Journal of Mathematical Sciences

, Volume 238, Issue 6, pp 862–869 | Cite as

On Dimensional Regularization in the Yang–Mills Theory

  • A. V. IvanovEmail author

We suggest an asymptotic approach to renormalization in the case of dimensional regularization. As an example, the quantum Yang–Mills theory in the four-dimensional space-time is considered. A formula for the renormalized effective action is derived by using the asymptotic behavior of the bare coupling constant. Then we discuss the dimensional transmutation, the process of renormalization, and the properties of the coupling constant.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. ’t Hooft and M. Veltman, “Regularization and renormalization of gauge fields,” Nucl. Phys. B, 44, 189–213 (1972).MathSciNetCrossRefGoogle Scholar
  2. 2.
    C. G. Bollini and J. J. Giambiaggi, “Lowest order ‘divergent’ graphs in υ-dimensional space,” Phys. Lett. B, 40, 566–568 (1972).CrossRefGoogle Scholar
  3. 3.
    S. E. Derkachev, A. V. Ivanov, and L. D. Faddeev, “Renormalization scenario for the quantum Yang–Mills theory in four-dimensional space-time,” Theoret. Math. Phys., 192, No. 2, 1134–1140 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press (2002).Google Scholar
  5. 5.
    S. Weinberg, The Quantum Theory of Fields, Cambridge Univ. Press (1999).Google Scholar
  6. 6.
    N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, Wiley (1980).Google Scholar
  7. 7.
    L. D. Faddeev and A. A. Slavnov, Gauge Fields: Introduction to Quantum Theory, Perseus Books (1991).Google Scholar
  8. 8.
    L. D. Faddeev, “Scenario for the renormalization in the 4D Yang–Mills theory,” Internat. J. Modern Phys. A, 31, 1630001 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    L. D. Faddeev, “Notes on divergences and dimensional transmutation in Yang–Mills theory,” Theoret. Math. Phys., 148, No. 1, 986–994 (2006).CrossRefzbMATHGoogle Scholar
  10. 10.
    D. J. Gross, “Applications of the renormalization group to high-energy physics,” in: Methods in Field Theory: Les Houches Session XXVIII, World Scientific (1981), pp. 141–250.Google Scholar
  11. 11.
    F. Olness and R. Scalise, “Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman E&M,” Amer. J. Phys., 79, 306–312 (2011).CrossRefGoogle Scholar
  12. 12.
    A. V. Ivanov, “About renormalization of the Yang–Mills theory and the approach to calculation of the heat kernel,” EPJ Web Conf., 158, 07004 (2017).CrossRefGoogle Scholar
  13. 13.
    V. Fock, “Die Eigenzeit in der klassischen und in der Quantenmechanik,” Phys. Z. Sowjetunion, 12, 404–425 (1937).zbMATHGoogle Scholar
  14. 14.
    L. D. Faddeev, Mass in quantum Yang–Mills theory: comment on a Clay Millenium problem, arXiv:0911.1013 (2009).Google Scholar
  15. 15.
    I. Y. Arefeva, L. D. Faddeev, and A. A. Slavnov, “Generating functional for the S-matrix in gauge-invariant theories,” Theoret. Math. Phys., 21, No. 3, 1165–1172 (1974).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of Steklov Institute of MathematicsSt. PetersburgRussia

Personalised recommendations