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(k) = - e2
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dd p tr [Pra(1 + ~)1' p] (2n'/ p2(p + k)2
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2 1 -e tr(1'p1'pYv1'a) (4n)d/2 B
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(d 2 (k) d/2-2 2' d)
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x [ - kpkvr
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(2 - ~) + D;v k 2r ( 1- ~)]
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my =
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m~ =
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The integral was performed using Eq. (8-9), where
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B(x, y) =
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0, in terms of Euler's B function
da ax - 1 (1 _ a)y-l = 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)
(8-106)
in agreement with Eq. (7-9), if we identify 2/1> with In(A2Im 2). With the conventions of Chap. 6, the euclidean proper two-point 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)
(8-107)
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 (8-106) 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. 7-7 and 7-10 are computed in the same way. The fermion
RENORMALIZATION
self-energy is
d",4
e "" - - - p - + finite terms
(41ll
(8-108)
Hence it requires a wave-function renormalization equal to
(Z2-1)
e 2 = --(41lY 8
(8-109)
As for the vertex function it reads
e3 2
d",4
(4n)
--2 -
+ finite terms
which leads to the counterterm
(ZI-1)
--(4n 8
(8-110)
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 two-loop diagrams of Fig. 8-17a1, a2, b. The insertion of the counterterms
(a'! )
Figure 8-17 Two-loop 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 self-energy r(2)(p) ofEq. (8-108):
The integral over p is performed using the identity (8-111) 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 d-4
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)d-4 pu( (4n)d 2 - d/2
x [ - 2k.kv1(4 - d)
+ o.vk21(3 -
d)(l - d/2)
+ k 2o. v1(4 - dJ]
dJ =
(8-112)
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
(8-113)
where constant terms have been omitted. The contribution r~,::) of the counterterms (Fig. 8-17 db d2 ) is also readily computed
(8-114)
Three remarks are in order. First, the contribution of the counterterms is transverse-this was obvious since it is essentially the one-loop diagram computed previously-whereas r~"J is not. Only the sum of all contributions of Fig. 8-17 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 self-energy diagram. As is seen from Eq. (8-114), 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 (8-59) 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. 8-17h. With the notations of Fig. 8-17, the amplitude reads rlJ'j(k) = _e where I aP., 4
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