{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"3ce9e777308edb36a66e1739b497ba3989d09ca0b89b42b5627ca87b6eb36598","created":"2026-10-03T07:18:08Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"aede74c7d50745fb60a485463166b0130caaeef1072b0a522d5cc643e8f4802e","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"plasma.sfqed-breakdown.ccf-all-orders","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"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.","posed_since":"1980","precise":"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.","problem_ref":null,"references":"","settled_by":"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_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.","title":"All-order high-$\\chi$ behavior of electron self-energy in a constant crossed 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