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bT' bA
In this operation, the sources K and L are passive spectators:
bG ibK bT' bK bG ibL bT' bL
(12-143)
The identities (12-140) and (12-141), expressed on
r, read
(12-144)
If the modified effective action
(12-145)
is introduced
d y ff2[A(y)]
(12-146)
Equations (12-144) and (12-145) take the simpler form
NONABELIAN GAUGE FIELDS
4 (if (if
(iA (iK
(if (if) + (i1J (iL (x) = 0
(iKI'
+ brj -
(if _ 0
(12-147)
The identities (12-147) have a universal form. They no longer involve any parameter affected by the renormalization, such as the coupling constant, and make no reference to the group structure, which has been hidden in the definition of the sources. Consequently, they apply as well to the action and are therefore suited to study the structure of counterterms.
It is easy to verify that the identities (12-144) and (12-145) are satisfied to lowest order. The first one expresses the invariance of the action under the Becchi-Rouet-Stora transformation
or[OJ
or[OJ
oA=-o(
01'/= - - o (
oij = ,l.S"(A)o(
The condition that the jacobian be equal to one reads (12-148) and is obviously satisfied. It is a good exercise to perform the analogous analysis in the abelian case, in a gauge such as S" = apAP + A2/2 which requires the introduction of ghosts.
12-4-3 Recursive Construction of the Counterterms
In the preceding considerations, the theory was implicitly dimensionally regular-
ized. We now want to renormalize it, in a way which preserves the previous identities. As in the one-loop computation of Sec. 12-3, it is convenient to perform the minimal renormalization defined in Sec. 8-4-4. In other words, we content ourselves with eliminating the divergent terms as the dimension d goes to 4. Order by order in Ii, we will write (12-149) where n~1 is computed by taking into account all lowest-order counterterms. More physical conditions may be substituted, provided they are consistent with the identities (12-144) and (12-145). Equation (12-147) tells us that all functionals depend on K and if only through the combination K - ifcp. We will use the compact notation
r 1* rz ==
d4 x
(ir1 (irz + (ir1(irz) (iA (iK (i1J ()L
(12-150)
The identity (12-147) satisfied by the power series in
I' = fro] + f[1] + f[Z] + ...
QUANTUM FIELD THEORY
reads to nth order
p+q=n
f[p]
* f[q] =
(12-151)
and we seek counterterms such that the renormalized
fir] satisfy
(12-152)
I fir] * f1q ] = 0
p+q=n
We proceed in a recursive way. To lowest order
fO =
I(A, 11, ij, K, L)
= d4 x [2 + (K - ijc/ sA - LSI1J
which of course satisfies (12-152). To first order, we have
fro] = f1 ] reduces to
+ K sA LSI1J
(12-153)
d4 x [2(A) - ij.A11
I * f[!] + f[l] * 1 = 0 {hf11 ]+f111 *I=o
dlV dlV
(12-154a) (12-154b)
Equation (12-154b) is simply the desired identity (12-152) for n = 1, while Eq. (12-154a) gives the structure of the first-order counterterm. It is suggested to modify 1 into (12-155) so as to cancel the divergent part. However, the recursive procedure works only if the renormalized action Ii satisfies itself:
11 * 11 = 0
as 1does. This is not the case for (12-155), as
(12-156)
-[1]
* Ii
-[1]
= r div
* rdiv
However, the right-hand side is of order
Ii I-
W. We are thus led to define r div + ~1
-[1]
where the extra piece ~1 is the integral of a local polynomial in the fields of degree four, of order W, and defined in such a way that (12-156) is satisfied. Of course, it does not affect quantities to order one, and hence preserves the finiteness and normalization condition of f111.
This complication is quite typical of symmetries involving nonlinear identities. The same phenomenon occurs, for instance, in the two-dimensional nonlinear (J model evoked at the end of Chap. 11.
The structure of ~1 will be deduced from the one of show that
(I -
r };. If we are able to
2 + 0(1 )
rdiv)(A,
-[1]
11, 11, K, L, g) - I(Ao, 110, 110, Ko, Lo, go)
(12-157)
NONABELIAN GAUGE FIELDS
where
Ao =
ZV 2 A
ZV21]
ijo = ZV 2 ij go
K o = Zjl 2 K
L o = Z}j2L
(12-158)
with Z3 = 1 + Z3/i, etc., a natural choice will be
11(A, 1], ij, K, L; g) =
This new action
I(Ao, 1]0, ijo, Ko, L o ; go)
- * 1-
rd4X Z3
11 will satisfy
[-1/2
bl(Ao, ... ; go) bl(Ao, .. .) -1/2 1/2 bl(Ao, ... ) bl(Ao, .. .)] bAo bK o +Z3 ZL bl]o bL o
If, moreover, (12-159) then the condition (12-156) will be fulfilled and the recursive proof can be pursued. The aim of the forthcoming technical discussion is therefore to prove (12-157) and (12-159).
We look for the general solution of the equation (12-160) satisfied by the divergent part ft~ to a given order n, when all counterterms of smaller order have been taken into account. The operation (J in (12-160), which is a generalization of the Becchi-Rouet-Stora transformation s, is nilpotent (12-161) This is easily seen if (J is written as
(J=--+--
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