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EIGHT
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RENORMALIZA TION
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Renormalization to all orders is at the heart of field theory. After an introduction devoted to a study of various regularization methods we present the BogoliubovZimmermann subtraction scheme. We give indications on the convergence proof of renormalized integrals, and study ultraviolet behavior (Weinberg's theorem) and massless theories. Renormalization of composite operators is briefly discussed. The interplay between gauge invariance and renormalization is dealt with in the last part of the chapter.
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8-1 REGULARIZATION AND POWER COUNTING 8-1-1 Introduction
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This chapter is devoted to a systematic study of the renormalization procedure. The underlying philosophy has already been presented in the preceding chapter, in the case of quantum electrodynamics. We have seen that divergences may be absorbed in a redefinition of the various parameters, mass, coupling constant, etc., of the theory. It is convenient to get rid of the difficulties specific to electrodynamics, namely, gauge invariance, and to study first the renormalization of a scalar theory. The new problems arising from the existence of symmetries will be considered at the end of this chapter for quantum electrodynamics, and in Chaps. 11 and 12 for other internal symmetries.
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RENORMALIZATION
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Figure 8-1 A divergent diagram and the associated counterterm.
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The nature and properties of renormalization were first formulated and studied by the founding fathers of quantum field theory, Tomonaga, Feynman, Dyson, Schwinger, etc. Important contributions were then made by Salam, Weinberg, Bogoliubov and Parasiuk, and Hepp. More recently, Zimmermann and his followers have further clarified the systematics of the renormalization operation, while Epstein and Glaser presented an axiomatic approach. The mathematical nature of the problem is clear. Divergences occur in perturbative computations because of lack of care in the multiplication of distributions. For instance, the amputated self-energy diagram of Fig. 8-1a is the illdefined expression [GF(YI - Y2)J 2. As it stands, this expression does not make sense; in momentum space, its Fourier transform is logarithmically divergent. As we saw in Chap. 7, such divergent expressions must be subtracted. Since we demand that the subtractions be local in configuration space, this operation amounts to a redefinition of the parameters of the lagrangian by an infinite amount. Therefore, we abandon the idea of using or observing the parameters of the initial lagrangian, the so-called "bare" quantities, and reexpress everything in terms of the finite "renormalized" and observable parameters. In the previous instance, we replace G}(YI - Y2) by
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II(YI - Y2) = [GF(YI - Y2)]2 - S(YI - Yz)
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where S is a distribution concentrated at the origin, chosen in such a way as to make the whole expression II meaningful. Here a term proportional to a b function suffices. In Fourier representation
[G (
)J2 FYI - Y2 -
is replaced, for instance, by the well-defined expression II Yl
Y2) -.
f ( - -f
d p iP'(Y'-Y2) (2n)4 e
d p e ip (Y'-Y2) (2n)4
d k 1 (2n)4 (k 2 _ m 2 + ie)[(p - k - m 2 + ieJ
d k (2n)4
_ m 2 + ieJ - (k 2 -
x [(k2 _ m2
+ ie)[(p1_ k)2
~2 + ie ]
(8-1)
and we may write formally
Strictly speaking, the constant A is infinite, but formally the second term in (8-1) may be regarded as coming from the zeroth-order counterterm depicted in Fig. 8-1h. By imposing a definite normalization condition on II, we may fix unambiguously the finite part of A. Let us emphasize that this renormalization
QUANTUM FIELD THEORY
procedure works because of the locality and reality of the subtraction and hence of the counterterm. This subtraction operation may be presented in a systematic way, as we shall see in the following. If only a finite number of types of additional terms is required in the lagrangian to eliminate the divergences of Green functions, the renormalized theory will depend only on a finite number of parameters. Such a theory is called renormalizable or super-renormalizable. It remains to prove that the renormalized integrals are indeed finite and satisfy the constraints of locality and unitarity. We shall evoke briefly these points. A slightly different approach, proposed by Dyson and Schwinger, relies on a set of integral equations between proper Green functions (Sec. 10-1). It also unfortunately involves infinite multiplicative renormalization constants at intermediate stages. Finally, the most orthodox procedure of Epstein and Glaser relies directly on the axioms of local field theory in configuration space. It is free of mathematically undefined quantities but hides the multiplicative structure of renormalization. The latter will lead under proper interpretation to the study of the renormalization group (Chap. 13).
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