Dispersion Sum Rules for Fixed-<span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>u</mi></math></span> <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>N</mi></math></span> Scattering Amplitudes

Yu-Chien Liu; Ian J. McGee

doi:10.1103/PhysRevD.3.183

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Dispersion Sum Rules for Fixed- $u$ $π N$ Scattering Amplitudes

Yu-Chien Liu and Ian J. McGee

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

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Phys. Rev. D 3, 183 – Published 1 January, 1971

DOI: https://doi.org/10.1103/PhysRevD.3.183

Abstract

We review the fixed $- u$ finite-energy sum rules for pion-nucleon scattering amplitudes. We also derive and test the continuous-dispersion sum rules for these amplitudes. The real part of the amplitudes is restricted to appear on the right-hand cut ( $N_{π} \to N_{π}$ ) only, because no information is available for the left-hand cut ( $π π \to N \bar{N}$ ). The main result is an evaluation of the $σ$ , $f$ , and $g$ coupling constants.

References (20)

S. Mandelstam, Rept. Progr. Phys. 22, 99 (1962) T. T. Chou and M. Dresden, Rev. Mod. Phys. 39, 143 (1967)
S. Mandelstam, Phys. Rev. 115, 1741 (1959)
G. F. Chew, The Dynamical S Matrix (Benjamin, New York, 1966)
D. Griffiths and W. Palmer, Phys. Rev. 161, 1606 (1967) D. S. Beder and J. Finkelstein, ibid. 160, 1363 (1967) G. G. Volkov, V. V. Ezhela, and V. S. Zamiralov, Yadern. Fiz. 8, 1028 (1968) [Soviet J. Nucl. Phys. 8, 593 (1969)]
V. Barger, C. Michael, and R. J. N. Phillips, Phys. Rev. 185, 1852 (1969) B. Kayser, Phys. Rev. D 1, 306 (1970)
V. Gribov, L. Okun', and I. Pomeranchuk, Zh. Eksperim. i Teor. Fiz. 45, 1114 (1963) [Soviet Phys. JETP 18, 769 (1964)] R. Carlitz and M. Kislinger, Phys. Rev. Letters 24, 186 (1970)
Y. C. Liu and I. J. McGee, Phys. Rev. D 1, 3123 (1970)
Omitted endnote
S. W. MacDowell, Phys. Rev. 116, 774 (1959)
Omitted endnote
J. Hamilton, in High Energy Physics, edited by E. H. S. Burhop (Academic, New York, 1967), Vol. 1
C. F. Chew, M. L. goldberger, F. E. Low, and Y. Nambu, Phys. Rev. 106, 1337 (1957)
A. Barbaro-Galtieri et al., Rev. Mod. Phys. 42, 87 (1970)
W. R. Frazer and J. R. Fulco, Phys. Rev. 117, 1603 (1960)
C. Lovelace, in Pion-Nucleon Scattering, edited by G. L. Shaw and D. Y. Wong (Interscience, New York, 1969)
D. J. Herndon, A. Barbaro-Galtieri, and A. H. Rosenfeld, LRL Report No. UCRL-20030 πN, 1970 (unpublished)
Omitted endnote
Omitted endnote
A. D. Martin and C. Michael, Phys. Letters 32B, 297 (1970)
F. Halzen et al., Phys. Letters 32B, 111 (1970)

[c1] S. Mandelstam, Rept. Progr. Phys. 22, 99 (1962) T. T. Chou and M. Dresden, Rev. Mod. Phys. 39, 143 (1967)

[c2] S. Mandelstam, Phys. Rev. 115, 1741 (1959)

[c3] G. F. Chew, The Dynamical S Matrix (Benjamin, New York, 1966)

[c4] D. Griffiths and W. Palmer, Phys. Rev. 161, 1606 (1967) D. S. Beder and J. Finkelstein, ibid. 160, 1363 (1967) G. G. Volkov, V. V. Ezhela, and V. S. Zamiralov, Yadern. Fiz. 8, 1028 (1968) [Soviet J. Nucl. Phys. 8, 593 (1969)]

[c5] V. Barger, C. Michael, and R. J. N. Phillips, Phys. Rev. 185, 1852 (1969) B. Kayser, Phys. Rev. D 1, 306 (1970)

[c6] V. Gribov, L. Okun', and I. Pomeranchuk, Zh. Eksperim. i Teor. Fiz. 45, 1114 (1963) [Soviet Phys. JETP 18, 769 (1964)] R. Carlitz and M. Kislinger, Phys. Rev. Letters 24, 186 (1970)

[c7] Y. C. Liu and I. J. McGee, Phys. Rev. D 1, 3123 (1970)

[c8] Omitted endnote

[c9] S. W. MacDowell, Phys. Rev. 116, 774 (1959)

[c10] Omitted endnote

[c11] J. Hamilton, in High Energy Physics, edited by E. H. S. Burhop (Academic, New York, 1967), Vol. 1

[c12] C. F. Chew, M. L. goldberger, F. E. Low, and Y. Nambu, Phys. Rev. 106, 1337 (1957)

[c13] A. Barbaro-Galtieri et al., Rev. Mod. Phys. 42, 87 (1970)

[c14] W. R. Frazer and J. R. Fulco, Phys. Rev. 117, 1603 (1960)

[c15] C. Lovelace, in Pion-Nucleon Scattering, edited by G. L. Shaw and D. Y. Wong (Interscience, New York, 1969)

[c16] D. J. Herndon, A. Barbaro-Galtieri, and A. H. Rosenfeld, LRL Report No. UCRL-20030 πN, 1970 (unpublished)

[c17] Omitted endnote

[c18] Omitted endnote

[c19] A. D. Martin and C. Michael, Phys. Letters 32B, 297 (1970)

[c20] F. Halzen et al., Phys. Letters 32B, 111 (1970)

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Dispersion Sum Rules for Fixed- $u$ $π N$ Scattering Amplitudes

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Dispersion Sum Rules for Fixed-u πN Scattering Amplitudes

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References (20)

Dispersion Sum Rules for Fixed- $u$ $π N$ Scattering Amplitudes