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Proof of chiral symmetry breaking in strongly coupled lattice gauge theory

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Abstract

We study chiral symmetry in the strong coupling limit of lattice gauge theory with staggered fermions and show rigorously that chiral symmetry is broken spontaneously in massless QED and the gauge-invariant Nambu-Jona-Lasinio model if the dimension of spacetime is at least four. The results for the chiral condensate as a function of the mass imply that the mean-field approximation is an upper bound for this observable which becomes exact as the dimension goes to infinity. For the model with gauge groupU(N),N=2,3,4, we prove that chiral long-range order exists at zero mass in four or more dimensions.

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References

  1. Nambu, Y.: Phys. Rev. Lett.4, 380 (1960)

    Article  ADS  Google Scholar 

  2. Nambu, Y., Jona-Lasinio, G.: Phys. Rev.122, 345 (1961)

    Article  ADS  Google Scholar 

  3. Kogut, J. B.: Rev. Mod. Phys.55, 775 (1983)

    Article  ADS  Google Scholar 

  4. Thirring, W.: Ann. Phys.3, 91 (1958)

    MATH  MathSciNet  Google Scholar 

  5. Mitter, P. K., Weisz, P. H.: Phys. Rev. D8, 4410 (1973)

    Article  ADS  Google Scholar 

  6. Gross, D., Neveu, A.: Phys. Rev. D10, 3235 (1974)

    ADS  Google Scholar 

  7. Mermin, N. D., Wagner, H.: Phys. Rev. Lett.17, 1133 (1966)

    ADS  Google Scholar 

  8. Dobrushin, R. L., Shlosman, S. B.: Commun. Math. Phys.42, 31 (1975)

    Article  MathSciNet  Google Scholar 

  9. Coleman, S.: Commun. Math. Phys.31, 259 (1973); actually this was first proven by Maison, D., Reeh, H.: Commun. Math. Phys.24, 67 (1971)

    Article  Google Scholar 

  10. Miransky, V. A.: Nuovo Cimento90A, 149 (1985)

    Google Scholar 

  11. Kogut, J. B., Dagotto, E., Kocic, A.: Phys. Rev. Lett.60, 772 (1988);61, 2416 (1988); Nucl. Phys. B317, 253, 271 (1989)

    ADS  Google Scholar 

  12. Göckeler, M., Horsley, R., Laermann, E., Rakow, P., Schierholz, G., Sommer, R., Wiese, U.-J.: Nucl. Phys. B334, 527 (1990)

    Article  ADS  Google Scholar 

  13. Hands, S., Kogut, J. B., Dagotto, E.: Nucl. Phys. B333, 551 (1990)

    Article  ADS  Google Scholar 

  14. Göckeler, M., Horsley, R., Laermann, E., Wiese, U.-J., Rakow, P., Schierholz, G., Sommer, R.: Phys. Lett.251B, 567 (1990)

    ADS  Google Scholar 

  15. Lüscher, M.: Nucl. Phys. B341, 341 (1990)

    Article  ADS  MATH  Google Scholar 

  16. Aoki, K. I.: Kyoto preprint RIFP-758 (1988)

  17. Rakow, P. E. L.: Nucl. Phys. B356, 27 (1991)

    Article  ADS  Google Scholar 

  18. Fröhlich, J., Simon, B., Spencer, T.: Commun. Math. Phys.50, 79 (1976)

    Article  Google Scholar 

  19. Fröhlich, J., Israel, R. B., Lieb, E. H., Simon, B.: Commun. Math. Phys.62, 1 (1978)

    Google Scholar 

  20. Gruber, C., Kunz, H.: Commun. Math. Phys.22, 133 (1971)

    Article  MathSciNet  Google Scholar 

  21. Heilmann, O. J., Lieb, E. H.: Commun. Math. Phys.25, 190 (1972)

    Article  MathSciNet  Google Scholar 

  22. Fisher, M. E., Stephenson, J.: Phys. Rev.132, 1411 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  23. Kogut, J., Susskind, L.: Phys. Rev. D11, 395 (1975) Banks, T., Kogut, J., Susskind, L.: Phys. Rev. D13, 1043 (1976) Banks, T., Raby, S., Susskind, L., Kogut, J., Jones, D. R. T., Scharbach, P. N., Sinclair, D.: Phys. Rev. D15, 1111 (1976) Susskind, L.: Phys. Rev. D16, 3031 (1977)

    ADS  Google Scholar 

  24. Sharatchandra, H., Thun, H., Weisz, P.: Nucl. Phys. B192, 205 (1981)

    Article  ADS  Google Scholar 

  25. Wilson, K.: Phys. Rev. D10, 2455 (1974)

    ADS  Google Scholar 

  26. Berezin, F. A.: The method of second quantization. New York: Academic Press 1966

    Google Scholar 

  27. Bałaban, T., Brydges, D., Imbrie, J., Jaffe, A.: Ann. Phys.158, 281 (1984)

    Google Scholar 

  28. Salmhofer, M.: PhD thesis, MPI-PAE-Th 80/90

  29. Lanford, O., Ruelle, D.: Commun. Math. Phys.13, 194 (1969)

    Article  MathSciNet  Google Scholar 

  30. Rossi, P., Wolff, U.: Nucl. Phys. B248, 105 (984) Wolff, U.: Nucl. Phys. B280, 680 (1987)

    Google Scholar 

  31. Samuel, S.: J. Math. Phys.21, 2695 (1980)

    ADS  MathSciNet  Google Scholar 

  32. Bars, I.: J. Math. Phys.21, 2678 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  33. Brézin, E., Gross, D.: Phys. Lett.97B, 120 (1980)

    ADS  Google Scholar 

  34. Kluberg-Stern, H., Morel, A., Napoly, O., Petersson, B.: Nucl. Phys. B190, 504 (1981) Kluberg-Stern, H., Morel, A., Petersson, B.: Nucl. Phys. B215, 527 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  35. Fisher, M. E.: Phys. Rev.124, 1664 (1961)

    ADS  MATH  MathSciNet  Google Scholar 

  36. Salmhofer, M., Seiler, E.: Lett. Math. Phys.21, 13 (1991)

    Article  MathSciNet  Google Scholar 

  37. Mermin, N. D.: J. Math. Phys.8, 1061 (1967)

    Article  Google Scholar 

  38. Kasteleyn, P. W.: Physica, Grav.27, 1209 (1961)

    Google Scholar 

  39. Temperley, H. N. V., Fisher, M. E.: Phil. Mag. Serie86, 1664 (1961)

    Google Scholar 

  40. Kasteleyn, P. W.: J. Math. Phys.4, 287 (1963)

    Article  MathSciNet  Google Scholar 

  41. McCoy, B., Wu, T. T.: The two-dimensional Ising model. Cambridge, MA: Harvard University Press 1973

    Google Scholar 

  42. Hartwig, R.: J. Math. Phys.7, 286 (1966)

    Article  MathSciNet  Google Scholar 

  43. Penrose, O., Lebowitz, J. L.: Commun. Math. Phys.39, 165 (1974)

    Article  MathSciNet  Google Scholar 

  44. Vladimirov, V. S.: Methods of the theory of functions of many complex variables. Cambridge, MA: M.I.T. Press 1966

    Google Scholar 

  45. Osterwalder, K., Seiler, E.: Ann. Phys.110, 440 (1978)

    MathSciNet  Google Scholar 

  46. 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

    Google Scholar 

  47. Reed, M., Simon, B.: Methods of modern mathematical physics, vol. 1. New York: Academic Press 1980

    Google Scholar 

  48. Erdélyi, A. (ed.): The Bateman manuscript project: higher transcendental functions, vol. 2. New York: McGraw-Hill 1953

    Google Scholar 

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Authors and Affiliations

  1. Max-Planck-Institut für Physik und Astrophysik, Werner-Heisenberg Institut, Föhringer Ring 6, W-8000, München 40, Federal Republic of Germany

    Manfred Salmhofer & Erhard Seiler

Authors
  1. Manfred Salmhofer
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  2. Erhard Seiler
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Communicated by J. Fröhlich

Address after August 1991: Mathematics Department, University of British Columbia, Vancouver, Canada V6T1Y4

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Salmhofer, M., Seiler, E. Proof of chiral symmetry breaking in strongly coupled lattice gauge theory. Commun.Math. Phys. 139, 395–431 (1991). https://doi.org/10.1007/BF02352501

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  • DOI: https://doi.org/10.1007/BF02352501

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