Note that the elog (Y^/fi2) term cancels, insuring that the singular term, while of order 1/e2, is nevertheless a constant, and hence a legitimate counterterm. To complete the proof (which will not be done here) the argument must be extended to graphs with more than one loop.

Overlapping Divergences

Finally, we consider the self-energy graphs and the problem of overlapping divergences. A general method for subtracting the divergences from any Feynman graph, including those with overlapping divergences, was developed by Bogoli-ubov and Parasiuk [BP 57, 80], Hepp [He 66], and Zimmerman [Zi 69, 71] and is referred to as the BPHZ method. For a general discussion of these methods, see Muta (1987). Here we will illustrate the results for the diagram shown in Fig. 16.11.

A good way to see how the BPHZ results are obtained is to begin with the Dyson equations for the self-energy. These equations are illustrated diagrammat-ically in Fig. 16.13. They are coupled, non-linear equations for the self-energy E and dressed vertex function T' expressed in terms of the dressed propagator A' (which depends on E) and the Bethe-Salpeter scattering amplitude, M, discussed in Chapter 12. However, this form of the equations is not convenient for the renormalization program because, for example, substitution of the renormalized vertex function into 16.13A would suggest only one subtraction of the counterterm Ao, which we know from our discussion in the previous section is incorrect. Two factors of A0 are needed. However, it is not necessary to write the equation for the self-energy in the form 16.13A; a completely equivalent form of this equation is shown diagrammatically in Fig. 16.14, where V is the kernel of the BS equation ■vhich the scattering amplitude M satisfies. (Recall the discussion in Chapter 12.) It is not difficult to prove that 16.14 is equivalent to 16.13A if Eq. (12.50) is used, and this proof is left as an exercise (Prob. 16.3).

Fig. 16.13 Diagrammatic representation of the Dyson equations for the vertex function and the self-energy (included in the propagator). These equations permit the determination of £ and T' if M is known.

Fig. 16.14 An alternative (and equivalent) equation for the self-energy which is more suitable for applications to renormalization.

Fig. 16.14 An alternative (and equivalent) equation for the self-energy which is more suitable for applications to renormalization.

Using Fig. 16.14, it is easy to see why the two subtraction terms shown in Fig. 16.9C and D are required in order to renormalize the fourth order self-energy. The kernel V must be at least of second order, and hence, to fourth order, 16.14B can only generate the overlapping diagram 16.8C (with a negative sign). However, Fig. 16.14A gives two of the overlapping diagrams 16.8C (one obtained from the product of the third order T in the left vertex and the first order F in the right vertex and another with these contributions interchanged) so that the sum of the two contributions is correct. But the counterterms are already of third order, so they can come only from Fig. 16.14A, and there are thus two such terms.

Next, note that the sixth order diagram in Fig. 16.11 arises, in the language of the Dyson equation, from three terms generated by Fig. 16.14A (two contributions of a fifth order T denoted by F(5) and shown in Fig. 16.15, with a first order I\ and one product of two third order T's) minus two terms generated by Fig. 16.14B (the two possible products of a third order T(3) with a first order T). The counterterms required for graph 16.11 follow from a consideration of the renormalization of r(5), as illustrated in Fig. 16.15. The full diagram 16.15A contains a subdivergence, which is regulated by the diagram with a counterterm shown in Fig 16.15B. These two diagrams then have an overall divergence, which requires the two subtractions illustrated in Figs. 16.15C and D (these two subtractions could be represented as a

Fig. 16.15 The 5th order vertex (A) with the graph containing the third order counter term (B). The counterterms generated by graphs (A) and (B) are shown in figures (C) and (D); the shaded box encloses the overall divergence.

Fig. 16.15 The 5th order vertex (A) with the graph containing the third order counter term (B). The counterterms generated by graphs (A) and (B) are shown in figures (C) and (D); the shaded box encloses the overall divergence.

Fig. 16.17 Diagrams in QED with overlapping divergences.

[self-energies need not be considered because of the Ward-Takahashi identities (11.137) and (16.76)], we may use a generalization of the relation (16.65),

where D = D1*" and S are the lowest order photon and quark propagators and r", D' = D and S' are the dressed vertex function and photon and electron propagators. With the counterterms included, the finite parts of rM, D', and S", denoted by P1, D, and S, are related to rM, D', and S' by the renormalization constants

These constants can then be absorbed into the charge (as we did in Chapter 11), and if the graph G has Np external quark lines and Ng external photon lines, then

This equation expresses G in terms of finite quantities and an overall multiplicative infinite renormalization constant. The finite part of G, GR, is then defined to be

Gr (V, D, S, e0) - (Z3)Nb/2 {Z2)Nf>2 G (V, D, S, e0) = Gs(t»,D,S,eR) .

This completes our proof that QED is renormalizable. Once it has been demonstrated that self-energy diagrams need not be explicitly considered and that therefore any diagram can be made finite by subtracting the infinities from its singular subdiagrams (self-energies or vertex parts which do not overlap) using counterterms defined in a lower order calculation, the proof is merely a matter of showing that the factors Z\, Z2, and Z3 can always be absorbed into the charge, except for some remaining overall factor associated with the external lines. This redefinition of the charge was already discussed in Chapter 11.

As an illustration of the usefulness of the Ward-Takahashi identity (16.76), in the next section we compute the vacuum polarization to fourth order. In addition to being an interesting example, the result is of practical importance. The final section of this chapter will discuss the renormalization of QCD.


As an illustration of the techniques we have developed, we will calculate the fourth order vacuum polarization in QED. There are three fourth order contributions,

corrections. Note that the vertex for the outgoing photon with polarization v is A "(fc.-g).

Now, note that the singular B{k2) parts of the (a) and (b) terms cancel exactly, and hence only finite contributions to the integrand remain. This tells us immediately that the fourth order result will only go like 1 /e (there are no 1 /e2 terms as there were in <p3 theory), so that the leading terms have the structure of Eq. (16.88) with ni = 1. Hence the leading log{-q2/^2) terms are finite. Accordingly, we can evaluate these terms by letting e —> 0 in all factors which multiply the 1/e singularity, except for the (—fi^/q2)' term, which gives the log. The final result therefore comes from only four terms: the Aj terms, two A2 terms, and the A3 terms. These will each be calculated in turn.

Consider the contribution from one of the Aj terms first. From Eq. (16.100), we see that and hence the Ai integral will diverge in four dimensions only if multiplied by a term which goes like fc3/fc6. Precisely such a term arises from the fc3 term in jyV (which did not contribute to the second order calculation because it was odd in fc), which now gives

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