Dispersive Sum-Rule Approach to <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mo>−</mo><mi>K</mi></math></span> Scattering

J. Michael McKisic

doi:10.1103/PhysRevD.2.531

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Dispersive Sum-Rule Approach to $π - K$ Scattering

J. Michael McKisic

Department of Physics, The Ohio State University, Columbus, Ohio 43210

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Phys. Rev. D 2, 531 – Published 1 August, 1970

DOI: https://doi.org/10.1103/PhysRevD.2.531

Abstract

The dispersive sum-rule method, originally developed by Fubini and Furlan, is applied to $π - K$ elastic scattering. Sum rules are derived for the $I = \frac{1}{2}$ and $I = \frac{3}{2}$ scattering amplitudes, and the isospin-antisymmetric combination of the $s$ -wave scattering lengths is calculated. These expressions contain terms involving the off-mass-shell kappa-kaon-pion coupling constant. By following a procedure introduced by Dashen and Weinstein, and assuming that the $SU (3) \times SU (3)$ symmetry-breaking part of the strong-interaction Hamiltonian transforms according to the $(3, 3^{*}) + (3^{*}, 3)$ representation of $SU (3) \times SU (3)$ , we evaluate the off-shell corrections. In the evaluation of the off-shell corrections, we obtain expressions for the postulated $κ$ -meson mass and decay width, consistent with a recent experimental indication. The $s$ -wave scattering lengths are consistent with other current-algebra and phenomenological-Lagrangian calculations, but smaller than those recently reported from calculations based on the leading term Veneziano model.

References (25)

M. Gell-Mann, Phys. Rev. 125, 1067 (1962) Physics 1, 63 (1964)
S. L. Adler and R. F. Dashen, Current Algebras and Applications to Particle Physics (Benjamin, New York, 1968)
S. Weinberg, Phys. Rev. Letters 17, 616 (1966)
H. J. Schnitzer and S. Weinberg, Phys. Rev. 164, 1828 (1967)
[4] I. S. Gerstein and H. J. Schnitzer, Phys. Rev. 175, 1876 (1968) R. Arnowitt, M. H. Friedman, P. Nath, and R. Suitor, ibid. 175, 1820 (1968)
S. Fubini and G. Furlan, Ann. Phys. (N. Y.) 48, 322 (1968)
Y. Nambu and J. J. Sakurai, Phys. Rev. Letters 18, 507 (1963)
[1] S. L. Glashow and S. Weinberg, Phys. Rev. Letters 20, 224 (1968) M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968)
R. Dashen and M. Weinstein, Phys. Rev. Letters 22, 1337 (1969)
G. Carbone, E. Donini, and S. Sciuto, Nuovo Cimento 58A, 688 (1968)
T. D. Lee, S. Weinberg, and B. Zumino, Phys. Rev. Letters 18, 129 (1967)
P. Carruthers, Introduction to Unitary Symmetry (Wiley, New York, 1966) P. Carruthers and J. P. Krisch, Ann. Phys. (N. Y.) 33, 1 (1965)
J. D. Bjorken, Phys. Rev. 148, 1467 (1966)
Omitted endnote
T. G. Trippe, C. Y. Chien, E. Malamud, J. Mellema, P. E. Schlein, W. E. Slater, D. H. Stork, and H. K. Ticho, Phys. Letters 28B, 203 (1968) (private communication from Professor Malamud to Professor Wada) D. J. Crennell, U. Karshon, K. W. Lai, J. S. O'Neall, and J. M. Scarr, Phys. Rev. Letters 22, 487 (1969)
D. W. McKay, J. M. McKisic, and W. W. Wada, Phys. Rev. 184, 1609 (1969) [Phys. Rev. D 1, 957(E) (1970)]
S. P. DeAlwis and D. A. Nutbrown, Nuovo Cimento 58, 876 (1968) D. H. Dahmen, K. D. Rothe, and L. Schülke, Nucl. Phys. B7, 472 (1968)
Omitted endnote
[8]
Y. Y. Lee, Nuovo Cimento 64A, 474 (1969) N. H. Fuchs and T. K. Kuo ibid. 64A, 382 (1969) J. Cleymans, ibid. 65A, 72 (1970)
([9])
L. K. Pande [Phys. Rev. Letters 23, 353 (1969)]
([3]) J. A. Cronin [Phys. Rev. 161, 1483 (1967)] H. Yabuki [ 170, 1410 (1968)] R. W. Griffith [ 176, 1705 (1968)]
K. Kawarabayashi, S. Kitakado, and H. Yabuki [Phys. Letters 28B, 432 (1969)] R. Arnowitt, P. Nath, Y. Srivastava, and M. H. Friedman [Phys. Rev. Letters 22, 1158 (1969)]
F. von Hippel and J. K. Kim, Phys. Rev. Letters 22, 740 (1969)

[c1] M. Gell-Mann, Phys. Rev. 125, 1067 (1962) Physics 1, 63 (1964)

[c2] S. L. Adler and R. F. Dashen, Current Algebras and Applications to Particle Physics (Benjamin, New York, 1968)

[c3] S. Weinberg, Phys. Rev. Letters 17, 616 (1966)

[c4] H. J. Schnitzer and S. Weinberg, Phys. Rev. 164, 1828 (1967)

[c5] [4] I. S. Gerstein and H. J. Schnitzer, Phys. Rev. 175, 1876 (1968) R. Arnowitt, M. H. Friedman, P. Nath, and R. Suitor, ibid. 175, 1820 (1968)

[c6] S. Fubini and G. Furlan, Ann. Phys. (N. Y.) 48, 322 (1968)

[c7] Y. Nambu and J. J. Sakurai, Phys. Rev. Letters 18, 507 (1963)

[c8] [1] S. L. Glashow and S. Weinberg, Phys. Rev. Letters 20, 224 (1968) M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968)

[c9] R. Dashen and M. Weinstein, Phys. Rev. Letters 22, 1337 (1969)

[c10] G. Carbone, E. Donini, and S. Sciuto, Nuovo Cimento 58A, 688 (1968)

[c11] T. D. Lee, S. Weinberg, and B. Zumino, Phys. Rev. Letters 18, 129 (1967)

[c12] P. Carruthers, Introduction to Unitary Symmetry (Wiley, New York, 1966) P. Carruthers and J. P. Krisch, Ann. Phys. (N. Y.) 33, 1 (1965)

[c13] J. D. Bjorken, Phys. Rev. 148, 1467 (1966)

[c14] Omitted endnote

[c15] T. G. Trippe, C. Y. Chien, E. Malamud, J. Mellema, P. E. Schlein, W. E. Slater, D. H. Stork, and H. K. Ticho, Phys. Letters 28B, 203 (1968) (private communication from Professor Malamud to Professor Wada) D. J. Crennell, U. Karshon, K. W. Lai, J. S. O'Neall, and J. M. Scarr, Phys. Rev. Letters 22, 487 (1969)

[c16] D. W. McKay, J. M. McKisic, and W. W. Wada, Phys. Rev. 184, 1609 (1969) [Phys. Rev. D 1, 957(E) (1970)]

[c17] S. P. DeAlwis and D. A. Nutbrown, Nuovo Cimento 58, 876 (1968) D. H. Dahmen, K. D. Rothe, and L. Schülke, Nucl. Phys. B7, 472 (1968)

[c18] Omitted endnote

[c19] [8]

[c20] Y. Y. Lee, Nuovo Cimento 64A, 474 (1969) N. H. Fuchs and T. K. Kuo ibid. 64A, 382 (1969) J. Cleymans, ibid. 65A, 72 (1970)

[c21] ([9])

[c22] L. K. Pande [Phys. Rev. Letters 23, 353 (1969)]

[c23] ([3]) J. A. Cronin [Phys. Rev. 161, 1483 (1967)] H. Yabuki [ 170, 1410 (1968)] R. W. Griffith [ 176, 1705 (1968)]

[c24] K. Kawarabayashi, S. Kitakado, and H. Yabuki [Phys. Letters 28B, 432 (1969)] R. Arnowitt, P. Nath, Y. Srivastava, and M. H. Friedman [Phys. Rev. Letters 22, 1158 (1969)]

[c25] F. von Hippel and J. K. Kim, Phys. Rev. Letters 22, 740 (1969)

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Dispersive Sum-Rule Approach to $π - K$ Scattering

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Dispersive Sum-Rule Approach to π−K Scattering

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Dispersive Sum-Rule Approach to $π - K$ Scattering