Higgs mass and vacuum stability in the Standard Model at NNLO

  • Giuseppe Degrassi
  • Stefano Di Vita
  • Joan Elias-Miró
  • José R. Espinosa
  • Gian F. Giudice
  • Gino Isidori
  • Alessandro Strumia
Open Access


We present the first complete next-to-next-to-leading order analysis of the Standard Model Higgs potential. We computed the two-loop QCD and Yukawa corrections to the relation between the Higgs quartic coupling (λ) and the Higgs mass (Mh), reducing the theoretical uncertainty in the determination of the critical value of Mh for vacuum stability to 1 GeV. While λ at the Planck scale is remarkably close to zero, absolute stability of the Higgs potential is excluded at 98 % C.L. for Mh< 126 GeV. Possible consequences of the near vanishing of λ at the Planck scale, including speculations about the role of the Higgs field during inflation, are discussed.


Higgs Physics Standard Model Beyond Standard Model 


  1. [1]
    ATLAS collaboration, G. Aad et al., Combined search for the standard model Higgs boson using up to 4.9 fb −1 of pp collision data at \(\sqrt {s} = {7}\;TeV\) with the ATLAS detector at the LHC, Phys. Lett. B 710 (2012) 49 [arXiv:1202.1408] [INSPIRE].ADSGoogle Scholar
  2. [2]
    CMS collaboration, S. Chatrchyan et al., Combined results of searches for the standard model Higgs boson in pp collisions at \(\sqrt {s} = {7}\;TeV\), Phys. Lett. B 710 (2012) 26 [arXiv:1202.1488] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Bounds on the fermions and Higgs boson masses in grand unified theories, Nucl. Phys. B 158 (1979) 295 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    P.Q. Hung, Vacuum instability and new constraints on fermion masses, Phys. Rev. Lett. 42 (1979) 873 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    M. Lindner, Implications of triviality for the standard model, Z. Phys. C 31 (1986) 295 [INSPIRE].ADSGoogle Scholar
  6. [6]
    M. Sher, Electroweak Higgs potentials and vacuum stability, Phys. Rept. 179 (1989) 273 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    B. Schrempp and M. Wimmer, Top quark and Higgs boson masses: Interplay between infrared and ultraviolet physics, Prog. Part. Nucl. Phys. 37 (1996) 1 [hep-ph/9606386] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    G. Altarelli and G. Isidori, Lower limit on the Higgs mass in the standard model: An Update., Phys. Lett. B 337 (1994) 141 [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    J. Casas, J. Espinosa and M. Quirós, Improved Higgs mass stability bound in the standard model and implications for supersymmetry, Phys. Lett. B 342 (1995) 171 [hep-ph/9409458] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    J. Casas, J. Espinosa and M. Quirós, Standard model stability bounds for new physics within LHC reach, Phys. Lett. B 382 (1996) 374 [hep-ph/9603227] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    G. Isidori, G. Ridolfi and A. Strumia, On the metastability of the standard model vacuum, Nucl. Phys. B 609 (2001) 387 [hep-ph/0104016] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    C. Burgess, V. Di Clemente and J. Espinosa, Effective operators and vacuum instability as heralds of new physics, JHEP 01 (2002) 041 [hep-ph/0201160] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    G. Isidori, V.S. Rychkov, A. Strumia and N. Tetradis, Gravitational corrections to standard model vacuum decay, Phys. Rev. D 77 (2008) 025034 [arXiv:0712.0242] [INSPIRE].ADSGoogle Scholar
  14. [14]
    N. Arkani-Hamed, S. Dubovsky, L. Senatore and G. Villadoro, (No) eternal inflation and precision Higgs physics, JHEP 03 (2008) 075 [arXiv:0801.2399] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    F. Bezrukov and M. Shaposhnikov, Standard model Higgs boson mass from inflation: two loop analysis, JHEP 07 (2009) 089 [arXiv:0904.1537] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    L.J. Hall and Y. Nomura, A finely-predicted Higgs boson mass from a finely-tuned weak scale, JHEP 03 (2010) 076 [arXiv:0910.2235] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    J. Ellis, J. Espinosa, G. Giudice, A. Hoecker and A. Riotto, The probable fate of the standard model, Phys. Lett. B 679 (2009) 369 [arXiv:0906.0954] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    J. Elias-Miro et al., Higgs mass implications on the stability of the electroweak vacuum, Phys. Lett. B 709 (2012) 222 [arXiv:1112.3022] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    L.N. Mihaila, J. Salomon and M. Steinhauser, Gauge coupling β-functions in the standard model to three loops, Phys. Rev. Lett. 108 (2012) 151602 [arXiv:1201.5868] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    K. Chetyrkin and M. Zoller, Three-loop β-functions for top-Yukawa and the Higgs self-interaction in the standard model, JHEP 06 (2012) 033 [arXiv:1205.2892] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    F. Bezrukov, M.Y. Kalmykov, B.A. Kniehl and M. Shaposhnikov, Higgs boson mass and new physics, arXiv:1205.2893 [INSPIRE].
  22. [22]
    D. Bennett, H.B. Nielsen and I. Picek, Understanding fine structure constants and three generations, Phys. Lett. B 208 (1988) 275 [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    C. Froggatt and H.B. Nielsen, Standard model criticality prediction: top mass 173 ± 5 GeV and Higgs mass 135 ± 9 GeV, Phys. Lett. B 368 (1996) 96 [hep-ph/9511371] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M. Shaposhnikov and C. Wetterich, Asymptotic safety of gravity and the Higgs boson mass, Phys. Lett. B 683 (2010) 196 [arXiv:0912.0208] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    M. Holthausen, K.S. Lim and M. Lindner, Planck scale boundary conditions and the Higgs mass, JHEP 02 (2012) 037 [arXiv:1112.2415] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    I. Masina and A. Notari, The Higgs mass range from standard model false vacuum inflation in scalar-tensor gravity, Phys. Rev. D 85 (2012) 123506 [arXiv:1112.2659] [INSPIRE].ADSGoogle Scholar
  27. [27]
    I. Masina and A. Notari, Inflation from the Higgs field false vacuum with hybrid potential, arXiv:1204.4155 [INSPIRE].
  28. [28]
    C. Ford, I. Jack and D. Jones, The standard model effective potential at two loops, Nucl. Phys. B 387 (1992) 373 [Erratum ibid. B 504 (1997) 551-552] [hep-ph/0111190] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    S.P. Martin, Two loop effective potential for a general renormalizable theory and softly broken supersymmetry, Phys. Rev. D 65 (2002) 116003 [hep-ph/0111209] [INSPIRE].ADSGoogle Scholar
  30. [30]
    S.P. Martin, Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings, Phys. Rev. D 70 (2004) 016005 [hep-ph/0312092] [INSPIRE].ADSGoogle Scholar
  31. [31]
    S.P. Martin, Evaluation of two loop self energy basis integrals using differential equations, Phys. Rev. D 68 (2003) 075002 [hep-ph/0307101] [INSPIRE].ADSGoogle Scholar
  32. [32]
    A. Sirlin and R. Zucchini, Dependence of the quartic coupling H(m) on M (h) and the possible onset of new physics in the Higgs sector of the standard model, Nucl. Phys. B 266 (1986) 389 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    G. Degrassi and P. Slavich, NLO QCD bottom corrections to Higgs boson production in the MSSM, JHEP 11 (2010) 044 [arXiv:1007.3465] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    J. Fleischer, M.Y. Kalmykov and A. Kotikov, Two loop self energy master integrals on-shell, Phys. Lett. B 462 (1999) 169 [hep-ph/9905249] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    S. Bethke, The 2009 world average of α s, Eur. Phys. J. C 64 (2009) 689 [arXiv:0908.1135] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    Tevatron Electroweak Working Group, CDF and D0 collaboration, Combination of CDF and D0 results on the mass of the top quark using up to 5.8 fb −1 of data, arXiv:1107.5255 [INSPIRE].
  38. [38]
    CMS collaboration, Measurement of the top quark mass in the muon + jets channel, PAS-TOP-11-015 (2011).
  39. [39]
    ALTAS collaboration, Top quark mass measurements at the ATLAS experiment, PHYS-SLIDE-2012-106 (2012).
  40. [40]
    D.J. Broadhurst, N. Gray and K. Schilcher, Gauge invariant on-shell Z 2 in QED, QCD and the effective field theory of a static quark, Z. Phys. C 52 (1991) 111 [INSPIRE].MathSciNetADSGoogle Scholar
  41. [41]
    K. Melnikov and T.v. Ritbergen, The three loop relation between the M S-bar and the pole quark masses, Phys. Lett. B 482 (2000) 99 [hep-ph/9912391] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    K. Chetyrkin and M. Steinhauser, The relation between the MS-bar and the on-shell quark mass at order α s, Nucl. Phys. B 573 (2000) 617 [hep-ph/9911434] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    R. Hempfling and B.A. Kniehl, On the relation between the fermion pole mass and MS Yukawa coupling in the standard model, Phys. Rev. D 51 (1995) 1386 [hep-ph/9408313] [INSPIRE].ADSGoogle Scholar
  44. [44]
    F. Jegerlehner and M.Y. Kalmykov, O(αα s) correction to the pole mass of the t quark within the standard model, Nucl. Phys. B 676 (2004) 365 [hep-ph/0308216] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    A.H. Hoang and I.W. Stewart, Top mass measurements from jets and the Tevatron top-quark mass, Nucl. Phys. Proc. Suppl. 185 (2008) 220 [arXiv:0808.0222] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    S. Fleming, A. H. Hoang, S. Mantry and I.W. Stewart, Vacuum instability and new constraints on fermion masses, Phys. Rev. D 77 (2008) 074010 [hep-ph/0703207] [INSPIRE]ADSGoogle Scholar
  47. [47]
    S. Moch, P. Uwer and A. Vogt, On top-pair hadro-production at next-to-next-to-leading order, Phys. Lett. B 714 (2012) 48 [arXiv:1203.6282] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    S. Alekhin, A. Djouadi and S. Moch, The top quark and Higgs boson masses and the stability of the electroweak vacuum, arXiv:1207.0980 [INSPIRE].
  49. [49]
    M. Veltman, The infrared-ultraviolet connection, Acta Phys. Polon. B 12 (1981) 437 [INSPIRE].Google Scholar
  50. [50]
    F. Bezrukov and M. Shaposhnikov, The standard model Higgs boson as the inflaton, Phys. Lett. B 659 (2008) 703 [arXiv:0710.3755] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    C. Burgess, H.M. Lee and M. Trott, Power-counting and the validity of the classical approximation during inflation, JHEP 09 (2009) 103 [arXiv:0902.4465] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    J. Barbon and J. Espinosa, On the naturalness of Higgs inflation, Phys. Rev. D 79 (2009) 081302 [arXiv:0903.0355] [INSPIRE].ADSGoogle Scholar
  53. [53]
    R.N. Lerner and J. McDonald, A unitarity-conserving Higgs inflation model, Phys. Rev. D 82 (2010) 103525 [arXiv:1005.2978] [INSPIRE].ADSGoogle Scholar
  54. [54]
    G.F. Giudice and H.M. Lee, Unitarizing Higgs inflation, Phys. Lett. B 694 (2011) 294 [arXiv:1010.1417] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    A. De Simone, M.P. Hertzberg and F. Wilczek, Running inflation in the standard model, Phys. Lett. B 678 (2009) 1 [arXiv:0812.4946] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    J. Elias-Miro, J.R. Espinosa, G.F. Giudice, H.M. Lee and A. Strumia, Stabilization of the electroweak vacuum by a scalar threshold effect, JHEP 06 (2012) 031 [arXiv:1203.0237] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    G.F. Giudice and A. Strumia, Probing high-scale and split supersymmetry with Higgs mass measurements, Nucl. Phys. B 858 (2012) 63 [arXiv:1108.6077] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    M. Cabrera, J. Casas and A. Delgado, Upper bounds on superpartner masses from upper bounds on the Higgs boson mass, Phys. Rev. Lett. 108 (2012) 021802 [arXiv:1108.3867] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    N. Arkani-Hamed and S. Dimopoulos, Supersymmetric unification without low energy supersymmetry and signatures for fine-tuning at the LHC, JHEP 06 (2005) 073 [hep-th/0405159] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    G. Giudice and A. Romanino, Split supersymmetry, Nucl. Phys. B 699 (2004) 65 [Erratum ibid. B 706 (2005) 65-89] [hep-ph/0406088] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    N. Arkani-Hamed, S. Dimopoulos, G. Giudice and A. Romanino, Aspects of split supersymmetry, Nucl. Phys. B 709 (2005) 3 [hep-ph/0409232] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    G.F. Giudice, M.A. Luty, H. Murayama and R. Rattazzi, Gaugino mass without singlets, JHEP 12 (1998) 027 [hep-ph/9810442] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    J.D. Wells, PeV-scale supersymmetry, Phys. Rev. D 71 (2005) 015013 [hep-ph/0411041] [INSPIRE].ADSGoogle Scholar
  64. [64]
    N. Arkani-Hamed, A. Delgado and G. Giudice, The well-tempered neutralino, Nucl. Phys. B 741 (2006) 108 [hep-ph/0601041] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    G. Giudice and R. Rattazzi, Living dangerously with low-energy supersymmetry, Nucl. Phys. B 757 (2006) 19 [hep-ph/0606105] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    Self-organized criticality: an explanation of 1/f noise, Phys. Rev. Lett. 59 (1987) 381 [INSPIRE].
  67. [67]
    S.R. Coleman and E.J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7 (1973) 1888 [INSPIRE].ADSGoogle Scholar

Copyright information

© SISSA 2012

Authors and Affiliations

  • Giuseppe Degrassi
    • 1
  • Stefano Di Vita
    • 1
  • Joan Elias-Miró
    • 2
  • José R. Espinosa
    • 2
    • 3
  • Gian F. Giudice
    • 4
  • Gino Isidori
    • 4
    • 5
  • Alessandro Strumia
    • 6
    • 7
  1. 1.Dipartimento di FisicaUniversità di Roma Tre and INFN — Sezione di Roma TreRomaItaly
  2. 2.IFAEUniversitat Autónoma de BarcelonaBarcelonaSpain
  3. 3.ICREA, Instituciò Catalana de Recerca i Estudis AvançatsBarcelonaSpain
  4. 4.CERN, Theory DivisionGeneva 23Switzerland
  5. 5.INFN, Laboratori Nazionali di FrascatiFrascatiItaly
  6. 6.Dipartimento di FisicaUniversità di Pisa and INFN — Sezione di PisaPisaItaly
  7. 7.National Institute of Chemical Physics and BiophysicsTallinnEstonia

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