represent exact electron states and propagator in a plane wave (Volkov states and propagator, respectively) [@Landau_b_4_1982].[]{data-label="FD_VC"}](Figure_3.pdf){width="0.4\columnwidth"}

The purpose of the present paper is to fill this gap and, indeed, to compute the one-loop vertex correction in an arbitrary plane wave for the case of two on-shell external electrons and an off-shell external photon. It is worth mentioning here that the computation of the vertex-correction function is not only important to evaluate the leading-order radiative corrections of strong-field QED processes. There is also a more fundamental reason related to the so-called Ritus-Narozhny conjecture [@Ritus_1970; @Narozhny_1979; @Narozhny_1980; @Morozov_1981] about the high-energy behavior of radiative corrections in strong-field QED in a constant crossed field. As we have mentioned, a constant crossed field is a constant and uniform electromagnetic field $F_0^{\mu\nu}=(\bm{E}_0,\bm{B}_0)$ such that the two field Lorentz-invariants $\bm{E}_0^2-\bm{B}_0^2$ and $\bm{E}_0\cdot\bm{B}_0$ vanish. Now, in a constant crossed field radiative corrections depend only on the Lorentz- and gauge-invariant quantum nonlinearity parameter $\chi_0=\sqrt{-(p_{\mu}F_0^{\mu\nu})^2}/mF_{cr}$ [@Mitter_1975; @Ritus_1985; @Ehlotzky_2009; @Reiss_2009; @Di_Piazza_2012], where $p^{\mu}$ is the four-momentum of the particle at hand and where the metric tensor $\eta^{\mu\nu}=\text{diag}(+1,-1,-1,-1)$ is employed. The Ritus-Narozhny conjecture states that at $\chi_0\gg 1$ the effective coupling of QED in a constant crossed field scales as $\alpha\chi_0^{2/3}$. 