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Prog. Theor. Phys. Vol. 25 No. 3 (1961) pp. 361-368

Integral Representation of Absorptive Part of Vertex Function

Kunio Yamamoto

Department of Physics, Osaka University, Osaka

(Received November 4, 1960)

Abstract:

On the basis of the Lorentz invariance, local commutativity and mass spectral conditions, it is shown that the absorptive part of the vertex function A(z₁, z₂, σ²) has the integral representation in the form
A(z₁, z₂, σ²) = \intdm₁dm₂dm₃ φ(σ, m₁, m₂, m₃) A^p (z₁, z₂, σ²; m₁, m₂, m₃),
provided that z₁ and z₂ are real negative, where A^p is that of the lowest order perturbation theory and m_i is the mass of the virtual particle. The vanishing region of the weight function φ is determined by the mass spectral conditions. As an immediate consequence of this representation, the usual proof of the dispersion relation of the vertex function is given. If we add the information derivable from the perturbation theory to this representation, we can say that the dispersion relation always holds and the threshold is not lower than the lowest threshold of the vertex function in the lowest order perturbation theory which satisfies the mass spectral condition. It seems to us that Jost's example has not this integral representation. Finally it is conjectured that the non-vanishing region of the weight function is narrowed by introducing the conservation of the nucleon number.

URL : http://ptp.ipap.jp/link?PTP/25/361/
DOI : 10.1143/PTP.25.361

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References:

1. G. Källén and A. Wightman, Kgl. Dan. Mat. -fys. Skrifter 1 (1958), No. 6.
2. R. Oehme, Phys. Rev. 117 (1960), 1151[APS].
3. R. Karplus, C. M. Sommerfield and E. H. Wichmann, Phys. Rev. 111 (1958), 1187[APS]; ibid. 114 (1959), 376[APS].
4. R. Jost. Helv. Phys. Acta 31 (1958), 263.
5. F. J. Dyson, Phys. Rev. 110 (1958), 1460[APS].
6. R. Oehme, Nuovo Cim. 13 (1959), 778.
7. G. Konisi and K. Yamamoto, Prog. Theor. Phys. 25 (1961), 461[PTP].

Citing Article(s) :

1. Progress of Theoretical Physics Vol. 25 No. 3 (1961) pp. 461-466 :

   Transition Amplitudes in Perturbation Theory and Dyson's Integral Representation

   Gaku Konisi and Kunio Yamamoto

2. Progress of Theoretical Physics Vol. 25 No. 4 (1961) pp. 720-721 :

   Double Dispersion Representation of Vertex Function

   Kunio Yamamoto

3. Progress of Theoretical Physics Vol. 25 No. 6 (1961) pp. 1056-1057 :

   Note on the Integral Representation of Absorptive Part of Vertex Function

   Kunio Yamamoto

4. Progress of Theoretical Physics Vol. 27 No. 1 (1962) pp. 133-138 :

   Some Analytic Properties of Absorptive Part of Scattering Amplitude

   Kunio Yamamoto

5. Progress of Theoretical Physics Vol. 27 No. 4 (1962) pp. 682-692 :

   The Mass Spectral Condition and Analytic Properties of the Vertex Function

   Gaku Konisi

6. Progress of Theoretical Physics Vol. 28 No. 2 (1962) pp. 258-264 :

   Use of the Requirement of Baryon Number Conservation in Quantum Field Theory

   Kunio Yamamoto

7. Progress of Theoretical Physics Vol. 28 No. 6 (1962) pp. 1080-1099 :

   Baryon Number Conservation and the Jost-Lehmann-Dyson Representatin

   Gaku Konisi and Kunio Yamamoto

8. Progress of Theoretical Physics Vol. 41 No. 1 (1969) pp. 252-263 :

   Connection between the Absorptive Part of the Scattering Amplitude and That of the Feynman Integral Associated with the Tetrahedron Graph

   Masatsugu Minami and Hideo Miyata

9. Progress of Theoretical Physics Vol. 42 No. 1 (1969) pp. 82-107 :

   Vertex Functions at Large Momentum Transfer. I

   Masatsugu Minami

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