Parametric Representation in .NET

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6-2-3 Parametric Representation
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We shall now consider an arbitrary proper diagram G in a scalar theory and compute the corresponding contribution I(G) as given by Feynman rules. We assume that G has no tadpole, i.e., has no internal line starting and ending at the same vertex. As before, I and V denote the number of internal lines and vertices, respectively. It is convenient to give an arbitrary orientation to each internal line. Define the incidence matrix {CVI}, with indices running over vertices and internal lines respectively, as if the vertex v is the starting point of the line I if the vertex v is the endpoint of the line I / if I is not incident on v This V x I matrix characterizes the topology of the diagram. We introduce the following definitions. A subdiagram G' of G is a subset of vertices and lines of G, such that no vertex is isolated; we do not assume that, given two vertices of G', all lines connecting them in G belong to G'. A tree on G will be a subdiagram containing all vertices of G but no loop. G may be reconstructed from one of its trees by addition oflines. From Eq. (6-69), we know that a tree has V-I internal lines. Its incidence matrix, a V x (V - 1) matrix, has a rank less or equal to V-I. Let us show that the rank is indeed V-I. If we discard an arbitrary vertex of the tree, then there exists a unique one-to-one correspondence between the lines lr, ... ,lv-l and the remaining vertices Vr, ... ,VV-l, such that Cvklk "# O. For any other correspondence, one Cvklk at least vanishes. This means that the corresponding (V - 1) x (V - 1) minor of the incidence matrix equals 1. This holds true for any such minor, and thus the incidence matrix of a tree diagram has rank V-I. For an arbitrary connected diagram, the condition L = 1+ 1 - V::> 0 implies that the V x I incidence matrix has a rank p less or equal to V. Since Lv Cvl = 0 for any 1= 1, ... , I, p :s:; V-I, and since C is obtained by the addition of further columns to the rank V-I matrix of a tree subdiagram, p = V-I.
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PERTURBATION THEORY
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Let us now consider G as a Feynman diagram contributing to some proper Green function r(plo ... , Pn). The P are incoming momenta, 2:~ Pi = 0, and we denote by Pv = 2: Pk, the sum of incoming momenta at the vertex v; clearly, 2::=1 Pv = 0. The contribution I G of G to if(Pi) = (2n)4c5 4(2:Pi)ir(Pi) depends (p) only on the P, provided of course that we consider a theory without derivative couplings. This quantity has the form
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I G(p) =
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qG) (2n)4c5 4(2: P )h(P) S(G) qG)
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= S(G)
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d kl ( i ) (2 )4 k2 _ 2 +.
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n (2n) c5
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2: evlkl
(6-83)
Here S(G) is the symmetry factor of the diagram and qG) stands for all factors pertaining to the vertices; for instance, q G) = ( - iAt in the A( <p4/4!) theory. The four-momentum kl has been oriented according to the orientation chosen along the lth internal line. The theory under study might involve various species of scalar fields-hence the subscripts on the masses mi. In a scalar theory with derivative couplings or if nonzero spin fields are introduced, polynomials in k would appear in the numerator of the integrand of (6-83). This is only a minor complication and will be illustrated in practical instances in the subsequent chapters. Next we use the following integral representations of the free scalar propagator and of the c5 4 function:
(6-84a) (6-84b)
In (6-84a), the integral converges at the upper limit owing to the presence of ie. This ie will be omitted in the sequel (m 2 may be regarded as having a small imaginary part). We insert the representations (6-84) into (6-83) and boldly interchange the order of integrations. This is illegitimate if the integrals are not absolutely convergent, which occurs frequently, but will be justified in Chap. 8 by the processes of regularization and renormalization. The integrations over each four-momentum kl may be carried out easily
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