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By I. T. Todorov, D. Ter Haar

ISBN-10: 0080165443

ISBN-13: 9780080165448

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By the operation of contracting a line to a point a diagram without tadpoles can be changed to a diagram containing a tadpole; the incidence matrix of this new graph is not defined. e. it flows out from i and also into i,). The quadratic form of such a diagram is related in the following manner to the quadratic form of the graph obtained from it by the removal (or equivalently, in this case, the contraction) of the line 1 (in accord with eq. , n). It can be shown (see [134], sect. 3) that A (cc, p), and hence Q (a, r), is a continuous function of the parameters a, including a on the boundary of the domain of integration (that is, A (a, p) approaches a well-defined limit when o„ — > 0).

9) is given by the extremum on the internal momenta of S91(a, k) = S n=1 a1k,2, . 3). This lemma plays an important role in the majorization of diagrams (Chapter 2). 37 Analytic Properties of Feynman Diagrams [Ch. 2). 2). 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.

From eq. ,n. * As an example we shall find the Euclidean region for the case of elastic scattering of a particle of mass m on a particle of mass M. 4) Pi =P3 = , Pi=Pá=m (M rm> 0). If we regard the masses as fixed, then from the vectors p' we can form two independent invariants which we will select from among the following three dependent invariants : t 12 s = (Pi + R2)2, t = (pi + P3)2 , u = (P2 + P3)2 ; 2 2 s+ t + u = pf +pi+p3 +p4= 2 (1 + m ). 5) * If two or more principal minors of the matrix (p,p') vanish, then this is not so.

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Analytic Properties of Feynman Diagrams in Quantum Field Theory by I. T. Todorov, D. Ter Haar


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