Skip to main content
SpringerLink
Log in

Renormalization of vacuum expectation values in spontaneously broken gauge theories

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We compute one-loop and two-loop β-functions for vacuum expectation values (VEVs) in gauge theories. In R ξ gauge the VEVs renormalize differently from the respective scalar fields. We focus particularly on the origin and behaviour of this difference and show that it can be interpreted as the anomalous dimension of a certain scalar background field, leading to simple direct computation and qualitative understanding. The results are given for generic as well as supersymmetric gauge theories. These complement the set of well-known γ- and β-functions of Machacek/Vaughn. As an application, we compute the β-functions for VEVs and tan β in the MSSM, NMSSM, and E6SSM.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. ALEPH, DELPHI, L3, OPAL, SLD collaboration, LEP Electroweak Working, SLD Electroweak, SLD Heavy Flavour groups, S. Schael et al., Precision electroweak measurements on the Z resonance, Phys. Rept. 427 (2006) 257 [hep-ex/0509008] [INSPIRE].

    ADS  Google Scholar 

  2. ATLAS collaboration, 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 (2012) 1 [arXiv:1207.7214] [INSPIRE].

    ADS  Google Scholar 

  3. CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].

    ADS  Google Scholar 

  4. G. ’t Hooft, Renormalizable Lagrangians for Massive Yang-Mills Fields, Nucl. Phys. B 35 (1971) 167 [INSPIRE].

    Article  ADS  Google Scholar 

  5. G. ’t Hooft and M. Veltman, Combinatorics of gauge fields, Nucl. Phys. B 50 (1972) 318 [INSPIRE].

    Article  MathSciNet  Google Scholar 

  6. B. Lee and J. Zinn-Justin, Spontaneously Broken Gauge Symmetries. 1. Preliminaries, Phys. Rev. D 5 (1972) 3121 [INSPIRE].

    ADS  Google Scholar 

  7. B. Lee and J. Zinn-Justin, Spontaneously broken gauge symmetries. 2. Perturbation theory and renormalization, Phys. Rev. D 5 (1972) 3137 [Erratum ibid. D 8 (1973) 4654] [INSPIRE].

  8. B. Lee and J. Zinn-Justin, Spontaneously broken gauge symmetries. 3. Equivalence, Phys. Rev. D 5 (1972) 3155 [INSPIRE].

    ADS  Google Scholar 

  9. C. Becchi, A. Rouet and R. Stora, The Abelian Higgs-Kibble Model. Unitarity of the S Operator, Phys. Lett. B 52 (1974) 344 [INSPIRE].

    ADS  Google Scholar 

  10. C. Becchi, A. Rouet and R. Stora, Renormalization of the Abelian Higgs-Kibble Model, Commun. Math. Phys. 42 (1975) 127 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. C. Becchi, A. Rouet and R. Stora, Renormalization of Gauge Theories, Annals Phys. 98 (1976) 287 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. E. Kraus, The Renormalization of the electroweak standard model to all orders, Annals Phys. 262 (1998) 155 [hep-th/9709154] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  13. E. Kraus and S. Groot Nibbelink, Renormalization of the electroweak standard model, hep-th/9809069 [INSPIRE].

  14. P.A. Grassi, The Abelian anti-ghost equation for the standard model in the ’t Hooft background gauge, Nucl. Phys. B 537 (1999) 527 [hep-th/9804013] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. P.A. Grassi, Renormalization of nonsemisimple gauge models with the background field method, Nucl. Phys. B 560 (1999) 499 [hep-th/9908188] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. W. Hollik, E. Kraus, M. Roth, C. Rupp, K. Sibold and D. Stöckinger, Renormalization of the minimal supersymmetric standard model, Nucl. Phys. B 639 (2002) 3 [hep-ph/0204350] [INSPIRE].

    Article  ADS  Google Scholar 

  17. E. Kraus and K. Sibold, Rigid invariance as derived from BRS invariance: The Abelian Higgs model, Z. Phys. C 68 (1995) 331 [hep-th/9503140] [INSPIRE].

    ADS  Google Scholar 

  18. B.S. DeWitt, Quantum Theory of Gravity. 3. Applications of the Covariant Theory, Phys. Rev. 162 (1967) 1239 [INSPIRE].

    Article  ADS  Google Scholar 

  19. H. Kluberg-Stern and J. Zuber, Renormalization of Nonabelian Gauge Theories in a Background Field Gauge. 1. Green Functions, Phys. Rev. D 12 (1975) 482 [INSPIRE].

    ADS  Google Scholar 

  20. H. Kluberg-Stern and J. Zuber, Renormalization of Nonabelian Gauge Theories in a Background Field Gauge. 2. Gauge Invariant Operators, Phys. Rev. D 12 (1975) 3159 [INSPIRE].

    ADS  Google Scholar 

  21. A. Dabelstein, The One loop renormalization of the MSSM Higgs sector and its application to the neutral scalar Higgs masses, Z. Phys. C 67 (1995) 495 [hep-ph/9409375] [INSPIRE].

    ADS  Google Scholar 

  22. P.H. Chankowski, S. Pokorski and J. Rosiek, Complete on-shell renormalization scheme for the minimal supersymmetric Higgs sector, Nucl. Phys. B 423 (1994) 437 [hep-ph/9303309] [INSPIRE].

    Article  ADS  Google Scholar 

  23. Y. Yamada, Two loop renormalization of tan beta and its gauge dependence, Phys. Lett. B 530 (2002) 174 [hep-ph/0112251] [INSPIRE].

    ADS  Google Scholar 

  24. P. Athron, D. Stöckinger and A. Voigt, Threshold Corrections in the Exceptional Supersymmetric Standard Model, Phys. Rev. D 86 (2012) 095012 [arXiv:1209.1470] [INSPIRE].

    ADS  Google Scholar 

  25. M.E. Machacek and M.T. Vaughn, Two Loop Renormalization Group Equations in a General Quantum Field Theory. 1. Wave Function Renormalization, Nucl. Phys. B 222 (1983) 83 [INSPIRE].

    Article  ADS  Google Scholar 

  26. M.E. Machacek and M.T. Vaughn, Two Loop Renormalization Group Equations in a General Quantum Field Theory. 2. Yukawa Couplings, Nucl. Phys. B 236 (1984) 221 [INSPIRE].

    Article  ADS  Google Scholar 

  27. M.E. Machacek and M.T. Vaughn, Two Loop Renormalization Group Equations in a General Quantum Field Theory. 3. Scalar Quartic Couplings, Nucl. Phys. B 249 (1985) 70 [INSPIRE].

    Article  ADS  Google Scholar 

  28. S.P. Martin and M.T. Vaughn, Two loop renormalization group equations for soft supersymmetry breaking couplings, Phys. Rev. D 50 (1994) 2282 [Erratum ibid. D 78 (2008) 039903] [hep-ph/9311340] [INSPIRE].

  29. Y. Yamada, Two loop renormalization group equations for soft SUSY breaking scalar interactions: Supergraph method, Phys. Rev. D 50 (1994) 3537 [hep-ph/9401241] [INSPIRE].

    ADS  Google Scholar 

  30. I. Jack and D. Jones, Soft supersymmetry breaking and finiteness, Phys. Lett. B 333 (1994) 372 [hep-ph/9405233] [INSPIRE].

    ADS  Google Scholar 

  31. A.D. Box and X. Tata, Threshold and flavor effects in the renormalization group equations of the MSSM: Dimensionless couplings, Phys. Rev. D 77 (2008) 055007 [Erratum ibid. D 82 (2010) 119904] [arXiv:0712.2858] [INSPIRE].

  32. M.-x. Luo, H.-w. Wang and Y. Xiao, Two loop renormalization group equations in general gauge field theories, Phys. Rev. D 67 (2003) 065019 [hep-ph/0211440] [INSPIRE].

    ADS  Google Scholar 

  33. O. Piguet and S. Sorella, Algebraic renormalization: Perturbative renormalization, symmetries and anomalies, Lect. Notes Phys. M 28 (1995) 1.

    MathSciNet  Google Scholar 

  34. F. Brandt, N. Dragon and M. Kreuzer, All consistent Yang-Mills anomalies, Phys. Lett. B 231 (1989) 263 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  35. R. Peccei and H.R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 (1977) 1440 [INSPIRE].

    Article  ADS  Google Scholar 

  36. D. Capper and D. Jones, The effective potential in the light cone gauge and supersymmetric Yang-Mills theories, Nucl. Phys. B 252 (1985) 718 [INSPIRE].

    Article  ADS  Google Scholar 

  37. H.E. Haber and G.L. Kane, The Search for Supersymmetry: Probing Physics Beyond the Standard Model, Phys. Rept. 117 (1985) 75 [INSPIRE].

    Article  ADS  Google Scholar 

  38. A. Freitas and D. Stöckinger, Gauge dependence and renormalization of tan beta in the MSSM, Phys. Rev. D 66 (2002) 095014 [hep-ph/0205281] [INSPIRE].

    ADS  Google Scholar 

  39. N. Baro, F. Boudjema and A. Semenov, Automatised full one-loop renormalisation of the MSSM. I. The Higgs sector, the issue of tan β and gauge invariance, Phys. Rev. D 78 (2008) 115003 [arXiv:0807.4668] [INSPIRE].

    ADS  Google Scholar 

  40. J.R. Ellis, J. Gunion, H.E. Haber, L. Roszkowski and F. Zwirner, Higgs Bosons in a Nonminimal Supersymmetric Model, Phys. Rev. D 39 (1989) 844 [INSPIRE].

    ADS  Google Scholar 

  41. S. King, S. Moretti and R. Nevzorov, Theory and phenomenology of an exceptional supersymmetric standard model, Phys. Rev. D 73 (2006) 035009 [hep-ph/0510419] [INSPIRE].

    ADS  Google Scholar 

  42. R.M. Fonseca, M. Malinsky, W. Porod and F. Staub, Running soft parameters in SUSY models with multiple U(1) gauge factors, Nucl. Phys. B 854 (2012) 28 [arXiv:1107.2670] [INSPIRE].

    Article  ADS  Google Scholar 

  43. F. Staub, Automatic Calculation of supersymmetric Renormalization Group Equations and Self Energies, Comput. Phys. Commun. 182 (2011) 808 [arXiv:1002.0840] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  44. F. Staub, T. Ohl, W. Porod and C. Speckner, A Tool Box for Implementing Supersymmetric Models, Comput. Phys. Commun. 183 (2012) 2165 [arXiv:1109.5147] [INSPIRE].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Kern- und Teilchenphysik, TU Dresden, Dresden, Germany

    Marcus Sperling, Dominik Stöckinger & Alexander Voigt

Authors
  1. Marcus Sperling
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominik Stöckinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander Voigt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Voigt.

Additional information

ArXiv ePrint: 1305.1548

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sperling, M., Stöckinger, D. & Voigt, A. Renormalization of vacuum expectation values in spontaneously broken gauge theories. J. High Energ. Phys. 2013, 132 (2013). https://doi.org/10.1007/JHEP07(2013)132

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2013)132

Keywords

Search

Navigation