Figure 8-11 Flow of hard momenta through a diagram, and the corresponding contracted diagram. in .NET framework

Creator PDF-417 2d barcode in .NET framework Figure 8-11 Flow of hard momenta through a diagram, and the corresponding contracted diagram.

Figure 8-11 Flow of hard momenta through a diagram, and the corresponding contracted diagram.
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QUANTUM FIELD THEORY
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Thus it behaves as a power b.wir in these momenta. Therefore w" is not affected. This is not surprising since wir is the homogeneity degree of the diagram.
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We may show even more. When a single momentum p vanishes, the others being nonzero and nonexceptional, Green functions remain finite.
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The reader will find no difficulty in extending the previous simple argument to that case and in showing that w lr is at most reduced by one unit.
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This crude argument has neglected possible ultraviolet problems. In order that renormalization does not upset the result it is mandatory to choose judicious normalization conditions. Subtractions at zero momentum must be avoided, since Green's functions are generally divergent at that point. In a zero-mass theory, it is safe to choose renormalization points at euclidean values of the momenta, for example, p2 = - /1 2 < 0, instead ofEq. (8-34) or (8-36). The necessity of introducing a non-zero normalization point implies that a theory where the physical mass parameters vanish involves nevertheless a mass scale /1. The independence of the physical quantities with respect to this arbitrary choice leads to renormalization group constraints, to be discussed later. The previous considerations apply as well to theories involving both massless and massive particles, such as quantum electrodynamics. Green's functions are finite at any nonzero and nonexceptional euclidean values of the momenta. When more than one external momentum vanishes a case-by-case analysis is required. In summary, if we let all (or some of) the internal masses of a Feynman diagram go to zero, we do not encounter any singularity, provided:
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1. All the vertices have degree four. 2. The external momenta are not exceptional. 3. There is at most one soft external momentum. 4. Renormalization has been carried out at some fixed euclidean point.
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What happens when external momenta are continued from euclidean to physical on-shell values is a harder question. A corollary of the previous theorem is of interest. Consider a proper two-point function and assume that analytic continuation to the minkowskian region may be performed, avoiding threshold singularities. The Green function remains finite, and so does its absorptive part. Owing to the Cutkosky rules (see Chap. 6), this means that any total decay rate to final states involving massless particles is finite. This result due to Kinoshita is to be compared with a theorem proved by Lee and Nauenberg. According to this theorem, any transition probability in a theory involving massless particles is finite, provided summation over degenerate states is performed. This is, of course, what we found in the examples of Chaps. 4 and 7, where we were mainly concerned with soft emission. It should be understood that additional divergences may occur when energetic, collinear, massless particles are produced. These cases are also included in the previous discussion.
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RENORMALIZA nON
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8-3-2 Ultraviolet Behavior and Weinberg's Theorem
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We look for a precise relation between the superficial degree of ultraviolet divergence w( G) and the behavior of Feynman integrals when all the external momenta are large. Rescaling all these momenta by a common large factor A is equivalent, on dimensional grounds, to dividing all internal masses by the same factor A. Therefore, the problem is closely related to the massless limit considered in the previous subsection. For simplicity, consider once again a scalar theory without derivative couplings. We select a Green function evaluated at euclidean external momenta and restrict ourselves, for the time being, to the ultraviolet convergent case. After integration over a global homogeneity variable, Eq. (8-20) becomes
I (P) = r[ -w(G)/2] G (4n L
f1 n d
(j(1 _" ) [LlXlml + Q(P, IX)] ",(G)/2 L.,IXI [2I'(1X>]Z
If the P are dilated, P -dP, the integral behaves as A"'(G) [remember that w(G) < 0 by assumption] provided that
n dlXl (j(1 -
LIXI)
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