About Feynman rules for an external field

[Sleuth]

Hi everybody, I'm new.
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.

 

[xepma]

An external field is represented by an external line. It is an incoming photon and it carries some momentum and polarization. As with all external lines, you label the line with a momentum $$p$$ and a spin $$s$$ (it will then hit some vertex where stuff like momentum conservation is imposed). They also contribute an overall factor: the line is external so there is some incoming and outcoming orientation, the polarization of the vector (notation: $$\epsilon_\mu(k)$$).

See for instance Griffiths - Introduction to Elementary particles, chapter 7.6 and example 7.4.

Note that this is the contribution of the scattering of a very specific photon (carrying momentum p and a polarization). We can average over the polarization, and impose some distribution on the momenta p - this gives a more realistic description of the external field.

 

[RedX]

I think you just multiply the amplitude with the Fourier transformation of the external field, taken at the 4-momentum value that conserves 4-momentum for the entire process. So if you have an incoming electron with momentum p, and outgoing electron with momentum p', then you would multiply your amplitude by A(p-p') or A(p'-p). Which one I think depends on your choice of metric, (+1,-1-1-1) or (-1,+1,+1,+1), and also how you define the Fourier transform of the external field (whether your formula is integral of A(q)e^(-iqx) or integral of A(q)e^(+iqx).