Since, apart from irrelevant prefactors, the energy of the particle enters radiative corrections only through $\chi_0$ at $\chi_0\gg 1$, the Ritus-Narozhny conjecture implies an asymptotic high-energy behavior of strong-field QED in a constant crossed field qualitatively different from the logarithmic one of QED in vacuum [@Jauch_b_1976; @Itzykson_b_1980; @Landau_b_4_1982; @Schwartz_b_2014]. The physical relevance of the Ritus-Narozhny conjecture is broadened by the so-called locally-constant field approximation (LCFA), stating that in the limit of low-frequency plane waves the probabilities of QED processes reduce to the corresponding probabilities in a constant crossed field averaged over the phase-dependent plane-wave profile [@Ritus_1985]. In Ref. [@Podszus_2019] we have investigated the one-loop mass and polarization operator to show that, if one first performs in the general expression of these quantities the high-energy limit, one indeed recovers the typical logarithmic behavior of QED as in vacuum (see also Ref. [@Ilderton_2019]). Below, we will also investigate the vertex correction within the LCFA, whereas the high-energy asymptotic will be presented elsewhere.

The paper is organized as follows: In Sec. \[Notation\] we introduce the basic notation of the paper. In Sec. \[VC\_General\] the general form of the vertex-correction function is derived by means of the operator technique. In Sec. \[VC\_GI\] the properties 