In QED, the Lagrangian with gauge-fixing terms is $$\tag{7.2}L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\xi(\partial^\sigma A_\sigma)^2 $$ (See Greiner field quantization), from which we can obtain the propagator to be (in momentum space)
$$\tag{3}D_F^{\mu\nu}(k)=\frac{-g^{\mu\nu}}{k^2+i\epsilon}+\frac{\xi-1}{\xi}\frac{k^\mu k^\nu}{(k^2+i\epsilon)^2}$$ (see Greiner field quantization page 190).
I understand that $\xi$ drops out at the end of the calculation of any QED scattering cross sections. My question is: how to see it explicitly and directly at the level of perturbation theory, using Feynman diagrams? (No path integrals.)
Greiner did such an explicit example in chapter 8 on top of page 247 for the simplest case of electron electron scattering up to 2nd order perturbation. In general I don't immediately see how to do it because for individual Feynman diagrams $\xi$ actually gives different values, it is only when we sum over different Feynman diagrams does the contribution of $\xi$ drop out.
$\textbf{EDIT}:$ I now realized this is more or less the Ward identity. Similar to how Peskin and Schroeder argues that for the purposes of computing S matrix elements we can make the abbreviation (7.75) on page 246. The proof of the Ward identity is diagrammatic.
