Skip to main content
SpringerLink
Log in

BPHZL renormalization of 1/N expansion and critical behaviour of the three-dimensional chiral field

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A modified soft mass renormalization scheme for 1/N expansion of the three dimensionalO(N)-invariant chiral field in the high- and low-temperature phases, as well as at the critical point, is constructed, free of infrared divergencies in each separate diagram. Generalized quantum chirality identities for composite operators are derived, from which the renormalizability of the model follows. The approach formulated here is applied to a rigorous analysis of the universal critical behaviour of theN-component chiral field in three dimensions.

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. Stanley, H.E.: Phase transitions and critical phenomena. Oxford: Cambridge University Press 1971

    Google Scholar 

  2. Weinberg, S.: Phys. Rev. Lett.18, 188 (1967)

    Google Scholar 

  3. Faddeev, L.D.: Dokl. Acad. Nauk SSSR210, 807 (1973) (in Russian)

    Google Scholar 

  4. Slavnov, A.A.: Theor. Math. Phys.10, 305 (1972) (in Russian)

    Google Scholar 

  5. Skyrme, T.R.H.: Proc. Roy, Sec. A247, 260 (1958); J. Math. Phys.12, 1735 (1971)

    Google Scholar 

  6. Faddeev, L.D.: In: Nonlinear, nonlocal, and nonrenormalizable field theories. Dubna, JINR 1976 (in Russian)

  7. Belavin, A.A., Polyakov, A.M.: Nucl. Phys. B123, 429 (1977)

    Google Scholar 

  8. Förster, D.: Nucl. Phys. B130, 38 (1977)

    Google Scholar 

  9. Polyakov, A.M.: Phys. Lett. B59, 79 (1975);

    Google Scholar 

  10. Migdal, A.A.: JETP Lett.69, 1457 (1975) (in Russian)

    Google Scholar 

  11. Bardeen, W.A., Lee, B.W., Shrock, R.E.: Phys. Rev. D14, 985 (1976)

    Google Scholar 

  12. Brezin, E., Zinn-Justin, J.: Phys. Rev. D14, 2615 (1976)

    Google Scholar 

  13. Aref'eva, I.Ya.: Theor. Math. Phys.29, 147 (1976) (in Russian)

    Google Scholar 

  14. Aref'eva, I.Ya.: Theor. Math. Phys.36, 24 (1978) (in Russian)

    Google Scholar 

  15. Polyakov, A.M.: Phys. Lett. B72, 224 (1977)

    Google Scholar 

  16. Aref'eva, I.Ya., Kulish, P.P., Nissimov, E.R., Pacheva, S.J.: LOMI preprint, E-1-1978

  17. Lüscher, M.: Nucl. Phys. B135, 1 (1978)

    Google Scholar 

  18. Zamolodchikov, A.B., Zamolodchikov, Al.B.: Nucl. Phys. B132, 48 (1977)

    Google Scholar 

  19. Witten, E.: Phys. Rev. D16, 2991 (1977)

    Google Scholar 

  20. Di Vecchia, P., Ferrara, S.: Nucl. Phys. B130, 93 (1977)

    Google Scholar 

  21. Alvarez, O.: Phys. Rev. D17, 1123 (1978)

    Google Scholar 

  22. Aref'eva, I.Ya., Krivoschekov, V.K., Medvedev, P.B.: ITEP preprint, 173 (1978)

  23. Bogoliubov, N.N., Shirkov, D.V.: Introduction to the theory of quantized fields. New York: Interscience 1959

    Google Scholar 

  24. Hepp, K.: Theorie de la renormalization. Berlin, Heidelberg, New York: Springer 1969

    Google Scholar 

  25. Brezin, E., Zinn-Justin, J.: Phys. Rev. Lett,36, 691 (1976); Phys. Rev. B14, 3110 (1976)

    Google Scholar 

  26. Hikami, S., Brezin, E.: J. Phys. A11, 1141 (1978)

    Google Scholar 

  27. Aref'eva, I.Ya.: Theor. Math. Phys.31, 3 (1977) (in Russian)

    Google Scholar 

  28. Aref'eva, I.Ya.: In: Proc. Leningrad Dept. Steclov Math. Inst. (eds. P.P. Kulish, V.N. Popov) (in Russian)77, 3 (1978)

  29. Aref'eva, I.Ya.: Ann. Phys. (N.Y.)117, 393 (1979)

    Google Scholar 

  30. Aref'eva, I.Ya.: Theor. Math. Phys.36, 159 (1978) (in Russian)

    Google Scholar 

  31. Lowenstein, J., Zimmenmann, W.: Nucl. Phys. B86, 77 (1975); Commun. Math. Phys.46, 105 (1976)

    Google Scholar 

  32. Lowenstein, J.: Commun. Math. Phys.47, 53 (1976)

    Google Scholar 

  33. Lowenstein, J.: Nucl. Phys. B96, 189 (1975)

    Google Scholar 

  34. Amati, D., Rouet, A.: CERN preprint TH.2468 (1978)

  35. Clark, T.: Max Planck Inst. preprint MPI-PAE/Pht 43/75

  36. Clark, T., Rouet, A.: MPI-PAE/Pth 17/75

  37. Zimmermann, W.: In: Lectures on elementary particles and quantum field theory, Vol. 1, p. 397 (eds. S. Deser, M. Grisaru, H. Pendleton). Cambridge: M.I. Press 1970

    Google Scholar 

  38. Zimmermann, W.: Ann. Phys. (N.Y.)77, 536, 569 (1973)

    Google Scholar 

  39. Anikin, S.A., Polivanov, M.C., Zavialov, O.I.: Fortschr. Phys.27, 459 (1977)

    Google Scholar 

  40. Lowenstein, J.: Phys. Rev. D4, 2281 (1971)

    Google Scholar 

  41. Lam, Y.M.P.: Phys. Rev. D6, 2145 (1972)

    Google Scholar 

  42. Gomes, M., Lowenstein, J.: Phys. Rev. D7, 550 (1973)

    Google Scholar 

  43. Jegerlehner, F., Schröer, B.: Acta Phys. Austriaca, Suppl.XI, 389 (1973)

    Google Scholar 

  44. Kadanoff, L.: Physics2, 263 (1966)

    Google Scholar 

  45. Kadanoff, L., Götze, W., Hamblen, D., Hecht, R., Lewis, E., Palciauskas, V., Raul, B., Swift, J.: Rev. Mod. Phys.39, 395 (1967)

    Google Scholar 

  46. Abe, R.: Prog. Theor. Phys.48, 1414 (1972)

    Google Scholar 

  47. Abe, R., Hikami, S.: Phys. Lett. A42, 419 (1973)

    Google Scholar 

  48. : Prog. Theor. Phys.49, 442 (1973)

    Google Scholar 

  49. Ma, S.: Rev. Mod. Phys.45, 589 (1973);

    Google Scholar 

  50. Wilson, K., Kogut, J.: Phys. Rep. C12, 75 (1974)

    Google Scholar 

  51. Brezin, E., Le Guillou, J.G., Zinn-Justin, J.: In: Phase transitions and critical phenomena (eds. C. Comb, M. Green). New York: Academic Press 1976

    Google Scholar 

  52. Symanzik, K.: DESY preprint 77/05

  53. Nissimov, E.R., Pacheva, S.J.: Theor. Math. Phys.41, 55, 220 (1979) (in Russian)

    Google Scholar 

  54. Zimmermann, W.: Commun. Math. Phys.15, 208 (1969)

    Google Scholar 

  55. Clark, T., Lowenstein, J.: Nucl. Phys. B113, 109 (1976)

    Google Scholar 

  56. Lowenstein, J., Zimmermann, W.: Commun. Math. Phys.44, 73 (1975)

    Google Scholar 

  57. Gomes, M., Lowenstein, J., Zimmermann, W.: Commun. Math. Phys.39, 81 (1974)

    Google Scholar 

  58. Jegerlehner, F., Schröer, B.: Nucl. Phys. B68, 461 (1974)

    Google Scholar 

  59. Jegerlehner, F.: Fortschr. Phys.23, 71 (1975)

    Google Scholar 

  60. Weinberg, S.: Phys. Rev.118, 838 (1960)

    Google Scholar 

  61. Lowenstein, J.: Commun. Math. Phys.24, 1 (1971)

    Google Scholar 

  62. Symanzik, K.: Lett. Nuovo Cimento3, 734 (1972)

    Google Scholar 

  63. Di Castro, C., Jona-Lasinio, G., Peliti, L.: Ann. Phys. N.Y.87, 327 (1974)

    Google Scholar 

  64. Brezin, E.: In: Methods in field theory (eds. R. Balian, J. Zinn-Justin). Amsterdam: North-Holland 1976

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, 117333, Moscow, USSR

    I. Ya. Aref'eva, E. R. Nissimov & S. J. Pacheva

Authors
  1. I. Ya. Aref'eva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. R. Nissimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. J. Pacheva
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by E. Brezin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aref'eva, I.Y., Nissimov, E.R. & Pacheva, S.J. BPHZL renormalization of 1/N expansion and critical behaviour of the three-dimensional chiral field. Commun.Math. Phys. 71, 213–246 (1980). https://doi.org/10.1007/BF01197293

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01197293

Keywords

Search

Navigation