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:


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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