By I. T. Todorov, D. Ter Haar
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Extra info for Analytic Properties of Feynman Diagrams in Quantum Field Theory
The set R 1 of primitive diagrams fir scalar meson scattering consists of the three diagrams with the topological structure of D 1 , D2 and D3 in Fig. 4, in which all lines are mesons. 6. The class R1 of primitive diagrams for scalar meson— nucleon scattering consists of the fourteen diagrams of Fig. 5. 5 (see appendix). It is also based on the methods presented in section 1 of this chapter (see ). 3. 1. The conjugate norms of a diagram We can reduce the number of majorizing diagrams by using some additional properties of the quadratic forms of diagrams.
On this basis the quadratic form of real momenta is majorized by a quadratic form of Euclidean momenta. 38 CHAPTER 2 Majorization of Feynman Diagrams 1. Principle of majorization. 1. The principle of majorization for a strongly connected diagram A connected diagram is called weakly connected if it contains an internal line such that when the line is cut the diagram is split into two parts; the vertices of the diagram are not to be affected by such a cut. A connected diagram which is not weakly connected is called strongly connected.
4 (with all possible enumeration of the external vertices). , 7 51 [Ch. 2 Analytic Properties of Feynman Diagrams 20 1 1 5 04 2 6 2 3 4 3 DI 1 6 5 D2 D3 6 5 D4 4 D6 05 D~ FtG. 4 2 0 4 10 10 • 3 2 o4 3 Da D2 D3 05 D6 D7 De D12 DII 013 Fio. 5 52 D4 D8 D11 R14 Majorization of Feynman Diagrams § 3] correspond to the nucleon—nucleon scattering (P i , p2 are the initial and — ps, —p 4 the final nucleon momenta). The diagrams Di [13, 24], Di [13, 42], Di [12, 34], Di [12, 43] correspond to anti-nucleon—nucleon scattering.