  By considering radiative corrections of up to 3rd-loop order, Ritus and
Narozhny conjectured that the proper expansion parameter for QED in a strong
constant crossed field is $g=\alpha\chi^{2/3}$, where the dynamical quantum
parameter $\chi=e\sqrt{-(Fp)^2}/m^3$ combines the particle momentum $p$ with
the external field strength tensor $F$. Here we present and discuss the first
non-perturbative result in this context, the resummed bubble-type polarization
corrections to the electron self-energy in a constant crossed field. Our
analysis confirms the relevance of the scaling parameter $g$ to the enhancement
of bubble-type radiative corrections. This parameter actually represents the
characteristic value of the ratio of the 1-loop polarization bubble to the
photon virtuality. After an all-order resummation we identify and discuss two
contributions to the self-energy with different formation regions and
asymptotic behavior for $g\gg1$. Whereas the breakdown of perturbation theory
occurs already for $g\gtrsim1$, the leading-order result remains dominant until
the asymptotic regime $g\gg 1$ is reached. However, the latter is specific to
processes like elastic scattering or photon emission and does not have to
remain true for general higher-order QED processes.
