Skip to main content
Log in

Critical exponents and large-order behavior of perturbation theory

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The principles of the recent calculations of critical exponents from three- and two-dimensional field theory are reviewed. They rely on the Callan-Symanzik equations, diagram calculations, and on the characterization of the asymptotic behavior of perturbation series at large order. We then present new results concerning the normalization of the large-order behavior.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. A. Baker, B. G. Nickel, M. S. Green, and P. I. Meiron,Phys. Rev. Lett. 36:1351 (1976); J. C. Le Guillou and J. Zinn-Justin,Phys. Rev. Lett. 39:95 (1977); G. A. Baker, B. S. Nickel, and P. I. Meiron, Saclay Preprint 77-39.

    Google Scholar 

  2. C. S. Lam,Nuovo Cimento 55:258 (1968); L. U. Lipatov,Zh. Eksp. Teor. Fiz. 72:411 (1977); E. Brézin, J. C. Le Guillou, and J. Zinn-Justin,Phys. Rev. D 15:1544 (1977).

    Google Scholar 

  3. K. Wilson and J. Kogut,Phys. Rep.12C:75 (1974); K. Wilson,Rev. Mod. Phys. 47:773 (1975); inPhase Transitions and Critical Phenomena, Vol. VI, C. Domb and M. S. Green, eds. (Academic, New York, 1976); S. K. Ma,Modern Theory of Critical Phenomena (Benjamin, Reading, Mass., 1976).

    Google Scholar 

  4. E. Brézin, J. C. deLe Guillou, and J. Zinn-Justin, inPhase Transitions and Critical Phenomena, Vol. VI, C. Domb and M. S. Green, eds. (Academic, New York, 1976); see also C. Di Castro,Nuovo Cimento Lett. 5:69 (1972); K. Symanzik (unpublished), as quoted in G. Mack,Lecture Notes in Physics, Vol. 17 (Springer Verlag, Berlin, 1973); P. K. Mitter,Phys. Rev. D 7:2927 (1973); B. Schroer,Phys. Rev. B 8:4200 (1973).

    Google Scholar 

  5. J. Glimm and A. Jaffe,Notes of the 1976 Cargèse Summer School.

  6. K. Symanzik,Comm. Math. Phys. 18:227 (1970); J. Callan,Phys. Rev. D 1:1541 (1971).

    Google Scholar 

  7. G. Parisi,Notes of the 1973 Cargèse Summer School.

  8. B. G. Nickel, Critical Exponents via theε expansion: A Study of Terms beyondε 3, Oxford Preprint (1974), unpublished.

  9. B. G. Nickel, D. I. Meiron, and G. A. Baker, Compilation of 2-pt and 4-pt Graphs, University of Guelph Report (1977); Evaluation of Series for Critical Exponents for aφ model in 3 and 2 Dimensions,Phys. Rev., to be published.

  10. B. G. Nickel,J. Math. Phys. 19:542 (1978).

    Google Scholar 

  11. F. J. Dyson,Phys. Rev. 85:631 (1952); J. S. Langer,Ann. Phys. (N. Y.) 41:108 (1967).

    Google Scholar 

  12. T. Banks, C. M. Bender, and T. T. Wu,Phys. Rev. D 8:3346 (1973).

    Google Scholar 

  13. G. Parisi,Phys. Lett. 67B:167 (1977); C. Itzykson, G. Parisi, and J. B. Zuber,Phys. Rev. Lett. 38:306 (1977); J. R. Klauder,Acta Phys. Austriaca Suppl. 11:341 (1973); E. M. Stein,Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, N.J., 1970); S. Coleman, V. Glaser, and A. Martin,Comm. Math. Phys. 58:211 (1978).

    Google Scholar 

  14. E. Brézin, J. C. Le Guillou, and J. Zinn-Justin,Phys. Rep. D 15:1558 (1977).

    Google Scholar 

  15. S. Graffi, V. Grecchi, and B. Simon,Phys. Lett. 32B:631 (1970); J. D. Eckmann, J. Magnen, and R. Sénéor,Comm. Math. Phys. 39:251 (1975).

    Google Scholar 

  16. J. Zittartz and J. S. Langer,Phys. Rev. 148:741 (1966); J. L. Gervais and B. Sakita,Phys. Rev. D 11:2943 (1975); V. E. Korepin, P. P. Kulish, and L. D. Faddeev,JETP Lett. 21:138 (1975).

    Google Scholar 

  17. C. Itzykson, G. Parisi, and J. B. Zuber,Phys. Rev. D 16, 996 (1977).

    Google Scholar 

  18. V. Glaser, H. Grosse, A. Martin, and W. Thirring, inStudies in Mathematical Physics, Essays in Honor of V. Bargman, E. Liebet al., eds. (Princeton Univ. Press, Princeton, N.J., 1976).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

On leave of absence from INFN, Frascati.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brézin, E., Parisi, G. Critical exponents and large-order behavior of perturbation theory. J Stat Phys 19, 269–292 (1978). https://doi.org/10.1007/BF01011726

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01011726

Key words

Navigation