Current Algebra and Gauge Theories. I

Steven Weinberg

doi:10.1103/PhysRevD.8.605

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Current Algebra and Gauge Theories. I

Steven Weinberg

Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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Phys. Rev. D 8, 605 – Published 15 July, 1973

DOI: https://doi.org/10.1103/PhysRevD.8.605

Abstract

The methods of current algebra are applied to the problem of calculating corrections to the symmetries of the strong interactions in renormalizable gauge theories of the weak and electromagnetic interactions. The strong interactions are described by a neutral-vector-gluon model, so that their symmetries are just the symmetries of the quark mass matrix, and are determined by the vacuum expectation values of the weakly coupled scalar fields. Corrections to these symmetries are calculated to all orders in the gluon coupling, but only to second order in the gauge coupling $e$ . After putting the results in a gauge-invariant form, it is found that all divergences cancel in the corrections to "natural" symmetries of the strong interactions. The weak interactions can produce corrections to strong-interaction symmetries of the same order of magnitude as the electromagnetic corrections, but such "order $α$ " effects occur only as corrections to the quark mass matrix, and therefore necessarily conserve parity, strangeness, charm, etc., and may produce only isovector corrections to isotopic-spin conservation. It is suggested that these weak-interaction effects of order $α$ are responsible for the nonelectromagnetic corrections to isotopic-spin conservation which seem to be needed in calculations of mass differences and $η$ decay.

References (36)

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H. Georgi and S. L. Glashow, Phys. Rev. D 6, 2977 (1972)
S. Weinberg, Phys. Rev. Lett. 29, 1698 (1972)
H. Georgi and S. L. Glashow, Phys. Rev. D 7, 2457 (1973)
S. Weinberg, Phys. Rev. D 7, 2887 (1973)
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S. Weinberg, Phys. Rev. Lett. 27, 1688 (1971) J. Schechter and Y. Ueda, Phys. Rev. D 5, 2846 (1972) W. F. Palmer, ibid. 6, 1190 (1972) M. Weinstein, ibid. 7, 1854 (1973) T. Hagiwara and B. W. Lee, ibid. 7, 459 (1973) D. A. Dicus and V. S. Mathur, ibid. 7, 525 (1973)
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[1–5] [5] Fayyazuddin and Riazuddin, Phys. Rev. D (to be published) A. Love and G. G. Ross, Nucl. Phys. (to be published)
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Omitted endnote
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[13]
[3,5] I. Bars and K. Lane, Phys. Rev. D, to be published [7] [9] [3,5]
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[8] M. Gell-Mann, M. L. Goldberger, N. M. Kroll, and F. E. Low, Phys. Rev. 179, 1518 (1969) S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970)
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Omitted endnote
M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954) N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Inter-science, New York, 1959)
[1]
[3]
H. Georgi and S. L. Glashow, Phys. Rev. Lett. 26, 1494 (1972)

[c1] S. Weinberg, Phys. Rev. Lett. 29, 388 (1972)

[c2] H. Georgi and S. L. Glashow, Phys. Rev. D 6, 2977 (1972)

[c3] S. Weinberg, Phys. Rev. Lett. 29, 1698 (1972)

[c4] H. Georgi and S. L. Glashow, Phys. Rev. D 7, 2457 (1973)

[c5] S. Weinberg, Phys. Rev. D 7, 2887 (1973)

[c6] K. Bardakci and M. B. Halpern, Phys. Rev. D 6, 696 (1972) I. Bars, M. B. Halpern, and M. Yoshimura, Phys. Rev. Lett. 29, 969 (1972) Phys. Rev. D 7, 1233 (1973) H. Georgi, ibid. 7, 1258 (1973) H. Georgi and T. Goldman, Phys. Rev. Lett. 30, 514 (1973) A. Pais (to be published).]

[c7] S. Weinberg, Phys. Rev. Lett. 27, 1688 (1971) J. Schechter and Y. Ueda, Phys. Rev. D 5, 2846 (1972) W. F. Palmer, ibid. 6, 1190 (1972) M. Weinstein, ibid. 7, 1854 (1973) T. Hagiwara and B. W. Lee, ibid. 7, 459 (1973) D. A. Dicus and V. S. Mathur, ibid. 7, 525 (1973)

[c8] G. Feinberg, P. Kabir, and S. Weinberg, Phys. Rev. Lett. 3, 527 (1959)

[c9] [4,5] [1,3,2] G. 't Hooft, Nucl. Phys. B35, 167 (1971)

[c10] [1–5] [5] Fayyazuddin and Riazuddin, Phys. Rev. D (to be published) A. Love and G. G. Ross, Nucl. Phys. (to be published)

[c11] [1] A. Duncan and P. Schattner, Phys. Rev. D 7, 1861 (1973) D. Z. Freedman and W. Kummer, ibid. 7, 1829 (1973)

[c12] S. L. Glashow, R. Jackiw, and S.-S. Shei, Phys. Rev. 187, 1916 (1969)

[c13] Omitted endnote

[c14] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961) J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127, 965 (1962), Sec. III K. Johnson, in Nuclear Physics and Particle Physics: The Ninth Latin American School of Physics, Santiago, Chile, 1967, edited by I. Saavedra (Benjamin, New York, 1968) H. Pagels, Phys. Rev. Lett. 28, 1482 (1972) Phys. Rev. D 7, 3689 (1973)

[c15] [13]

[c16] [3,5] I. Bars and K. Lane, Phys. Rev. D, to be published [7] [9] [3,5]

[c17] P. W. Higgs, Phys. Lett. 12, 132 (1964) Phys. Rev. Lett. 13, 508 (1964) Phys. Rev. 145, 1156 (1966) F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964) G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, ibid. 13, 585 (1964) T. W. B. Kibble, Phys. Rev. 155, 1554 (1967)

[c18] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973) [3,5] [5]

[c19] R. N. Mohapatra and P. Vinciarelli, Phys. Rev. D 8, 481 (1973) Fayyazuddin and Riazuddin (unpublished) W. Kummer and K. D. Lane, Phys. Rev. D 7, 1910 (1973)

[c20] [8] M. Gell-Mann, M. L. Goldberger, N. M. Kroll, and F. E. Low, Phys. Rev. 179, 1518 (1969) S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970)

[c21] S. Coleman and S. L. Glashow, Phys. Rev. Lett. 6, 423 (1961) Phys. Rev. 134, B671 (1964)

[c22] A. Zee, Phys. Rep. 3C, 127 (1972)

[c23] D. G. Sutherland, Phys. Lett. 23, 384 (1966) J. S. Bell and D. Sutherland, Nucl. Phys. B4, 315 (1968)

[c24] K. G. Wilson, Phys. Rev. 179, 1499 (1969) [12]

[c25] R. Jackiw, R. Van Royen, and G. B. West, Phys. Rev. D 2, 2473 (1970) H. Pagels, Phys. Rev. 185, 1990 (1969)

[c26] R. Jackiw and H. J. Schnitzer, Phys. Rev. D 5, 2008 (1972) ibid.7, 3116 (1973) J. Gunion, this issue, 8, 517 (1973)

[c27] S. Weinberg, Phys. Rev. D 7, 1068 (1973) [5] L.-F. Li, Phys. Rev. D 7, 3815 (1973)

[c28] K. Fujikawa, B. W. Lee, and A. I. Sanda, Phys. Rev. D 6, 2923 (1972) Y.-P. Yao, 7, 1647 (1973) G.'t Hooft and M. Veltman, Nucl. Phys. B50, 318 (1972) T. Appelquist, J. Carazzone, T. Goldman, and H. R. Quinn, Phys. Rev. D, to be published [5]

[c29] B. W. Lee, Nucl. Phys. B9, 649 (1969) J.-L. Gervais and B. W. Lee, B12, 627 (1969)

[c30] S. Weinberg, Phys. Rev. 118, 838 (1960) J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965), Sections 19.10, 19.11, and 19.14

[c31] K. Wilson, unpublished report [24] W. Zimmermann, in Lectures on Elementary Particles and Quantum Field Theory (M.I.T. Press, Cambridge, Mass., 1970) [30] C. Callan, Phys. Rev. D 5, 3202 (1972)

[c32] Omitted endnote

[c33] M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954) N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Inter-science, New York, 1959)

[c34] [1]

[c35] [3]

[c36] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 26, 1494 (1972)

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Current Algebra and Gauge Theories. I

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Current Algebra and Gauge Theories. II. Non-Abelian Gluons

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