Communications in Mathematical Physics

, Volume 109, Issue 3, pp 437–480 | Cite as

Construction and Borel summability of infrared Φ44 by a phase space expansion

  • J. Feldman
  • J. Magnen
  • V. Rivasseau
  • R. Sénéor
Article

Abstract

We construct the thermodynamic limit of the critical (massless) φ4 model in 4 dimensions with an ultraviolet cutoff by means of a “partly renormalized” phase space expansion. This expansion requires in a natural way the introduction of effective or “running” constants, and the infrared asymptotic freedom of the model, i.e. the decay of the running coupling constant, plays a crucial rôle. We prove also that the correlation functions of the model are the Borel sums of their perturbation expansion.

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References

  1. 1.
    Glimm, J., Jaffe, A.: Positivity of the ϕ34 Hamiltonian. Phys.21, 327 (1973)Google Scholar
  2. 2.
    Feldman, J.: The λϕ34 field theory in a finite volume. Commun. Math. Phys.37, 93 (1974)Google Scholar
  3. 3.
    Feldman, J., Osterwalder, K.: The Wightman axioms and the mass gap for weakly coupled (ϕ4)3 quantum field theories. Ann. Phys.97, 80 (1976)Google Scholar
  4. 4.
    Magnen, J., Seneor, R.: The infinite volume limit of the ϕ34 model. Ann. Inst. Henri Poincaré24, 95 (1976)Google Scholar
  5. 5.
    Magnen, J., Sénéor, R.: Phase space cell expansion and Borel summability for the Euclidean ϕ34 theory. Commun. Math. Phys.56, 237 (1977)Google Scholar
  6. 6.
    Magnen, J., Sénéor, R.: The infrared behaviour of (∇Φ)34. Ann. Phys.152, 130 (1984)Google Scholar
  7. 7.
    De Calan, C., Rivasseau, V.: Local existence of the Borel transform in Euclidean Φ44. Commun. Math. Phys.82, 69 (1981)Google Scholar
  8. 8.
    Rivasseau, V.: Construction and Borel summability of planar 4-dimensional Euclidean field theory. Commun. Math. Phys.95, 445 (1984)Google Scholar
  9. 9.
    Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Bounds on completely convergent Euclidean Feynman graphs. Commun. Math. Phys.98, 273 (1985)Google Scholar
  10. 10.
    Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Bounds on renormalized Feynman graphs. Commun. Math. Phys.100, 23 (1985)Google Scholar
  11. 11.
    Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Massive Gross-Neveu model: A rigorous perturbative construction. Phys. Rev. Lett.54, 1479 (1985). A renormalizable field theory: the massive Gross-Neveu model in two dimensions. Commun. Math. Phys.103, 67 (1985)Google Scholar
  12. 12.
    't Hooft, G.: Lecture presented at Cargèse Summer School. September 1983Google Scholar
  13. 13.
    Magnen, J., Rivasseau, V.: The Lipatov argument for Φ34 perturbation theory. Commun. Math. Phys.102, 59 (1985)Google Scholar
  14. 14.
    Feldman, J., Rivasseau, V.: Preprint Ecole Polytechnique (1985). Ann. Inst. Henri Poincaré (to appear)Google Scholar
  15. 15.
    Magnen, J., Nicolò, F., Rivasseau, V., Sénéor, R.: A Lipatov bound for Φ44 Euclidean field theory. Commun. Math. Phys. (in press)Google Scholar
  16. 16.
    Wilson, K.: Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior. Phys. Rev. B4, 3184 (1974)Google Scholar
  17. 17.
    Kogut, J., Wilson, K.: The normalization group and the ε expansion. Phys. Rep.12C, 75 (1975)Google Scholar
  18. 18.
    Freedman, B., Smolensky, P., Weingarten, D.: Monte Carlo evaluation of the continuum limit of (φ4)4 and (φ4)3. Phys. Lett.113B, 481 (1982)Google Scholar
  19. 19.
    Wegner, F., Riedel, E.: Logarithmic corrections to the molecular-field behavior of critical and tricritical systems. Phys. Rev. B7, 248 (1973)Google Scholar
  20. 20.
    Brezin, E., Zinn-Justin, J.: Renormalization of the nonlinear σ-model in 2 + ε dimensions-application to Heisenberg ferromagnets. Phys. Rev. B13, 251 (1976)Google Scholar
  21. 21.
    Gawedzki, K., Kupiainen, A.: Massless lattice φ44 theory: A nonperturbative control of a renormalizable model. Phys. Rev. Lett.54, 92 (1985); Massless lattice φ44 theory: Rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys.99, 197 (1985)Google Scholar
  22. 22.
    Gawedzki, K., Kupiainen, A.: Proceedings of Les Houches Summer School (1984)Google Scholar
  23. 23.
    Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Proceedings of Les Houches Summer School (1984)Google Scholar
  24. 24.
    Aizenman, M.: Geometric analysis of Φ4 fields and Ising models, Parts I and II. Commun. Math. Phys.86, 1 (1982)Google Scholar
  25. 25.
    Fröhlich, J.: On the triviality of λϕd4 theories and the approach to the critical point ind ≧ 4 dimensions. Nucl. Phys. B200 (FS4), 281 (1982)Google Scholar
  26. 26.
    Aragao de Carvalho, C., Canaciolo, S., Fröhlich, J.: Polymers andg|φ|4 theory in four dimensions. Nucl. Phys. B215, 209 (1983)Google Scholar
  27. 27.
    Brydges, D., Federbush, P.: A new form of the Mayer expansion in classical statistical mechanics. J. Math. Phys.19, 2064 (1978)Google Scholar
  28. 28.
    Brydges, D.: Proceedings of Les Houches Summer School 1984Google Scholar
  29. 29.
    Battle III, G.A., Federbush, P.: A phase cell cluster expansion for Euclidean field theories. Ann. Phys.142, 95 (1982)Google Scholar
  30. 30.
    Mack, G., Pordt, A.: Convergent perturbation expansions for Euclidean quantum field theory. Commun. Math. Phys.97, 267 (1985)Google Scholar
  31. 31.
    Sokal, A.: An improvement of Wilson's theorem on Borel summability. J. Math. Phys.21, 261 (1980)Google Scholar
  32. 32.
    Gallavotti, G., Nicolò, F.: Renormalization theory in four-dimensional scalar fields (I) and (II). Commun. Math. Phys.100, 545 (1985) and101, 247 (1985)Google Scholar
  33. 33.
    Cooper, A., Feldman, J., Rosen, L.: Legendre transforms andr-particle irreducibility in quantum field theory: the formal power series framework. Ann. Phys.137, 213 (1981)Google Scholar
  34. 34.
    Gallavotti, G.: Elements of mechanics. Berlin, Heidelberg, New York: Springer 1983Google Scholar
  35. 35.
    Gawedzki, K., Kupiainen, A.: Preprint IHES (1985)Google Scholar
  36. 36.
    Felder, G.: Construction of a non-trivial planar field theory with ultraviolet stable fixed point. Commun. Math. Phys.102, 139 (1985)Google Scholar
  37. 37.
    de Calan, C., Faria da Vega, P., Magnen, J., Sénéor, R.: In preparationGoogle Scholar
  38. 38.
    Glimm, J., Jaffe, A., Spencer, T.: In: Constructive field theory. G. Velo, A. Wightman (eds.). Lecture Notes, Vol. 25, Berlin, Heidelberg, New York: Springer 1973Google Scholar
  39. 39.
    Iagolnitzer, D., Magnen, J., Bros, J.: Preprint Ecole PolytechniqueGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • J. Feldman
    • 1
  • J. Magnen
    • 1
  • V. Rivasseau
    • 1
  • R. Sénéor
    • 1
  1. 1.Centre de Physique ThéoriqueEcole PolytechniquePalaiseau CedexFrance

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