All-order high-$\chi$ behavior of electron self-energy in a constant crossed field
In plain words
Theory says each extra layer of quantum correction grows with field strength as $\alpha$ times $\chi^{2/3}$, where $\chi$ measures the field the electron feels in its own frame. The question is what the sum of all layers gives once that combination reaches 1.
Precise statement
In a constant crossed field (E perpendicular to B, $|E|=|B|$), the n-loop contributions to the electron mass operator and photon polarization operator at quantum parameter $\chi=(e\hbar/(m^{3}c^{4}))\sqrt{-(F p)^{2}} >> 1$ are conjectured to scale as $(\alpha \chi^{2/3})^{n}$, implying breakdown of the loop expansion at $\chi \sim \alpha^{-3/2} \sim 1600$. Determine the asymptotic form of the full (all-order) mass operator for $\alpha \chi^{2/3} >\sim 1$: whether the bubble-chain (polarization-insertion) class dominates at every loop order, whether the resulting series is Borel summable, and the functional form of Re and Im of the mass shift. An answer is a controlled closed asymptotic expression or a demonstration that no expansion in $\alpha \chi^{2/3}$ exists.
What would settle it
An all-order calculation or controlled resummation of the constant-crossed-field mass operator at $\alpha \chi^{2/3} >\sim 1$, cross-checked against explicit three-loop and higher results.
Status in the literature
Unverified note
Mironov, Meuren and Fedotov (Phys. Rev. D, 2020) resummed the bubble-type polarization corrections to the mass operator and found that the leading-order term stays dominant until $\alpha \chi^{2/3} >> 1$, a feature specific to this class; no complete all-order result is known as of 2026.