Foundations of Physics

, Volume 44, Issue 9, pp 932–952 | Cite as

Standard Model Gauge Couplings from Gauge-Dilatation Symmetry Breaking

  • Kosuke Odagiri


It is well known that the self-energy of the gauge bosons is quadratically divergent in the Standard Model when a simple cutoff is imposed. We demonstrate phenomenologically that the quadratic divergences in fact unify. The unification occurs at a surprisingly low scale, \(\Lambda _\mathrm {u}\approx 4\times 10^7\) GeV. Suppose now that there is a spontaneously broken rotational symmetry between the space-time coordinates and gauge theoretical phases. The symmetry-breaking pattern is such that the gauge bosons arise as the massless Goldstone bosons, whereas the dilatonic mode acts as the massive (Higgs) boson, whose vacuum expectation value determines the gauge couplings. In this case, the quadratic divergences or the tadpoles of the gauge boson self-energy should indeed unify because these divergences need to be cancelled by a universal dilatonic contribution, assuming dynamical symmetry breaking. If there is dynamical symmetry breaking, we are in principle able to calculate the value of the gauge couplings as well as the scale hierarchy \(\Lambda _\mathrm {cut}/\Lambda _\mathrm {u}\). We perform this calculation by adopting a naive quartic symmetry-breaking potential which unfortunately violates local gauge invariance. Using tadpole-cancellation and dilatonic self-energy conditions, the value of \(\Lambda _\mathrm {cut}\) is then found to be approximately \(4\times 10^{18}\) GeV in the Feynman gauge and \(5\times 10^{15}\) GeV in the Landau gauge. The cancellation of an anomaly in the dilaton self-energy requires that the number of fermionic generations equals three. The symmetry-breaking needs to be driven by some other mass-generating mechanism such as electroweak symmetry breaking. Our estimation for \(\Lambda _\mathrm {u}\) is of the correct order if \(\Lambda _\mathrm {cut}\approx 5\times 10^{15}\) GeV.


Standard model Dynamical symmetry breaking Gauge principle Kaluza–Klein theory 



The author thanks B. R. Webber, whose insightful remarks dispelled the confusion regarding cancellations that was present in an earlier version of this work. The author also thanks the members of his research group for their encouragement. It would not have been possible to work on this topic without their generosity and support.


  1. 1.
    Nambu, Y.: New Theories in Physics. In: Ajduk, Z., Pokorski, S., Trautman, A. (eds.) Proceedings of the XI International Symposium on Elementary Particle Physics, Kazimierz, Poland, 1988, pp. 1–10. World Scientific, Singapore (1990)Google Scholar
  2. 2.
    Bardeen, W.A., Hill, C.T., Lindner, M.: Phys. Rev. D 41, 1647 (1990)ADSCrossRefGoogle Scholar
  3. 3.
    Cvetič, G.: Rev. Mod. Phys. 71, 513 (1999). arXiv:hep-ph/9702381
  4. 4.
    Appelquist, T., Chodos, A., Freund, P.G.O.: Modern Kaluza–Klein Theories. Addison-Wesley, California (1987)Google Scholar
  5. 5.
    Coleman, S.: Chapter 3 of Aspects of Symmetry. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
  6. 6.
    Gribov, V.N.: Phys. Lett. B 336, 243 (1994). arXiv:hep-ph/9407269
  7. 7.
    Odagiri, K.: arXiv:1112.6036.
  8. 8.
    Das, D., Odagiri, K.: Phys. Rev. D 86, 055019 (2012). arXiv:1201.3968
  9. 9.
    Giudice, G.F., Rattazzi, R., Wells, J.D.: Nucl. Phys. B544, 3 (1999). arXiv:hep-ph/9811291
  10. 10.
    Han, T., Lykken, J.D., Zhang, R.-J.: Phys. Rev. D59, 105006 (1999). arXiv:hep-ph/9811350
  11. 11.
    Georgi, H., Glashow, S.L.: Phys. Rev. Lett. 32, 438 (1974)ADSCrossRefGoogle Scholar
  12. 12.
    Arason, H., et al.: Phys. Rev. D 46, 3945 (1992)ADSCrossRefGoogle Scholar
  13. 13.
    Georgi, H., Quinn, H.R., Weinberg, S.: Phys. Rev. Lett. 33, 451 (1974)ADSCrossRefGoogle Scholar
  14. 14.
    Hagiwara, K., et al. (Particle Data Group): Phys. Rev. D 66, 010001 (2002)Google Scholar
  15. 15.
    J. Beringer et al. (Particle Data Group): Phys. Rev. D 86 (2012) 010001; 2013 online updates at

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Electronics and Photonics Research InstituteNational Institute of Advanced Industrial Science and TechnologyTsukubaJapan

Personalised recommendations