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I have a question about virtual corrections of an external fermion line.

According to the LSZ reduction formula and what some other people say, we neglect the self energies of external legs and multiply the wavefunction therefor just with a squareroot of the wave function renormalization constant.

So considering the feynman rules in momentum space this would giive for an incoming electron:

u → √z u

So in the end I would get just the Born diagram multplied by a term which is proportioanl to 1/ε plus some finite parts ( depending on your renormalization scheme)

So, my question is: How do I get rid of that divergence? Someone told me it is connected to the vertex correction. And in principal one does not compute that diagram at all. It will be considered in the vertex correction.

But I do not understand that.

I also found an somehow wrong and old fashion approach for external self energies. The problem there is the "on shell propagator". But there it is easier for me to see how I can cancel the divergence by adding a counterterm. But in the end that diagram should also still be divergent, right? And someone told me that I have to introduce a factor of 1/2 by hand to get the correct result like that one with the LSZ formula.

So, my questions are:

How to apply LSZ for external self energies? What do I have to compute for example for e+ + e- → μ+ + μ- ?

How do I consider there the "external leg self energies"?

What are the differences between LSZ and the old approach? Are the counterterms for the vertex correction the same?

How do I get rid of the divergence introduced by squareroot of Z?

In most of the books they introduce LSz before renormalization and never use it for an NLO computation. I know that it is good for the transition from greens functions to matrix elements.

Is there any book, paper, reference where they really compute a full NLO process with those factors of Z^(1/2) ?

Thanks a lot!

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# Application of LSZ reduction in NLO computations (external legs)

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