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These expressions reduce to those computed for quantum electrodynamics in Chap. 7, if we set C j = 1, C = 0, g2j4n = IX. Finally, the modifications induced by an internal fermion loop (Fig. 12-8) may be expressed as
(i2(A) fermlons (iZ(F)
(iZ(F)[i (i Z 3(F)
tr (FJlV pLY)]
(i Z l(F)
(i Z 4(F)
= _ Tj g2 ~~
16n 3
(12-123)
The reader will have no difficulty in working out the case where scalar fields are coupled to the gauge vectors.
NONABELIAN GAUGE FIELDS
Figure 12-7 Fermion-vector vertex.
12-3-4 One-Loop Renormalization
We have discovered that all counterterms have the same structure as monomials of the initial lagrangian. This is not completely sufficient to insure that the renormalized and bare lagrangians enjoy the same symmetry properties. Omitting matter fields for a while, we found explicitly
+ [)!i' =
tr {iz 3(01lA v - ovAIl)(OIl A V- OV All)
- gZ l(OIlA v - ovAIl)[AIl, AV]
+ ~ (0 A)2
Z4[A Il , Av][AIl, AV]}
(12-124) We define the bare fields and parameters according to
= ZV2 A
= ZV21]
= ZV 2ij
(12-125)
gO=ZlZ:;3/2g
Ao=AZ:;l
Then !i' + [)!i' may be regarded as the initial lagrangian !i'(Ao, 1]0, ijo; go, Ao) written in terms of bare quantities provided the following identities hold:
Z4 Zl Zl Z3 Zl Z3
(12-126a)
A glance at the expressions (12-114) to (12-118) suffices to convince ourselves that they are satisfied to the one-loop order. These identities, which generalize the relation Z 1 = Z2 of quantum electrodynamics, express the fact that the coupling' constant re!lormalizations of the cubic, quartic, and l]ijA vertices coincide, i.e., that the universality of the coupling is preserved by renormalization. In the
Figure 12-8 Fermion contributions to the two-, three-, and four-point functions.
QUANTUM FIELD THEORY
presence of matter fields, fermions, say, we demand moreover that
Z4 Zl Zl Z3
.......
--;;;:;---
Zl Z3
(12-126b)
This relation, too, is satisfied by the expressions (12-121) to (12-123). Finally, we write the coupling constant renormalization in the presence of fermion fields:
g2 A2 1 - --(1J-C - j Tf ) In 2 16n
(12-127)
Notice that the gauge dependence has disappeared. For future use, 21e has been replaced by In (A21f.12).
12-4 RENORMALIZATION
This section deals with the renormalization of nonabelian gauge theories with an unbroken symmetry. What happens as the local symmetry is spontaneously broken will be examined later. The issue is to know whether the remarkable features of gauge theories, in particular the universality of the coupling constant, will be preserved by the renormalization. A regularization is introduced at intermediate stages; in practice, the dimensional regularization is the most convenient. The desired properties will follow from Ward identities, first derived in this context by Slavnov and by Taylor. The reader may find the repeated use of such identities in Chaps. 8, 11, and here rather cumbersome. However, nonabelian gauge theories are more intricate and require a sophisticated analysis.
12-4-1 Slavnov-Taylor Identities
We start from the generating functional
eG(J)
E0(A) det A exp
[i fd
ff' -
~ ff'2 + J . A) ]
(12-128)
where A is the variation of ff' with respect to a gauge transformation
= DOa
off' = AOa
We are going to exploit the property that the measure E0(A) det A is invariant under this transformation, even when Oa itself depends on A. In other words, if
A= A'
+ oA
ff"(A') = ff'[ A(A')] = ff'(A')
+ off'
(12-129)
then
E0(A) det A
~A) =
E0(A') det A F(A')
NONABELIAN GAUGE FIELDS
The proof is straightforward. We write t.(A) for det A(A):
fq&(A)t.~A) f fq&(A)q&(A')q&(g)t.~A)o(A' f (g-!A')t.~g-!A')o[fJ"(A') f
q&(A)q&(A')t.j'-(A)o(A' - A
+ oA)
- gA)t.J',(A)O[fJ',(g A) - fJ'(A)] - fJ'(g-!A')]
q&(g)q&(A')t. ...
q&(A')t.J',(A')
where the invariance of the measure q&(A) and of t.y;(A) under a transformation A .... gA has been used,
Consequently,
eG(J)
2t1(A) det A exp [i
d4x (se -
~ ff2 + J . A -
AffA b!X
+ J. Db!X) ]
The content of this identity is most easily explored if we take b!X = A- i bw, This corresponds to a nonlocal gauge transformation, which translates ff by bw, To lowest order in bw, we get
f 2t1(A) det A f d4x(Aff -
J. DA- i )bW exp [i
d4x(se - iff 2
+ J. A)]
or, after substitution of b!ibJ for A,
{AffaD
bJ~X)J -
4 d y Jb(Y)DbcG
~ )A~i (y,X; ~ ~)}
eG(J)
(12-130)
where summation over repeated indices is understood, The expression
_ Gca(y, x) - Aca 1 ( y, x, ~~) eG(J) i (jJ
(12-131)
may be regarded as the ghost propagator in the presence of the source J. The Slavnov-Taylor identities finally read
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