By I. T. Todorov, D. Ter Haar
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Extra info for Analytic Properties of Feynman Diagrams in Quantum Field Theory: International Series of Monographs in Natural Philosophy
4). The second can be rewritten in the form (M — m)2
In this way we replace the vertex O by n vertices into each of which enter exactly three lines. , for example, , ch. 21, p. 237). Translator's note. The method replaces c Hence the diagram 0' by 7t goes into D 41 Analytic Properties of Feynman Diagrams [Ch. 2 gram D that has been so obtained majorizes the diagram D'. By an analogous method we can deal with vertices into which not only meson lines enter, taking into account conservation of baryon number and strangeness. Consider now the class R of strongly connected diagrams involving external nucleons and pions only such that into each vertex there enter either two baryon lines and one meson line or four meson lines.
COROLLARY 1. All strongly connected diagrams for the self-energy of a scalar meson are majorized by a diagram of type D, in which both internal lines are mesons (of mass m). COROLLARY 2. The domain of analyticity of any strongly connected diagram for the self-energy consists of the complex p 2 plane with a cut along the real axis for p 2 (M + m)2. From this, with an application of the Cauchy theorem, follows the validity of the well-known K~llen—Lehmann representation (see, for example  and Appendix to Chapter 4 below) for the nucleon Green's function to any order in perturbation theory.