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ASYMPTOTIC BEHAVIOR
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point,u with the result that Eq. (13-71) reads
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[,u :,u
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+ f3(g) :g - ny(g)]r(n)(p, g,,u) =
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It is also clear that the derivation generalizes to other renormalizable theories. The f3(g) function becomes a vector field in the case of several dimensionless coupling constants. In gauge theories, abelian or not, the Green functions depend in general on the gauge parameter A, and a new term ((g, A)OIoA shows up on the left-hand side ofEq. (13-71), where
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((g'A)=mo:A(~,gO'AO)1
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A = Z3AO
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(13-74)
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A,go.A.o
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As Ao is related to the wave-function renormalization Z3 of the gauge field through
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we have
((g, A) = 2Ay(g, A)
in terms of the anomalous dimension y = im(ojom) In Z3. In nonabelian theories, the latter, as well as anomalous dimensions of other matter fields, depends on A. This is not the case for f3(g), at least for certain normalization conditions, as noticed in Chap. 12. Equation (13-71) may be compared to the analogous result (13-28) in electrodynamics. Note that there exists a relation between f3 and the anomalous dimension of the electrodynamic field y. This is a consequence of the Ward identity: e2 = e6Z3. The previous analysis also generalizes to functions involving composite operators. Each set of operators with the same dimension is multiplicatively renormalized by a renormalization matrix Z [see (8-69)]. The proper functions (of a massless theory, for simplicity) satisfy
[,u :,u
with
+ f3(g) :g - ny(g)]r:;; + Yij(g)r:;; =
yij(g)
(13-75)
o -,u o,u Zik Igo(Z -1 )kj
The rationale for the discrepancy of sign between (13-67) and (13-75) will appear soon.
13-2-3 CalIan-8ymanzik Coefficients to Lowest Orders
For the <p4 theory we could make use of the results given up to order h2 in Chap. 9. It is perhaps more instructive to derive f3 and y from the counterterms of the lagrangian. To be specific we assume here a gaussian regularization
+ m2
fro drx ea
a (p2+m2)
drxe- a (p2+ m 2) =e _ _~ _~ 2 ljA2 p2 + m
-(p2+ m 2)jA2
and compute r(2)(O), (djd p2)r(2)(O), and r(4)(O) as functions of go, rna, and A. We will not omit the tadpole terms which must cancel at the end of the calculation. For our purpose it is sufficient to compute r(2)(O) up to terms in h; we recall that in <p4 theory there is no wave-function renormalization to this order. We display the Feynman diagrams to the right of the corresponding expressions. As in previous chapters y denotes Euler's constant. Terms written as constant are meant to be independent of A.
QUANTUM FIELD THEORY
We find -m 5
1 go Z Z AZ Z - { - 2 (4n)Z [A - mo In ~ - mo(1 m6
qZ)(O)=
-1 driZ) az-(O) = 1 Z AZ { - p go 4 (In + constant) 12 (4n) mo
3 g5 A + - - - ( In ~ - y - 1 - In 2) 2 (4n)Z m6 Z g3 A - 3 ~)4 - (AZ)Z - In ~z (y + In 2) + constant ] In ~Z (4n 2 mo mo Z 3 g6 [( AZ)Z A - 4" (4n)4 In m5 - 2(y + 1 + In 2) In m5
-eX o >()
(13-76)
+ (y + 1 + In 2 ]
g6 (AZ -In ~ + Y _ 1) 4 (4n)4 m6 m5
According to (13-59) we have
dr(Z)
-Z-l
Therefore Eq. (13-76) yields Z, g, and m as functions of mo, go, and A. We substitute mZ for m6, keep only terms up to order gZ, and use the notations
I'/. =
(4n)Z
(13-77)
It follows that
Z-l = 1 + 1'/.5 (In A: + constant) + ... 12 m
I'/.
= 1'/.0Zz
{1 -
~ 1'/.0 (In ~:
- y - 1 - In 2)
(13-78)
+1'/.5[~(In ~:X -In ~: (~y+~+~In2)+constant]+ ...}
ASYMPTOTIC BEHAVIOR
Terms proportional to A2 have fortunately disappeared, leaving only logarithms. Applying the definition (13-66),
P(g) - 2(4n)2 =
all. I aIn A2
m,iXo
= 2a
aIn Z 2 [ aIn A2 + Z ao
+ a6 (~ In ~ 2
~ y - ~ - ~ In 2 2 2
2) + ...J
(13-79a)
The factor Z2 may be replaced by unity to this order. Reexpressing ao in terms of a we obtain
Similarly,
y(g) = a In A2 In Z <5(g)
Im,go
= 12(4n)4 + ...
(13-79b) (13-79c)
-y(g)
1 g2 12 (4n)4
+ ...
All coefficients are finite, as expected. Moreover, the Euler constant y and In 2 which appeared at intermediate stages as a result of our regularization scheme have disappeared.
Exercises
(a) Check that P(g) is independent of the conventions to order 1i2. (b) Study the modifications arising from an internal O(n) symmetry group.
(c) Derive from (13-70) the Callan-Symanzik equations satisfied by the functions V,rr(<P) and Zeff(<P) of Chap. 9 [Eq. (9-116)] and verify that they are obeyed perturbatively by the expressions obtained in (9-132).
We can also use the results of Chap. 12 on gauge fields to get the corresponding P function. To lowest order the relation between renormalized and bare couplings was found to be
g6 g = go [1 + (4n)2
(11 3
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