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Prog. Theor. Phys. Vol. 21 No. 1 (1959) pp. 135-150

[ Full Text PDF : FREE ACCESS (1087K) ]

On the Validity of Dispersion Relations in Perturbation Theory

Noboru Nakanishi

Department of Physics, Kyoto University, Kyoto

(Received September 10, 1958)

Abstract:

The dispersion relations with the ordinary threshold energies are proved in every order of perturbation expansion for π-π, π-K, π-N, π-Λ, π-Ξ K-K, N-N, Λ-Λ, N-Ξ, K-N, K-Λ forward scatterings. A counter-example is presented for N-Λ, N-Σ, Σ-Σ, Λ-Ξ, Σ-Ξ scatterings. Non-forward scatterings also are treated.


URL : http://ptp.ipap.jp/link?PTP/21/135/
DOI : 10.1143/PTP.21.135

[ Full Text PDF : FREE ACCESS (1087K) ] Citation:

References:

  1. M. Gell-Mann, M. L. Goldberger and W. E. Thirring, Phys. Rev. 95 (1954), 1612[APS].
  2. M. L. Goldberger, Phys. Rev. 97 (1955), 508[APS]; ibid. 99 (1955), 979[APS].
  3. N. N. Bogoliubov, B. V. Medvedev and M. K. Polivanov, Problems of the Theory of Dispersion Relations (1957, Princeton) unpublished.
    See also K. Symanzik, Phys. Rev. 105 (1957), 743[APS].
  4. H. J. Bremermann, R. Oehme and J. G. Taylor, Phys. Rev. 109 (1958), 2178[APS].
    J. G. Taylor, to be published.
  5. R. Oehme, Phys. Rev. 111 (1958), 1430[APS].
  6. R. Karplus, C. Sommerfield and F. Wichmann, Phys. Rev. 111 (1958), 1187[APS].
  7. Y. Nambu, Nuovo Cim. 9 (1958), 610, ibid. 6 (1957), 1064.
  8. K. Symanzik, Prog. Theor. Phys. 20 (1958), 690[PTP].
  9. N. Nakanishi, Prog. Theor. Phys. 17 (1957), 401[PTP].
    The proof in p. 404 should be supplemented by some words. To prove R ≡ 0, the basis momenta should be chosen so as to contain not only pk but also all independent ones among pν1′, pν2′, … pνn′. Then the circuit along which pk flows contains none of ν1′, ν2′, …, νn′.
  10. S. Matsuyama and H. Miyazawa, Prog. Theor. Phys. 19 (1958), 517[PTP].

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 22 No. 1 (1959) pp. 128-144 :
    Ordinary and Anomalous Thresholds in Perturbation Theory
    Noboru Nakanishi
  2. Progress of Theoretical Physics Vol. 22 No. 2 (1959) pp. 247-273 :
    Electromagnetic Structure of the Nucleon. I
    Kichiro Hiida, Noboru Nakanishi, Yukihisa Nogami and Masayuki Uehara
  3. Progress of Theoretical Physics Vol. 22 No. 3 (1959) pp. 351-372 :
    Electromagnetic Structure of the Nucleon. II
    Kichiro Hiida, Noboru Nakanishi, Yukihisa Nogami and Masayuki Uehara
  4. Progress of Theoretical Physics Vol. 23 No. 5 (1960) pp. 942-956 :
    Dispersion Relations in Nucleon-Nucleon Scattering
    Yasuo Hara and Hironari Miyazawa
  5. Progress of Theoretical Physics Vol. 25 No. 2 (1961) pp. 277-289 :
    New Analyses of Anomalous Imaginary Part
    Jun Otokozawa
  6. Progress of Theoretical Physics Vol. 25 No. 2 (1961) pp. 296-297 :
    Validity of the Integral Representations for the Vertex Part in Perturbation Theory
    Noboru Nakanishi
  7. Progress of Theoretical Physics Vol. 25 No. 5 (1961) pp. 753-779 :
    Dispersion Relations for Kaon-Nucleon Scattering and KKππ Interaction
    Keiji Igi
  8. Progress of Theoretical Physics Vol. 27 No. 1 (1962) pp. 133-138 :
    Some Analytic Properties of Absorptive Part of Scattering Amplitude
    Kunio Yamamoto
  9. Progress of Theoretical Physics Vol. 27 No. 6 (1962) pp. 1193-1198 :
    Dispersion Relations for the Elastic Scattering Amplitude with the Anomalous Imaginary Part
    Jun Otokozawa

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