ZERO-MASS LIMIT, ASYMPTOTIC BEHAVIOR, AND WEINBERG'S THEOREM in VS .NET

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8-3 ZERO-MASS LIMIT, ASYMPTOTIC BEHAVIOR, AND WEINBERG'S THEOREM
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Until now, we have sidestepped all problems related to massless particles. The object of this section is to present an heuristic discussion on some aspects of these questions and their relation with Weinberg's theorem. The latter deals with the behavior of Feynman diagrams as external momenta become very large.
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8-3-1 Massless Theories
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When dealing with massless particles in the internal propagators, we face the risk of encountering new divergences. We are going to show that nothing of this sort happens as long as we keep away from some particular values of the external momenta. In the euclidean (i.e., Wick-rotated) version of the theory, when all vertices of the lagrangian have a dimension four (or higher) the proper functions are finite at any nonzero and nonexceptional value of the external momenta. Nonexceptional momenta are configurations such that no partial sum of the incoming momenta Pi vanishes. The integrations remain finite for small values of the internal momenta because the external momenta provide a lower cutoff.
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This result breaks down in theories containing super-renormalizable couplings. For instance, the diagram of Fig. 8-lOa with one cp2 vertex (or, equivalently, a, cp2 insertion of zero momentum) is
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Figure 8-10 Infrared-divergent diagram,
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infrared divergent:
This amplitude could also be considered as a six-point function in a then it is evaluated for an exceptional configuration.
theory (Fig. 8-lOb), but
A rigorous proof requires careful slicings of the integration domain, very much as for the theorem of Sec. 8-1-4. Here we shall just give a simple argument based on power counting, discarding possible ultraviolet problems.
Infrared power counting amounts to finding out how many internal momenta may become soft in the diagram while preserving momentum conservation at each vertex. Let us consider an N-point function with nonvanishing external momenta, and assume that a definite flow through the diagram has been chosen for these hard external momenta. Since the latter are not exceptional, any internal line irrigated by them cannot have a vanishing momentum, as all loop momenta go to zero. Moreover, the same property shows that these hard internal lines form a connected pattern on the diagram (the heavy lines in the example of Fig. 8-11). Consequently, as far as infrared behavior is concerned, we can contract them into a single vertex. To this vertex are attached N hard external lines and i internal lines, with i 2: 2 since the diagram is one-particle irreducible. Let I, L, V3 , and V4 denote the number of internal lines, loops, three- and four-point vertices respectively in the contracted diagram. We have the usual topological relations
L = 1+ 1 - (V3 N
+ 21 =
+ V4 + 1) + 4V4 + (N + i)
Since, by assumption, all vertices of the lagrangian have degree four, one power of momentum is attached to each three-point vertex. Therefore, the degree of homogeneity of the contracted diagram, which gives the superficial degree of infrared divergence when all internal loop momenta are simultaneously smaIl, is Wir = 4L - 21 + V3
= i 2:
At least heuristicaIly, this ensures infrared convergence. We might wonder whether relaxing the hypothesis that all internal loop momenta after contraction are smaIl would not upset the previous power counting. For instance, let us assume that a number IH :s; I of internal momenta are kept hard and that they form a pattern with LH loops and V3H and V4H vertices of each species. It would seem that wir is changed into wir = wir - ~wir:
~wir =
+ V3H -
and that we might be in trouble. Fortunately, this is not the case. The hard pattern may be considered as a diagram (possibly disconnected), the external lines of which are all soft by construction.
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