Skip to main content
Log in

Construction and Borel summability of planar 4-dimensional euclidean field theory

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

Abstract

We use the methods of [1] to show that the planar part of the renormalized perturbation theory forϕ 44 -euclidean field theory is Borel-summable on the asymptotically free side of the theory. The Borel sum can therefore be taken as a rigorous definition of theN→∞ limit of a massiveN×N matrix model with a +tr 4 interaction, hence with “wrong sign” ofg. Our construction is relevant for a solution of the ultra-violet problem for planar QCD. We also propose a program for studying the structure of the “renormalons” singularities within the planar world.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. de Calan, C., Rivasseau, V.: Local existence of the Borel transform in EuclideanΦ 44 . Commun. Math. Phys.82, 69 (1981)

    Google Scholar 

  2. Gross, D., Neveu, A.: Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D10, 3235 (1974)

    Google Scholar 

  3. Lautrup, B.: On high order estimates in QED. Phys. Lett.69B, 109 (1977)

    Google Scholar 

  4. 't Hooft, G.: Lectures given at Ettore Majorana School, Erice, Sicily (1977)

  5. Parisi, G.: Singularities of the Borel transform in renormalizable theories. Phys. Lett.76B, 65 (1978)

    Google Scholar 

  6. Parisi, G.: The Borel transform and the renormalization group. Phys. Rep.49, 215 (1979)

    Google Scholar 

  7. 't Hooft, G.: Is asymptotic freedom enough? Phys. Lett.109B, 474 (1982)

    Google Scholar 

  8. 't Hooft, G.: On the convergence of planar diagram expansion. Commun. Math. Phys.86, 449 (1982)

    Google Scholar 

  9. 't Hooft, G.: Rigorous construction of planar diagram field theories in four dimensional Euclidean space. Commun. Math. Phys.88, 1 (1983)

    Google Scholar 

  10. 't Hooft, G.: Borel summability of a four-dimensional field theory. Phys. Lett.119B, 369 (1982)

    Google Scholar 

  11. Rivasseau, V.: Rigorous construction and Borel summability for a planar four dimensional field theory. Phys. Lett.137 B, 98 (1984)

    Google Scholar 

  12. Graffi, S., Grecchi, V., Simon, B.: Borel summability: application to the anharmonic oscillator. Phys. Lett. B32, 631 (1970)

    Google Scholar 

  13. Eckmann, J., Magnen, J., Sénéor, R.: Decay properties and Borel summability for the Schwinger functions inP(φ)2 theories. Commun. Math. Phys.39, 251 (1975)

    Google Scholar 

  14. Magnen, J., Sénéor, R.: Phase space cell expansion and Borel summability for the Euclideanϕ 43 theory. Commun. Math. Phys.56, 237 (1977)

    Google Scholar 

  15. Renouard, P.: Analyticité et sommabilité “de Borel” des fonctions de Schwinger du modèle de Yukawa en dimensiond=2. I and II. Ann. Inst. H. Poincaré,27, 237 (1977);31, 235 (1979)

    Google Scholar 

  16. Eckmann, J., Epstein, H.: Time-ordered products and Schwinger functions, Commun. Math. Phys.64, 95 (1979); Borel summability of the mass and theS-Matrix inϕ 4 models68, 245 (1979)

    Google Scholar 

  17. Dyson, F.J.: Divergence of perturbation theory in quantum electrodynamics. Phys. Rev.85, 631 (1952)

    Google Scholar 

  18. Jaffe, A.: Divergence of perturbation theory for bosons, Commun. Math. Phys.1, 127 (1965)

    Google Scholar 

  19. de Calan, C., Rivasseau, V.: The perturbation series forΦ 43 field theory is divergent. Commun. Math. Phys.83, 77 (1982)

    Google Scholar 

  20. Watson, G.: Phil. Trans. R. Soc. London Ser. A211, 279 (1912); see also Hardy, G.: Divergent Series, London: Oxford U.P. 1949

    Google Scholar 

  21. Nevanlinna, F.: Ann. Acad. Sci. Fenn. Ser. A12, 3 (1919)

    Google Scholar 

  22. Sokal, A.: An improvement of Watson's theorem on Borel summability. J. Math. Phys.21, 261 (1980)

    Google Scholar 

  23. Rivasseau, V., Speer, E.: The Borel transform in Euclideanϕ 4 v ; Local existence for Rev<4. Commun. Math. Phys.72, 293 (1980)

    Google Scholar 

  24. Gallavotti, G., Rivasseau, V.:φ 4 field theory in dimension 4. A modern introduction to its unsolved problems. Ann. Inst. H. Poincaré40, 185 (1984)

    Google Scholar 

  25. Brydges, D., Fröhlich, J., Sokal, A.: A new proof of the existence and non-triviality of the continuumϕ 42 andϕ 43 quantum field theories. Commun. Math. Phys.91, 141 (1983)

    Google Scholar 

  26. Aizenman, M.: Proof of the triviality ofϕ 4 field theory and some mean-field features of Ising models ford>4. Phys. Rev. Lett.47, 1 (1981)

    Google Scholar 

  27. Aizenman, M.: Geometric analysis ofΦ 4 fields and Ising models. Parts I and II. Commun. Math. Phys.86, 1 (1982)

    Google Scholar 

  28. Fröhlich, J.: On the triviality ofλϕ 4 d theories and the approach to the critical point ind ≧4 dimensions. Nucl. Phys. B200 (FS4), 281 (1982)

    Google Scholar 

  29. 't Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B72, 461 (1974)

    Google Scholar 

  30. Migdal, A.A.: Properties of the loop average in QCD, Ann. Phys.126, 279 (1980)

    Google Scholar 

  31. Makeenko, Y.M., Migdal, A.A.: Quantum chromodynamics as dynamics of loops. Nucl. Phys. B188, 269 (1981)

    Google Scholar 

  32. Eguchi, T., Kawai, H.: Reduction of dynamical degrees of freedom in the largeN gauge theory. Phys. Rev. Lett.48, 1063 (1982)

    Google Scholar 

  33. Bhanot, G., Heller, U., Neuberger, H.: The quenched Eguchi-Kawai model. Phys. Lett.113B, 47 (1982)

    Google Scholar 

  34. Parisi, G.: A simple expression for planar field theories. Phys. Lett.112B, 463 (1982)

    Google Scholar 

  35. Gross, D., Kitazawa, Y.: A quenched momentum prescription for largeN-theories. Nucl. Phys. B206, 440 (1982)

    Google Scholar 

  36. Lipatov, L.N.: Calculation of the Gell-Mann-Low function in scalar theory with strong nonlinearity. Sov. Phys. JETP44, 1055 (1976), and Divergence of the perturbation theory series and the quasi-classical theory. JETP45, 216 (1977)

    Google Scholar 

  37. Brézin, E., Le Guillou, J.C., Zinn-Justin, J.: Perturbation theory at large order. I. Theφ 2N interaction, and II. Role of the vacuum instability. Phys. Rev. D15, 1544, 1558 (1977)

    Google Scholar 

  38. Breen, S.: PhD thesis and “Leading large order asymptotics forϕ 42 perturbation theory.” Commun. Math. Phys.92, 197 (1983)

    Google Scholar 

  39. Koplik, J., Neveu, A., Nussinov, S.: Some aspects of the planar perturbation series. Nucl. Phys. B123, 109 (1977)

    Google Scholar 

  40. Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys.59, 35 (1978)

    Google Scholar 

  41. Brydges, D., Sokal, A., Spencer, T.: Private communication

  42. Bergère, M., Zuber, J.B.: Renormalization of Feynman amplitudes and parametric integral representation. Commun. Math. Phys.35, 113 (1974)

    Google Scholar 

  43. Bergère, M., Lam, Y.M.P.: Bogolubov-Parasiuk theorem in theα-parametric representation. J. Math. Phys.17, 1546 (1976)

    Google Scholar 

  44. Symanzik, K.: Small-distance-behaviour analysis and Wilson expansions. Commun. Math. Phys.23, 49 (1971); Infrared singularities and small-distance-behaviour analysis.34, 7 (1973)

    Google Scholar 

  45. Bogoliubov, N., Parasiuk,: Acta Math.97, 227 (1957)

    Google Scholar 

  46. Hepp, K.: Proof of the Bogoliubov-Parasiuk theorem on renormalization. Commun. Math. Phys.2, 301 (1966)

    Google Scholar 

  47. Zimmermann, W.: Convergence of Bogoliubov's method for renormalization in momentum space. Commun. Math. Phys.15, 208 (1969)

    Google Scholar 

  48. Speer, E.: Generalized Feynman amplitudes. Princeton, NJ: Princeton University Press 1969

    Google Scholar 

  49. Renormalization theory. Proceedings of 1975 Erice Summer School. Velo, G., Wightman, A. (eds.)

  50. de Calan, C., David, F., Rivasseau, V.: Renormalization in the complete Mellin representation of Feynman amplitudes. Commun. Math. Phys.78, 531 (1981)

    Google Scholar 

  51. Landau, L., et al.: Collected papers of L. D. Landau. New York: Gordon and Breach 1965

    Google Scholar 

  52. Pomeranchuk, I., Sudakov, V., Ter Martirosyan, K.: Vanishing of renormalized charges in field theories with point interaction. Phys. Rev.103, 784 (1956)

    Google Scholar 

  53. Redmond, P.J.: Elimination of ghosts in propagators. Phys. Rev.112, 1404 (1958)

    Google Scholar 

  54. Bogoliubov, N.N., Logunov, A.A., Shirkov, D.V.: The method of dispersion relations and perturbation theory. Sov. Phys. JETP10, 574 (1960)

    Google Scholar 

  55. Crutchfield, W.Y.: Phys. Rev. D19, 2370 (1979)

    Google Scholar 

  56. David, F.: Non-perturbative effects and infrared renormalons within theI/N expansion of theO(N) non-linear sigma model. Nucl. Phys. B209, 433 (1982)

    Google Scholar 

  57. Bergère, M., David, F.: In preparation

  58. Wilson, K.: Non-Lagrangian models of current algebra. Phys. Rev.179, 1499 (1969)

    Google Scholar 

  59. Shifman, M.A., Vainshtein, A.I., Zakharov, V.I.: QCD and resonance physics. Theoretical foundations. Nucl. Phys. B147, 385–534 (1979)

    Google Scholar 

  60. David, F.: Private communication

  61. Glimm, J., Jaffe, A.: Positivity of theφ 43 Hamiltonian. Fortschr. Phys.21, 327 (1973)

    Google Scholar 

  62. de Calan, C., Rivasseau, V.: Comment on [1]. Commun. Math. Phys.91, 265 (1983)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by A. Jaffe

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rivasseau, V. Construction and Borel summability of planar 4-dimensional euclidean field theory. Commun.Math. Phys. 95, 445–486 (1984). https://doi.org/10.1007/BF01210833

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01210833

Keywords

Navigation