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