plane wave (Volkov states and propagator, respectively) [@Landau_b_4_1982].[]{data-label="FD_NCS_MO_PO"}](Figure_5.pdf){width="\columnwidth"}

If we indicate as $i\mathcal{M}^{(1)}_{s,s',\mu}(p,p',q)e_l^{*\,\mu}(q)$ the amplitude of the one-loop radiative corrections to nonlinear Compton scattering represented by the diagrams in Figs. \[FD\_VC\] and \[FD\_NCS\_MO\_PO\], the gauge invariance of QED implies that $\mathcal{M}^{(1)}_{s,s',\mu}(p,p',q)q^{\mu}=0$ [@Peskin_b_1995]. Since the contribution corresponding to Fig. \[FD\_NCS\_MO\_PO\].c is by itself gauge invariant [@Baier_1976_b; @Meuren_2013], by summing the contributions from Fig. \[FD\_VC\] and from Figs. \[FD\_NCS\_MO\_PO\].b and \[FD\_NCS\_MO\_PO\].c, we obtain \[see also Eq. (\[Gamma\_q\_0\])\] $$\begin{split}
i\mathcal{M}^{(1)}_{s,s',\mu}(p,p',q)q^{\mu}&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\\
&\times\bar{U}_{s'}(p',x)\left\{\gamma^{\lambda}\left[\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}e^{i(qx)}-e^{i(qx)}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\right]\gamma_{\lambda}\right.\\
&\quad+\hat{q}e^{i(qx)}\frac{1}{\hat{\Pi}(\phi)-m+i0}\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}\\
&\quad\left.+\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}\frac{1}{\hat{\Pi}(\phi)-m+i0}\hat{q}e^{i(qx)}\right\}U_s(p,x).
\end{split}$$ Now, by using the fact that $[\hat{\Pi}(\phi)-m]U_s(p,x)=[\hat{\Pi}(\phi)-m]U_{s'}(p',x)=0$, we first replace $\hat{q}$ with $\hat{\Pi}(\phi)+\hat{q}-m$ ($m-\hat{\Pi}(\phi)+\hat{q}$) in the third (fourth) line of this equation and then we move the exponential $\exp[i(qx)]$ to the left of all other operators by exploiting the identity in Eq. (\[Trans\_X\]). The result is $$\begin{split}
i\mathcal{M}^{(1)}_{s,s',\mu}(p,p',q)q^{\mu}&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{e^{i(qx)}}{k^2-\kappa^2+i0}\\
&\times\bar{U}_{s'}(p',x)\left\{\gamma^{\lambda}\left[\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}-\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\right]\gamma_{\lambda}\right.\\
&\quad\left.+\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}-\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}\gamma_{\lambda}\right\}U_s(p,x),
\end{split}$$ which indeed vanishes identically. This result indicates that gauge invariance implies that the component $\Gamma_q(p,p',q;\phi)$ can be ignored as it will always be compensated by the corresponding contributions arising from the mass operators (some properties of $\Gamma_q(p,p',q;\phi)$ are discussed in the appendix).

At this point, we can consider the other component $\Gamma_-(p,p',q;\phi)$, whose structure is particularly easy. In fact, starting from Eq. (\[Gamma\_f\]), we have that 