Compton scattering, the constant $Z^{(\xi)}$ can be absorbed in the renormalization of the electric charge exactly as in vacuum [@Itzykson_b_1980]. Thus, we conclude that the gauge-dependent part of the vertex correction can be absorbed in the renormalization of the electric charge and below we will continue to work in the Lorenz gauge.

Convergence properties of the vertex-correction function {#VC_CP}
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Analogously as the corresponding quantity in vacuum, the quantity $\Gamma^{\mu}(p,p',q;\phi)$ is logarithmically divergent in the ultraviolet, as it can be ascertained from the integral in $d^4k$ in Eq. (\[Gamma\_2\]) (see, e.g., the book [@Itzykson_b_1980] for the analysis of the vacuum case). Now, if we imagine to expand the exact Volkov propagators in powers of the external field (see also Fig. \[FD\_VC\]), it is clear that, since the divergence of the corresponding vacuum amplitude is logarithmic, all resulting terms depending on the field are ultraviolet convergent because the loop contains at least three vacuum electron propagators apart from the photon propagator. It is important to stress here that this does not imply that the whole field-dependent part of $\Gamma_{s,s',l}(p,p',q)$ is ultraviolet convergent because the terms dependent on the field exclusively through the external electron states are still logarithmically divergent. For this 