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The function QjPV(k) is the Fourier transform of the current-current Green function. To any order, current conservation leads to the condition
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which is formally satisfied, since
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d p 1 (2n)4 tr y p _ ~ _ m
+ ie -
+ ie
where we have replaced ~ by (p - m + ie) - Cp - ~ - m + ie) and used the cyclic invariance of the trace. If the integral were convergent a translation of integration variable p -+ p + k in the first term would insure that condition (7-5) is fulfilled. While this manipulation is meaningless for the original expression, the same steps are justified for the regularized tensor, showing that we have a sensible regularization. To evaluate (7-4) we use the parametric representation introduced in the previous chapter with the additional complication arising from the numerators
RADIATIVE CORRECTIONS
in the fermion propagators. We therefore write
wpv(k, m, A) = -4e 2
L~:~4 O~f O~2 + O~1 O~~ - gpv (O~l O~2 + m2)]
too dIY.1 too dIY.2 exp {i {IY.1(p2 0
m2) + IY.2[(P - k - m 2]
Z2 o(p - k)}) + J1 C(m -+ Asm) 11=z2=0 Auxiliary four-vectors Zl and Z2 have been introduced to generate the integrand
+ Zl
in (7-4) by differentiation. Integrating over p and performing the required derivatives we obtain
iIY. wpv(k, m, A) = -n
foo foo (dIY.1 dIY.2 ({( 2IY.1IY.2 kpk v + +
0 0 IY.1 IY.2 IY.1 IY.2
[ IY.1IY.2k2 gpv (IY.1 + IY.2
+ J1
C,(m -+ Asm )
The polynomial in k appearing in the integral can be rearranged to read
2(k pk v - g pvk ) (
2 IY.1 IY.2
IY.1
+ IY.2 -
g pv m - (
IY.1 IY.2
IY.1
+ IY.2 k 2 -
IY.1
i + IY.2)
Our calculation has remained covariant but wpv does not explicitly exhibit current conservation, which requires it to be proportional to the combination kpkvgpvk2. The second term in the above expression does not have this structure and its contribution reads g pvLlw, with
uW A _
iIY. n
foo foo
dIY.1 dIY.2 2 ~ C s [IY.1IY.2k2 2 ~ (IY.1 + IY.2) s=O (IY.1 + IY.2)
IY.1
+ IY.2
+ ms
exp {i [- m;(IY.1
+ IY.2) +
IY.1 IY.2 IY.1 IY.2
+ IY.2)
IY.1IY.2
where by convention Co = 1, mi5 = m2, m; = A;m 2. This can also be written
Llw =
iIY. n
foo foo (dIY. dIY. 2)3 ip -;- -1 I 0 1
0 0 IY.1
+ IY.2
up P s=O
Cs exp { ip [ -m;(IY.1
+ IY.1 + IY.2
J}1 p=l
For fixed ms the constants Cs are chosen in such a way that the integrals converge in the vicinity of IY.1, IY.2 -+ o. The convergence for large IY. is insured by the is prescription implicit in the masses. We may then interchange derivatives
QUANTUM FIELD THEORY
on p with integration. The expression
oo foo (!Xl +!X2 P I c, exp {iP [-m;(!Xl + !X2) + !Xl!Xl+ !X2 k 2J} d!Xl d!X;3 !X2 fo
is then found to be P independent as shown by a change of variable !Xi --+ P - 1 !Xi. Therefore ~w vanishes. The remaining term of the vacuum polarization tensor reads
(7-6)
csexp{i[-m;(!X l +
!X2)
!Xl!X2 k 2 !Xl+!X2
Using the homogeneity properties we introduce a factor 1 = S~ dp 8(p - !Xl - !X2) under the integral sign and change the variables according to !Xi --+ P!Xi' We obtain
_ 2!Xflfl d!Xl d!X2 8(1 - !Xl - !X2)!Xl!X2 fOOdP W(k 2, m, A) = n 0 0 0 P
e1p[ -m,
+ rt lrt 2k
The integral over P looks logarithmically divergent at the point P = O. Having factorized two powers of momentum in passing from wPV to W we have decreased the degree of divergence from two to zero. This is an example of the close relationship between the degree of divergence and the dimensionality of integrals. If we choose the coefficients Cs in such a way that
L c, = 1 + s= 1 Cs = L s= 0
(7-7)
the regularized integral over P will be convergent. Furthermore, let us pick k such that k2 < 4m 2 , which corresponds to the threshold of pair creation. Since !Xl,!X2 are between zero and one and satisfy !Xl + !X2 = 1, it follows that !Xl!X2 :s:: {. Then (m; - !Xl !X2k2) is positive and the integration contour in the complex p plane can be rotated by - nl2 in such a way that the integral reads lim
q->O
foo dp
s= 0
e-P(m;-rtl(l2k2)
11-+0s=0
c,(-e-Pln p /
r[ms
CilCi2
+ fOO dPe-Pln p ) 0
Condition (7-7) eliminates the dangerous In 11 term in the integrated expression. For fixed k 2 < 4m 2 and m; = A;m 2 --+ 00 the result takes the form - [In (m 2 - !Xl!X2k2)
+ S~l
C s In m; ] = - [In
(1 - !Xl:~k2) + S~l
C s In A; ]
where we have neglected k2 as compared to
m;. Let us define A such that
RADIATIVE CORRECTIONS
Figure 7-2 The complex k 2 plane for the vacuum polarization. The arrow indicates how the physical region is reached above threshold.
(7-8)
We end up with the following expression for the regularized vacuum polarization:
w(k 2, m, A) =
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