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k 13) - In m2 + In [ 1 - 13(1 - 13) m 2] }
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The remaining integral can readily be performed, leading to the analytic expression 2 2m W(k 2 m A) = - -IX { - In A2 + - + 2 ( 1 + - -) [(4m2 - 1)1/2 -2 1 -, , 3n m 3 k2 k2 4m2 x arccot ( y - 1)1/2 - 1]}
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The calculation has been carried out under the assumption k 2 < 4m 2 . The corresponding function may be continued in the complex k 2 plane. The values for k 2 > 4m 2 are obtained by taking a limiting value from above the cut, starting at the point k 2 = 4m 2 (Fig. 7-2). The discontinuity across the cut is w(k 2 + is) - w(e - is) = 2i 1m w(k 2 + is). This can also be derived using the last integral representation given above by setting f3 = (1 - u)/2, integrating by parts, and changing the variable to u = (1 - 4m 2/k'2)1/2. We find that
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1 ( 4m2)1/2( 2m2) k'2 _ k2 1 - 7 1+ 7
(7-10)
a once-subtracted dispersion relation exhibiting the above analytic properties, with 1m w(k 2 m)
ex(1 - - 4m2)1/2(1 + - 2m2) 3 e e
(7-11)
This absorptive part is independent of the regularizing cutoff A and hence cannot be affected by renormalization. It coincides with our earlier calculation of pair production by an external field, Eq. (4-105), expressed as the square of the corresponding amplitude (Fig. 7-3) and given to this order by
The only effect ofregularization has been to provide the constant (ex/3n) In (A 2 /m 2 ), which is of course divergent if we let A go to infinity.
Figure 7-3 The probability for pair production in an external field giving the discontinuity of the vacuum polarization tensor.
QUANTUM FIELD THEORY
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Figure 7-4 Photon propagator in terms of the vacuum polarization.
Before applying renormalization, let us return to the original expressions (7-1) and (7-2). We do not really intend to compute the photon propagator like that! The contribution G[ll added to G[OJ would produce a double pole besides the simple one. When we were discussing the ordering of perturbation theory according to the number of loops we had in mind the one-particle irreducible Green functions, the inverse propagator for the case at hand. The zero-loop contribution to this quantity is G~~J-l and its one-loop contribution is - wW, where we have added the suffix 1 to indicate that it was evaluated to order one: G~~J-l - wW = i[(k 2 - p.2)gpv - (1 - A)kpk v + (gpvk2 - k pk v)w[1J] (7-12) Inverting this expression we obtain the propagator represented diagrammatically by a sum, each term of which is a string of bubbles (Fig. 7-4). If more generally we succeed in showing to all orders that the vacuum polarization tensor has the form of a scalar function multiplying the combination (g pv k 2 - kpk v), then Eq. (7-12) with the index one omitted will give the general relation between the propagator and w = w[1J + W[2J + ... as 'G _ gpv - [1 + w(k 2)]k pkv/.u 2 kpkv 1 (7-13) I pv k2[1 + w(k2)] _ .u2 +.u2 k2 - .u 2/A
Equation (7-13) can be compared with formula (5-80). At the time we were ignoring mass renormalization as well as divergences. To be more precise, assume that the denominator in (7-13) admits an integral representation of the form (7-14)
If w continues to satisfy a dispersion relation of the type (7-10) (with a threshold at the origin as J12 ---> 0, instead of 4m 2 as it is to lowest order), we shall have the relation
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