Ryder QFT 2nd ed. page 341 misprint?

[melezmelez]

Eq. 9.124 has 3 steps. The first one gives 0 propagator for ε goes to 0. The final step, the one with Z gives infinity. Can anyone post the correct steps? Is there anything wrong in previews equations? It is important for me to learn the Ryder's book method. I have an exam in 10 days

 

[vanhees71]

I do not understand, where you have a problem here. Of course, the one-loop polarization tensor (photon selfenergy) is UV divergent. In Ryder's regularization scheme (dimensional regularization) this manifests itself in the fact that there's a term $\propto 1/\epsilon$ which diverges for the physical value $\epsilon \rightarrow 0$. Thus you need a counter term to compensate (at least) this divergency, and as Ryder nicely shows, this counter term leads to a photon-field renormalization. At the end everything is finite by expressing physical quantities in terms of the finite dressed wave functions, masses, and couplings.