADSCrossRefGoogle Scholar
  101. 101.
    Faisst, M., Kühn, J.H., Veretin, O.: Pole versus MS mass definitions in the electroweak theory. Phys. Lett. B 589, 35 (2004)ADSCrossRefGoogle Scholar
  102. 102.
    Taylor, J.C.: Gauge Theories of Weak Interactions, Cambridge Monographs on Mathematical Physics, p. 167. Cambridge University Press, Cambridge (1976)Google Scholar
  103. 103.
    Kraus, E.: Renormalization of the Electroweak standard model to all orders. Ann. Phys. 262, 155 (1998)ADSzbMATHCrossRefGoogle Scholar
  104. 104.
    Jegerlehner, F.: Variations on Photon Vacuum Polarization, arXiv:1711.06089 [hep-ph]
  105. 105.
    Jegerlehner, F.: Renormalization scheme dependence of electroweak radiative corrections. In: Radiative corrections: results and perspectives. Proceedings, NATO Advanced Research Workshop, Brighton, UK, July 10–14, 1989, Dombey N., Boudjema, F. (eds.) NATO Sci. Ser. B 233, 185 (1990)Google Scholar
  106. 106.
    Politzer, H.D.: Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30, 1346 (1973)ADSCrossRefGoogle Scholar
  107. 107.
    Gross, D., Wilczek, F.: Ultraviolet behavior of nonabelian gauge theories. Phys. Rev. Lett. 30, 1343 (1973)ADSCrossRefGoogle Scholar
  108. 108.
    Louis, J., et al.: String theory: an overview. Lect. Notes Phys. 721, 289 (2007). 323ADSMathSciNetzbMATHCrossRefGoogle Scholar
  109. 109.
    Glashow, S.L., Iliopoulos, J., Maiani, L.: Weak interactions with Lepton-Hadron symmetry. Phys. Rev. D 2, 1285 (1970)ADSCrossRefGoogle Scholar
  110. 110.
    Czakon, M., Gluza, J., Jegerlehner, F., Zrałek, M.: Confronting electroweak precision measurements with new physics models. Eur. Phys. J. C 13, 275 (2000)ADSCrossRefGoogle Scholar
  111. 111.
    Wilson, K.G.: Renormalization group and critical phenomena. 1. Renormalization group and the Kadanoff scaling picture. Phys. Rev. B 4, 3174 (1971)ADSzbMATHCrossRefGoogle Scholar
  112. 112.
    Wilson, K.G.: Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior. Phys. Rev. B 4, 3184 (1971)ADSzbMATHCrossRefGoogle Scholar
  113. 113.
    Jegerlehner, F.: An introduction to the theory of critical phenomena and the renormalization group, Preprint, ZIF Universität, Bielefeld, p. 158 (May 1976). Lausanne Lectures
  114. 114.
    Jegerlehner, F.: On the construction of renormalized field theories from cutoff and lattice models. Phys. Rev. D 16, 397 (1977)ADSMathSciNetCrossRefGoogle Scholar
  115. 115.
    Ahn, C.R., Peskin, M.E., Lynn, B.W., Selipsky, S.B.: Delayed unitarity cancellation and heavy particle effects in \(e^+ e^- \rightarrow W^+ W^-\). Nucl. Phys. B 309, 221 (1988)ADSCrossRefGoogle Scholar
  116. 116.
    Yang, C.N., Mills, R.L.: Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. 96, 191 (1954)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    ’t Hooft, G.: Renormalization of massless Yang-Mills fields. Nucl. Phys. B 33, 173 (1971)ADSCrossRefGoogle Scholar
  118. 118.
    ’t Hooft, G.: Renormalizable lagrangians for massive Yang-Mills fields. Nucl. Phys. B 35, 167 (1971)ADSCrossRefGoogle Scholar
  119. 119.
    ’t Hooft, G., Veltman, M.: Combinatorics of gauge fields. Nucl. Phys. B 50, 318 (1972)ADSMathSciNetCrossRefGoogle Scholar
  120. 120.
    Lüscher, M., Weisz, P.: Is there a strong interaction sector in the standard lattice Higgs model? Phys. Lett. B 212, 472 (1988)ADSCrossRefGoogle Scholar
  121. 121.
    Lüscher, M., Weisz, P.: Scaling laws and triviality bounds in the lattice \(\phi ^4\) theory. 1. One component model in the symmetric phase. Nucl. Phys. B 290, 25 (1987)ADSMathSciNetCrossRefGoogle Scholar
  122. 122.
    Lüscher, M., Weisz, P.: Scaling laws and triviality bounds in the lattice \(\phi ^4\) theory. 2. One component model in the phase with spontaneous symmetry breaking. Nucl. Phys. B 295, 65 (1988)ADSMathSciNetCrossRefGoogle Scholar
  123. 123.
    Lang, C.B.: On the continuum limit of \(D=4\) lattice \(\phi ^4\) theory. Nucl. Phys. B 265, 630 (1986)ADSCrossRefGoogle Scholar
  124. 124.
    Callaway, D.J.E.: Triviality pursuit: can elementary scalar particles exist? Phys. Rep. 167, 241 (1988)ADSCrossRefGoogle Scholar
  125. 125.
    Bass, S.D.: Emergence in particle physics. Acta Phys. Polon. B 48, 1903 (2017)ADSCrossRefGoogle Scholar
  126. 126.
    Jegerlehner, F.: The vector boson and graviton propagators in the presence of multipole forces. Helv. Phys. Acta 51, 783 (1978)MathSciNetGoogle Scholar
  127. 127.
    Veltman, M.J.G.: Perturbation theory of massive Yang-Mills fields. Nucl. Phys. B 7, 637 (1968)ADSCrossRefGoogle Scholar
  128. 128.
    Llewellyn Smith, C.H.: High-energy behavior and gauge symmetry. Phys. Lett. B 46, 233 (1973)ADSCrossRefGoogle Scholar
  129. 129.
    Bell, J.S.: High-energy behavior of tree diagrams in gauge theories. Nucl. Phys. B 60, 427 (1973)ADSCrossRefGoogle Scholar
  130. 130.
    Cornwall, J.M., Levin, D.N., Tiktopoulos, G.: Uniqueness of spontaneously broken gauge theories, Phys. Rev. Lett. 30, 1268 (1973) [Erratum-ibid. 31, 572 (1973)]; Derivation of gauge invariance from high-energy unitarity bounds on the \(S\)-matrix. Phys. Rev. D 10, 1145 (1974) [Erratum-ibid. D 11, 972 (1975)]Google Scholar
  131. 131.
    Jegerlehner, F.: Search for anomalous gauge boson couplings. Nucl. Phys. Proc. Suppl. 37B, 129 (1994)ADSCrossRefGoogle Scholar
  132. 132.
    Jegerlehner, F.: The ’Ether world’ and elementary particles. In: Theory of Elementary Particles. Lüst, G. Weigt, Wiley-VCH, Berlin, (1998), p. 386 particles* 386–392 [arXiv:hep-th/9803021]
  133. 133.
    Jegerlehner, F.: The hierarchy problem of the electroweak Standard Model revisited, arXiv:1305.6652 [hep-ph]; The hierarchy problem and the cosmological constant problem in the Standard Model, arXiv:1503.00809 [hep-ph]
  134. 134.
    Lüscher, M.: Chiral gauge theories revisited. Subnucl. Ser. 38, 41 (2002)Google Scholar
  135. 135.
    Lüscher, M.: Lattice regularization of chiral gauge theories to all orders of perturbation theory. JHEP 0006, 028 (2000)MathSciNetzbMATHCrossRefADSGoogle Scholar
  136. 136.
    Elitzur, S.: Impossibility of spontaneously breaking local symmetries. Phys. Rev. D 12, 3978 (1975)ADSCrossRefGoogle Scholar
  137. 137.
    Hamada, Y., Kawai, H., Kawana, K.: Natural solution to the naturalness problem: the universe does fine-tuning. PTEP 2015, 123B03 (2015)Google Scholar
  138. 138.
    Coleman, S.R., Weinberg, E.J.: Radiative corrections as the origin of spontaneous symmetry breaking. Phys. Rev. D 7, 1888 (1973)ADSCrossRefGoogle Scholar
  139. 139.
    Ginzburg, V.L., Landau, L.D.: Zh. Eksp. Teor. Fiz. 20, 1064 (1950), English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546; Teor. Fiz. 32, 1442 (1957); Pitaevskii, L.P.: Statistical Physics: Theory of the Condensed State (Landau-Lifshitz Course of Theoretical Physics Vol. 9) (Pergamon, Oxford, 1980)Google Scholar
  140. 140.
    Cooper, L.N.: Bound electron pairs in a degenerate Fermi gas. Phys. Rev. 104, 1189 (1956)ADSzbMATHCrossRefGoogle Scholar
  141. 141.
    Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Microscopic theory of superconductivity. Phys. Rev. 106, 162 (1957)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  142. 142.
    Gor’kov, L.P.: Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity. Sov. Phys. JETP 9, 1364 (1959)MathSciNetzbMATHGoogle Scholar
  143. 143.
    Abrikosov, A.A.: On the magnetic properties of superconductors of the second group. Zh. Eksp. Teor. Fiz. 32, 1442 (1957) [Sov. Phys. JETP 5 1174 (1957)]Google Scholar
  144. 144.
    Masina, I.: Higgs boson and top quark masses as tests of electroweak vacuum stability. Phys. Rev. D 87, 053001 (2013)ADSCrossRefGoogle Scholar
  145. 145.
    Tang, Y.: Vacuum stability in the standard model. Mod. Phys. Lett. A 28, 1330002 (2013)ADSMathSciNetCrossRefGoogle Scholar
  146. 146.
    Hoang, A.H.: The top mass: interpretation and theoretical uncertainties, arXiv:1412.3649 [hep-ph]
  147. 147.
    Degrassi, G., Sirlin, A.: Gauge dependence of basic electroweak corrections of the standard model. Nucl. Phys. B 383, 73 (1992)ADSCrossRefGoogle Scholar
  148. 148.
    Fang, Z.Y., Lopez Castro, G., Lucio, J.L., Pestieau, J.: Effective \(SU(2)_L\times U(1)\) theory and the Higgs boson mass. Mod. Phys. Lett. A 12, 1531 (1997)ADSCrossRefGoogle Scholar
  149. 149.
    Kirzhnits, D.A.: Weinberg model in the hot universe, JETP Lett. 15, 529 (1972) [Pisma Zh. Eksp. Teor. Fiz. 15, 745 (1972)]Google Scholar
  150. 150.
    Dolan, L., Jackiw, R.: Symmetry behavior at finite temperature. Phys. Rev. D 9, 3320 (1974)ADSCrossRefGoogle Scholar
  151. 151.
    Weinberg, S.: Gauge and global symmetries at high temperature. Phys. Rev. D 9, 3357 (1974)ADSCrossRefGoogle Scholar
  152. 152.
    Kirzhnits, D.A., Linde, A.D.: Symmetry behavior in gauge theories. Ann. Phys. 101, 195 (1976)ADSCrossRefGoogle Scholar
  153. 153.
    Dine, M., Leigh, R.G., Huet, P.Y., Linde, A.D., Linde, D.A.: Towards the theory of the electroweak phase transition. Phys. Rev. D 46, 550 (1992)ADSCrossRefGoogle Scholar
  154. 154.
    Weinberg, S.: Perturbative calculations of symmetry breaking. Phys. Rev. D 7, 2887 (1973)ADSCrossRefGoogle Scholar
  155. 155.
    Weinberg, S.: Mass of the Higgs Boson. Phys. Rev. Lett. 36, 294 (1976)ADSCrossRefGoogle Scholar
  156. 156.
    Ford, C., Jones, D.R.T.: The Effective potential and the differential equations method for Feynman integrals. Phys. Lett. B 274, 409 (1992) Erratum: [Phys. Lett. B 285, 399 (1992)]Google Scholar
  157. 157.
    Ford, C., Jack, I., Jones, D.R.T.: The standard model effective potential at two loops. Nucl. Phys. B 387, 373 (1992) Erratum: [Nucl. Phys. B 504, 551 (1997)]Google Scholar
  158. 158.
    Ford, C., Jones, D.R.T., Stephenson, P.W., Einhorn, M.B.: The effective potential and the renormalization group. Nucl. Phys. B 395, 17 (1993)ADSCrossRefGoogle Scholar
  159. 159.
    Kastening, B.M.: Renormalization group improvement of the effective potential in massive \(\phi ^4\) theory. Phys. Lett. B 283, 287 (1992)ADSCrossRefGoogle Scholar
  160. 160.
    Martin, S.P.: Three-loop standard model effective potential at leading order in strong and top Yukawa couplings. Phys. Rev. D 89(1), 013003 (2014)ADSCrossRefGoogle Scholar
  161. 161.
    Nakano, H., Yoshida, Y.: Improving the effective potential, multimass problem and modified mass dependent scheme. Phys. Rev. D 49, 5393 (1994)ADSCrossRefGoogle Scholar
  162. 162.
    Burgess, C.P., Di Clemente, V., Espinosa, J.R.: Effective operators and vacuum instability as heralds of new physics. JHEP 0201, 041 (2002)ADSMathSciNetCrossRefGoogle Scholar
  163. 163.
    Straumann, N.: The mystery of the cosmic vacuum energy density and the accelerated expansion of the universe. Eur. J. Phys. 20, 419 (1999)CrossRefGoogle Scholar
  164. 164.
    Volovik, G.E.: Vacuum energy: quantum hydrodynamics versus quantum gravity, JETP Lett. 82, 319 (2005) [Pisma Zh. Eksp. Teor. Fiz. 82, 358 (2005)]Google Scholar
  165. 165.
    Sola, J.: Cosmological constant and vacuum energy: old and new ideas. J. Phys. Conf. Ser. 453, 012015 (2013). [arXiv:1306.1527 [gr-qc]]CrossRefGoogle Scholar
  166. 166.
    Weinberg, D.H., White, M.: Dark enegy, review 27. In: Tanabashi, M., et al. (Particle Data Group), Review of Particle Physics, Phys. Rev. D 98, 010001 (2018)Google Scholar
  167. 167.
    Cabibbo, N.: Unitary symmetry and leptonic decays. Phys. Rev. Lett. 10, 531 (1963)ADSCrossRefGoogle Scholar
  168. 168.
    Kobayashi, M., Maskawa, K.: CP violation in the renormalizable theory of weak interaction. Prog. Theor. Phys. 49, 652 (1973)ADSCrossRefGoogle Scholar
  169. 169.
    Weinberg, S.: Baryon and Lepton nonconserving processes. Phys. Rev. Lett. 43, 1566 (1979)ADSCrossRefGoogle Scholar
  170. 170.
    Buchmüller, W., Wyler, D.: Effective lagrangian analysis of new interactions and flavor conservation. Nucl. Phys. B 268, 621 (1986)ADSCrossRefGoogle Scholar
  171. 171.
    Grzadkowski, B., Iskrzynski, M., Misiak, M., Rosiek, J.: Dimension-six terms in the standard model lagrangian. JHEP 1010, 085 (2010)ADSzbMATHCrossRefGoogle Scholar
  172. 172.
    Ade, P.A.R., et al.: [Planck Collaboration], Planck 2013 results. XXII. Constraints on inflation, Astron. Astrophys. 571, A22 (2014)Google Scholar
  173. 173.
    Lyth, D.H.: What would we learn by detecting a gravitational wave signal in the cosmic microwave background anisotropy? Phys. Rev. Lett. 78, 1861 (1997)ADSCrossRefGoogle Scholar
  174. 174.
    Aoki, H., Iso, S.: Revisiting the naturalness problem—who is afraid of quadratic divergences? Phys. Rev. D 86, 013001 (2012)ADSCrossRefGoogle Scholar
  175. 175.
    Blanke, M., Giudice, G.F., Paradisi, P., Perez, G., Zupan, J.: Flavoured naturalness. JHEP 1306, 022 (2013)ADSCrossRefGoogle Scholar
  176. 176.
    Tavares, G.M., Schmaltz, M., Skiba, W.: Higgs mass naturalness and scale invariance in the UV. Phys. Rev. D 89, 015009 (2014)ADSCrossRefGoogle Scholar
  177. 177.
    Masina, I., Quiros, M.: On the veltman condition, the hierarchy problem and high-scale supersymmetry. Phys. Rev. D 88, 093003 (2013)ADSCrossRefGoogle Scholar
  178. 178.
    Bian, L.: Renormalization group equation, the naturalness problem, and the understanding of the Higgs mass term. Phys. Rev. D 88, 056022 (2013)ADSCrossRefGoogle Scholar
  179. 179.
    Jegerlehner, F.: The SM as a low energy effective theory and the role of the Higgs in the early universe, Lectures at IFJ-PAN, Krakow, October 14–27, (2014),
  180. 180.
    Corning, P.: The re-emergence of emergence: a venerable concept in search of a theory. Complexity 7(6), 18–30 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  181. 181.
    Veltman, M.: The importance of radiative corrections, In: Radiative corrections: Results and Perspectives. Proceedings, NATO Advanced Research Workshop, Brighton, UK, July 10–14, 1989, Dombey, N., Boudjema, F. (eds.) NATO Sci. Ser. B 233, 1 (1990)Google Scholar
  182. 182.
    Geng, C.Q., Marshak, R.E.: Uniqueness of quark and lepton representations in the standard model from the anomalies viewpoint. Phys. Rev. D 39, 693 (1989)ADSCrossRefGoogle Scholar
  183. 183.
    Babu, K.S., Mohapatra, R.N.: Is there a connection between quantization of electric charge and a majorana neutrino? Phys. Rev. Lett. 63, 938 (1989)ADSCrossRefGoogle Scholar
  184. 184.
    Minahan, J.A., Ramond, P., Warner, R.C.: A comment on anomaly cancellation in the standard model. Phys. Rev. D 41, 715 (1990)ADSCrossRefGoogle Scholar
  185. 185.
    Rudaz, S.: Electric charge quantization in the standard model. Phys. Rev. D 41, 2619 (1990)ADSCrossRefGoogle Scholar
  186. 186.
    Jegerlehner, F.: Renormalizing the standard model. In: Cvetic, M., Langacker, P., Teaneck, N.J. (eds.) Testing the Standard Model—TASI-90 Proceedings, p. 916. World Scientific, Singapore (1991)Google Scholar

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Authors and Affiliations

  1. 1.Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Deutsches Elektronen-Synchrotron (DESY)ZeuthenGermany

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