Abstract
The Euclidean massive Gross-Neveu model in two dimensions is just renormalizable and asymptotically free. Thanks to the Pauli principle, bare perturbation theory with an ultra-violet cut-off (and the correct ansatz for the bare mass) is convergent in a disk, whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant. Therefore, the theory can be fully constructed in a perturbative way. It satisfies the O.S. axioms and is the Borel sum of the renormalized perturbation expansion of the model
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Mitter, P.K., Weisz, P.H.: Asymptotic scale invariance in a massive Thirring model withU(n) symmetry. Phys. Rev. D8, 4410 (1973)
Gross, D. Neveu, A.: Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D10, 3235 (1974)
Coleman, S.: Quantum sine-Gordon equation as a massive Thirring model. Phys. Rev. D11, 2088 (1975);
Fröhlich, J., Seiler, E.: The massive Thirring-Schwinger model (QED2): Convergence of perturbation theory and particle structure. Helv. Phys. Acta49, 889 (1976)
Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Massive Gross-Neveu model: A rigorous perturbative construction. Phys. Rev. Lett.54, (1479) (1985)
't Hooft, G.: On the convergence of planar diagram expansion. Commun. Math. Phys.86, 449 (1982) and Rigorous construction of planar diagram field theories in four dimensional Euclidean space. Commun. Math. Phys.88, 1 (1983)
Rivasseau, V.: Construction and Borel summability of planar 4-dimensional Euclidean field theory. Commun. Math. Phys.95, 445 (1984)
Gallavotti, G., Nicolò, F.: Renormalization theory in four-dimensional scalar fields, I and II. Commun. Math. Phys.100, 545 and101, 247 (1985)
Gawedzki, K., Kupiainen, A.: Massless lattice φ 44 theory: a nonperturbative control of a renormalizable model. Phys. Rev. Lett.54, 92 (1985) and IHES preprint (1984)
Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: “Infrared Φ 44 , preprint Ecole Polytechnique, and contribution to the proceedings of Les Houches summer school (1984)
Gawędzki, K., Kupiainen, A.: Harvard University preprint (1984)
Felder, G., Gallavotti, G.: Perturbation theory and non-renormalizable scalar fields. Commun. Math. Phys.102, 549–571 (1985)
Felder, G.: Work in preparation on planar −gϕ 44+ɛ
Gawedzki, K., Kupiainen, A.: Preprints IHES and Helsinki University (1985)
Brydges, D.: Lectures at the 1984 “Les Houches” summer school (to be published)
Seiler, E.: Gauge theories as a problem of constructive quantum field theory and statistical mechanics. Lecture Notes in Physics, Vol. 159. Berlin, Heidelberg, New York: Springer 1982
Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Bounds on completely convergent Euclidean Feynman graphs. Commun. Math. Phys.98, 273 (1985) and Bounds on renormalized graphs. Commun. Math. Phys.100, 23 (1985)
Mack, G., Pordt, A.: Convergent perturbation expansions for Euclidean quantum field theory. Commun. Math. Phys.97, 267 (1985)
Zimmermann, W.: Convergence of Bagoliubov's method for renormalization in momentum space. Commun. Math. Phys.15, 208 (1969)
Rivasseau, V.: In preparation
de Calan, C., Rivasseau, V.: Local existence of the Borel transform in Euclidean Φ 44 . Commun. Math. Phys.82, 69 (1981)
Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. Inst. H. Poincaré19, 211 (1973)
Sokal, A.: An improvement of Watson's theorem on Borel summability. J. Math. Phys.21, 261 (1980)
Wetzel, W.: Two-loop β-function for the Gross-Neveu model. Phys. Lett.153 B, 297 (1985)
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Communicated by K. Osterwalder
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Feldman, J., Magnen, J., Rivasseau, V. et al. A renormalizable field theory: The massive Gross-Neveu model in two dimensions. Commun.Math. Phys. 103, 67–103 (1986). https://doi.org/10.1007/BF01464282
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DOI: https://doi.org/10.1007/BF01464282