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(k) =  e2 in VS .NET
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da ax  1 (1 _ a)yl = r(x)r(y) r(x + y) Defining I> = 4  d we are led to
d",4 ""  !:.. [~ 3n
In k: J.i.
+ constant + O(I ]( Dpvk2  kpk v) (8106) in agreement with Eq. (79), if we identify 2/1> with In(A2Im 2). With the conventions of Chap. 6, the euclidean proper twopoint function is the opposite of the inverse propagator. Therefore, after introduction of the counterterm a 2 (Z3  1) [1] =    the renormalized vacuum polarization reads, in four dimensions, rW = (Dpak2  k pk a)W[I](k2) (8107) The constant is an uninteresting combination of n, l' (Euler constant), etc. Notice that we have used a new type of normalization conditions. Instead of fixing the value of w(k 2 ) at some point, we have decided to subtract from (8106) the pole term only. We shall adhere to this convenient prescription, referred to as the minimal renormalization. Such a procedure satisfies automatically the Ward identities. The other two diagrams of Fig. 77 and 710 are computed in the same way. The fermion RENORMALIZATION
selfenergy is
d",4 e ""    p  + finite terms
(41ll
(8108) Hence it requires a wavefunction renormalization equal to
(Z21) e 2 = (41lY 8
(8109) As for the vertex function it reads
e3 2 d",4 (4n) 2  + finite terms
which leads to the counterterm
(ZI1) (4n 8
(8110) We verify that the identity ZI = Z2 is satisfied to this order, and that Z1, Z2, Z3 are the same as in the massive minkowskian theory with the identification 2/8 = In (A 2/ m2). This computation was clearly unnecessary! We now turn to the twoloop diagrams of Fig. 817a1, a2, b. The insertion of the counterterms (a'! ) Figure 817 Twoloop contributions to the vacuum polarization. The contributions of order ~ counterterms are also depicted. QUANTUM FIELD THEORY
of order h is also depicted. The two diagrams al and a2 obviously give equal contributions. We first compute r~"J = n"d) + r~d) in terms of the fermion selfenergy r(2)(p) ofEq. (8108): The integral over p is performed using the identity (8111) and the parametric machinery. This yields ddp (p + k).pv __ (2n)d (p + k)2(p2 d/2  (4n)d/2 1(3  d/2) 1_ 1 k2 d4 x [ k.kv1(4  d) + iD.vk21(3  and after some algebra r(a) k) = 8e4(d  2) B(d/2  1, d/2)B(d/2, d  2)(k 2)d4 pu( (4n)d 2  d/2 x [  2k.kv1(4  d) + o.vk21(3  d)(l  d/2) + k 2o. v1(4  dJ] dJ =
(8112) Let us expand this expression near d = 4, 1(4  d) = l/e  l' + O(e), 1(3 We find
l/e + (1'  1), etc.
_OPuk2[_~+~(1'+lnk2 _2)_~ln2k2 +ln J12 (_1'+2)+"']} e2 e J12 2 J12
(8113) where constant terms have been omitted. The contribution r~,::) of the counterterms (Fig. 817 db d2 ) is also readily computed (8114) Three remarks are in order. First, the contribution of the counterterms is transversethis was obvious since it is essentially the oneloop diagram computed previouslywhereas r~"J is not. Only the sum of all contributions of Fig. 817 will be transverse. Second, we see that in the sum r(a) + r(a') the divergent terms do not cancel. This was expected since the diagram still requires an overall subtraction. At first sight, it seems surprising that the dominant terms 1/e 2 (or In 2 A2) do not cancel, since the internal divergences have been subtracted by the counterterms. But the renormalized fermion selfenergy behaves as In p2 at large p2; hence its insertion in a superficially divergent diagram gives rise RENORMALIZATION
to a In 2
divergence
The coefficient of the In (k 2/ Ji2) term depends on the normalization condition for the selfenergy diagram. As is seen from Eq. (8114), changing the counterterm (Z2  1)[1] by a finite amount results in a modification of the In (k 2/ Ji2) term (and of the neglected constant term). However, this dependence will disappear in the complete expression of rap. This illustrates the fact that the renormalization group equation (859) is only satisfied by a Green function to a given order, but not by individual diagrams. We proceed to the much more cumbersome computation of Fig. 817h. With the notations of Fig. 817, the amplitude reads rlJ'j(k) = _e where I aP., 4

