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where the commuting and anticommuting variables have been collectively denoted x and B respectively; in our problem {Xi} = {A, L}, {Bi} = {'1, K}. Then (12-161) results from the identity
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(12-162)
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(a1 a1 a a a1 a1 a a a1 a1 a a a1 a1 a a) aXi aXj aBi aB j aBi aB jaXi aXj aXi aB jaBi aXj aBi aXj aXi aBj
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a~j (:~i:!) a~j - a~j (:~i :!) a~j
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The first bracket vanishes as a consequence of anticommutativity and the two remaining terms owing to Eq. (12-162). On the other hand, any gauge-invariant functional Rinv(A) of A only satisfies
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DR inv 01 d4 X - - - = OA oK
ORinv d4 x--(D"'1)a(x) = 0 OA"a(x)
QUANTUM FlliLD THEORY
Therefore an expression of the form
R = Rinv(A)
+ (JR'
(12-163)
is a solution of (J R = O. It is possible to show that this is the general solution of the equation for I'~'i;(A), even when additional sources coupled to gauge-invariant composite operators are added to the initial lagrangian. In the present case, however, we can prove it by inspection, using the fact that power counting and ghost number conservation restrict the form of I'~'i; to
In this expression, /(A) is of dimension four, /:;."ab is of dimension one, and thus at most linear in A, and dab, = -da,b are numbers. If we assume that the global symmetry is unbroken (by the choice of
gauge), then
/:;."ab
aiJ"Oab
+ fJgCab,A",
dab, = ygCab,
with a, fJ, and y being pure numbers. Inserting these expressions into (12-160) yields
fJ=y " iJ/(A) Dab iJA"b
+ g(fJ -
iJf! a)Cab,A", iJA"b = 0
A particular solution to the second equation is
iJf! /(A) = (fJ - a)A"a iJA"a
and the general solution is obtained through the addition of a gauge-invariant functional of A, of degree four, and thus a multiple of f! (A):
/(A) = af! (A)
+ (fJ -
iJf! a)A"a iJA"a
To summarize, I'~'i; has the form
I'i'i; =
d x [af! (A)
+ (fJ -
a)A"a :::a
+ a(K
- if</J)"a(D"'1)a
+ (fJ -
a)g(K - fi</J)"aCab,A",'1b
+ fJ~LaCab''1b~,J
(12-164)
where a, a, and fJ are of order W. Simple algebra and the use of the homogeneity property of f! ,
iJf! iJf! 2f! = A"a(x) - - - g iJA"a(x) iJg
enable us to rewrite I'i'i~ as
(12-165) which is the desired result. All counterterms arise from a renormalization of the parameters of the initial action. Moreover, A and L are renormalized in the same way. If, according to the recursion
NONABELIAN GAUGE FIELDS
hypothesis, we write the action renormalized up to order n - 1 as
t-I = I(Zj:~_IA, Zj:~-IIJ, Zj:~_lij, Zj:~_IK, Zj:~_IL; Zg,n-Ig)
then we have just proved that In
In-I - ft~
+ O(W+ I )
(12-166)
I(Zj:~A, Zj:~IJ, Zj:~ij, Z:l:~K, Zj:;L; Zg,ng)
ZI/2 3,n
with
ZI/2 _ I 3,n
(f3 - ex + 2 ~)
ZI/2 =Z1/2 1 _~ 3,n 3.n~ 2 Zg,n This completes the inductive proof. Zg,n-I
+ :2
We have shown that it is possible to renormalize nonabelian gauge theories, while preserving gauge invariance expressed through the Slavnov-Taylor identities (12-144) and (12-145). Happily the whole operation boils down to wave-function and coupling constant renormalizations, We generate finite Green functions if we use the action
J R(A, 1], ij, K, L; g, A) = J(Ao, 1]0, ijo, Ko, L o ; go, Ao)
(12-167)
with the same notations as in (12-158) and ZL = Z3, Ao = Z3"lA, which takes into account the nonrenormalization of the gauge term (A/2)~(A)2 (for a linear function ~). As a consequence of (12-167), the Green functions are multiplicatively renormalized: (12-168) and satisfy the identities (12-144) and (12-145). After completion of this proof, we are of course entitled to drop the auxiliary sources K and L in the two preceding equations (12-167) and (12-168). The previous considerations extend to cases involving matter fields, coupled in a minimal way. If fermion fields are present, we assume for the time being that the couplings involve no As matrix (this point will be investigated in Sec. 12-4-5). Dimensional regularization makes the theory finite, the Becchi-RouetStora transformation and the identities (12-147) can be generalized, and, as expected from our one-loop computation, equations analogous to (12-167) and (12-168) hold. The crucial feature is, of course, the universality of coupling constant renormalization.
The compact formulation of Ward identities may have obscured simple facts. We emphasize that the results found in the one-loop computation are mere consequences of the Eqs. (12-147). For instance, if the counterterms Z" Z" Z4 are reinstated as in (12-124), it is easy to see that (12-167)
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