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QUANTUM FIELD THEORY in .NET
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while ordinary dimensional analysis implies
(1364) This enables us to rewrite (1363) as
 2 [~ omo(A/m, g)] 1 L1,b (Zl/2 <p, mo, go ) '" mo um
(1365) Let us anticipate the fact that the dimensionless coefficients on the lefthand side have a finite limit as A 4 00 with g and m fixed. Again from dimensional arguments they can only depend on g in this limit. We set (1366) 0 y(g)=mlnZ ,go 2 om m (A ) AlnZ (A) 1 oA ,go 0 m
(1367) As the notation indicates the derivatives are taken at fixed go. Of course, A 4 00 is implied, which means that in perturbative calculations all terms vanishing for infinite cutoff are to be neglected. According to the analysis of Chap. 8, the mass insertion is multiplicatively renormalizable. There exists a constant Z"" such that Z""lL1,b(Zl/2<p, mo, go) lL1,R(<p, m, g) (1368) with lL1,R(<p, m, g) finite. We define
'() 1 +ug
Z"'" mo
~ omo(A/m, go) '" um
(1369) assuming the limit to exist, for a welldefined normalization prescription of the insertion. Dropping the suffix R we obtain the CallanSymanzik equation in the final form QUANTUM FIELD THEORY
Comparing this expression to the incorrect formula (1353) which neglected renormalization, we see that it differs by the appearance of terms involving the coefficients P(g), y(g), and b(g). Before proving their finiteness, we comment on their interpretation. First of all, they are similar to the anomalies of chiral in variance, modifying the classical Ward identity. We may look at y(g) as playing the role of a coupling dependent addition to the field dimension. The term in P(g)ajog is a remnant of the presence of the dimensional cutoff A in the relation between g and go. lt implies that an infinitesimal scale transformation has to be accompanied by a small change in coupling constant. Finally, b(g) could be absorbed a posteriori in a finite renormalization of the mass insertion. To prove that these coefficients are finite we expand (1370) in powers of cp (even powers for the cp4 theory). In momentum space we obtain nl 1
L Pk' OPk r(n)(p) + P(g) og r(n)(p) + {4  + y(g)]}r(n\p) (1371) For definiteness we complete the normalization conditions (1359) by (1372) which is verified to order h O, and is sufficient to ensure the finiteness of Applying Eq. (1371) in the vicinity of p = 0 for n = 2 we get two relations: y(g) y(g) _ [1 op
+ b(g) = 0 + b(g)] ori2)(~; p2) I
p2=O
(1373) which prove that y(g) and b(g) are finite. From (1371) it then follows that P(g) is also well defined. We emphasize that in general these coefficients depend on the normalization conditions. In practical calculations it will sometimes be advantageous to use (1366), (1367), and (1369), which relate them to the divergences of perturbation theory. This shows in fact that to lowest order (hi), P and y do not depend on any convention. The above presentation relies heavily on the bare theory with its infinite cutoff A. A scrupulous reader might suspect that it is possible to avoid such a detour, and derive Eq. (1370) within the finite renormalized theory. The price of such an approach is the lack of intuitive appeal. On the other hand, the CallanSymanzik equations may serve as a basis for the construction of the renormalized theory. We have emphasized the interpretation corresponding to the modifications of broken scale invariance Ward identities. This is in fact the useful aspect of the CallanSymanzik equations. They may, however, also be interpreted as renormalization group equations. This is achieved, for instance, in the massless theory by shifting the dilatation factor from the momenta to the arbitrary subtraction

