Skip to main content
SpringerLink
Log in

Asymptotic behavior of the β function in the ϕ4 theory: A scheme without complex parameters

  • Order, Disorder, and Phase Transition in Condensed System
  • Published:
Journal of Experimental and Theoretical Physics Aims and scope Submit manuscript

Abstract

The previously-obtained analytical asymptotic expressions for the Gell-Mann-Low function β(g) and anomalous dimensions in the ϕ4 theory in the limit g → ∞ are based on the parametric representation of the form g = f(t), β(g) = f 1(t) (where t ∝ g −1/20 is the running parameter related to the bare charge g 0), which is simplified in the complex t plane near a zero of one of the functional integrals. In this work, it has been shown that the parametric representation has a singularity at t → 0; for this reason, similar results can be obtained for real g 0 values. The problem of the correct transition to the strong-coupling regime is simultaneously solved; in particular, the constancy of the bare or renormalized mass is not a correct condition of this transition. A partial proof has been given for the theorem of the renormalizability in the strong-coupling region.

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. I. M. Suslov, Zh. Éksp. Teor. Fiz. 134(3), 490 (2008) [JETP 107 (3), 413 (2008)].

    Google Scholar 

  2. I. M. Suslov, Zh. Éksp. Teor. Fiz. 135(6), 1129 (2009) [JETP 108 (6), 980 (2009)].

    Google Scholar 

  3. T. D. Lee, Phys. Rev. 95, 1329 (1954).

    Article  MATH  ADS  Google Scholar 

  4. G. Källen and W. Pauli, Mat.-Fyz. Medd.-K. Dan. Vidensk. Selsk. 30(7) (1955).

  5. C. M. Bender, S. F. Brandt, J.-H. Chen, and Q. Wang, Phys. Rev. D: Part. Fields 71, 025014 (2005).

    ADS  Google Scholar 

  6. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002); C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).

    Article  MathSciNet  Google Scholar 

  7. S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and Co., Evanston, Illinois, United States, 1961), Chap. 12.

    Google Scholar 

  8. G. Barton, Introduction to Advanced Field Theory (John, Willey, New York, United States, 1963), Chap. 12.

    MATH  Google Scholar 

  9. N. N. Bogolyubov and D. V. Shirkov, Quantum Fields (Nauka, Moscow, 1976; Benjamin/Cummings, Reading, Massachusetts, United States, 1982).

    Google Scholar 

  10. A. A. Pogorelov and I. M. Suslov, Zh. Éksp. Teor. Fiz. 133(6), 1277 (2008) [JETP 106 (6), 1118 (2008)].

    Google Scholar 

  11. A. A. Vladimirov and D. V. Shirkov, Usp. Fiz. Nauk 129(2), 407 (1979) [Sov. Phys.—Usp. 22 (10), 860 (1979)].

    MathSciNet  Google Scholar 

  12. G. A. Baker, Jr., B. G. Nickel, and D. I. Meiron, Phys. Rev. Lett. 36, 1351 (1976); Phys. Rev. B: Solid State 17, 1365 (1978); J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39, 95 (1977); Phys. Rev. B: Condens. Matter 21, 3976 (1980).

    Article  ADS  Google Scholar 

  13. C. Domb, in Phase Transitions and Critical Phenomena, Ed. by C. Domb and M. S. Green (Academic, New York, 1974), Vol. 3.

    Google Scholar 

  14. C. M. Bender, F. Cooper, G. S. Guralnik, and D. H. Sharp, Phys. Rev. D: Part. Fields 19, 1865 (1979).

    ADS  Google Scholar 

  15. C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D: Part. Fields 23, 2976 (1981); Phys. Rev. D: Part. Fields 23, 2999 (1981); Phys. Rev. D: Part. Fields 24, 2683 (1981).

    ADS  Google Scholar 

  16. P. Castoldi and C. Schomblond, Nucl. Phys. B 139, 269 (1978).

    Article  ADS  Google Scholar 

  17. R. Benzi, G. Martinelli, and G. Parisi, Nucl. Phys. B 135, 429 (1978).

    Article  ADS  Google Scholar 

  18. M. Frasca, arXiv:0909.2428.

  19. S. Ma, Modern Theory of Critical Phenomena (Benjamin/Cummings, Reading, Massachusetts, United States; Mir, Moscow, 1980).

    Google Scholar 

  20. E. Brezin, J. C. Le Guillou, and J. Zinn-Justin, in Phase Transitions and Critical Phenomena, Ed. by C. Domb and M. S. Green (Academic, New York, 1976), Vol. VI.

    Google Scholar 

  21. C. M. Bender and S. Boettcher, Phys. Rev. D: Part. Fields 48, 4919 (1993).

    ADS  Google Scholar 

  22. F. J. Dyson, Phys. Rev. 75, 1736 (1949); G.’ t Hooft, Nucl. Phys. B 35, 167 (1971).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. I. M. Suslov, Zh. Éksp. Teor. Fiz. 127(6), 1350 (2005) [JETP 100 (6), 1188 (2005)].

    Google Scholar 

  24. I. M. Suslov, Zh. Éksp. Teor. Fiz. 116(2), 369 (1999) [JETP 89 (2), 197 (1999)].

    Google Scholar 

  25. G. ’t Hooft, in Proceedings of the 1977 International School of the Whys of Subnuclear Physics, Erice, Trapani, Sicily, Italy, July 23–August 10, 1977, Ed. by A. Zichichi (Plenum, New York, 1979).

    Google Scholar 

  26. M. Beneke, Phys. Rep. 317(Sect. 2.4), 1 (1999).

    Article  ADS  Google Scholar 

  27. I. M. Suslov, Zh. Éksp. Teor. Fiz. 126(3), 542 (2004) [JETP 99 (3), 474 (2004)].

    MathSciNet  Google Scholar 

  28. M. Moshe and J. Zinn-Justin, Phys. Rep. 385, 69 (2003).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov, Dokl. Akad. Nauk SSSR 95, pp. 497, 773, 1177 (1954).

    MATH  Google Scholar 

  30. H. Kleinert and V. Schulte-Frohlinde, Critical Properties of ϕ 4 Theories (World Sci., Singapore, 2001).

    Google Scholar 

  31. A. M. Polyakov, Nucl. Phys. B 120, 429 (1977).

    Article  MathSciNet  ADS  Google Scholar 

  32. E. Gildener and A. Patrasciouiu, Phys. Rev. D: Part. Fields 16, 423 (1977).

    ADS  Google Scholar 

  33. R. F. Dashen and H. Neuberger, Phys. Rev. Lett. 50, 1897 (1983).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kapitza Institute for Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 119334, Russia

    I. M. Suslov

Authors
  1. I. M. Suslov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. M. Suslov.

Additional information

Original Russian Text © I.M. Suslov, 2010, published in Zhurnal Éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2010, Vol. 138, No. 3, pp. 508–523.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Suslov, I.M. Asymptotic behavior of the β function in the ϕ4 theory: A scheme without complex parameters. J. Exp. Theor. Phys. 111, 450–465 (2010). https://doi.org/10.1134/S1063776110090153

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063776110090153

Keywords

Search

Navigation