Four-Point and Five-Point Veneziano-Type Formulas on the Basis of Spectral Representations

Noboru Nakanishi

doi:10.1103/PhysRevD.2.288

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Four-Point and Five-Point Veneziano-Type Formulas on the Basis of Spectral Representations

Noboru Nakanishi^*

Applied Mathematics Department, Brookhaven National Laboratory, Upton, Long Island, New York 11973

^*On leave of absence from Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan.

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Phys. Rev. D 2, 288 – Published 15 July, 1970

DOI: https://doi.org/10.1103/PhysRevD.2.288

Abstract

The Veneziano-type formula for the scattering amplitude $F (t, s)$ is derived in the form of an integral representation under the following assumptions: (1) $F (t, s)$ has a fixed- $s$ unsubtracted dispersion relation, and (2) for $s$ fixed, $F (t, s)$ has equidistant simple poles only. The crossing symmetry, $F (t, s) = F (u, s)$ , is seen to be nothing more than a relation specifying the characters of the weight function. It is demonstrated that the Veneziano-type integral representation is more convenient than the conventional Veneziano series. A Veneziano-type integral representation for the five-point function is derived under the following assumptions, which are natural extensions of the above two assumptions for the four-point function: (1′) The five-point function has a certain double-spectral representation, and (2′) it has equidistant simple poles only in the two relevant variables.

References (14)

Omitted endnote
S. Matsuda, Phys. Rev. 185, 1811 (1969)
N. N. Khuri, Phys. Rev. 185, 1876 (1969)
Omitted endnote
K. M. Bitar, Phys. Rev. 186, 1424 (1969)
G. Tiktopoulos, Phys. Letters 31B, 138 (1970)
L. Schwartz, Théorie des Distributions (Hermann & Cie, Paris, 1950), Chap. 2
S. Mandelstam, Phys. Rev. Letters 21, 1724 (1968)
Omitted endnote
L. F. Cook, Jr. and J. Tarski, J. Math. Phys. 3, 1 (1962) O. I. Zav'yalov and V. P. Pavlov, Zh. Eksperim. i Teor. Fiz. 44, 1500 (1963) [Soviet Phys. JETP 17, 1009 (1963)]
N. Nakanishi, J. Math. Phys. 3, 1139 (1962)
[11]
K. Bardakci and H. Ruegg, Phys. Letters 28B, 342 (1968) M. A. Virasoro, Phys. Rev. Letters 22, 37 (1969)
Z. Koba and H. B. Nielsen, Nucl. Phys. B12, 517 (1969)

[c1] Omitted endnote

[c2] S. Matsuda, Phys. Rev. 185, 1811 (1969)

[c3] N. N. Khuri, Phys. Rev. 185, 1876 (1969)

[c4] Omitted endnote

[c5] K. M. Bitar, Phys. Rev. 186, 1424 (1969)

[c6] G. Tiktopoulos, Phys. Letters 31B, 138 (1970)

[c7] L. Schwartz, Théorie des Distributions (Hermann & Cie, Paris, 1950), Chap. 2

[c8] S. Mandelstam, Phys. Rev. Letters 21, 1724 (1968)

[c9] Omitted endnote

[c10] L. F. Cook, Jr. and J. Tarski, J. Math. Phys. 3, 1 (1962) O. I. Zav'yalov and V. P. Pavlov, Zh. Eksperim. i Teor. Fiz. 44, 1500 (1963) [Soviet Phys. JETP 17, 1009 (1963)]

[c11] N. Nakanishi, J. Math. Phys. 3, 1139 (1962)

[c12] [11]

[c13] K. Bardakci and H. Ruegg, Phys. Letters 28B, 342 (1968) M. A. Virasoro, Phys. Rev. Letters 22, 37 (1969)

[c14] Z. Koba and H. B. Nielsen, Nucl. Phys. B12, 517 (1969)

Physical Review D

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Four-Point and Five-Point Veneziano-Type Formulas on the Basis of Spectral Representations

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