Reduction in the Number of Coupling Parameters

  • W. Zimmermann

Abstract

A method is developed for reducing the formulation of massless models with several independent couplings to a description in terms of a single coupling parameter. The original as well as the reduced system are supposed to be renormalizable and invariant under the renormalization group. For most models there are, if any, only a finite number of reductions possible including those which correspond to symmetries of the system. The reduction method leads to a consistent formulation of the reduced model in any order of perturbation theory even in cases where it is difficult to establish a symmetry in higher orders. An example where no symmetry seems to be involved is the interaction of a spinor field with a pseudoscalar field. For this model the reduction method determines the quartic coupling constant uniquely as a function of the Yukawa coupling constant. The Wess-Zumino model is an exceptional case for which the reduction method admits an infinite number of solutions besides the supersymmetric one.

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References

  1. 1.
    Callan, C.: Broken scale invariance in scalar field theory. Phys. Rev. D2, 1541–1547 (1970)ADSCrossRefGoogle Scholar
  2. 2.
    Symanzik, K.: Small distance behavior in field theory and power counting. Commun. Math. Phys. 18, 227–246 (1970)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Mandelstam, S.: Light-cone superspace and the ultraviolet finiteness of the N = 4 model. Nucl. Phys. B213, 149–168 (1983)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Chang, N.P.: Eigenvalue conditions and asymptotic freedom for Higgs scalar gauge theories. Phys. Rev. D10, 2706–2709 (1974)ADSGoogle Scholar
  5. 5.
    Chang, N.P., Das, A., Perez-Mercader, J.: Asymptotically free SU(5) model with three generations. Phys. Rev. D22, 1829–1832 (1980)ADSGoogle Scholar
  6. 6.
    Oehme, R., Zimmermann, W.: Relation between effective couplings for asymptotically free models. Commun. Math. Phys. (to appear)Google Scholar
  7. 7.
    Oehme, R., Sibold, K., Zimmermann, W.: Renormalization group equations with vanishing lowest order of the primary β-function (to be published in Phys. Lett.)Google Scholar
  8. 8.
    Gross, D., Wilczek, F.: Asymptotically free gauge theories. I. Phys. Rev. D 8, 3633–3652 (1973)ADSGoogle Scholar
  9. 9.
    Wess, J., Zumino, B.: A Lagrangian model invariant under supergauge transformations. Phys. Lett. 49 B, 52–54 (1974)Google Scholar
  10. 10.
    Clark, T., Piguet, O., Sibold, K.: Supercurrents, renormalization and anomalies. Nucl. Phys. B143, 445–484 (1978)ADSCrossRefGoogle Scholar
  11. 11.
    Maison, D.: Determination of correct β-functions for super Yang-Mills theories using a supersymmetry violating renormalization scheme. Preprint Werner-Heisenberg-Institut fur Physik MPI/PTh 43/84 (to be published)Google Scholar
  12. 12.
    Suzuki, M.: On instability of asymptotic freedom of supergauge Yang-Mills theories. Nucl. Phys. B83, 269–275 (1974)ADSCrossRefGoogle Scholar
  13. 13.
    Zimmermann, W.: The renormalization group of the model of A4-coupling in the abstract approach of quantum field theory. Commun. Math. Phys. 76, 39–64 (1980)ADSCrossRefGoogle Scholar
  14. 14.
    Oehme, R., Zimmermann, W.: Analyticity of effective coupling and propagators in massless models of quantum field theory. Commun. Math. Phys. 85, 363–379 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Symanzik, K.: On some massless superrenormalizable and non-renormalizable theories. In: Lecture Notes in Physics, Vol. 39, pp. 101–106, H. Araki (ed.). Berlin, Heidelberg, New York: Springer 1975Google Scholar
  16. 16.
    Bandelloni, G., Becchi, C., Blasi, A., Collina, R.: Renormalization of models with radiative mass generation. Commun. Math. Phys. 67, 147–178 (1978)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Piguet, O., Sibold, K.: Renormalizing supersymmetry without auxiliary fields. Preprint Université de Genève, UGYA-DPT 1984/10-443 (to be published)Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1985

Authors and Affiliations

  • W. Zimmermann
    • 1
  1. 1.Max-Planck-Institut für Physik und Astrophysik, Werner-Heisenberg-Institut für PhysikMünchen 40Federal Republic of Germany

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