Application of the Mandelstam Representation to Photoproduction of Pions from Nucleons

James Stutsman Ball

doi:10.1103/PhysRev.124.2014

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Application of the Mandelstam Representation to Photoproduction of Pions from Nucleons

James Stutsman Ball^*

University of California, San Diego, La Jolla, California

^*This work was done in part under the auspices of the U. S. Atomic Energy Commission while the author was at the Lawrence Radiation Laboratory, Berkeley, California.

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Phys. Rev. 124, 2014 – Published 15 December, 1961

DOI: https://doi.org/10.1103/PhysRev.124.2014

Abstract

The Mandelstam representation is applied to the invariant amplitudes for photoproduction. By treating gauge invariance as a subsidiary condition, it is shown that the fixed-momentum-transfer dispersion relations of Chew, Goldberger, Low, and Nambu (CGLN) are probably valid without subtractions for the (-) amplitudes while a three-pion resonance would perhaps require a subtraction in the (+) amplitudes. The two-pion resonance will certainly require a subtraction for the (0) amplitudes, but to a good approximation the contribution of the two-pion intermediate state is found to produce a simple additive correction to the CGLN (0) formula. The strength of this new term is determined by a parameter $Λ$ , which has been introduced elsewhere in treating the photon, three-pion problem. Otherwise, the form of the new term can be expressed in terms of nucleon electromagnetic form factors. Finally, the photoproduction amplitudes are calculated in the threshold region, and an estimate of the size of $Λ$ is made.

References (24)

G. F. Chew and S. Mandelstam, Phys. Rev. 119, 467 (1960)
S. Mandelstam, Phys. Rev. 112, 1344 (1958) ibid.115, 1741 (1959) ibid.115, 1752 (1959)
H. S. Wong, Phys. Rev. Letters 5, 70 (1960) private communication M. Gourdin and A. Martin, Nuovo cimento 16, 78 (1960)
W. R. Frazer and J. R. Fulco, Phys. Rev. 117, 1603 (1960)
D. Y. Wong, Phys. Rev. Letters 2, 406 (1959) H. P. Noyes and D. Y. Wong, ibid. 3, 191 (1959) M. L. Goldberger, M. Grisaru, S. W. MacDowell, and D. Y. Wong, Phys. Rev. 120, 2250 (1960)
Omitted endnote
G. F. Chew, M. L. Goldberger, F. E. Low, and Y. Nambu, Phys. Rev. 106, 1345 (1957)
K. M. Watson, Phys. Rev. 95, 228 (1954)
M. Jacob and G. C. Wick, Ann. phys. 7, 404 (1959)
D. Hall and A. S. Wightman, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 31, No. 5 (1957)
S. Mandelstam, Phys. Rev. Letters 4, 84 (1960)
S. W. MacDowell, Phys. Rev. 116, 774 (1959)
G. F. Chew and F. E. Low, Phys. Rev. 101, 1579 (1956)
G. F. Chew, Ann. Rev. Nuclear Sci. 9, 29 (1959)
W. R. Frazer and J. R. Fulco, Lawrence Radiation Laboratory (private communication) S. Frautschi and D. Walecka, Department of Physics, University of California (private communication)
W. R. Frazer and J. R. Fulco, Phys. Rev. 117, 1609 (1960)
G. F. Chew, Phys. Rev. Letters 4, 142 (1960)
R. Oehme and J. G. Taylor, Phys. Rev. 113, 371 (1959)
G. F. Chew and D. Y. Wong, Lawrence Radiation Laboratory (private communication)
H. Y. Chiu and E. L. Lomon, Ann. phys. 6, 50 (1959)
S. W. Barnes, B. Rose, G. Giacomelli, J. Ring, K. Miyake, and K. Kinsey, Phys. Rev. 117, 238 (1960)
V. I. Goldansky, B. B. Govorkov, and R. G. Vassikov, Soviet Phys.—JETP 7, 37 (1960)
A. Barbaro, E. L. Goldwasser, and D. Carlson-Lee, Bull. Am. Phys. Soc. 4, 273 (1959) M. Beneventano, G. Bernardini, D. Carlson-Lee, G. Stoppini, and L. Tau, Nuovo cimento 4, 323 (1956)
C. S. Robinson, "Tables of cross sections for $π^{+}$ production from hydrogen, according to the theory of Chew, Goldberger, Low, and Nambu," University of Illinois Report, May 22, 1959 (unpublished)

[c1] G. F. Chew and S. Mandelstam, Phys. Rev. 119, 467 (1960)

[c2] S. Mandelstam, Phys. Rev. 112, 1344 (1958) ibid.115, 1741 (1959) ibid.115, 1752 (1959)

[c3] H. S. Wong, Phys. Rev. Letters 5, 70 (1960) private communication M. Gourdin and A. Martin, Nuovo cimento 16, 78 (1960)

[c4] W. R. Frazer and J. R. Fulco, Phys. Rev. 117, 1603 (1960)

[c5] D. Y. Wong, Phys. Rev. Letters 2, 406 (1959) H. P. Noyes and D. Y. Wong, ibid. 3, 191 (1959) M. L. Goldberger, M. Grisaru, S. W. MacDowell, and D. Y. Wong, Phys. Rev. 120, 2250 (1960)

[c6] Omitted endnote

[c7] G. F. Chew, M. L. Goldberger, F. E. Low, and Y. Nambu, Phys. Rev. 106, 1345 (1957)

[c8] K. M. Watson, Phys. Rev. 95, 228 (1954)

[c9] M. Jacob and G. C. Wick, Ann. phys. 7, 404 (1959)

[c10] D. Hall and A. S. Wightman, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 31, No. 5 (1957)

[c11] S. Mandelstam, Phys. Rev. Letters 4, 84 (1960)

[c12] S. W. MacDowell, Phys. Rev. 116, 774 (1959)

[c13] G. F. Chew and F. E. Low, Phys. Rev. 101, 1579 (1956)

[c14] G. F. Chew, Ann. Rev. Nuclear Sci. 9, 29 (1959)

[c15] W. R. Frazer and J. R. Fulco, Lawrence Radiation Laboratory (private communication) S. Frautschi and D. Walecka, Department of Physics, University of California (private communication)

[c16] W. R. Frazer and J. R. Fulco, Phys. Rev. 117, 1609 (1960)

[c17] G. F. Chew, Phys. Rev. Letters 4, 142 (1960)

[c18] R. Oehme and J. G. Taylor, Phys. Rev. 113, 371 (1959)

[c19] G. F. Chew and D. Y. Wong, Lawrence Radiation Laboratory (private communication)

[c20] H. Y. Chiu and E. L. Lomon, Ann. phys. 6, 50 (1959)

[c21] S. W. Barnes, B. Rose, G. Giacomelli, J. Ring, K. Miyake, and K. Kinsey, Phys. Rev. 117, 238 (1960)

[c22] V. I. Goldansky, B. B. Govorkov, and R. G. Vassikov, Soviet Phys.—JETP 7, 37 (1960)

[c23] A. Barbaro, E. L. Goldwasser, and D. Carlson-Lee, Bull. Am. Phys. Soc. 4, 273 (1959) M. Beneventano, G. Bernardini, D. Carlson-Lee, G. Stoppini, and L. Tau, Nuovo cimento 4, 323 (1956)

[c24] C. S. Robinson, "Tables of cross sections for $π^{+}$ production from hydrogen, according to the theory of Chew, Goldberger, Low, and Nambu," University of Illinois Report, May 22, 1959 (unpublished)

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Application of the Mandelstam Representation to Photoproduction of Pions from Nucleons

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