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(710) a oncesubtracted dispersion relation exhibiting the above analytic properties, with 1m w(k 2 m) ex(1   4m2)1/2(1 +  2m2) 3 e e
(711) 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. (4105), expressed as the square of the corresponding amplitude (Fig. 73) 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 73 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 74 Photon propagator in terms of the vacuum polarization.
Before applying renormalization, let us return to the original expressions (71) and (72). 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 oneparticle irreducible Green functions, the inverse propagator for the case at hand. The zeroloop contribution to this quantity is G~~Jl and its oneloop contribution is  wW, where we have added the suffix 1 to indicate that it was evaluated to order one: G~~Jl  wW = i[(k 2  p.2)gpv  (1  A)kpk v + (gpvk2  k pk v)w[1J] (712) Inverting this expression we obtain the propagator represented diagrammatically by a sum, each term of which is a string of bubbles (Fig. 74). 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. (712) 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 (713) I pv k2[1 + w(k2)] _ .u2 +.u2 k2  .u 2/A Equation (713) can be compared with formula (580). At the time we were ignoring mass renormalization as well as divergences. To be more precise, assume that the denominator in (713) admits an integral representation of the form (714) If w continues to satisfy a dispersion relation of the type (710) (with a threshold at the origin as J12 > 0, instead of 4m 2 as it is to lowest order), we shall have the relation

