I am following Peskin and Schroeder and consider for simplicity only the $\phi^4$ theory in 4D.
The field strength renormalization factor seems to first appear p.214 and be defined by $$Z= \left| \left\langle \Omega \mid \phi (0) \mid 0\right\rangle \right|^2$$ where $\mid 0\rangle$ is the vacuum of the free theory. It then reappears (7.44) p.228 and p.324 but I do not have the impression it is computed anywhere so how do we know it is infinite?
It seems to be related to the "vacuum energy shift" (p.324) but I wish for an explicit expression.
EDIT: I have been on stackexchange for a while, mainly in mathematics and it never happened there that my question was shut down, while it did several times in physics. Let me say out loud that most people ask questions because they are interested and did not find the answer. (Also the "exercise" criterion is completely irrelevant, many as is my case have finished their studies long ago and just want to understand things in their free time!!) And above all, this closing question policy DOES NOT IMPROVE THE QUALITY OF THE SITE, as a rapid comparison with the maths side will show.
Now for the "duplicate problem": do you unkwown moderator Sir consider a formula for QED given without detail as a satisfying answer to my question for the $\phi^4$ theory? or in the second formula, just a Laurent expansion as an answer? Is this the quality of the answers that you are trying to maintain? Or the comment of the form, it's written in this book that I do not have at my disposal at the moment?
