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Renormalization of the Higgs model: Minimizers, propagators and the stability of mean field theory

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Abstract

We study the effective actionsS (k) obtained byk iterations of a renormalization transformation of the U(1) Higgs model ind=2 or 3 spacetime dimensions. We identify a quadratic approximationS (k) Q toS (k) which we call mean field theory, and which will serve as the starting point for a convergent expansion of the Green's functions, uniformly in the lattice spacing. Here we show how the approximationsS (k) Q arise and how to handle gauge fixing, necessary for the analysis of the continuum limit. We also establish stability bounds onS (k) Q , uniformly ink. This is an essential step toward proving the existence of a gap in the mass spectrum and exponential decay of gauge invariant correlations.

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References

  1. Bałaban, T., Imbrie, J., Jaffe, A.: Exact renormalization group for gauge theories, 1983 Cargese Lectures, Progress in Gauge Field Theories, pp. 79–104. G. 't Hooft et al. (ed.). New York: Plenum Press 1984

    Google Scholar 

  2. Brydges, D., Fröhlich, J., Seiler, E.: On the construction of quantized gauge fields. I. General results. Ann. Phys.121, 227–284 (1979). II. Convergence of the lattice approximation. Commun. Math. Phys.71, 159–205 (1980). III. The two-dimensional abelian Higgs model without cutoffs. Commun. Math. Phys.79, 353–400 (1981)

    Google Scholar 

  3. Bałaban, T.: (Higgs)2, 3 quantum fields in a finite volume. I. A lower bound. Commun. Math. Phys.85, 603–626 (1983). II. An upper bound. Commun. Math. Phys.86, 555–594 (1982)

    Google Scholar 

  4. King, C.: The U(1) Higgs model. I. The continuum Limit, HUTMP 84/B 167. II. The infinite volume limit. HUTMP 84/B 168

  5. Bałaban, T., Brydges, D., Imbrie, J., Jaffe, A.: The mass gap for Higgs models on a unit lattice. Ann. Phys.158, 281–319 (1984)

    Google Scholar 

  6. Bałaban, T.: Propagators and renormalization transformations for lattice gauge theories. I. Commun. Math. Phys.95, 17–40 (1984). II. Commun. Math. Phys.96, 223–250 (1984)

    Google Scholar 

  7. Bałaban, T.: Regularity and decay of lattice Green's functions. Commun. Math. Phys.89, 571–597 (1983)

    Google Scholar 

  8. Federbush, P.: A phase cell approach to Yang-Mills theory. I. Small field modes. University of Michigan preprint

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Author notes

  1. Tadeusz Bałaban

    Present address: Department of Mathematics, University of Michigan, 48109, Ann Arbor, MI, USA

Authors and Affiliations

  1. Lyman Laboratory of Physics, Harvard University, 02138, Cambridge, MA, USA

    Tadeusz Bałaban, John Imbrie & Arthur Jaffe

Authors
  1. Tadeusz Bałaban
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  2. John Imbrie
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  3. Arthur Jaffe
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Additional information

Communicated by A. Jaffe

Dedicated to the memory of Kurt Symanzik

Supported in part by the National Science Foundation under Grant PHY 82-03669

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Bałaban, T., Imbrie, J. & Jaffe, A. Renormalization of the Higgs model: Minimizers, propagators and the stability of mean field theory. Commun.Math. Phys. 97, 299–329 (1985). https://doi.org/10.1007/BF01206191

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  • DOI: https://doi.org/10.1007/BF01206191

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