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I'm approaching to QFT in these months and I have a couple of questions about Feynman rules.

The most of the books I have read (or tried to) explain feynman rules telling what you have to do when you have an internal or external line in a graph, and when you have a vertex, but I wasn't able to find a complete treatment and justification of what you have to do when you consider an external field.

For example what happens when I want to study the scattering of an electron with an external electric of magnetic field? Let's say we are in QED: do I simply have to multiply the vertex for the external field (say ieA^\mu \gamma_mu)? and why? do I have to integrate over the momenta of the external fields? can I use the conservation of momenta on the modified vertex in the same way?

I understand that maybe this is a really trivial question, but I would like to find someone explaining this in a complete and not-misleading way.

Thank you all

S.