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Let's consider a physical process in QED: \begin{equation} e^+ e^- \rightarrow \mu^+ \mu^- \end{equation}

I have to calculate the $\hat{S}$ matrix element at $O(\alpha^2)$ (tree level + one loop corrections). I don't understand what Feynman diagrams I have to consider. I'm sure I have to take into account:

  • one tree level diagram
  • two box diagrams
  • one diagram with the photon self energy
  • two diagrams with vertex corrections (triangle)

Do I have to add diagrams with external leg corrections (fermion self energies)? Which counterterm diagrams I have to consider? I suppose that if I have to exclude the external leg corrections I won't consider $\delta_2$ counterterm diagrams.

I've studied QFT on Peskin&Shroeder. On pag. 113 of this book it is written that diagrams with a loop on an external leg must be excluded, and only amputated diagrams must be considered (figure on top of pag.114). But on pag. 229 it is written that the $\hat S$ matrix element is the sum of amputated diagrams times a factor proportional to field strength renormalization (due to the LSZ formula). Then, on the next page, this factor is used to justify the subtraction made to remove UV divergences of the vertex.

I'm confused: I would justify this subtraction with the $\delta_1$ counterterm coming from the renormalized perturbation theory. How do I take into account field strength renormalization factor in my physical process? Is the field strength renormalization factor essential to cancel some divergences of my diagrams or it is redundant if I consider $\delta_1$ counterterm diagrams?

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The answer to your question depends on which computation strategy you want to follow either \sl{renormalized perturbation theory} or {\sl bare perturbation theory}. Actually the results of both are the same (of course once you start with one of it, you should stick to it up to the end).

Actually, the loops on the external lines have to be accounted for, however, the chapter 4.6 of P&S (in particular formula (4.103)) is an introduction into the computation of $S$-matrix elements, therefore the consideration of the loops was just postponed to a later section. In chapter 7.2 in formula (7.45) the loops in the external lines are accounted for via the $Z$-factors. When towards the end of this section it is explained that $Z$-factors cancel out against UV-divergences of vertices, it is meant in bare perturbation theory. This actually means that you have to use the formula (7.45) including the $Z$-factors in bare perturbation theory. To make it clear, in bare perturbation theory are no counter terms (actually sometimes mixed forms of perturbation theory are also used, for instance in Bjorken&Drell. They use in their book on QFT (the 2. book) a kind of mass correction counterterm, whereas for the other renormalisations, charge and field strength, no counterterms are used). So this should clear up your confusion.

However, using renormalized perturbation theory, all renormalizations are done with counterterms. In this case, formula (7.45) is again substituted by the original one (4.103) without $Z$-factors and the renormalisations are done by the introduction of additional diagrams representing the counterterms in the perturbation series. This is explained in P&S in chapter 10.2. So it is up to you to choose which strategy you prefer. For your $e^+\,e^- \rightarrow \mu^+\,\mu^-$ process computation in $O(\alpha^2)$ order, I guess, it is meant to use renormalized perturbation theory (then loops on the external legs does not need to be corrected). Actually, in the meantime renormalized perturbation theory is preferred for most perturbation theory calculations as it is easier to apply. So, I guess, your list of different diagrams is in this sense complete (actually I can't guarantee you this with 100% as I am not a routine Feynman diagram calculator).

I also struggled in understanding P&S, actually it is not a very pedagogical textbook. I already read many respective comments on this. I hope this answer is satisfactory, any comments are welcome.

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  • $\begingroup$ Thank you for the answer, now the situation is more clear. Can you suggest me other books on QFT? $\endgroup$ – Syn Oct 23 '17 at 20:51
  • $\begingroup$ Probably the best books on QFT are those I've not read. If you read several ones, you get QFT presented from different points of view which helps. A rather good source is: Aitchison-Hey Gauge theories in particle physics. It is not about pure QFT-formalism, It's written for particle physicists. I don't know if it is in all respects fully rigorous. Others might be better in that respect: Schwartz and Srednicki. Srednicki is not easy to read neither. He requires a high level of understanding. Lee's QFT in a nutshell is interesting and makes thirst to read more, but is not rigorous. $\endgroup$ – Frederic Thomas Oct 25 '17 at 11:44

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