Fig. 16.8 The three diagrams which contribute to the fourth order self-energy, with their diverging subdiagrams enclosed in a shaded box. Diagrams (A) and (B) have nested divergences, while (C) has overlapping divergences.

the subtraction by a factor of two as needed, giving

The troublesome e 1 log(-q2//j,2) term has been canceled, the infinite part is constant, and the renormalization program can be carried through.

We conclude this section with a brief review of the fourth order self-energy calculation which we have just completed. Evaluation of the self-energy requires the calculation of the diagrams shown in Fig. 16.8. The subdiagrams with divergences are enclosed by a shaded box, and in diagrams A and B these subdiagrams (with divergences arising from an internal self-energy) are nested within the overall divergence of the skeleton diagram. Each of those subdivergences is removed by counterterms associated with them, drawn in Fig. 16.9A and B. The final sum of these four diagrams does not completely reduce the divergence to the strength of the original skeleton, which is —■ 1 /e, but it does cancel the serious

EB + = <72-f-rg + - (constant) + ■ • • 6(47r)b e

Fig. 16.9 The graphs with counterterms which must be added to the graphs shown in Fig. 16.8 in order to give a finite result.

Fig. 16.9 The graphs with counterterms which must be added to the graphs shown in Fig. 16.8 in order to give a finite result.

e log(-<?2/m ) term, giving a new singular self-energy of the form with a new constant counterterm of the form

where the Ci are constants.

The third diagram, Fig. 16.8C, has an overlapping structure; the subdivergences are enclosed by shaded boxes which overlap. There are two subdivergences associated with either p —> oo, k finite, or k —> oo, p finite (in the notation of Fig. 16.6). Because there are two subdivergences (even though there is only a single diagram), we must remove each of these subdivergences, which requires the two counterterms shown in Fig. 16.9C and D. It is the generation of two (or more) counterterms by a single diagram associated with the presence of overlapping divergences which complicates the general proof of renormalizability. In the next section we discuss some features of the problem in the general case. After this, we turn to a discussion of QED.

We now return to the general discussion which was interrupted at the end of Sec. 16.1. As we observed there, the proof of renormalizability requires that we demonstrate, order-by-order in perturbation theory, that a finite number of counterterms can be introduced (one for each term in the original Lagrangian) which will render all Feynman diagrams of that order finite. If this procedure can be shown to work to all orders, we will have proved that a theory is renormalizable.

The first step in a general proof is to identify those diagrams or subdiagrams which are divergent. For superficially renormalizable theories with a <p3 structure (i.e., with an interaction Lagrangian which is a product of only three fields), the divergent diagrams are usually limited to the self-energy and vertex corrections, and the counterterms will be defined by these graphs. An explicit example of how these terms are defined was given for 4? theory in the last section. In fact, if the counterterms are to have the same structure as the original terms which occurred in the Lagrangian, they must be limited to self-energies [which generate counterterms associated with the kinetic energy and mass terms in the free Lagrangian; recall Eq. (16.30)] and vertex corrections [which generate counterterms associated with the interaction term; recall Eq. (16.14)]. If there are <fi4-type terms, as there are in QCD, additional four-point counterterms can be expected to be present.

Fig. 16.10 Examples of graphs with insertions and their corresponding skeletons.

The proof that a superficially renormalizable theory with a structure is, in tact, renormalizable proceeds in two stages. First, we discuss all diagrams with three or more external particles (i.e., diagrams which are not self-energies), and then we consider self-energies. Examples of nth order diagrams with three or more external particles are given in Fig. 16.10 and an example of a sixth order self-energy diagram is given in Fig. 16.11. From the previous discussion, we know that the divergences can only come from subdiagrams which contain vertex or self-energy insertions or from the overall diagram if it is a vertex correction or self-energy. In both of these figures, these diverging parts are enclosed by a shaded box. For graphs with three or more external particles, the singularities from vertex insertions and self-energies do not overlap, and the graphs have a well-defined skeleton, which is the diagram with all self-energy or vertex insertions removed (or collapsed to a point). Examples of diagrams (A) and their skeletons (B) are shown in Fig. 16.10. The proof that such diagrams cannot have overlapping divergences (except for those contained inside of self-energy insertions) will not be given; its truth may seem evident from the examination of many examples, and a discussion can be found in Vol. 2 of Bjorken and Drell (1964). However, as previously emphasized, self-energy diagrams can have overlapping divergences. In the example shown in Fig. 16.11, there are four diverging subgraphs and several

where Ao and Bo are the counterterms [calculated to lowest order in (f>3 theory in Eq. (16.24)]. Then, if the skeleton of G is not a vertex correction, the finite part of the graph G will be defined to be gr(i\a) = gs (r'-r0l a (£-£„))

In words: the finite part of a diagram with vertex or self-energy insertions is obtained by using renormalized vertex or self-energy insertions. The proof that Gr is finite will be regarded as more or less self-evident: if the skeleton Gs is finite (which it is if Gs is neither a vertex correction nor a self-energy), then it is permissible to take the limit e —> 0 before doing the final integration, and the finiteness of GR follows from the finite behavior of f - r0 and E - E0.

However, if Gs is a vertex correction, GR as defined in (16.68) may require a new additional overall subtraction, and in this case the finite part will be defined to be

G„(r, A) = Gs (r' - r0, A (E - E0)) - Go , (16.69)

where Go is a new, higher order counterterm. The proof that this GR is finite for vertex functions is less obvious. It requires that we show that the infinite part of Gs (r' - r0, A (E - E0)) is a constant (independent of momenta) and hence is a legitimate counterterm. In view of the calculations carried out in Sec. 16.2 it may be useful to sketch a more general demonstration of this point.

Consider the insertion of a self-energy bubble in an nth order single-loop diagram, as shown in Fig. 16.12A. The corresponding skeleton (with counterterm) is shown in Fig. 16.12B. In massless 03 theory, these two diagrams have the form

J (2ttYJ

where e = 6 - d and /(e) can be read from Eq. (16.24)

p is the four-momentum of the line with the insertion, and f dy is the Feynman parameterized integral over the other n - 1 internal lines. Using (16.36), we can write

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