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where [fin(P) is finite as the cutoff is removed and D is a polynomial in the P and the m of degree less or equal to w( G).
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8-2 RENORMALIZATION 8-2-1 Normalization Conditions and Structure of the Counterterms
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We shall now come to grips with the renormalization operation itself. Let us once again repeat the general ideas and emphasize the outcome of this operation. Our aim is to express the proper Green functions in terms of renormalized Feynman integrals associated with the initial diagrams. This may be achieved by means of three equivalent procedures. In the first approach, the one presented in Chap. 7, we add to the initial lagrangian a formal series (in h) of counterterms. This in turn amounts to an order-by-order redefinition of the parameters of the theory: the "bare" parameters appearing in the lagrangian are implicit functions of the renormalized ones. The former are unobservable and divergent as the regularization is removed, while the latter are the real finite parameters of the theory, mass, coupling constants, etc. Finally, we will see in the next subsections that these two procedures are equivalent to an algorithm of subtraction of the integrand. This operation, due to Bogoliubov, has the merit of providing, diagram by diagram, a finite result without any intermediate recourse to a regularization. We will try to juggle as skilfully as possible with these three equivalent approaches and to use the most appropriate one for each problem. For instance, we shall first discuss the recursive construction of the counterterms and then use the relation between bare and renormalized theories to stress the multiplicative character of the renormalization process. Finally, Bogoliubov's subtraction opera-
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QUANTUM FIELD THEORY
tion will offer a convenient framework for an (heuristic) proof of the fundamental convergence theorem. The construction of counterterms proceeds by induction. We assume that the theory has been made finite up to a given order hL - 1 (i.e., a number L - 1 of/oops), through the introduction of judicious counterterms. According to power counting [Eqs. (8-13) to (8-18)J, to the next order hL , only a finite number of proper functions have a nonnegative superficial degree of divergence. Except for possible cancellations arising from the existence of symmetries, we associate to each of them a local monomial in the fields and their derivatives. It is, of course, a Lorentz scalar; its structure reflects the nature of the Green function since it must contribute itself to the process under study and its extra contribution to this function to that order must cancel its divergence. There remains, of course, a finite arbitrariness. After introduction of a regularization, the coefficients of the counterterms are completely determined by normalization conditions, imposed on the superficially divergent Green functions. If these normalization conditions are satisfied to lowest order (the tree approximation as implied by the initial Lagrangian), demanding that the Green functions satisfy them order by order fixes unambiguously not only the infinite but also the finite part of the counterterms. For instance, in the renormalizable <p4 theory of a scalar field of mass m, the two-point function r(2) is quadratically divergent [w(r(2) = 2]. We demand that the renormalized r~) satisfies (8-30) This is a natural condition for the physical mass, since the interpretation of the theory in terms of particles requires that the complete propagator G~)(p2) = i[r}l)(p2)]-1 has a pole of residue i at p2 = m2. To order L, the regularized function q;~ already renormalized up to order L - 1 reads
rrt;l(P2)
q~)(p2, m2, A2)
+ p2 ~lr(2)(m2, N) + ~2r(2)(m2, A2)
(8-31)
where q~)(p2, m2, A2) is finite as N -+ 00, whereas ~lr(2)(m2, A2) behaves at worst as a power of In (A2jm 2), and ~2r(2) as A2 [times In P (A2jm 2)]. The counterterm to order L reads
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