Current Algebra, <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mover><mrow><mi>K</mi></mrow><mrow><mi>¯</mi></mrow></mover><mrow><mi>l</mi><mn>3</mn><mn></mn></mrow><mrow></mrow><mrow></mrow><mrow><mn>0</mn></mrow></mmultiscripts></mrow></math></span> Form Factors, and Radiative <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mover><mrow><mi>K</mi></mrow><mrow><mi>¯</mi></mrow></mover><mrow><mi>l</mi><mn>3</mn><mn></mn></mrow><mrow></mrow><mrow></mrow><mrow><mn>0</mn></mrow></mmultiscripts></mrow></math></span> Decay

Harold W. Fearing; Ephraim Fischbach; Jack Smith

doi:10.1103/PhysRevD.2.542

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Current Algebra, ${\bar{K}}_{l 3}^{0}$ Form Factors, and Radiative ${\bar{K}}_{l 3}^{0}$ Decay

Harold W. Fearing^*

Ephraim Fischbach^†

Jack Smith

The Institute for Theoretical Physics, State University of New York, Stony Brook, New York 11790

The Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmakr
The Institute for Theoretical Physics, State University of New York, Stony Brook, New York 11790

The Institute for Theoretical Physics, State University of New York, Stony Brook, New York 11790

^*Air Force Office of Scientific Research Postdoctoral Fellow, 1968-1969. Address after Sept. 1, 1970: Los Alamos Scientific Laboratory, Los Alamos, N. M.
^†Present address.

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Phys. Rev. D 2, 542 – Published 1 August, 1970

DOI: https://doi.org/10.1103/PhysRevD.2.542

Abstract

The complete gauge-invariant matrix element for the decays ${\bar{K}}^{0} \to π^{+} l^{-} \bar{ν} γ$ ( $l = e$ or $μ$ ) is derived using the soft-photon theorems of Low and of Adler and Dothan. These theorems, along with several corollaries of them, are reviewed in detail and their application demonstrated by reference to the radiative ${\bar{K}}_{l 3}^{0}$ decay mode. The square of the matrix element is calculated using the theorem of Burnett and Kroll, and is compared with the result of a direct computer evaluation of the appropriate traces. Structure-dependent terms are discussed, and the dominant terms among those linear in the photon energy are estimated in the softpion and kaon limits. Results for the radiative photon spectra are given, together with the decay rates for a specific value of the minimum photon energy $E_{0}$ .

References (26)

E. Fischbach and J. Smith, Phys. Rev. 184, 1645 (1969) H. W. Fearing, E. Fischbach, and J. Smith, Phys. Rev. Letters 24, 189 (1970) N. Cabibbo in Proceedings of the Thirteenth International Conference on High-Energy Physics, Berkeley, 1966 (University of California Press, Berkeley, 1967), p. 34 C. T. Murphy in University of Michigan Bubble Chamber Group Research Note No. 58/66 (unpublished)
F. E. Low, Phys. Rev. 110, 974 (1958)
S. L. Adler and Y. Dothan, Phys. Rev. 151, 1267 (1966)
T. H. Burnett and N. M. Kroll, Phys. Rev. Letters 20, 86 (1968)
J. S. Bell and R. Van Royen, Nuovo Cimento 60A, 62 (1969)
E. Nyman, Phys. Rev. 170, 1628 (1968) R. Ferrari, Nuovo Cimento 48A, 898 (1968) R. Ferrari and M. Rosa-Clot, ibid. 56A, 582 (1968)
J. Steinberger, in Proceedings of the Topical Conference on Weak Interactions, CERN Report No. 69-7, 1969, p. 291 (unpublished) D. Dorfan, J. Enstrom, D. Raymond, M. Schwartz, S. Wojcicki, D. H. Miller, and M. Paciotti, Phys. Rev. Letters 19, 987 (1967) S. Bennett, D. Nygren, H. Saal, J. Steinberger, and J. Sutherland, ibid. 19, 993 (1967) ibid.19, 997 (1967) S. Bennett, D. Nygren, H. Saal, J. Sutherland, J. Steinberger, and K. Kleinknecht, Phys. Letters 27B, 244 (1968)
G. R. Evans, M. Golden, J. Muir, K. J. Peach, I. A. Budakov, H. W. K. Hopkins, W. Krenz, F. A. Nezrick, and R. G. Worthington, Phys. Rev. Letters 23, 427 (1969)
[2] S. Barshay and T. Yao, Phys. Rev. 171, 1708 (1968) H. Feshbach and D. R. Yennie, Nucl. Phys. 37, 150 (1962)
H. Chew, Phys. Rev. 123, 377 (1961)
J. Pestieau, Phys. Rev. 160, 1555 (1967)
N. Brene, M. Roos, and A. Sirlin, Nucl. Phys. B6, 255 (1968)
M. Gell-Mann, Physics 1, 63 (1964)
Omitted endnote
G. W. Intemann, Phys. Rev. 181, 1866 (1969)
W. A. Bardeen and W. K. Tung, Phys. Rev. 173, 1423 (1968)
Omitted endnote
Omitted endnote
C. G. Callan and S. B. Treiman, Phys. Rev. Letters 16, 153 (1966) R. Oehme, ibid. 16, 215 (1966)
Omitted endnote
M. Gourdin, Unitary Symmetry (North-Holland, Amsterdam, 1967), p. 100
R. E. Marshak, Riazuddin, and C. P. Ryan, Theory of Weak Interactions in Particle Physics (Wiley-Interscience, New York, 1969), pp. 359-363
J. Bernstein, Elementary Particles and Their Currents (W. H. Freeman and Co., San Francisco, 1968), p. 272 H. T. Nieh, Phys. Rev. Letters 19, 43 (1967) S. L. Glashow, H. J. Schnitzer, and S. Weinberg, ibid. 19, 139 (1967)
Fayyazuddin and Riazuddin, Phys. Rev. Letters 18, 715 (1967)
S. Weinberg, Phys. Rev. Letters 18, 507 (1967)
J. W. Cronin in Proceedings of the Fourteenth International Conference on High-Energy Physics, Vienna, 1968, edited by J. Prentki and J. Steinberger (CERN, Geneva, 1968), p. 284

[c1] E. Fischbach and J. Smith, Phys. Rev. 184, 1645 (1969) H. W. Fearing, E. Fischbach, and J. Smith, Phys. Rev. Letters 24, 189 (1970) N. Cabibbo in Proceedings of the Thirteenth International Conference on High-Energy Physics, Berkeley, 1966 (University of California Press, Berkeley, 1967), p. 34 C. T. Murphy in University of Michigan Bubble Chamber Group Research Note No. 58/66 (unpublished)

[c2] F. E. Low, Phys. Rev. 110, 974 (1958)

[c3] S. L. Adler and Y. Dothan, Phys. Rev. 151, 1267 (1966)

[c4] T. H. Burnett and N. M. Kroll, Phys. Rev. Letters 20, 86 (1968)

[c5] J. S. Bell and R. Van Royen, Nuovo Cimento 60A, 62 (1969)

[c6] E. Nyman, Phys. Rev. 170, 1628 (1968) R. Ferrari, Nuovo Cimento 48A, 898 (1968) R. Ferrari and M. Rosa-Clot, ibid. 56A, 582 (1968)

[c7] J. Steinberger, in Proceedings of the Topical Conference on Weak Interactions, CERN Report No. 69-7, 1969, p. 291 (unpublished) D. Dorfan, J. Enstrom, D. Raymond, M. Schwartz, S. Wojcicki, D. H. Miller, and M. Paciotti, Phys. Rev. Letters 19, 987 (1967) S. Bennett, D. Nygren, H. Saal, J. Steinberger, and J. Sutherland, ibid. 19, 993 (1967) ibid.19, 997 (1967) S. Bennett, D. Nygren, H. Saal, J. Sutherland, J. Steinberger, and K. Kleinknecht, Phys. Letters 27B, 244 (1968)

[c8] G. R. Evans, M. Golden, J. Muir, K. J. Peach, I. A. Budakov, H. W. K. Hopkins, W. Krenz, F. A. Nezrick, and R. G. Worthington, Phys. Rev. Letters 23, 427 (1969)

[c9] [2] S. Barshay and T. Yao, Phys. Rev. 171, 1708 (1968) H. Feshbach and D. R. Yennie, Nucl. Phys. 37, 150 (1962)

[c10] H. Chew, Phys. Rev. 123, 377 (1961)

[c11] J. Pestieau, Phys. Rev. 160, 1555 (1967)

[c12] N. Brene, M. Roos, and A. Sirlin, Nucl. Phys. B6, 255 (1968)

[c13] M. Gell-Mann, Physics 1, 63 (1964)

[c14] Omitted endnote

[c15] G. W. Intemann, Phys. Rev. 181, 1866 (1969)

[c16] W. A. Bardeen and W. K. Tung, Phys. Rev. 173, 1423 (1968)

[c17] Omitted endnote

[c18] Omitted endnote

[c19] C. G. Callan and S. B. Treiman, Phys. Rev. Letters 16, 153 (1966) R. Oehme, ibid. 16, 215 (1966)

[c20] Omitted endnote

[c21] M. Gourdin, Unitary Symmetry (North-Holland, Amsterdam, 1967), p. 100

[c22] R. E. Marshak, Riazuddin, and C. P. Ryan, Theory of Weak Interactions in Particle Physics (Wiley-Interscience, New York, 1969), pp. 359-363

[c23] J. Bernstein, Elementary Particles and Their Currents (W. H. Freeman and Co., San Francisco, 1968), p. 272 H. T. Nieh, Phys. Rev. Letters 19, 43 (1967) S. L. Glashow, H. J. Schnitzer, and S. Weinberg, ibid. 19, 139 (1967)

[c24] Fayyazuddin and Riazuddin, Phys. Rev. Letters 18, 715 (1967)

[c25] S. Weinberg, Phys. Rev. Letters 18, 507 (1967)

[c26] J. W. Cronin in Proceedings of the Fourteenth International Conference on High-Energy Physics, Vienna, 1968, edited by J. Prentki and J. Steinberger (CERN, Geneva, 1968), p. 284

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Current Algebra, ${\bar{K}}_{l 3}^{0}$ Form Factors, and Radiative ${\bar{K}}_{l 3}^{0}$ Decay