F ddk fl J w0

-k2-q2x( 1 -i)]3+£/2 x I (gVA + <?VA) (1 - 2x) - ff'V(3 - 4x)

where the k^ku terms were reduced to g^k2/4 using (C.9). Doing the k integration [setting e = 0 everywhere except in the (-/i2/<j2) factor] gives the following leading term:

24tt2

Note that this does not have the gauge invariant form of Eq. (16.81), and hence it must be canceled.

The cancellation comes from the other terms proportional to A2. Using k = | (k+ + k-), the numerator for the first of these terms reduces to

= k»k% tr {7" It + 7" K } + k»kl tr {7" }

where terms of 0(q2) can be dropped because they are not of leading order. Since A2(k,q) — \2{k, -q), the second of these terms, which differs only by fj, *-* u.

16.5 FOURTH ORDER VACUUM POLARIZATION 567 may be added, and after the trace has been taken, the first two A2 terms become

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