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leads to renormalized Green functions satisfying (8-96). We remind the reader that and (8-98)
QUANTUM FIELD THEORY
Moreover, we assume that, to this order, the Ward identities imply that and
(8-99)
To the next order h + 1, the Green functions are still divergent. We introduce a gauge-invariant regularization such as the one exhibited in Sec. 8-4-2 to regularize the theory and compute the Green functions using .,2"[L], that is, taking account of all lower-order counterterms. Because of the structure of .,2"[L], we observe that these regularized functions r~~g+ 1] satisfy the identities derived in Sec. 8-4-l. Using these identities and the normalization conditions (8-96), it is straightforward to verify that the new counterterms needed to order L + 1 have the same structure as in lower orders [namely, that no (A 2)2 counterterm is necessary] and that they still enjoy (8-99).
For instance
Z[L+
=!..8p
[r(2)[L+
Il(.J.) _
r(2)[L+
Il('/')] _
Ji-m
= _!..8p
r(2)[L+
Il(.J.)
IJi-m _
and Finally,
Z\L+ Ilyp
[A~/ Il(p, p) - A~~;gll(p, p)] Ji~m
A1~r;gll(p, p) IJi~m
follows from Eq. (8-88).
In the renormalized theory we end up with functions related to the bare regularized ones through
r}fn,p)(Pl, Pb" .,p~, Pm qb ,qp, m, /1, e, A)
I' Zp/2 zn r(2n,p)( Pb"" qp, mo, /10, eO, /1,0, A) I 1 1m 3 2 reg
A-HY;)
(8-100)
These renormalized Green functions satisfy the Ward identities as a trivial consequence of this multiplicative character of renormalization. It is important to emphasize the role of the identity Z1 = Z2 in the charge renormalization eo = eZ31/2. If several species of charged particles (electron, muon, etc.) are coupled to the electromagnetic field, the identities z\e) = Z<;l, zy) = Z<t), ... , guarantee that renormalization is universal. The concept of charge universality only makes sense owing to this identity.
It would be more appropriate to say that the ratio of renormalized to bare charge is independent
of the type of charged particle, since in this restricted framework there is no natural explanation for charge quantization. This is not the case for certain unified models of weak and electromagnetic interactions, where the electromagnetic gauge invariance corresponds to a subgroup of a larger simple group of invariance (see Chap. 11).
In summary the method described in this section is to express the consequences of a symmetry in terms of Ward identities and to verify their consistency with renormalization. It will again be encountered in different contexts in the following: chiral symmetry, nonabelian gauge symmetries, etc.
RENORMALIZATION
8-4-4 Two-Loop Vacuum Polarization
The computation of the two-loop vacuum polarization will be presented in massless euclidean quantum electrodynamics (/1 = m = 0), using dimensional regularization.
It is rather instructive in several respects:
(a) It provides an example of features specific to higher-order corrections, i.e., not visible at the
one-loop order.
(b) It shows how dimensional regularization may be applied in a case involving spinors. (c) It is convincing evidence that the renormalization program does work, even when overlapping
divergences are present.
(d) It illustrates the statements concerning massless theories and asymptotic behavior. The zero-mass
computation may also be regarded as yielding the asymptotic (large-k) behavior of the vacuum polarization w(k 2 ) in massive quantum electrodynamics. (e) It serves as a test of the general results derived in the previous subsections, namely, the transversity of the vacuum polarization tensor. (f) Finally, it exhibits an interesting property of the vacuum polarization, namely, unexpected cancellations at large momentum. We shall elaborate on this point at the end of the calculation. Since we are working in the euclidean version of the theory, with a dimensional regularization, let us first list some useful formulas. As in Eq. (8-11), the antihermitian matrices satisfy
{Y., Yv}
20.v
(8-101)
and, by convention, we choose [compare with (8-11 c)] tr I
(8-102)
All standard identities for contractions and traces of Y matrices then follow (d is the euclidean space dimension):
YPYP
~2 =
YPY.YP = (d - 2)y. YPY.YvYP
-(d - 4)y.yv
+ 40.v
(8-103)
YPY.YvYuYP = (6 - d)yuYvY. - 2(d - 4)(0.vYu - o.uYv + ovuY.)
tr Y.Yv
-40.v
tr Y.YvYPYu = 4(0.vopu - o.povu + 0MOVP) tr Y.YvYpYuY,Yv
-4(0.vopuow - perm)
In euclidean space, the action of massless quantum electrodynamics reads
(8-104)
In the Feynman gauge, .Ie
1, the perturbative rules derived from e- 1E read
Fermion propagator
Photon propagator
.J.l
(j)P.
QUANTUM FIELD THEORY
Vertex
and, of course, a minus sign for each fermion loop. When working in dimension d # 4, the charge e acquires a dimension (4 - d)/2, as already noticed in Sec. 8-1-2. If J.i. is an arbitrary mass scale, we write
(8-105)
where e' is dimensionless. We shall use this expression whenever we expand our results near d = 4, in order to recover correct homogeneity properties. In this way the unavoidable mass scale creeps into the massless theory. As a warming up, we first recalculate the one-loop vacuum polarization, fermion self-energy, and vertex in this formalism. The one-loop self-energy of Fig. 7-5 is
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