Electromagnetic mass divergence in QED

Question

In CED the electromagnetic mass of a point-like electron diverges as $1/r_0 $ ( a self-induction effect) and in QED it diverges logarithmically (Weisskopf et al). But the latter result is not exact, but a one-loop result. Does anybody know how the electromagnetic mass diverges in QED when all loops are taken into account exactly?

Comments

Knowing this to all orders would amount to having solved QED exactly. 

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Thanks, Arnold. Still, for a charge there is something like a Landau pole. Is there something similar for the mass?

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I never heard of a Landau pole for the bare mass. 

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Neither did I. ;-)