Self-Consistent Pomeranchukon Singularities. I

J. B. Bronzan

doi:10.1103/PhysRevD.4.1097

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Self-Consistent Pomeranchukon Singularities. I

J. B. Bronzan

Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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Phys. Rev. D 4, 1097 – Published 15 August, 1971

DOI: https://doi.org/10.1103/PhysRevD.4.1097

Abstract

We formulate a functional self-consistency equation for the Pomeranchukon pole trajectory. The only ingredients are unitarity in the $t$ channel (the channel of the Pomeranchukon) and the specification of $α (0)$ . The three solutions studied here are: $α (0) = 1$ , cut dominant; $α (0) = 1 - ε$ , cuts dominant; and $α (0) = 1$ , pole dominant. We further provide expressions for the $t$ -channel partial-wave amplitudes in terms of a small number of free parameters; these expressions are adequate for the calculation of the high- $s$ behavior of $σ_{tot}$ , $\frac{Re F (s, 0)}{Im F (s, 0)}$ , and the diffraction peak in the domain $| \frac{t [\ln (\frac{s}{s_{0}})]}{[\ln \ln (\frac{s}{s_{0}})]} | ≲ constant$ . Three prominent conclusions are: (1) multi-Pomeranchukon phase space plays a leading role in determining the relative pole-cut strength; (2) there are fixed cuts in the vacuum partial-wave amplitude that accumulate at $j = 1$ even when $α (0) = 1 - δ$ ; (3) Schwarz trajectories $α (t) = 1 + γ t^{\frac{1}{2}} + \dots$ have a special status.

References (27)

M. Froissart, Phys. Rev. 123, 1053 (1961)
J. V. Allaby et al., Phys. Letters 30B, 500 (1969)
H. Cheng and T. T. Wu, Phys. Rev. Letters 24, 1456 (1970)
S. Mandelstam, Nuovo Cimento 30, 1127 (1963) ibid.30, 1148 (1963)
V. N. Gribov, I. Ya. Pomeranchuk, and K. A. Ter-Martirosyan, Phys. Rev. 139, B184 (1965)
A. A. Anselm, Ya. I. Azimov, G. S. Danilov, I. T. Dyatlov, and V. N. Gribov, Ann. Phys. (N.Y.) 37, 227 (1966) J. B. Bronzan and C. E. Jones, Phys. Rev. 160, 1494 (1967)
Omitted endnote
T. T. Wu and C. N. Yang, Phys. Rev. 137, B708 (1965) N. Byers and C.N. Yang, ibid. 142, 976 (1966) T. T. Chou and C. N. Yang, in Proceedings of the Second International Conference on High-Energy Physics and Nuclear Structure, Rehovoth, Israel, 1967, edited by G. Alexander (North-Holland, Amsterdam, The Netherlands, 1967), pp. 348-359 Phys. Rev. 170, 1591 (1968) Phys. Rev. Letters 20, 1213 (1968) Phys. Rev. 175, 1832 (1968) J. Benecke, T. T. Chou, C. N. Yang, and E. Yen, ibid. 188, 2159 (1969)
G. F. Chew and A. Pignotti, Phys. Rev. 176, 2112 (1968) G. F. Chew, M. L. Goldberger, and F. E. Low, ibid. 180, 1577 (1969) M. L. Goldberger, C.-I Tan, and J. M. Wang, ibid. 184, 1920 (1969)
R. C. Hwa, Phys. Rev. D 1, 1790 (1970)
J. H. Schwarz, Phys. Rev. 167, 1342 (1968) T. Sawada, ibid. 165, 1848 (1968) R. E. Mickens, Nuovo Cimento 56A, 799 (1968)
V. N. Gribov, I. Yu. Kobsarev, V. D. Mur, L. B. Okun, and V. S. Popov, Phys. Letters 32B, 129 (1970) A. A. Anselm, G. S. Danilov, I. T. Dyatlov, and E. M. Levin, Yadern. Fiz. 11, 896 (1970) [Soviet J. Nucl. Phys. 11, 500 (1970)] J. Finkelstein, Phys. Rev. Letters 24, 172 (1970) R. Oehme, Phys. Rev. D 3, 3217 (1971)
V. N. Gribov, Zh. Eksperim. i Teor. Phys. 53, 654 (1967) [Soviet Phys. JETP 26, 414 (1968)]
V. N. Gribov and A. A. Migdal, Yadern. Fiz. 8, 1002 (1968) [Soviet J. Nucl. Phys. 8, 583 (1969)] V. N. GribovA. A. MigdalYadern. Fiz.8, 1213 (1968) [Soviet J. Nucl. Phys.8, 703 (1969)] V. N. GribovA. A. MigdalZh. Eksperim. i Teor. Phys. 55, 1498 (1968) [Soviet Phys. JETP 28, 784 (1969)] V. N. Gribov, Yadern. Fiz. 9, 424 (1969) [Soviet J. Nucl. Phys. 9, 246 (1969)]
R. Oehme, in Strong Interactions and High Energy Physics, edited by R. G. Moorhouse (Oliver and Boyd, London, 1964), p. 148
R. Oehme, Phys. Letters 30B, 414 (1969)
R. Oehme, Phys. Letters 32B, 573 (1970) Phys. Rev. D 2, 801 (1971) J. S. Ball and F. Zachariasen, Phys. Rev. Letters 23, 346 (1969) P. Kaus and F. Zachariasen, Phys. Rev. D 1, 2962 (1970) J. S. Ball, G. Marchesini, and F. Zachariasen, Phys. Letters 32B, 583 (1970)
[5]
Omitted endnote
[16]
D. Amati, S. Fubini, and A. Stanghellini, Nuovo Cimento 26, 896 (1962)
G. F. Chew and W. R. Frazer, Phys. Rev. 181, 1914 (1969)
J. Finkelstein and M. Jacob, Nuovo Cimento 56A, 681 (1968)
K. J. Foley, R. S. Jones, S. J. Lindenbaum, W. A. Love, S. Ozaki, E. D. Plattner, C. A. Quarles, and E. H. Willen, Phys. Rev. 181, 1775 (1969)
S. Sugiyama, in Proceedings, United States-Japan Seminar on Differential and Functional Equations, edited by W. A. Harris, Jr. and Y. Sibuya (Benjamin, New York, 1967)
A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 8
Omitted endnote

[c1] M. Froissart, Phys. Rev. 123, 1053 (1961)

[c2] J. V. Allaby et al., Phys. Letters 30B, 500 (1969)

[c3] H. Cheng and T. T. Wu, Phys. Rev. Letters 24, 1456 (1970)

[c4] S. Mandelstam, Nuovo Cimento 30, 1127 (1963) ibid.30, 1148 (1963)

[c5] V. N. Gribov, I. Ya. Pomeranchuk, and K. A. Ter-Martirosyan, Phys. Rev. 139, B184 (1965)

[c6] A. A. Anselm, Ya. I. Azimov, G. S. Danilov, I. T. Dyatlov, and V. N. Gribov, Ann. Phys. (N.Y.) 37, 227 (1966) J. B. Bronzan and C. E. Jones, Phys. Rev. 160, 1494 (1967)

[c7] Omitted endnote

[c8] T. T. Wu and C. N. Yang, Phys. Rev. 137, B708 (1965) N. Byers and C.N. Yang, ibid. 142, 976 (1966) T. T. Chou and C. N. Yang, in Proceedings of the Second International Conference on High-Energy Physics and Nuclear Structure, Rehovoth, Israel, 1967, edited by G. Alexander (North-Holland, Amsterdam, The Netherlands, 1967), pp. 348-359 Phys. Rev. 170, 1591 (1968) Phys. Rev. Letters 20, 1213 (1968) Phys. Rev. 175, 1832 (1968) J. Benecke, T. T. Chou, C. N. Yang, and E. Yen, ibid. 188, 2159 (1969)

[c9] G. F. Chew and A. Pignotti, Phys. Rev. 176, 2112 (1968) G. F. Chew, M. L. Goldberger, and F. E. Low, ibid. 180, 1577 (1969) M. L. Goldberger, C.-I Tan, and J. M. Wang, ibid. 184, 1920 (1969)

[c10] R. C. Hwa, Phys. Rev. D 1, 1790 (1970)

[c11] J. H. Schwarz, Phys. Rev. 167, 1342 (1968) T. Sawada, ibid. 165, 1848 (1968) R. E. Mickens, Nuovo Cimento 56A, 799 (1968)

[c12] V. N. Gribov, I. Yu. Kobsarev, V. D. Mur, L. B. Okun, and V. S. Popov, Phys. Letters 32B, 129 (1970) A. A. Anselm, G. S. Danilov, I. T. Dyatlov, and E. M. Levin, Yadern. Fiz. 11, 896 (1970) [Soviet J. Nucl. Phys. 11, 500 (1970)] J. Finkelstein, Phys. Rev. Letters 24, 172 (1970) R. Oehme, Phys. Rev. D 3, 3217 (1971)

[c13] V. N. Gribov, Zh. Eksperim. i Teor. Phys. 53, 654 (1967) [Soviet Phys. JETP 26, 414 (1968)]

[c14] V. N. Gribov and A. A. Migdal, Yadern. Fiz. 8, 1002 (1968) [Soviet J. Nucl. Phys. 8, 583 (1969)] V. N. GribovA. A. MigdalYadern. Fiz.8, 1213 (1968) [Soviet J. Nucl. Phys.8, 703 (1969)] V. N. GribovA. A. MigdalZh. Eksperim. i Teor. Phys. 55, 1498 (1968) [Soviet Phys. JETP 28, 784 (1969)] V. N. Gribov, Yadern. Fiz. 9, 424 (1969) [Soviet J. Nucl. Phys. 9, 246 (1969)]

[c15] R. Oehme, in Strong Interactions and High Energy Physics, edited by R. G. Moorhouse (Oliver and Boyd, London, 1964), p. 148

[c16] R. Oehme, Phys. Letters 30B, 414 (1969)

[c17] R. Oehme, Phys. Letters 32B, 573 (1970) Phys. Rev. D 2, 801 (1971) J. S. Ball and F. Zachariasen, Phys. Rev. Letters 23, 346 (1969) P. Kaus and F. Zachariasen, Phys. Rev. D 1, 2962 (1970) J. S. Ball, G. Marchesini, and F. Zachariasen, Phys. Letters 32B, 583 (1970)

[c18] [5]

[c19] Omitted endnote

[c20] [16]

[c21] D. Amati, S. Fubini, and A. Stanghellini, Nuovo Cimento 26, 896 (1962)

[c22] G. F. Chew and W. R. Frazer, Phys. Rev. 181, 1914 (1969)

[c23] J. Finkelstein and M. Jacob, Nuovo Cimento 56A, 681 (1968)

[c24] K. J. Foley, R. S. Jones, S. J. Lindenbaum, W. A. Love, S. Ozaki, E. D. Plattner, C. A. Quarles, and E. H. Willen, Phys. Rev. 181, 1775 (1969)

[c25] S. Sugiyama, in Proceedings, United States-Japan Seminar on Differential and Functional Equations, edited by W. A. Harris, Jr. and Y. Sibuya (Benjamin, New York, 1967)

[c26] A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 8

[c27] Omitted endnote

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Self-Consistent Pomeranchukon Singularities. I

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Self-Consistent Pomeranchukon Singularities. II

References (27)

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