# About feynman rules for an external field

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.

## Answers and Replies

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.

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).