In this chapter we return to the subject of renormalization, or the removal of infinities from field theories. An introductory discussion of this topic was presented in Chapter 11, but because this is a problem of central importance, we will now discuss it in somewhat greater depth.

The principal goal of this chapter is to introduce some of the main issues and develop the language to the point where the interested student is equipped to pursue the literature. After defining the problem and studying <fi3 theory as an example, we discuss the renormalization of QED, emphasizing the role played by gauge invariance in the form of the Ward-Takahashi identities. The chapter concludes with a brief discussion of the renormalization of QCD.


Before defining what is meant by a renormalizable theory, look at the ultraviolet behavior of a typical Feynman diagram which arises from the perturbative expansion of the theory. Diverging diagrams will be evaluated using dimensional regularization, introduced in Chapter 11. For simplicity, we will first discuss theories with no derivatives in the interaction term; the discussion will be extended to derivative interactions later. Then a typical diagram will have

I — the number of internal loops tib = the number of internal spin zero boson lines (16.1)

rip = the number of internal fermion lines and it is easy to see that if the momenta in all of its loops become large at the same time, then the overall divergence of all the integrals is determined by the quantity

where d is the dimension of space-time. This quantity is sometimes referred to as the superficial degree of divergence, or as the overall divergence of the diagram,

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