By I. T. Todorov, D. Ter Haar
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Extra resources for Analytic Properties of Feynman Diagrams in Quantum Field Theory
O',)m2 — o' 1M 2 + Sb1(q - m). n * Translator's note. As an example, consider the diagrams D and D' below (equal masses, scalar lines). 2). 1). Hence in this example, D and D' have the same domain of analyticity. This is not surprising because we can regard the lines 1, 2, 3 of D as a vertex renormalization of the vertex a of D'. a 1 3 44 D D' Majorization of Feynman Diagrams § 1] If we set ai a i = k m t , k — i (a i + ap +i . + ap) M2 then p ~y M2 — m }rR 7 hD. 3, QD < QD' • Hence G E(D') GE(D).
10) Remark. We note that the first singularity along the rea1r2 axis for the diagram D of Fig. 2 is the branch point (M + m)2. This is the square of the sum of the masses on the two internal lines, after the cut of which the diagram is split into two parts. This property is not accidental. 4). 2. 1. 4. The class R 1 of primitive diagrams for the set R of strongly connected diagrams of the meson-nucleon vertex part consists of the two diagrams D1 and D2 shown in Fig. 3a. ~ ~1 D2 D1 (b) (a) FiG. 10), one must show that there exists a (distribution valued) limit of the function G(z), when z approaches one side of the cut.
11) v=1 where 91 is extremized on the f = 1 — n + 1 independent internal momenta. The momenta k v can be assigned to any f internal lines, the removal of which does not violate the connectedness of the diagram. Renumbering the lines, we can assume that the first f lines have the above property. We consider the following set of ! vector equations for the vectors k1. , k J = t,, where t j are arbitrarily given four-dimensional vectors. This system has a unique solution which is a linear combination of the external momenta R ; and the arbitrary internal momenta t.
Analytic Properties of Feynman Diagrams in Quantum Field Theory by I. T. Todorov, D. Ter Haar