Finite-Energy Sum Rules and Their Application to <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 at Fixed <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>u</mi></math></span>

Boris Kayser

doi:10.1103/PhysRevD.1.306

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Finite-Energy Sum Rules and Their Application to $π N$ Scattering at Fixed $u$

Boris Kayser

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

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Phys. Rev. D 1, 306 – Published 1 January, 1970

DOI: https://doi.org/10.1103/PhysRevD.1.306

Abstract

The finite-energy sum rule, and a class of sum rules which can be used to probe the existence of fixed poles, are obtained for amplitudes whose left- and right-hand cuts are not related by crossing symmetry. The finite-energy sum rule is evaluated for each of four independent $π N$ amplitudes with $u$ fixed at ${(1236 M e V)}^{2}$ , both sides of the resultant four sum rules being obtained from the properties of the low-energy $π N$ resonances. Results are presented for three choices of end point: ${(1356 M e V)}^{2}$ , ${(1808 M e V)}^{2}$ , and ${(2313 M e V)}^{2}$ . For the intermediate end point, all four sum rules work. For the highest one, however, they all fail. These results, while pointing to a failure of the resonance dominance approximation above 1800 MeV, give us a new confirmation of Regge high-energy behavior on the basis of low-energy data alone. In particular, they verify in some detail the relation predicted by Reggeism between the high-energy, fixed- $u$ behavior of the amplitudes and the low-energy $u$ -channel resonances. They also show that for $u = {(1236 MeV)}^{2}$ , all the $π N$ amplitudes have Regge behavior on the average (duality) above 1800 MeV. The finite-energy sum rules are shown to be violated in a fictitious universe where the lowest particle on each of the leading $π N$ Regge trajectories is accompanied by a degenerate partner of opposite parity.

References (38)

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B. Kayser, Phys. Rev. Letters 21, 1292 (1968)
G. Chew, Phys. Rev. Letters 9, 233 (1962)
Omitted endnote
S. Mandelstam and L. Wang, Phys. Rev. 160, 1490 (1967)
(Dolen, Horn, and Schmid, [1])
Omitted endnote
Omitted endnote
J. Schwarz, Phys. Rev. 159, 1269 (1967)
G. Chew, M. Goldberger, F. Low, and Y. Nambu, Phys. Rev. 106, 1337 (1957)
C. Chiu and M. DerSarkissian, Nuovo Cimento 55A, 396 (1968) V. Barger, C. Michael, and R. Phillips, Phys. Rev. 185, 1852 (1969)
Omitted endnote
Omitted endnote
N. Barash-Schmidt et al., Rev. Mod. Phys. 41, 109 (1969)
J. Sakurai, Phys. Rev. Letters 17, 1021 (1966)
V. Barger and D. Cline, Phys. Rev. 155, 1792 (1967)
V. Singh, Phys. Rev. 129, 1889 (1963)
B. Desai, Phys. Rev. Letters 17, 498 (1966)
C. Chiu and J. Stack, Phys. Rev. 153, 1575 (1967)
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([15])
[22] A. Rosenfeld et al., University of California Radiation Laboratory Report No. UCRL-8030-Rev., 1968 (unpublished)
Schmid, [2]
Omitted endnote
[15] A. Brody et al., Phys. Rev. Letters 22, 1401 (1969), [22]
Omitted endnote
C. Schmid, [2]
Omitted endnote
S. Kormanyos, A. Krisch, J. O'Fallon, K. Ruddick, and L. Ratner, Phys. Rev. Letters 16, 709 (1966)
F. Ned Dikmen, Phys. Rev. Letters 18, 798 (1967)
A. Carroll et al., Phys. Rev. Letters 20, 607 (1968)
Omitted endnote
Omitted endnote
Omitted endnote
Chew et al., [11]
[D. Beder and J. Finkelstein, Phys. Rev. 160, 1363 (1967)]

[c1] R. Dolen, D. Horn, and C. Schmid, Phys. Rev. Letters 19, 402 (1967) Phys. Rev. 166, 1768 (1968) A. Logunov, L. Soloviev, and A. Tavkhelidze, Phys. Letters 24B, 181 (1967) L. Balázs and J. Cornwall, Phys. Rev. 160, 1313 (1967) K. Igi and S. Matsuda, Phys. Rev. Letters 18, 625 (1967)

[c2] [1] F. Gilman, H. Harari, and Y. Zarmi, Phys. Rev. Letters 21, 323 (1968) S. Mandelstam, Phys. Rev. 166, 1539 (1968) D. Gross, Phys. Rev. Letters 19, 1303 (1967) C. Schmid, ibid. 20, 628 (1968) V. Barger and R. Phillips, ibid. 21, 865 (1968)

[c3] B. Kayser, Phys. Rev. Letters 21, 1292 (1968)

[c4] G. Chew, Phys. Rev. Letters 9, 233 (1962)

[c5] Omitted endnote

[c6] S. Mandelstam and L. Wang, Phys. Rev. 160, 1490 (1967)

[c7] (Dolen, Horn, and Schmid, [1])

[c8] Omitted endnote

[c9] Omitted endnote

[c10] J. Schwarz, Phys. Rev. 159, 1269 (1967)

[c11] G. Chew, M. Goldberger, F. Low, and Y. Nambu, Phys. Rev. 106, 1337 (1957)

[c12] C. Chiu and M. DerSarkissian, Nuovo Cimento 55A, 396 (1968) V. Barger, C. Michael, and R. Phillips, Phys. Rev. 185, 1852 (1969)

[c13] Omitted endnote

[c14] Omitted endnote

[c15] N. Barash-Schmidt et al., Rev. Mod. Phys. 41, 109 (1969)

[c16] J. Sakurai, Phys. Rev. Letters 17, 1021 (1966)

[c17] V. Barger and D. Cline, Phys. Rev. 155, 1792 (1967)

[c18] V. Singh, Phys. Rev. 129, 1889 (1963)

[c19] B. Desai, Phys. Rev. Letters 17, 498 (1966)

[c20] C. Chiu and J. Stack, Phys. Rev. 153, 1575 (1967)

[c21] V. Barger, in Proceedings of the 1967 International Theoretical Physics Conference on Particles and Fields, Rochester, New York, 1967, edited by C. R. Hagen (Wiley-Interscience, Inc., New York, 1968), p. 655

[c22] A. Donnachie, R. Kirsopp, and C. Lovelace, Phys. Letters 26B, 161 (1968)

[c23] ([15])

[c24] [22] A. Rosenfeld et al., University of California Radiation Laboratory Report No. UCRL-8030-Rev., 1968 (unpublished)

[c25] Schmid, [2]

[c26] Omitted endnote

[c27] [15] A. Brody et al., Phys. Rev. Letters 22, 1401 (1969), [22]

[c28] Omitted endnote

[c29] C. Schmid, [2]

[c30] Omitted endnote

[c31] S. Kormanyos, A. Krisch, J. O'Fallon, K. Ruddick, and L. Ratner, Phys. Rev. Letters 16, 709 (1966)

[c32] F. Ned Dikmen, Phys. Rev. Letters 18, 798 (1967)

[c33] A. Carroll et al., Phys. Rev. Letters 20, 607 (1968)

[c34] Omitted endnote

[c35] Omitted endnote

[c36] Omitted endnote

[c37] Chew et al., [11]

[c38] [D. Beder and J. Finkelstein, Phys. Rev. 160, 1363 (1967)]

Physical Review D

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Finite-Energy Sum Rules and Their Application to $π N$ Scattering at Fixed $u$

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Finite-Energy Sum Rules and Their Application to πN Scattering at Fixed u

Abstract

References (38)

Finite-Energy Sum Rules and Their Application to $π N$ Scattering at Fixed $u$