Journal of High Energy Physics

, 2018:87 | Cite as

Renormalization of the multi-Higgs-doublet Standard Model and one-loop lepton mass corrections

  • W. GrimusEmail author
  • M. Löschner
Open Access
Regular Article - Theoretical Physics


Motivated by models for neutrino masses and lepton mixing, we consider the renormalization of the lepton sector of a general multi-Higgs-doublet Standard Model with an arbitrary number of right-handed neutrino singlets. We propose to make the theory finite by \( \overline{\mathrm{MS}} \) renormalization of the parameters of the unbroken theory. However, using a general Rξ gauge, in the explicit one-loop computations of one-point and two-point functions it becomes clear that — in addition — a renormalization of the vacuum expectation values (VEVs) is necessary. Moreover, in order to ensure vanishing one-point functions of the physical scalar mass eigenfields, finite shifts of the tree-level VEVs, induced by the finite parts of the tadpole diagrams, are required. As a consequence of our renormalization scheme, physical masses are functions of the renormalized parameters and VEVs and thus derived quantities. Applying our scheme to one-loop corrections of lepton masses, we perform a thorough discussion of finiteness and ξ-independence. In the latter context, the tadpole contributions figure prominently.


Beyond Standard Model Higgs Physics Neutrino Physics 


Open Access

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  1. [1]
    P. Minkowski, μeγ at a rate of one out of 109 muon decays?, Phys. Lett. B 67 (1977) 421 [INSPIRE].
  2. [2]
    T. Yanagida, Horizontal symmetry and masses of neutrinos, in Proceedings of “Workshop on unified theory and baryon number in the universe”, Tsukuba, Japan, (1979), O. Sawata and A. Sugamoto eds., KEK report 79-18, Tsukuba, Japan, (1979) [Conf. Proc. C 7902131 (1979) 95] [INSPIRE].
  3. [3]
    S.L. Glashow, The future of elementary particle physics, in Quarks and leptons, Proceedings of the Advanced Study Institute, Cargèse, Corsica, France, (1979), M. Lévy et al. eds., Plenum, New York, U.S.A., (1980) [NATO Sci. Ser. B 61 (1980) 687] [INSPIRE].
  4. [4]
    M. Gell-Mann, P. Ramond and R. Slansky, Complex spinors and unified theories, in Supergravity, D.Z. Freedman and F. van Nieuwenhuizen eds., North Holland, Amsterdam, The Netherlands, (1979) [Conf. Proc. C 790927 (1979) 315] [arXiv:1306.4669] [INSPIRE].
  5. [5]
    R.N. Mohapatra and G. Senjanović, Neutrino mass and spontaneous parity violation, Phys. Rev. Lett. 44 (1980) 912 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    W. Grimus and H. Neufeld, Radiative neutrino masses in an SU(2) × U(1) model, Nucl. Phys. B 325 (1989) 18 [INSPIRE].
  7. [7]
    A. Pilaftsis, Radiatively induced neutrino masses and large Higgs neutrino couplings in the Standard Model with Majorana fields, Z. Phys. C 55 (1992) 275 [hep-ph/9901206] [INSPIRE].
  8. [8]
    W. Grimus and L. Lavoura, One-loop corrections to the seesaw mechanism in the multi-Higgs-doublet Standard Model, Phys. Lett. B 546 (2002) 86 [hep-ph/0207229] [INSPIRE].
  9. [9]
    P.S.B. Dev and A. Pilaftsis, Minimal radiative neutrino mass mechanism for inverse seesaw models, Phys. Rev. D 86 (2012) 113001 [arXiv:1209.4051] [INSPIRE].
  10. [10]
    M. Fink and H. Neufeld, Neutrino masses in a conformal multi-Higgs-doublet model, arXiv:1801.10104 [INSPIRE].
  11. [11]
    W. Grimus and L. Lavoura, Soft lepton flavor violation in a multi Higgs doublet seesaw model, Phys. Rev. D 66 (2002) 014016 [hep-ph/0204070] [INSPIRE].
  12. [12]
    W. Grimus, L. Lavoura, O.M. Ogreid and P. Osland, A precision constraint on multi-Higgs-doublet models, J. Phys. G 35 (2008) 075001 [arXiv:0711.4022] [INSPIRE].
  13. [13]
    G.C. Branco, L. Lavoura and J.P. Silva, CP violation, Int. Ser. Monogr. Phys. 103 (1999) 1 [INSPIRE].
  14. [14]
    G. ’t Hooft, Renormalizable Lagrangians for massive Yang-Mills fields, Nucl. Phys. B 35 (1971) 167 [INSPIRE].
  15. [15]
    K. Fujikawa, B.W. Lee and A.I. Sanda, Generalized renormalizable gauge formulation of spontaneously broken gauge theories, Phys. Rev. D 6 (1972) 2923 [INSPIRE].
  16. [16]
    Y.-P. Yao, Quantization and gauge freedom in a theory with spontaneously broken symmetry, Phys. Rev. D 7 (1973) 1647 [INSPIRE].ADSGoogle Scholar
  17. [17]
    M. Sperling, D. Stöckinger and A. Voigt, Renormalization of vacuum expectation values in spontaneously broken gauge theories, JHEP 07 (2013) 132 [arXiv:1305.1548] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. Fox, W. Grimus and M. Löschner, Renormalization and radiative corrections to masses in a general Yukawa model, Int. J. Mod. Phys. A 33 (2018) 1850019 [arXiv:1705.09589] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    W. Grimus, P.O. Ludl and L. Nogués, Mass renormalization in a toy model with spontaneously broken symmetry, arXiv:1406.7795 [INSPIRE].
  20. [20]
    K.I. Aoki, Z. Hioki, M. Konuma, R. Kawabe and T. Muta, Electroweak theory. Framework of on-shell renormalization and study of higher order effects, Prog. Theor. Phys. Suppl. 73 (1982) 1 [INSPIRE].
  21. [21]
    W. Grimus and M. Löschner, Revisiting on-shell renormalization conditions in theories with flavor mixing, Int. J. Mod. Phys. A 31 (2017) 1630038 [Erratum ibid. A 32 (2017) 1792001] [arXiv:1606.06191] [INSPIRE].
  22. [22]
    J. Fleischer and F. Jegerlehner, Radiative corrections to Higgs decays in the extended Weinberg-Salam model, Phys. Rev. D 23 (1981) 2001 [INSPIRE].
  23. [23]
    F. Jegerlehner, Electroweak radiative corrections in the Higgs sector, in Proceedings of “Topical Conference on Radiative Corrections in SU(2)L × U(1), Trieste, Italy, (1983), pg. 237.Google Scholar
  24. [24]
    S. Weinberg, Perturbative calculations of symmetry breaking, Phys. Rev. D 7 (1973) 2887 [INSPIRE].
  25. [25]
    A. Denner, L. Jenniches, J.-N. Lang and C. Sturm, Gauge-independent \( \overline{MS} \) renormalization in the 2HDM, JHEP 09 (2016) 115 [arXiv:1607.07352] [INSPIRE].
  26. [26]
    M. Krause, R. Lorenz, M. Mühlleitner, R. Santos and H. Ziesche, Gauge-independent renormalization of the 2-Higgs-doublet model, JHEP 09 (2016) 143 [arXiv:1605.04853] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    M. Krause, D. Lopez-Val, M. Mühlleitner and R. Santos, Gauge-independent renormalization of the N2HDM, JHEP 12 (2017) 077 [arXiv:1708.01578] [INSPIRE].
  28. [28]
    M.P. Bento, H.E. Haber, J.C. Romão and J.P. Silva, Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds, JHEP 11 (2017) 095 [arXiv:1708.09408] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    I. Schur, Ein Satz über quadratische Formen mit komplexen Koeffizienten (in German), Amer. J. Math. 67 (1945) 472.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    A. Denner, H. Eck, O. Hahn and J. Küblbeck, Compact Feynman rules for Majorana fermions, Phys. Lett. B 291 (1992) 278 [INSPIRE].
  31. [31]
    V. Dūdėnas and T. Gajdosik, Gauge dependence of tadpole and mass renormalization for a seesaw extended 2HDM, Phys. Rev. D 98 (2018) 035034 [arXiv:1806.04675] [INSPIRE].
  32. [32]
    D. Aristizabal Sierra and C.E. Yaguna, On the importance of the 1-loop finite corrections to seesaw neutrino masses, JHEP 08 (2011) 013 [arXiv:1106.3587] [INSPIRE].
  33. [33]
    G. Passarino and M.J.G. Veltman, One loop corrections for e + e annihilation into μ + μ in the Weinberg model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].
  34. [34]
    M. Böhm, A. Denner and H. Joos, Gauge theories of the strong and electroweak interaction, Teubner, Stuttgart, Germany, (2001) [INSPIRE].
  35. [35]
    T. Hahn and M. Pérez-Victoria, Automatized one loop calculations in four-dimensions and D-dimensions, Comput. Phys. Commun. 118 (1999) 153 [hep-ph/9807565] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.University of Vienna, Faculty of PhysicsViennaAustria

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