Communications in Mathematical Physics

, Volume 172, Issue 1, pp 143–186 | Cite as

The short distance behavior of (φ4)3

  • D. Brydges
  • J. Dimock
  • T. R. Hurd


We consider theφ34 quantum field theory on a torus and study the short distance behavior. We reproduce the standard result that the singularities can be removed by a simple mass renormalization. For the resulting model we give anLp bound on the short distance regularity of the correlation functions. To obtain these results we develop a systematic treatment of the generating functional for correlations using a renormalization group method incorporating background fields.


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  1. [Ba83] Balaban, T.: Ultraviolet stability in field theory. Theφ 34 field theory. In: Fröhlich, J., editor, Scaling and Self-similarity in Physics. Boston: Birkhäuser, 1983Google Scholar
  2. [BCG+80] Benfatto, G., Cassandro, M., Gallavotti, G., Nicolò, F., Olivieri,E., Presutti, E., Scacciatelli, E.: On the ultraviolet stability in the Euclidean scalar field theories. Commun. Math. Phys.71, 95–130 (1980)CrossRefGoogle Scholar
  3. [BDH93] Brydges, D. C., Dimock, J., Hurd, T. R.: Weak perturbations of Gaussian measures. In: J.S. Feldman, R. Froese, L.M. Rosen, editors, Mathematical Quantum Field Theory I:: Field Theory and Many-Body Theory, CRM Proceedings & Lecture Notes, Providence, Rhode Island: American Mathematical Society, 1993Google Scholar
  4. [BF83] Battle, G. A., Federbush, P.: A phase cell cluster expansion for a hierarchicalφ 34 model. Commun. Math. Phys.88, 263–293 (1983)CrossRefGoogle Scholar
  5. [BFS83] Brydges, D., Fröhlich, J., Sokal, A.: A new proof of the existence and nontriviality of the continuumφ 24 andφ 34 quantum field theories. Commun. Math. Phys.91, 141–186 (1983)CrossRefGoogle Scholar
  6. [BK93] Brydges, D., Keller, G.: Correlation functions of general observables in dipole-type systems I: accurate upper bounds. Helv. Phys. Acta67, 43–116 (1994)Google Scholar
  7. [Bry92] Brydges, D.: Functional integrals and their applications. Technical report, Ecole Polytechnique Federale de Lausanne, 1992Google Scholar
  8. [BY90] Brydges, D., Yau, H. T.: Grad φ perturbations of massless Gaussian fields. Commun. Math. Phys.129, 351–392 (1990)CrossRefGoogle Scholar
  9. [DH91] Dimock, J., Hurd, T. R.: A renormalization group analysis of the Kosterlitz-Thouless phase. Commun. Math. Phys.137, 263–287 (1991)Google Scholar
  10. [DH92a] Dimock, J., Hurd, T. R.: A renormalization group analysis of correlation functions for the dipole gas. J. Stat. Phys.66 1277–1318 (1992)CrossRefGoogle Scholar
  11. [DH92b] Dimock, J., Hurd, T. R.: A renormalization group analysis of QED. J. Math. Phys.33, 814–821 (1992)Google Scholar
  12. [DH93] Dimock, J., Hurd, T. R.: Construction of the two-dimensional sine-Gordon model for β<8π. Commun. Math. Phys.156, 547–580 (1993)CrossRefGoogle Scholar
  13. [FMRS86] Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: The massive Gross-Neveu model: A rigorous perturbative construction. Commun. Math. Phys.103 67–103 (1986)CrossRefGoogle Scholar
  14. [FO76] Feldman, J., Osterwalder, K.: The Wightman axioms and the mass gap for weakly coupled (φ 4)3 quantum field theories. Ann. Phys.97, 80–135 (1976)CrossRefGoogle Scholar
  15. [GJ73] Glimm, J., Jaffe, A.: Positivity of theφ 34 Hamiltonian. Fortschr. Phys.21, 327–376 (1973)Google Scholar
  16. [GK85] Gawędzki, K., Kupiainen, A.: Massless latticeφ 44 theory: Rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys.99, 197–252 (1985)CrossRefGoogle Scholar
  17. [GK86] Gawędzki, K., Kupiainen, A.: Asymptotic freedom beyond perturbation theory. In: K. Osterwalder, R. Stora, (eds.) Critical Phenomena, Random Systems, Gauge Theories. Les Houches 1984. Amsterdam: North-Holland, 1986Google Scholar
  18. [MRS93] Magnen, J., Rivasseau, V., Sénéor, R.: Construction of YM4 with an infra-red cutoff. Commun. Math. Phys.155, 325–383 (1993)Google Scholar
  19. [MS77] Magnen, J., Sénéor, R.: Phase space cell expansion and Borel summaility for the Euclideanφ 34 theory. Commun. Math. Phys.56, 237–276 (1977)CrossRefGoogle Scholar
  20. [Riv91] Rivasseau, Y.: From perturbative to constructive renormalization. Princeton, N.J.: Princeton University Press, 1991Google Scholar
  21. [Wat89] H. Watanabe. Block spin approach toφ 34. J. Stat. Phys.54, 171–190 (1989)CrossRefGoogle Scholar
  22. [Wil87] Williamson, C.: A phase cell cluster expansion forφ 34. Ann. Phys.175, 31–63 (1987)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • D. Brydges
    • 1
  • J. Dimock
    • 2
  • T. R. Hurd
    • 3
  1. 1.Dept. of MathematicsUniversity of VirginiaCharlottesvilleUSA
  2. 2.Dept. of MathematicsSUNY at BuffaloBuffaloUSA
  3. 3.Dept. of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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