PiG = PlG= in VS .NET

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L yEV (-Ty) G n v L n (-Ty) G
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is the solution ofEq. (8-45). First, we remark that to the empty forest corresponds the term G. Second, if G is a renormalization part, we may associate to every V two forests of the U type, U1 = V and U2 = {G} u V. Therefore Eq. (8-49) leads in this case to PlG = (1 - TG)PiG whereas if w(G) < 0, PlG = PiG by definition of the V. In either case, this is in agreement with Eqs. (8-41) and (8-42). To prove that Eq. (8-49) is indeed the solution to Eq. (8-45), we proceed by induction, assuming that it has been proved for arbitrary diagrams up to a given number of loops L - 1. If G has L loops, let us write Eq. (8-45) and insert Eq. (8-49) for each ~Y' We get
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(8-50)
YanYb= 0
The notations are cumbersome but obvious. It is then a matter of a simple inspection to check that Eq. (8-50) generates all the terms of (8-49), corresponding to forests of extremal elements Yb' .. , Ys' By definition, an extremal element of a forest is an element Y not included in any other one of the forest. The summation over the disjoint sets {Yb"" Ys} generates all the forests of G once and only once.
We have been discrete about the choice of normalization conditions. Since the nature of the Taylor subtraction depends on this choice, the most convenient one is the intermediate renormalization defined above, i.e., the subtraction at zero momentum. When a different point is chosen, for instance, in massless theories where subtracting at zero momentum is forbidden (see Sec. 8-3-1), some care is required to maintain Lorentz in variance. Supplementary subtractions, i.e., beyond the degree w(G) prescribed by power counting, may be performed consistently in an analogous way. We shall not develop this point here, and refer the reader to the references (see also some remarks at the end of Sec. 8-2-6). The construction of [Jl has involved only disjoint renormalization parts, i.e., without any common vertex or line. As a point of consistency, let us show that the renormalized integrand for an articulate (or one-vertex reducible) diagram may be factorized (Fig. 8-6). As an example, we consider the diagram of Fig. 8-6b of the four-dimensional cp4 theory. The forests U are
Hence
QUANTUM FIELD THEORY
Figure 8-6 Articulate (one-vertex reducible) diagrams.
For such a diagram, TG = T y , T". Since G, Yl, and Yz have the same degree of divergence, and since a Taylor expansion of degree w does not affect a polynomial of degree w, we have
Thus,
[JilG
also reads
[JilG =
(1 - TyJ.J'y,(1 - T").J',, =
[Jil"[Jil,,
This proof is readily generalized. It is straightforward to verify that in the case where all the renormalization parts of G are nested, Eq. (8-49) reduces to the product of operators (1 - T,) over all the y. The general case consists in expanding the same product and omitting all terms corresponding to overlapping subdiagrams. These two statements are a mere generalization of the properties found in the particular examples at the end of Sec. 8-2-2.
The result expressed in Eq. (8-49) is a major achievement but is not completely satisfactory until we prove that it leads to a convergent integral. Especially embarrassing is the case of overlapping divergences. Is the prescription contained in Eq. (8-49) really sufficient to provide a finite expression in such instances As we shall see, the answer is "yes." The proof of this result will render the use of an intermediate regularization unnecessary since the subtraction of the integrand leads to a convergent Feynman integral. However, it is often more convenient to deal with regularized amplitudes rather than with the cumbersome expression (8-49). Before giving a sketchy proof of convergence, we will first discuss the form of the subtractions in the parametric representation.
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