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(12-132)
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It is difficult to express in a compact way the Slavnov-Taylor- identities (12-132) on one-particle irreducible functions. A transformation discovered by Becchi, Rouet, and Stora enables us to reach this goal. The ghost fields are reintroduced in the action
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(12-133)
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QUANTUM FIELD THEORY
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It is easy to show that this action is invariant under a combined transformation of the variables
bAl'a(X) {
Dl'ab(X)'1b(X)b(
= sAb(
(12-134)
b'ija(X) = A~[ A(x)Jb( b'1a(x) =
=sijb(
~ Cabc'1b(X)'1c(x)b( =s'1b(
This transformation is local, in contrast to the one encountered above. It introduces an x-independent anticommuting parameter b( and mixes commuting and anticommuting variables.
The invariance of I is easy to prove. First 2 is invariant since oA is a special gauge transformation. Second, we observe that
0(- ~ 9'2) a
(oYi)A1)
as a consequence of the anticommutation of 1) and 0(, Finally,
o(A1))
0 [::: (D"1))a]
The first term vanishes because (D'1)MD"1))a is antisymmetric in the interchange of (J1.a) and (vb), while it is easy to see that
o(D"1))
(12-135)
as a consequence of(12-134) and of the Jacobi identity. For later use, we also mention that similarly (12-136) The Becchi-Rouet-Stora transformation s is defined as the right derivative of (12-134) with respect to 0(, that is, sA = D1), s1)a = -gCabc1)b1)c/2. Equations (12-135) and (12-136) imply that S2 A = 0, S21) = O.
This invariance implies identities between Green functions. Let us first show how the Slavnov-Taylor identity (12-130) is recovered. We start from
E0(A, '1, ij)ija(x) exp
[i (1 +
d y J.
which results from the odd character in the ghost variables of the integrand. We then perform a change of integration variables of the form (12-134); such a change does not affect the integral and has a jacobian equal to one, as is easily checked. Hence
E0(A, '1, ij) [Affa(X)
+ ija(x)
d4 y iJb(y)(D'1)b(y) ] exp
[i (1 +
d y J.
(12-137)
NONABELIAN GAUGE FlliLDS
The integration over 11, if entails the substitution
ifa(X)l1b(Y) --+ iA- 1 ba(Y, x)
and leads to (12-130).
12-4-2 Identities for Proper Functions
The identities are first expressed on connected functions. It is convenient to introduce sources not only for the ghost fields but also for the composite operators involved in the transformation (12-134). We write
eG(J,a,K,L) =
f~(A,
11, if) exp [i
d4 y ( ~ -
~ ff2 -
ifAI1
(12-138)
+ J. A + ~11 + if~ + KsA G(J)
LSI1)]
G(J,~,~, K, L) [~=~=K=L=O
Here Klla(y) and La(Y) are local sources coupled to sAlla(y) = (D Il I1)a(Y) and -sl1a(y) = gCabcl1b(Y)l1c(y)/2, and are therefore anticommuting or commuting objects respectively. More precisely, if 11 is given the ghost number y = -1 (and if, + 1), K and L have y = 1 and 2. For power counting sA and SI1, and hence K and L, have dimension two. We assume hereafter that the function ff is linear in A: (12-139) For instance, the generalized Feynman gauges correspond to Ilab = oll[)ab' The use of a nonlinear gauge condition ff would require the introduction in (12-138) of an extra source coupled to it. After a change of variables of the form (12-134), we get
f~(A, f
11, if)
d x (J. sA
~SI1 - Aff~)(x) exp [i
d y (... )]
since sA and SI1 are also invariant. This is rewritten as
d x [J' i:K -
~ i:L - A~ff C:J)](x) eG(J,~,~,K,L) = 0
The operator of functional differentiation is linear owing to hypothesis (12-139). Hence
f [J [)~ - ~ [)~ -A~ff(:J)]G(J,~,~,
K, L) = 0
(12-140)
In the preceding section, the identity (12-130) was supplemented by an equation of motion for the ghost propagator [Eq. (12-132)]' The analog here is obtained by expressing that the functional integral (12-138) is invariant under an infinitesimal
QUANTUM FIELD THEORY
change ij --+ ij
+ bij where bij is arbitrary. We get the local relation
f~(A,
11, ij)( - AI1
+ ~)(x) exp
d4 y (- .. )]
or, since AI1 = sff = <p' sA:
(~ -
<p ib~}X) eG(J,a,K,L) = 0
For connected functions, this reads
<p ibK(x) G(J, ~, ~, K, L) = ~(x)
(12-141)
Equations (12-140) and (12-141) must now be translated in terms of proper functions. This is achieved after a Legendre transformation
r(A, 11, ij, K, L)
= - iG(J, ~, ~,K, L) -
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