QUANTUM FIELD THEORY in .NET

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QUANTUM FIELD THEORY
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Figure 13-4 Mass insertion to the one-loop order.
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mediate renormalization at zero momentum: dr1f\p2) I dp p'=o
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(13-59)
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where m is proportional but not equal to the physical mass. Using a regularization with a large cutoff A and adding the required counterterms to the lagrangian, we obviously modify the Ward identities of scale transformations. The relation between bare and renormalized irreducible Green's functions is (13-60) where on the right-hand side the cutoff A appears in the wave-function renormalization Z, the bare mass mo, and bare coupling go, but the combination r R has a finite limit where A -+ CX) with m and g held fixed. Ordinary dimensional analysis implies that the dependence on A of the quantities Z, go, m61m2 is only through the combination A21m 2. The bare generating functional of mass insertions is (13-61) We are now in a position to study scale anomalies. A variation of mo at fixed go and A implies variations of the renormalized parameters g and m satisfying ogo ogo o = dg o = -;- dm + ~ dg um og From (13-60) and (13-61),
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2 mo r Ll (zl/2cp,mo,go) = dm om +dg og
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(13-62)
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1 -2 d(lnZ)
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d x cp(x) <5:(X)]rR (cp,m,g)
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(13-63)
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ASYMPTOTIC BEHAVIOR
while ordinary dimensional analysis implies
(13-64)
This enables us to rewrite (13-63) as
- 2 [~ omo(A/m, g)] 1 L1,b (Zl/2 <p, mo, go ) '" mo um
(13-65)
Let us anticipate the fact that the dimensionless coefficients on the left-hand 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 (13-66) 0 y(g)=-m-lnZ -,go 2 om m
(A ) --A-lnZ (A) 1 oA -,go 0 m
(13-67)
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)
(13-68)
with lL1,R(<p, m, g) finite. We define
'() 1 +ug
Z"'" mo
~ omo(A/m, go) '" um
(13-69)
assuming the limit to exist, for a well-defined normalization prescription of the insertion. Dropping the suffix R we obtain the Callan-Symanzik equation in the final form
QUANTUM FIELD THEORY
Comparing this expression to the incorrect formula (13-53) 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 (13-70) in powers of cp (even powers for the cp4 theory). In momentum space we obtain
n-l 1
L Pk' OPk r(n)(p) + P(g) -og r(n)(p) + {4 -
+ y(g)]}r(n\p)
(13-71)
For definiteness we complete the normalization conditions (13-59) by (13-72) which is verified to order h O, and is sufficient to ensure the finiteness of Applying Eq. (13-71) 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
(13-73)
which prove that y(g) and b(g) are finite. From (13-71) 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 (13-66), (13-67), and (13-69), 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. (13-70) within the finite renormalized theory. The price of such an approach is the lack of intuitive appeal. On the other hand, the Callan-Symanzik 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 Callan-Symanzik 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
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