which completes the proof of (11.93).
While all of these identities have been derived for integral d, the final results are expressed as functions of d which can be analytically continued into the complex d plane. Hence, from now on, we will think of d as a continuous variable.
Finally, we are ready to return to the integral (11.67). We will evaluate it following the steps we have just discussed. First, we write the integral in d .lmensions, and start off by assuming that d < 2, so that the integral is convergent and everything is well defined. Then we combine the two propagators using the first of the identities (11.79), ddp [m2g^ + 2p^p" - p»qu - q"pu - g^p ■ (p - q)} (2tr)d (m2 - p2 - it) (m2 - {p - q)2 -it)
Next we complete the square in the denominator by introducing p = k + qx. Because the integration is over all of space-time, adding or subtracting a fixed four-vector to p does not change the volume of integration, and the new integral is
[m2 — k2 — q2x( 1 — x) — ie] where the transformed numerator is
N'1*" =m2g>+ 2^^ + (g"Jfc" + k^q") {2x - 1) - 2q>iql/x{l --g^(k + qx)-(k-q(l-x)) .
Next, drop terms odd in k, and use the identity (11.93), k^k" -> g^k2/d, to reduce N'*" to
Now we may evaluate the integrals over k using the identity (11.84) and the identity (11.93) for the term proportional to k2 in the numerator,
Using the properties of the F-function, we see immediately that the coefficient of the g^ term is zero! This is a nice feature of dimensional regularization; it respects the gauge invariance of the theory. Other regularization methods give the same result, but only after considerable labor. The remaining term is gauge invariant and has the form we anticipated in Eq. (11.69). Extracting the scalar part, nd(<72), gives
In this expression, the singularity which exists for d = 4 dimensions appears as a pole in n<j. This pole corresponds to a logarithmic singularity in the original integral; the quadratic divergence has disappeared because it was contained in the gauge violating g*1" term, which integrated to zero. This means that the vacuum polarization is now well defined for all d < 4, and the physical result can be obtained from the limit d —* 4. The scalar vacuum polarization will now be written as the sum of two terms, an (infinite) constant corresponding to its value at q2 = 0 and a (finite) term obtained by subtracting the integral (11.96) at q1 = 0. This gives
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