By I. T. Todorov, D. Ter Haar
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Additional resources for Analytic Properties of Feynman Diagrams in Quantum Field Theory
3. Properties of the quadratic form Q (cc, p) for real and Euclidean momenta In the following we study the domain in the space of real momenta p in which the function Q (a, p) of a given diagram is non-zero for all non-negative a with positive sum. 3, in this domain the amplitude of the diagram is an analytic function of p. 2. If eq. 17) holds for some real momenta p, and if the diagram contains at least one closed loop L, then Q (a , r) < 0 for a„ 0, S a„ > 0. 23) = x„ = 0; then on the basis of Proof.
In other words, L(r) is a normt on the 4 (n — 1)-dimensional Euclidean space 13„ _ . The Symanzik theorem  is based on certain properties of the conjugate norm (see ). 4) i=1 subject to L(r) < 1. 22). We mention without proof the following elementary property of conjugate norms. Let L1(p), L 1(c) and L2(r), L2(c) be two pairs of mutually conjugate norms. The inequality L1(p) L2(r) holds for all p e and only if the inverse inequality L 1(x) < L 2(c) holds for the conjugate norms for all Euclidean x.
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).
Analytic Properties of Feynman Diagrams in Quantum Field Theory by I. T. Todorov, D. Ter Haar