the case under consideration with an incoming and an outgoing electron, the other possible cases being proved in an analogous way. First, we start back from Eq. (\[Gamma\_2\]) and we apply the same procedure to prove the Ward identity [@Mitter_1975; @Morozov_1981]. From the second equality in Eq. (\[Gamma\_2\]) and from the definition of $\Gamma^{\mu}(p,p',q;\phi)$ in Eq. (\[Gamma\^mu\]), we obtain $$\Gamma_q(p,p',q;\phi)=-ie^2\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}\hat{q}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}.$$ now, by writing $\hat{q}=\hat{\Pi}(\phi)+\hat{k}-m-[\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m]$ it is clear that we can express $\Gamma_q(p,p',q;\phi)$ as the difference of two terms containing only one propagator in the plane wave: $$\label{Gamma_q_0}
\Gamma_q(p,p',q;\phi)=-ie^2\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\gamma^{\lambda}\left[\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}-\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\right]\gamma_{\lambda}.$$ At this point, we observe that in computing, for example, the one-loop radiative corrections to nonlinear Compton scattering (see the leading-order diagram in Fig. \[FD\_NCS\_LO\], for the kinematical situation corresponding to the case under study)

![The leading-order Feynman diagram corresponding to nonlinear Compton scattering of an off-shell photon. The double lines represent exact electron states in a plane wave (Volkov states) [@Landau_b_4_1982].[]{data-label="FD_NCS_LO"}](Figure_4.pdf){width="0.4\columnwidth"}

we also have to include the remaining diagrams listed in Fig. \[FD\_NCS\_MO\_PO\].

![The one-loop Feynman diagrams corresponding, together with the one-loop vertex correction in Fig. \[FD\_VC\], to the leading-order radiative corrections of nonlinear Compton scattering of an off-shell photon. The double lines represent exact electron states and propagator in a 