Abstract
Renormalizable quantum field theories whose perturbation expansions are described by planar Feynman diagrams only, such as SU(∞) gauge theory, are considered in 4 dimensional Euclidean space. For studying asymptotic properties of the perturbation series one might wish to isolate first all those planar diagrams that do not contain any ultraviolet divergent subgraphs. In this paper it is proved that this infinite set of diagrams, when summed, converges within a finite radius of convergence for the coupling constant.
Similar content being viewed by others
References
't Hooft, G.: In: The whys of subnuclear physics, Erice 1977, Zichichi, A. (ed.). New York, London: Plenum Press 1979, p. 943
't Hooft, G.: Is asymptotic freedom enough? Phys. Lett.109B, 474 (1982)
't Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B72, 461 (1974)
Dyson, F.: Divergence of perturbation theory in quantum electrodynamics. Phys. Rev.85, 631 (1952)
Riddell, R.J., Jr., The number of Feynman diagrams. Phys. Rev.91, 1243 (1953)
Koplik, J., Neveu, A., Nussinov, S.: Some aspects of the planar perturbation series. Nucl. Phys. B123, 109 (1977)
Tuttle, W.T.: Can. J. Math.14, 21 (1962)
Symanzik, K.: Euclidean quantum field theory. I. Equations for a scalar model. J. Math. Phys.7, 510 (1966)
De Calan, C., Rivasseau, V.: Local existence of the Borel transform in euclidean Φ 44 . Commun. Math. Phys.82, 69 (1981)
Rivasseau, V.: Talk presented at the Conference on Statistical Mechanics and Quantum Field Theory, Rutgers University, USA, May 1980 (unpublished)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
't Hooft, G. On the convergence of planar diagram expansions. Commun.Math. Phys. 86, 449–464 (1982). https://doi.org/10.1007/BF01214881
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01214881