Spontaneous Breakdown of Chiral Symmetry with Linear Realizations for Asymptotic Fields

L. Bessler; T. Muta; H. Umezawa; D. Welling

doi:10.1103/PhysRevD.2.349

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Spontaneous Breakdown of Chiral Symmetry with Linear Realizations for Asymptotic Fields

L. Bessler, T. Muta^*, H. Umezawa, and D. Welling

Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

^*Permanent address: Department of Physics, Kyoto University, Japan.

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Phys. Rev. D 2, 349 – Published 15 July, 1970

DOI: https://doi.org/10.1103/PhysRevD.2.349

Abstract

A method for considering the spontaneous breakdown of chiral symmetry without recourse to a particular Lagrangian model is presented. The physical or asymptotic fields are assigned to linear representations of chiral ${SU}_{3} \times {SU}_{3}$ together with appropriate $c$ -number addition to the scalar fields. This $c$ -number addition manifests the spontaneous breakdown of the symmetry. Expansion of the interpolating fields in terms of asymptotic fields is introduced, which, together with the spontaneously broken chiral symmetry and the arbitrariness in the choice of interpolating fields, produces sum rules relating leptonic, semileptonic, and strong-coupling constants of $0^{\pm}$ and $1^{\pm}$ mesons. Results similar to those of current algebra are obtained with some notable differences. Among the most important is the appearance of the soft-meson amplitude as a consequence of our mechanism of spontaneous breakdown. We are thereby led, in an exceedingly simple way, to generalized soft-meson sum rules. In particular, new results are given for the semileptonic decays of the $K$ meson, and generalized relations among strong-coupling constants and leptonic decays of $1^{\pm}$ mesons are derived.

Comments & Replies

Chiral Symmetry and the Kuo Transformation

S. Okubo and V. S. Mathur

Phys. Rev. D 2, 394 (1970)

References (20)

K. Nishijima, Phys. Rev. 133, B204 (1964) High Energy Physics and Elementary Particles (International Atomic Energy Agency, Vienna, 1965), p. 137
M. Gell-Mann and M. Lévy, Nuovo Cimento 16, 705 (1960) Y. Nambu, Phys. Rev. Letters 4, 380 (1960)
S. L. Adler and R. F. Dashen, Current Algebras and Applications to Particle Physics (Benjamin, New York, 1968) B. Renner, Current Algebras and Their Applications (Pergamon, Oxford, England, 1968)
S. Weinberg, Phys. Rev. Letters 18, 188 (1967) J. Schwinger, Phys. Letters 24B, 473 (1967)
S. Weinberg, Phys. Rev. 166, 1568 (1968)
S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177, 2239 (1969) C. G. Callan, Jr., S. Coleman, J. Wess, and B. Zumino, ibid. 177, 2247 (1969)
S. Kamefuchi, L. O'Raifeartaigh, and A. Salam, Nucl. Phys. 28, 529 (1961) L. Bessler, Phys. Rev. 184, 1523 (1969)
H. Umezawa, University of Wisconsin-Milwaukee Report No. UWM-4867-68-6, 1968 (unpublished)
T. Muta and H. Umezawa, University of Wisconsin-Milwaukee Report No. UWM-4867-69-4, 1969 (unpublished) M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968) S. L. Glashow and S. Weinberg, Phys. Rev. Letters 20, 224 (1968)
G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, in Advances in Particle Physics, edited by P. L. Cool and R. E. Marshak (Interscience, New York, 1968), Vol. 2 P. B. Kantor and J. L. Pientenpol, Phys. Rev. Letters 21, 241 (1968) A. Salam and J. Strathdee, International Center for Theoretical Physics Report No. IC/68/109, 1968 (unpublished) G. Cicogna, F. Strocchi, and R. Vergara Caffarelli, Phys. Rev. Letters 22, 497 (1969) J. Honerkamp, Universität Bonn Report, 1969 (unpublished)
R. Hagedorn, Introduction to Field Theory and Dispersion Relations (Pergamon, Oxford, England, 1964), p. 27 L. Leplae, R. N. Sen, and H. Umezawa, Progr. Theoret. Phys. (Kyoto) Suppl. Extra No., 637 (1965)
S. L. Adler, Phys. Rev. 137, B1022 (1965)
H. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimento 1, 205 (1955)
Omitted endnote
C. G. Callan and S. B. Treiman, Phys. Rev. Letters 16, 153 (1966) V. S. Mathur, S. Okubo and L. K. Pandit, ibid. 16, 371 (1966)
([15]) S. Weinberg, Phys. Rev. Letters 17, 336 (1966)
N. M. Kroll, T. D. Lee, and B. Zumino, Phys. Rev. 157, 1376 (1967) T. D. Lee and B. Zumino, ibid. 163, 1667 (1967)
M. Gell-Mann and F. Zachariasen, Phys. Rev. 124, 953 (1961)
S. Weinberg, Phys. Rev. Letters 18, 507 (1967)
L. Leplae, R. N. Sen, and H. Umezawa, Nuovo Cimento 49, 1 (1967) R. N. Sen and H. Umezawa, ibid. 50, 53 (1967)

[c1] K. Nishijima, Phys. Rev. 133, B204 (1964) High Energy Physics and Elementary Particles (International Atomic Energy Agency, Vienna, 1965), p. 137

[c2] M. Gell-Mann and M. Lévy, Nuovo Cimento 16, 705 (1960) Y. Nambu, Phys. Rev. Letters 4, 380 (1960)

[c3] S. L. Adler and R. F. Dashen, Current Algebras and Applications to Particle Physics (Benjamin, New York, 1968) B. Renner, Current Algebras and Their Applications (Pergamon, Oxford, England, 1968)

[c4] S. Weinberg, Phys. Rev. Letters 18, 188 (1967) J. Schwinger, Phys. Letters 24B, 473 (1967)

[c5] S. Weinberg, Phys. Rev. 166, 1568 (1968)

[c6] S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177, 2239 (1969) C. G. Callan, Jr., S. Coleman, J. Wess, and B. Zumino, ibid. 177, 2247 (1969)

[c7] S. Kamefuchi, L. O'Raifeartaigh, and A. Salam, Nucl. Phys. 28, 529 (1961) L. Bessler, Phys. Rev. 184, 1523 (1969)

[c8] H. Umezawa, University of Wisconsin-Milwaukee Report No. UWM-4867-68-6, 1968 (unpublished)

[c9] T. Muta and H. Umezawa, University of Wisconsin-Milwaukee Report No. UWM-4867-69-4, 1969 (unpublished) M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968) S. L. Glashow and S. Weinberg, Phys. Rev. Letters 20, 224 (1968)

[c10] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, in Advances in Particle Physics, edited by P. L. Cool and R. E. Marshak (Interscience, New York, 1968), Vol. 2 P. B. Kantor and J. L. Pientenpol, Phys. Rev. Letters 21, 241 (1968) A. Salam and J. Strathdee, International Center for Theoretical Physics Report No. IC/68/109, 1968 (unpublished) G. Cicogna, F. Strocchi, and R. Vergara Caffarelli, Phys. Rev. Letters 22, 497 (1969) J. Honerkamp, Universität Bonn Report, 1969 (unpublished)

[c11] R. Hagedorn, Introduction to Field Theory and Dispersion Relations (Pergamon, Oxford, England, 1964), p. 27 L. Leplae, R. N. Sen, and H. Umezawa, Progr. Theoret. Phys. (Kyoto) Suppl. Extra No., 637 (1965)

[c12] S. L. Adler, Phys. Rev. 137, B1022 (1965)

[c13] H. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimento 1, 205 (1955)

[c14] Omitted endnote

[c15] C. G. Callan and S. B. Treiman, Phys. Rev. Letters 16, 153 (1966) V. S. Mathur, S. Okubo and L. K. Pandit, ibid. 16, 371 (1966)

[c16] ([15]) S. Weinberg, Phys. Rev. Letters 17, 336 (1966)

[c17] N. M. Kroll, T. D. Lee, and B. Zumino, Phys. Rev. 157, 1376 (1967) T. D. Lee and B. Zumino, ibid. 163, 1667 (1967)

[c18] M. Gell-Mann and F. Zachariasen, Phys. Rev. 124, 953 (1961)

[c19] S. Weinberg, Phys. Rev. Letters 18, 507 (1967)

[c20] L. Leplae, R. N. Sen, and H. Umezawa, Nuovo Cimento 49, 1 (1967) R. N. Sen and H. Umezawa, ibid. 50, 53 (1967)

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Spontaneous Breakdown of Chiral Symmetry with Linear Realizations for Asymptotic Fields

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Chiral Symmetry and the Kuo Transformation

References (20)

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