Model for <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mo>−</mo><mi>π</mi></math></span> Scattering Satisfying Analyticity and Crossing Symmetry

J. W. Moffat

doi:10.1103/PhysRevD.3.1222

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Model for $π - π$ Scattering Satisfying Analyticity and Crossing Symmetry

J. W. Moffat^*,†

Department of Physics, University of Toronto, Toronto, Canada

^*Supported in part by the National Research Council of Canada.
^†Part of this work was done while the author was a visitor at Northern Illinois University, Dekalb, Ill. 60115 and Argonne National Laboratory, Argonne, Ill. 60439.

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Phys. Rev. D 3, 1222 – Published 1 March, 1971

DOI: https://doi.org/10.1103/PhysRevD.3.1222

Abstract

A crossing-symmetric model for $π - π$ scattering is presented which satisfies the Mandelstam representation, has a finite number of resonances associated with an exchange-degenerate trajectory that turns over at high energies, and which has Regge asymptotic behavior in all channels. The Pomeranchukon amplitude is nonresonating and has background cuts. The total amplitude satisfies the Adler condition. Satellites are included that eliminate all the odd-daughter (ghost) resonances. The double-spectral functions are calculated and shown to have, except for the lack of curvature, the correct boundaries determined by elastic unitarity. The structure of the second-sheet singularities is briefly discussed. The $π - π$ scattering lengths are calculated and found to be consistent with those obtained from current algebra, when terms of order ${m_{π}}^{2}$ and unitarity corrections are neglected.

Comments & Replies

$η \to 3 π$ and $K \to 3 π$ Decays in a Model Satisfying Analyticity, Crossing Symmetry, and Approximate Unitarity

Charles Picciotto and Donald Smith

Phys. Rev. D 6, 1450 (1972)

References (28)

S. Mandelstam, Phys. Rev. 112, 1344 (1958)
G. F. Chew, S-Matrix Theory of Strong Interactions (Benjamin, New York, 1962) G. F. Chew and S. Mandelstam, Phys. Rev. 119, 467 (1960)
J. W. Moffat, Phys. Rev. 121, 926 (1961) B. H. Bransden and J. W. Moffat, Nuovo Cimento 21, 505 (1961) ibid.23, 598 (1962) G. F. Chew and S. Mandelstam, ibid. 19, 752 (1961)
G. Veneziano, Nuovo Cimento 57A, 190 (1968)
J. W. Moffat, Nuovo Cimento 64A, 485 (1969)
R. C. Johnson and J. W. Moffat, Toronto report, 1969 (unpublished) I. O. Moen and J. W. Moffat, Nuovo Cimento Letters 3, 473 (1970)
R. Dolen, D. Horn, and C. Schmid, Phys. Rev. 166, 1768 (1968) Kwan Wu Lai and J. Louie, Nucl. Phys. B19, 205 (1970)
R. H. Graham and R. C. Johnson, Phys. Rev. 188, 2362 (1969) Phys. Rev. D 2, 2114(E) (1970)
J. Shapiro and J. Yellin, Yadern. Fiz. 11, 443 (1970) [Soviet J. Nucl. Phys. 11, 247 (1970)] J. Shapiro, Phys. Rev. 179, 1345 (1969) A. Yahil, ibid. 185, 1786 (1969) C. Lovelace, Phys. Letters 28B, 265 (1968)
E. Del Giudice and G. Veneziano, Nuovo Cimento Letters 3, 363 (1970)
S. L. Adler, Phys. Rev. 137, B1022 (1965) ibid.139, B163 (1965)
J. W. Moffat, Toronto report, 1970 (unpublished)
Particle Data Group, Rev. Mod. Phys. 41, 1 (1969)
I. O. Moen (private communication)
F. Cerulus and A. Martin, Phys. Letters 8, 80 (1964)
G. G. Beznogikh et al., Phys. Letters 30B, 274 (1969)
H. Harari, Phys. Rev. Letters 20, 1395 (1968) P. G. O. Freund, ibid. 20, 235 (1968)
R. J. Eden, Phys. Rev. 121, 1567 (1961) P. Landshoff, J. C. Polkinghorne, and J. G. Taylor, Nuovo Cimento 19, 939 (1961) R. J. Eden, P. Landshoff, J. C. Polkinghorne, and J. G. Taylor, Phys. Rev. 122, 307 (1961) D. Atkinson, Nucl. Phys. B7, 375 (1968) ibid.B13, 415 (1969)
J. G. Taylor, Nuovo Cimento 22, 92 (1961)
A. O. Barut, The Theory of the Scattering Matrix (MacMillan, London, 1967), p. 214
Y. Nambu, Phys. Rev. Letters 4, 380 (1960) J. Bernstein, S. Fubini, M. Gell-Mann, and W. Thirring, Nuovo Cimento 17, 757 (1960)
S. Weinberg, Phys. Rev. Letters 17, 616 (1966)
([22])
A. Martin, Nuovo Cimento 42, 930 (1966) ibid.47, 265 (1967)
B. H. Bransden, P. G. Burke, J. W. Moffat, R. G. Moorhouse, and D. Morgan, Nuovo Cimento 30, 207 (1963) S. Mandelstam, Ann. Phys. (N. Y.) 21, 302 (1963) N. F. Bali, G. F. Chew, and S. Y. Chiu, Phys. Rev. 150, 1352 (1966) P. D. B. Collins and R. C. Johnson, ibid. 177, 2472 (1969) ibid.185, 2020 (1969)
J. W. Moffat, Nuovo Cimento Letters 2, 773 (1969) A. O. Barut and J. W. Moffat, Phys. Rev. D 1, 532 (1970)
S. Fubini and G. Veneziano, Nuovo Cimento 64A, 811 (1969) K. Kikkawa, B. Sakita, and M. A. Virasoro, Phys. Rev. 184, 1701 (1969)
T. H. Burnett, D. J. Gross, A. Neveu, J. Scherk, and J. H. Schwarz, Phys. Letters 32B, 115 (1970)

[c1] S. Mandelstam, Phys. Rev. 112, 1344 (1958)

[c2] G. F. Chew, S-Matrix Theory of Strong Interactions (Benjamin, New York, 1962) G. F. Chew and S. Mandelstam, Phys. Rev. 119, 467 (1960)

[c3] J. W. Moffat, Phys. Rev. 121, 926 (1961) B. H. Bransden and J. W. Moffat, Nuovo Cimento 21, 505 (1961) ibid.23, 598 (1962) G. F. Chew and S. Mandelstam, ibid. 19, 752 (1961)

[c4] G. Veneziano, Nuovo Cimento 57A, 190 (1968)

[c5] J. W. Moffat, Nuovo Cimento 64A, 485 (1969)

[c6] R. C. Johnson and J. W. Moffat, Toronto report, 1969 (unpublished) I. O. Moen and J. W. Moffat, Nuovo Cimento Letters 3, 473 (1970)

[c7] R. Dolen, D. Horn, and C. Schmid, Phys. Rev. 166, 1768 (1968) Kwan Wu Lai and J. Louie, Nucl. Phys. B19, 205 (1970)

[c8] R. H. Graham and R. C. Johnson, Phys. Rev. 188, 2362 (1969) Phys. Rev. D 2, 2114(E) (1970)

[c9] J. Shapiro and J. Yellin, Yadern. Fiz. 11, 443 (1970) [Soviet J. Nucl. Phys. 11, 247 (1970)] J. Shapiro, Phys. Rev. 179, 1345 (1969) A. Yahil, ibid. 185, 1786 (1969) C. Lovelace, Phys. Letters 28B, 265 (1968)

[c10] E. Del Giudice and G. Veneziano, Nuovo Cimento Letters 3, 363 (1970)

[c11] S. L. Adler, Phys. Rev. 137, B1022 (1965) ibid.139, B163 (1965)

[c12] J. W. Moffat, Toronto report, 1970 (unpublished)

[c13] Particle Data Group, Rev. Mod. Phys. 41, 1 (1969)

[c14] I. O. Moen (private communication)

[c15] F. Cerulus and A. Martin, Phys. Letters 8, 80 (1964)

[c16] G. G. Beznogikh et al., Phys. Letters 30B, 274 (1969)

[c17] H. Harari, Phys. Rev. Letters 20, 1395 (1968) P. G. O. Freund, ibid. 20, 235 (1968)

[c18] R. J. Eden, Phys. Rev. 121, 1567 (1961) P. Landshoff, J. C. Polkinghorne, and J. G. Taylor, Nuovo Cimento 19, 939 (1961) R. J. Eden, P. Landshoff, J. C. Polkinghorne, and J. G. Taylor, Phys. Rev. 122, 307 (1961) D. Atkinson, Nucl. Phys. B7, 375 (1968) ibid.B13, 415 (1969)

[c19] J. G. Taylor, Nuovo Cimento 22, 92 (1961)

[c20] A. O. Barut, The Theory of the Scattering Matrix (MacMillan, London, 1967), p. 214

[c21] Y. Nambu, Phys. Rev. Letters 4, 380 (1960) J. Bernstein, S. Fubini, M. Gell-Mann, and W. Thirring, Nuovo Cimento 17, 757 (1960)

[c22] S. Weinberg, Phys. Rev. Letters 17, 616 (1966)

[c23] ([22])

[c24] A. Martin, Nuovo Cimento 42, 930 (1966) ibid.47, 265 (1967)

[c25] B. H. Bransden, P. G. Burke, J. W. Moffat, R. G. Moorhouse, and D. Morgan, Nuovo Cimento 30, 207 (1963) S. Mandelstam, Ann. Phys. (N. Y.) 21, 302 (1963) N. F. Bali, G. F. Chew, and S. Y. Chiu, Phys. Rev. 150, 1352 (1966) P. D. B. Collins and R. C. Johnson, ibid. 177, 2472 (1969) ibid.185, 2020 (1969)

[c26] J. W. Moffat, Nuovo Cimento Letters 2, 773 (1969) A. O. Barut and J. W. Moffat, Phys. Rev. D 1, 532 (1970)

[c27] S. Fubini and G. Veneziano, Nuovo Cimento 64A, 811 (1969) K. Kikkawa, B. Sakita, and M. A. Virasoro, Phys. Rev. 184, 1701 (1969)

[c28] T. H. Burnett, D. J. Gross, A. Neveu, J. Scherk, and J. H. Schwarz, Phys. Letters 32B, 115 (1970)

Physical Review D

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Model for $π - π$ Scattering Satisfying Analyticity and Crossing Symmetry

Abstract

Comments & Replies

$η \to 3 π$ and $K \to 3 π$ Decays in a Model Satisfying Analyticity, Crossing Symmetry, and Approximate Unitarity

References (28)

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Model for π−π Scattering Satisfying Analyticity and Crossing Symmetry

Abstract

Comments & Replies

η→3π and K→3π Decays in a Model Satisfying Analyticity, Crossing Symmetry, and Approximate Unitarity

References (28)

Model for $π - π$ Scattering Satisfying Analyticity and Crossing Symmetry

$η \to 3 π$ and $K \to 3 π$ Decays in a Model Satisfying Analyticity, Crossing Symmetry, and Approximate Unitarity