In QED, we are taught about the one-loop corrections and the counter-terms to the photon propagator, as well as the one loop corrections and the counter-terms to the fermion propagator. We are also taught about the one-loop correction to the photon-fermion-fermion vertex.
This is all nice and well, but I want to make sure that I understand how to apply these notions to a real interaction. Consider, for instance a $e^-\mu^-\rightarrow e^-\mu^-$ collision. I consider this interaction because one needs only one tree-level diagram in order to write down the Feynman amplitude.
We wish to draw all the diagrams contributing to the Feynman amplitude at the loop-level. According to my understanding, I will modify the tree-level diagram such that my loop-level diagrams include
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Four fermion self-energy diagrams: in those diagrams an additional virtual photon will be added on the tree-level diagram, whose (virtual photon's) beginning and end will lie on the same external electron/muon.
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Four counter-term diagrams for the fermion propagator
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Two vertex diagrams, one for the electron and one for the muon.
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Two photon self energy diagrams
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Photon self-energy counter-term diagram
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Box diagrams: these are obtained by the tree-level diagram by adding one virtual photon that connects (a) the initial electron with the initial muon and (b) the initial electron with the final muon.
So, in total, in the Feynman amplitude I need to consider $4+4+2+2+1+2=15$ diagrams of which I only need to calculate the two vertex diagrams and the two box diagrams, as well as the photon self-energy diagrams and its respective counter term diagram, because the remaining diagrams (electron self-energy) are negated by the respective counter-term diagrams.
Are my considerations correct? Any comment/suggestion will be helful.
Best Answer
Using FeynArts to generate all diagrams for the given process, excluding Tadpole contributions, results in:
While the generation of counterterm diagrams yields
So basically you missed the vertex counter terms when you count all the diagrams. Furthermore, as also mentioned in the comments, the loop correction to the outer legs only add up to zero with the counter term
insertions on the outer legs in the on-schell scheme. Then these corrections are "pushed into" the vertex counter terms. To calculate them you also need to compute the lepton self energies and the photon self energy!
For a nice discussion on the one-loop renormalization of QED in the
on-shell scheme see for example "Gauge Theories of the Strong and Electroweak Interaction" by Böhm, Denner and Joos.
Furthermore, since the photon is massless there probably are IR-Divergencies so you need to include real radiation diagrams.