Graduate Trace of Numerator in QED vacuum polarization

[Elmo]

TL;DR

Basically this :
Why do we have to take the trace of the numerator when calculating the vacuum polarization loop ?

Sorry I just typed out my query .For some reason I can't seem to find the buttons for attaching files on this thread.

When writing the QED vacuum polarization loop, the numerator ,consisting momenta slashed + m from the fermion propagators and the two gamma matrices, has a trace over all of it.
Yet we do not take traces in other loop diagrams like fermion self energy or vertex correction.
Couldn't figure out why. Some clarification on it will be most helpful.
My best (and very vague) guess is that it has got something to do with the spinor completeness relation.
For reference see page 308 of Schwartz.

 

[Elmo]

actually here is that particular page.

 

[vanhees71]

You take traces (in Dirac-spinor space) if you have loops consisting of fermion lines only, as in this one-loop example where you calculate the 2nd-order contribution to the photon self-energy (or "vacuum polarization"). It's also clear that you need a Dirac trace, because the result must be usual complex-valued tensor components not some matrix in Dirac space. For the electron-self energy the analogous diagram has one fermion and one photon propgator in the loop, and there's thus no trace in Dirac space, and indeed the result must be a matrix in Dirac space.

Formally you get these Feynman rules (including the additional sign for a closed purely fermionic loop, also applicable in the calculation of the photon-self-energy diagram discussed here) of course from Wick's theorem.